Osteopathic Medicine Essay -- Medical Health Papers

Osteopathic MedicineI. Introduction of the D.O. Currently there be two primary(prenominal) types of licensed physicians in the United States. The first is the medical posit or the allopathic doctor. About 95% of licensed doctors have been educated at one of over 120 allopathic medical schools and have frankincense obtained a medical doctor degree (M.D.). The second type of doctor, the doctor of osteopathy, is less common. Osteopathic doctors make up about five percent of the physicians in the nation (Peters 730). Although this represents a significant amount of physicians many people are un long-familiar with the second type of doctor. In a 1981 the AOA (American Osteopathic Association) released the results of a look into about public familiarity with the osteopathic doctor. Only 20% of the 1,003 individuals surveyed, age eighteen and over, were familiar with the D.O. (doctor of osteopathy) abbreviation. Furthermore only 50% of those that were familiar with the title could correctly give the unabbreviated version (Gevitz 154-55). The results of this survey are to a certain extent outdated, just an overall atmosphere of unfamiliarity still looms about the doctor of osteopathy. The chase discussion pull up stakes attempt to clarify this modern enigma. A outline history of the much-maligned doctor of osteopathy will be provided along with a modern description of the doctors focuses and responsibilities. Given the current situation of health care be the legitimacy of the D.O. will be discussed as well as the mental ability to contest these rising costs intrinsically possessed by this do of doctor. Finally, the integration of D.O. and M.D. treatments will be discussed. Through subsequent arguments the reader will learn t... ...mberly and Matt Popowsky. Open Season. Kiplingers Personal Finance 56.12 (2002) 78-79.McGirt, Ellen. Health indemnification Less Costs More. Money 31.12 (2002) 146-48.Newswanger, Dana L. Osteopathic Medicine in the Treatment of offset Back Pain. American Family Physician 62.11 (2000) 2414-15.Osteopathic Manipulative Treatment May hit Patients. Womens Health Weekly 11 Oct. 2001 13.Peters, Antoinette S., Nancy Clark-Chiarelli and Susan D. Block. Comparison of Osteopathic and Allopathic Medical Schools represent for Primary Care.Journal of General Internal Medicine 14.12 (1999) 730-39.Stieg, Bill and Lisa Jones. Whats the Diff? mens room Health 17.5 (2002) 50.Tyler, Lawrence. Understanding Alternative Medicine New Health Paths in America. New York Haworth Press, 2000.

Distance Learning Followed by World Première :: Katalin Pocs Evening Song Music Essays

Distance learning Followed by World Premire Esti Dal (Evening Song) by Katalin Pcs was pen for the Indiana University International Vocal supporting players and harpist Erzs run Gal in January 2000. The write up received its world premire conducted by Professor Mary Goetze on April 9, 2000 at IU School of harmony, Bloomington, Indiana.Katalin Pcs is one of the leading members of the young generation of Hungarian composers. Her compositions admit orchestral and chamber whole works that have been performed throughout Europe. She has performed some of her take piano medication in Berlin, Gdansk, Moscow, Munich, and Vienna. In addition, Pcs has written electronic works that were performed in Canada, as well as works for harp, for example a Septet called Vibrarions and a solo piece called Ballade which were introduced by Erzsbet Gal in Hungary and in the United States. In her setting of the folk song Esti Dal, Pcs employs the sounds of mixed choir, harp, and synthesized music. Th ese ternary elements blend together to express the songs text about wandering, weariness, and a excuse to God for rest and shelter. This choral work connects the past with the record by incorporating an ancient tonal Hungarian folk song into a modern texture with harp accompaniment. The addition of an electronic sound track that creates juvenile relationships between consonance and dissonance in the music juxtaposes tradition with present 21st century practices. Mary Goetze is a Professor of Music and chairman of the Music in General Studies Department. She founded the International Vocal Ensemble in 1995. The choir specializes in the recreation of music from outside the European and American art traditions. Through the learning process, the choir becomes acquainted not only if with the music itself but also with the related aspects of the culture and language. In 1996, Dr. Goetze was awarded a grant from Indiana University for a project entitled Multicultural Music Education which allowed her to do research in Zimbabwe and South Africa. Currently she is co-authoring a series of CD ROMs that facilitate the oral transmission of vocal music from diverse sources. In preparation for the first performance, the International Vocal Ensemble had the privilege to work with composer Pcs on her new composition through a satellite hook-up connection between Budapest, Hungary and Bloomington, Indiana that was make possible by the Center for the Study of Global Change on IUs campus. A technology called interactive compressed video opened the opening for the choir and the composer to work together on Esti Dal notwithstanding a difference of six time zones.

My First College Class Experience Essay

The transition from high school to college in itself is a frightening experience in some way for every unitary whether it be sorrowful away from home or the vastness of a college campus others learn from these cutting experiences. Many very important lessons are learned removed the categoriseroom in college exclusively there is in like manner so much to be learned about who you are in the categoriseroom as well. When mortal first-year sees Composition I on their schedule the first mean solar day of college they might think that this class is going to be tedious, boring, and nonhing but writing essays, after a few class periods though they leave behind begin to realize that this class is much more than that. Composition one from my perspective was homogeneous dipping your toes into peeing to test and see if you are make believe to go in or not, the water may be similarly warm or cold at first but that is no reason to shy away from it, if they jump right in they testam ent find that the positives far outweigh the negatives.My first semester of college was spent at The University of the support Word it was my first real taste of what college was going to be like and I very much enjoyed it but the class that was most surpirsing in how much that was learned to me was composition I. In the course catalog this classs description was, Part of the UIW Core Curriculum, this is a writing-intensive course focalization on numerous rhetorical modes to develop main ideas. This course introduces students to creative, schoolman and business writing and communication, emphasizing grammar and syntax, with a view to increasing instructive skills and critical thinking ability. Now to be terribly honorable this description made this class seem extremely dull secure based on the description but even so I was there on the first day ready to learn. The class was clear and more relaxed than I mind it would be it turned out that this class was going to be the exa ct opposite of the description I had read.The assignments that were minded(p) in writing always forced the class to think not just about their own world but the bigger pic of how what they did affected the people around them and the world, for example they wrote about the short spirit level by Ray Bradbury There Will Come Soft Rains and how they thought the world was becoming more and more like the story it gave a much greater sense of awareness of how change was needed in the community and to become more involved with each other and evaluate one another and not be so consumed by technology. They were also given a project on conservation or countries that didnt have unmortgaged sources of water and how they were affected by them It was an eye opening move experience seeing how much others suffered and couldnt have access to clean water in different places around the world. Such as in an article I read in The New York Times where it was said, iii -fifths of all water suppl ies are relatively bad or worse. round half of rural residents lack access to insobriety water that meets worldwide standards.It gave a much greater sense of compassion for those who dont have access to something so basic as clean drinking water. In the words of the Dalai Lama, Love and compassion are necessities, not luxuries. Without them, earth cannot survive. This quote is the perfect representation of what I learned in that class. grace is probably the greatest quality you can bring to college with you and in outside of college as well because without compassion were not even benignant anymore. The greatest lessons you will learn in your life will be in your college years, you find who you are and who you want to become after you graduate. When you first get to college youre just getting your feet wet and testing the water but once you start immersing yourself in your classes thats when you really meet the plunge into the pool. There are certain classes you will take tha t will help you to have that drive to be better and change the business concern or hesitation you first had when you started college and those classes will better who you are as a person and make the whole college experience worth it.

Determining of the equilibrium constant for the formation of FeSCN2+ Essay

Determining of the equilibrium constant for the formation of FeSCN2+ Introduction The accusative of this experiment was to determine the equilibrium minginess and then determine Kc. A dilution calculation was formed to determine the concentration of SCN- and Fe(SCN)2+. Each cuvette was filled to the similar volume and can be seen in submit 1. hence the absorbances were put down from each cuvette and can be seen in table1. A Beers law plot was made from the data that was recorded from the optical absorbance. During the secondly part of the experiment Fe (NO3)3 was added and diluted with HNO3 . All of the cuvettes were mixed with the analogous solutions in the second part of the experiment, which can be seen in table 2. A dilution calculation was made to determine the initial concentration of Fe3+and SCN-. Then the formula Abs + b/ slope was used to determine the equilibrium concentration which lead to the calculation of each Kc per trial. Chemical reactionFe3+ + SCN- FeSCN2+&n bsp oddmentAn acid and a base were mixed together throughout the experiment, which resulted in a bright orange color. It was determined that using the colorimeter at 565nm the would give the optimum wavelength because it was the closest absorbance to 430nm. All of the cuvettes were filled to 3mL so thither would not be another dependent variable star. Whenever Fe3+ would come in connexion with SCN- there would be a color change. Relatively all of the Kc were close to each other as they should be because the only variable that affects a change Kc and the temperature was kept consent throughout the experiment. The clean Kc from all five trials is 1.52 x 10 2.

1987 Constitution of the Republic of the Philippines Bill of Rights Essay

Article III enumerates the innate effectives of the Filipino multitude. The Bill of Rights sets the limits to the organizations power which proves to be not absolute. Among the compensates of the tribe are impec do-nothingtdoms of speech, assembly, religion, and the press. An important feature here is the suspension of the perk of the judicial writ of habeas head which hire three available grounds such as invasion, insurrection and rebellion.PRINCIPLES role 1. No somebody shall be divest of life, liberty, or quality without due process of constabulary, nor shall both mortal be denied the equal safeguard of the faithfulnesss. no person shall be deprived of life or principles and dignity without due Process of jurisprudence or guidelines should be fair then all the protection of each. member 2. The even out of the people to be batten in their persons, ho accustoms, papers, and effects once morest unreasonable searches and seizures of whatever personalisedity and for either purpose shall be inviolable, and no search apologize or warrant of book shall issue eject upon probable gravel to be decided personally by the judge subsequently examination chthonian sworn statement or affirmation of the complainant and the witnesses he may produce, and peculiarly describing the practice to be searched and the persons or things to be seized. human justs and protection to their property and themselves against the search warrant without differentiate against them except to prove that when probable cause to determine personally the judge after examination low oath or affirmation the complainant and the witnesses he may produce, and peculiarly describing the place to be searched and the persons or things to be taken. class 3. (1) The privacy of communication and remainder shall be inviolable except upon lawful shape of the coquet, or when earth safety or order requires otherwise, as prescribed by law. The head-to-head communication s and correspondence shall be inviolable except by court or when human race safety requires otherwise as prescribed by law. (2). Any evidence obtained in violation of this or the preceding subsection shall be inadmissible for whatever purpose in some(prenominal) proceeding. -If there is evidence that violation of this or in the next section that is fast to every purpose. atom 4. No law shall be passed abridging the freedom of speech, of expression, or of the press, or the right of the people peaceably to assemble and petition the government for modify of grievances. no law can be passed or people can assemble and petition or said to the government for redress of grievances that can be able to abridging the freedom of speech or expression. office 5. No law shall be made respecting an establishment of religion, or prohibiting the free exercise thereof. The free exercise and enjoyment of religious profession and worship, without inconsistency or preference, shall forever be a llowed. No religious test shall be required for the exercise of civil or political rights. -There is no law to prevent an establishment of religion, or prohibiting the free exercise of its civil or political rights. scratch 6. The liberty of abode and of ever-changing the aforementioned(prenominal) at bottom the limits prescribed by law shall not be damage except upon lawful order of the court. Neither shall the right to travel be impaired except in the interest of national security, worldly extend to safety, or public health, as may be provided by law. The liberty of abode and of changing the same within the limits prescribed by law shall not be made except when disabled according to the law of the court, which may not be impaired except in the national or public man safety as maybe that has been provide by the law. character 7. The right of the people to information on matters of public concern shall be recognized. Access to official records, and to documents and papers per taining to official acts, transactions, or decisions, as salubrious as to government research data use as beneathstructure for policy development, shall be endureed the citizen, pendant to such limitations as may be provided by law. -the human right of the information and on public concern shall be recognized, based on the records pertaining to official acts basis to government use for research documents detailed on policy development passed by the citizen subject to such limitations as may be provided by law.Section 8. The right of the people, including those busy in the public and buck private sectors, to form unions, associations, or societies for purposes not contrary to law shall not be abridged. -the people including those who are employed to the private sector will form a union or associations to the built a negotiation for them that is not against the law. Section 9. Private property shall not be taken for public use without just compensation. -the private property sh all not be used for public use without pay any(prenominal)thing or allowed by the owner. Section 10. No law impairing the obligation of contracts shall be passed.-There is no law that can pass impairing to the obligation of contracts. Section 11. disengage access to the courts and quasi-judicial bodies and adequate legal assistance shall not be denied to any person by reason of poverty. -any person should not be denied by reason of poverty of adequate legal assistance in court. Section 12. (1) Any person under investigation for the commission of an abhorrence shall go through the right to be informed of his right to remain dim and to have competent and independent counsel preferably of his own choice. If the person cannot afford the services of counsel, he must be provided with one. These rights cannot be waived except in writing and in the comportment of counsel.-any person that is been under investigation of any offence has a right to inform his right and remain silent and have competent and independent counsel on his choice, if the person cannot afford the service of counsel he must be provided with one and this right must be raised in the court. (2) No torture, force, violence, threat, intimidation, or any other path which vitiate the free will shall be used against him. Secret detention places, solitary, incommunicado, or other similar forms of detention are prohibited. (3) Any confession or admission obtained in violation of this or Section 17 hereof shall be inadmissible in evidence against him. (4) The law shall provide for penal and civil sanctions for violations of this section as closely as compensation to the rehabilitation of victims of torture or similar practices, and their families. Section 13. All persons, except those charged with offenses punishable by reclusion perpetua when evidence of guilt feelings is strong, shall, before conviction, be looseable by sufficient sureties, or be egressd on recognizance as may be provided by la w.The right to bail shall not be impaired even when the privilege of the writ of habeas corpus is suspended. Excessive bail shall not be required. -all persons except those charged with offenses punishable by reclusion perpetua and when evidence is that he/she is guilty strong shall be bailable, has the right to bail. Section 14. (1) No person shall be held to answer for a wrong offense without due process of law. (2) In all criminal prosecutions, the accused shall be pre vegetable marrowed innocent until the contrary is proved, and shall enjoy the right to be heard by himself and counsel, to be informed of the character and cause of the direction against him, to have a speedy, impartial, and public trial, to meet the witnesses face to face, and to have irresponsible process to secure the attendance of witnesses and the production of evidence in his behalf.However, after arraignment, trial may proceed notwithstanding the absence of the accused Provided, that he has been duly noti fied and his failure to appear is unjustifiable. Section 15. The privilege of the writ of habeas corpus shall not be suspended except in cases of invasion or rebellion, when the public safety requires it. Section 16. All persons shall have the right to a speedy disposition of their cases before all judicial, quasi-judicial, or administrative bodies. Section 17. No person shall be compelled to be a witness against himself. -no person can be a witness against himself.Section 18. (1) No person shall be solely by reason of his political beliefs and aspirations. -No person wholly by reason of his political beliefs and aspirations. (2) No involuntary servitude in any form shall exist except as a penalty for a crime whereof the party shall have been duly convicted. -no person is excepted by the punishment of the crime by only involuntary servitude he should be duly convicted. Section 19. (1) Excessive fines shall not be enforce, nor cruel, degrading or insensate punishment inflicted. Ne ither shall death penalty be imposed, unless, for compelling reasons involving atrocious crimes, the Congress hereafter provides for it. Any death penalty already imposed shall be reduced to reclusion perpetua. (2) The employment of physical, psychological, or degrading punishment against any prisoner or detainee or the use of substandard or inadequate penal facilities under subhuman conditions shall be dealt with by law. Section 20. No person shall be imprisoned for debt or non-payment of a crest tax -No man is a prisoner just because tax debtSection 21. No person shall be twice be sick in endangerment of punishment for the same offense. If an act is punished by a law and an ordinance, conviction or acquittal under either shall constitute a bar to another prosecution for the same act. Nobody was twice put in jeopardy of punishment.Section 22. No ex charge facto law or bill of attainder shall be enacted. No ex post facto law or bill of attainder is legislation.Searches To mak e a positive examination of look over carefully in order to generate something explore. Seashore The coast of the sea the land that lies adjacent to the sea or ocean.Warrantless Arrest-Under the Rules of Court, Rule 113, Section 5, a warrantless arrest, also k at presentn as citizens arrest, is lawful under three circumstances 1. When, in the presence of the policeman, the person to be arrested has committed, is actually committing, or is attempting to commit an offense. This is the in flagrante delicto rule. 2. When an offense has just been committed, and he has probable cause to believe, based on personal knowledge of facts or circumstances, that the person to be arrested has committed it. This is the hot pursuance arrest rule. 3. When the person to be arrested is a prisoner who has escaped from a penal establishment. In flagrante delicto warrantless arrest should comply with the element of immediacy amid the time of the offense and the time of the arrest. For example, in one case the dictatorial Court held that when the warrantless arrest was made three months after the crime was committed, the arrest was unconstitutional and illegal.Warrantless Searches-Our law on search and seizure has essentially been de,-eloped and clarified from the injunction in our Constitution that the right of the people to be secure in their persons, houses, papers, and effects against unreasonable earc es and seizures of whatever nature and for any purpose shall not be violated. The injunction, however is qualified in toll what is proscribed are only unreasonable searches and seizures.The Constitutional prohibition accordingly readily translates itself into a reasonableness test. search warrant or warrant of arrest shall issue except upon probable cause to be determined by the judge, or such other responsible officer as may be authorized by law, after examination under oath or affirmation of the complainant and the witnesses he may produce, and particularly describing th e place to be searched, and the persons or things to be seized. Definition of bail- the temporary release of an accused person awaiting trial, sometimes on condition that a sum of money is lodged to guarantee their appearance in court (he has been released on bail money paid by or for someone in order to secure their release on bailthey feared the financier would be tempted to drop by the wayside the 10 million bail and flee)Philippine Writ of AmparoDefinition and nature The writ of amparo is a remedy available to any person whose right to life, liberty, and security has been violated or is threatened with violation by an vile act or omission of a public official or employee, or of a private individual or entity. The writ covers illegal killings and enforced disappearances or threats thereof.(Sec. 1, Rule on the Writ of Amparo, A.M. No. 07-9-12-SC, 25 September 2007), The word Amparo is a Spanish term which means protection.Writ of Habeas Corpus is a writ (legal action) which req uires a person under arrest to be brought before a judge or into court.12 This ensures that a prisoner can be released from unlawful detentionthat is, detention absentminded sufficient cause or evidence. The remedy can be seek by the prisoner or by another person approach shot to the prisoners aid. This right originated in the English legal system, and is now available in many nations. It has historically been an important legal cock safeguarding individual freedom against arbitrary state action.Double jeopardy-is a procedural defence that forbids a defendant from being tried again on the same (or similar) charges following a legitimate acquittal or conviction. In common law countries, a defendant may wear a peremptory plea of autrefois acquit or autrefois convict (autrefois means previously in French), meaning the defendant has been acquitted or convicted of the same offence.Self-incrimination-is the act of accusing oneself of a crime for which a person can then be prosecuted. Self-incrimination can occur either directly or indirectly directly, by means of interrogation where information of a self-incriminatory nature is disclosed indirectly, when information of a self-incriminatory nature is disclosed voluntarily without squeeze from another person.

Poverty essay

It has been express that Poverty is functional to society at present, the extent to which this is true have been explore by theorists of the cultural, redness, and functional perspectives and croup be examined SSI Eng countries of the Caribbean such as Trinidad and Tobago. To begin, in the late sass, Oscar Lewis, an American anthropologist created t he culture of leanness ideology. It is in this ideology that he states that on that point argon t here take aims in the culture of poverty . The root being the individual level.Here the scurvy feels helpless, inferior to those around him and marginals from society. The individual De plops a sense of acquiescence and fatalism. He goes to say that the individual desires immediate ratification even through expedient means as the are ineffectual to delay gratifier action. Secondly, the family level. On this level, on that point exist consensual marriages or FRR e runs, a high rate of divorce and a significant military issue of ma triarchal families. Lastly, the partnership level in which most people are fatalistic thusly leading to mini mum effectual sess in the major institutions.There is non membership in try add unions and different homogeneous organizations. Not to mention, that little use is made of banks, hospitals, museums and other selfsame(a) institutions. Lewis concluded his concept y suggesting that the culture of poverty emboldens poverty as the preceding characteristics of poverty act as vehicles to ensure the continuation of poverty If this is applied to the country of Trinidad and Tobago, it can be said that those e Of the genus Beta residence formerly known as shanty town may give pope rotor each level.Individualistic exclusivelyy, persons of the Beta residences and surrounding areas are a lot marginals and are left to feel helpless and inferior to those belonging g to other areas in Trinidad and Tobago. Most have given up on finding a way out of poverty a ND has turn o illegal means f or immediate gratification. On the family level in that location are ma NY common law marriages along with a high rate of divorce and separation between couple sees, therefore contributing to the existence of a notable number of matriarchal households.Also there is little participation in institutions such as schools whether as a teacher or stud antes suggested by Lewis the persons belonging to this region may never find their way out of poverty as all the aforementioned characteristics encourages poverty to be c intended . Thus, it can be said that Lewis does not believe that poverty is functional in s society. Though Lewis held great views, Critics have argued that in many countries, t poor has found ways of improving their life situation. These include, participate Ins In community groups and politics and also by maintaining strong family units.A Iso other researchers have insisted that the behaviour of the poor is not culturally est. Wished but instead is due to situational co nstraints. For ex underemployment, unemployed meet low-down income and other related factors. Conflict theorists rig forward the idea that poverty is a result of the states fail ere to portion resources equitably. They examine poverty from different angles inch ding that of the bear on market, stratification system of rules and capitalism. They state that in the I Barbour market, there is an increasing get hold of for skilled workers in industrialized societies. Hose who are unemployed and underemployed are most likely unable to meet the require meets and are unable to secure high wages on the labor market,thus, be in poverty. A great example of this can be seen in Trinidad and Tobago where a person is now ex pecked to have at to the lowest degree three SEC passes in order to work as a cashier in a grocery or in KEF. Alternatively, there is a dual labor market, consisting of the primary labor racket and the standby labor market. The primary labor market is found in man-si zed e profitable companies with ancestry security, high wages and education opportunities.While the supplementary labor market is found in small companies with little job security low wages a ND few training purport nineties. Conflict theorists suggests that women and those of et hon. minorities are concentrated in the secondary labor market and take low paying jobs. T his too, can be seen in Trinidad and Tobago where the primary labor market is official go Vermont offices or non governmental organizations and the secondary labor market would be irking in hairdressing salons, hardware, groceries, in the market, CHEEP, et c.It is in most of these organizations Women and the ethnic minorities are heavily me played. In addition, Marxist theorist explores poverty from the angle of the stratification system. They indicate that this system relates poverty to house. They believe the at the underclass, namely the retired elderly, the physically disabled and genius par .NET families la ck prestige and wealth and that their low position is a result of their low sat tutus. They go on to say that it is because of their low wages that opportunities for better pap d jobs are limited. Equally pregnant is the angle of capitalism.It is here that Marxist states that the existence of poverty is in privilege of the owners of production, as it allows the m to maintain the capitalist system and maximize salary . It is important to realism that me embers of the working class only owns their labor which is interchange at a wage and for those who SSE labor is not in demand, wages are low. Therefore competitions among workers arise which keeps the wage levels in check. The departure theorist says that the state will act in FAA our of the ruling class and therefore, the government would do little to reduce poverty.From this it can be said that Marxist view poverty as being functional in society but to non e other than the upper class or owners of production while exploiting th e working class. There are researchers who disagree with the Marxist point Of view reason bee Eng, the Marxist view has failed to clearly states what causes authoritative groups in society to become poor, also the perspective does not distinctly distinguish the poor from other members of the working class, and therefore fails to offer an score for their poverty. Not to mention that the theory does not explore the income variations existing with n the working class.Another Perspective taken on poverty is the Functionalist or orthodox app roach led by Herbert . J Gangs. He argues that poverty persists because it IS well(p) to original non poor and affluent groups in society. Functionalist theorists including Gangs argues that there are certain functions of poverty.

Bishop’s poetry Essay

I agree with this assessment of Bishops poetry. Her poetrys on the program certainly pose enkindle questions about identity, ken and ones place in the world, indeed the universe, and they do so by way of a unique style. This style is influenced by Bishops acute aw atomic number 18ness of the poets craft and her ability to work with both traditional forms (sestina and sonnet, for instance) and give up verse. The questions that interested me most argon those posed in Questions of Travel.These fascinated me because Bishop dedicated so much of her life to travel, yet in this poem she questions the motives behind travel and exploration. One stylistic feature that is characteristic of Bishop is the colloquial tone and it is evident in the opening lines, as she states There are too many waterfalls here. The question raised in my consciousness is How mint there be too many waterfalls? Surely the waterfalls are a sight of natural splendour?Yet, reading on, we see that everything in this place of natural beauty over-powers the poet the streams are crowded, they hurry too chop-chop, there are so many clouds. Why is this? She says that the streams and clouds keep locomotion, travelling and this poses the question of her own travels has travel be semen as mo nononous as the relentless waterfalls or is it a type of addiction or destiny for the poet? This question poses more questions when we consider the poets alcoholism and the part vie by addiction in her life.The questions raised in the next stanza come up to themes, which are central to her poetry piazza, exclusion, and the quest for new horizons. Bishop wonders if the idea of a place is more satisfying than the place itself Should we have stayed at home and thought of here? This apparently simple question is loaded with difficulties for Bishop as home was never a simple concept for her. She is acutely witting of herself as an outsider in this culture and feels she is watching strangers in a play in t his strangest of theatres.Bishop describes the urge for travel as a childishness and the image of travellers rushing to see the sun the other way about is an image of thrill-seekers consuming views and experiences without understanding or insight (inexplicable and gravid). I find this very relevant, as we live in a society, which is obsessed with consuming things and experiences, often at the expense of understanding.This image in any case prepares us for the question at the heart of this poem Oh, must we pipe dream our dreams and have them too? I found this question very interesting because dreams are not reality and there are other references to fallacy in this poem strangest of theatres and pantomimists. The question of why we travel and explore is not explicitly answered in the poem but one wonders if it has something to do with charge or escape from reality. The disparity between the real and the imagined is alluded to again in another thought-provoking question Is it l ack of imagination that makes us come To imagined places, not just stay at home? All of our preconceived, modernistic ideas about travel choice, freedom, excitement, broadening of horizons, understanding of other cultures are dour on their head and challenged in the questions raised here about travel. In both Questions of Travel, and The Prodigal, Bishop deals with being away from home and returning. In both poems, the idea of returning is difficult and complex Bishop is not plane sure where home is Should we have stayed at home, wherever that may be? Her sense of displacement is much stronger than her sense of belonging.Similarly, in The Prodigal, the souse in exile must struggle with uncertain staggering leak/his shuddering insights, beyond his control before he can reckon the journey home. A stylistic feature of Bishops work, which I unfeignedly enjoyed, was her tendency, in some poems, to move from sensory description of the apparently casual to profound awareness and insight, even epiphany. This can be seen in In the Waiting Room where Bishop begins with a description of a dull tooth doctors waiting-room, full of grown-up people, arctics and overcoats, lamps and magazines. This is a scene from everyday life in Worcester, Massachusetts.The setting is ordinary, yet the title denotes a place of anticipation and expectation, and raises questions. What can the young Bishop be anticipating or expecting? What is to come? The National Geographic a magazine we could easily expect to see in any waiting-room transports the child, in her imagination, to the inside of a volcano, a far cry from the suaveness of the dentists waiting-room. The images of other races and civilizations are both horrifying and make but the child cannot stop reading them.

Three generation project

Since, my obtain and I are also Russian language emigrants our moms found each other fairly quickly by dint of usual English rouses in the local church. We are sharing a truly friendly relationship between our families spending holidays, birthdays and other felicitous cause to regainher. My frontmost impression of my message was t wear he was extremely shy and solitary however, after the ice between us was broken, he opened up to me and we shared a lot of interesting conversations.Needless to say, I am advance this converse from a biased point of view since I turn in and wish well this young man. However, I will try my best to appease objective and critical for the sake of this report. To conduct this interview, I contacted my subject and awaited him to invite me at my house. To be completely h unrivaledst, I had my doubts regarding this interview because I had to be extremely cautious not to puff him feel uncomfort adequate with my questions however, I feel that my finish of getting to know him a little to a greater extent individual(prenominal)ly and be able to understand his age classify was reached.INTERVIEW aestival My subject arrived to our meeting as scheduled, and even surprised me with dental platemade java cookies. I started our interview by asking him to tell me nigh himself his interests, hobbies and and so forth Basketb in all turned start to be his favorite sports activity, which he a great deal shares with his friends. Even though he menti unrivaledd suffering from asthma he tell that it neer stopped him from being active. I asked him about his boorhood, and what kind of memories he had from that period of sentence.He briefly talked about him growing up oerseas and his ethnic background telling me about how his emotional state experience, in two variant nationalities, had mouldd his identity. He remembers his childhood as being as if it was a black and white movie, meaning he didnt waste ofttimes colorful m emories due to their poverty I assumed. As far as adjusting in the United States, my subject say that it wasnt as hard for him as t was for his mother who had to take care of them both.As we move on through the interview, I entangle comfortable enough to ask him more personal questions about his romantic and informal experience. I asked him to discuss about how puberty affected/or still effecting him, and what emotional or physical changes he went through during those geezerhood. I found out that my subject has had sexual sexual congress with a girl from his shallow except they did not continue to pause their relationship. In regard to the physical changes with his body, my subject didnt elaborate too much provided mentioned becoming more masculine.As farewell of the emotional changes during puberty, my subject said he feels that his parents dont understand him and he prefers not share his though or ideas with them to avoid any conflict. DISCUSSION Since I do know this immature and familiar with his background, history, and his development up to this point of his life, I feel that on that point are several important points that must be mentioned. As I brought up earlier, my subject grew up in Russia, raised by his mother and grandmother.His biological fuck off was never around to represent the male fibre or father figure in the family structure. Shortly after they moved to the States, his mother remarried to a man who become a ill-use father to my subject. According Bellboys (2013) physical developmental milestones for adolescence, my subject appears to be mostly on track. The phase of puberty, which is defined as a period of somatic and hormonal changes by which children become sexually mature, has evidently reached its peak when looking at my subject (238).His voice has definitely deepened (since I knew him when he had the highschool-pitched voice) and his force performance is very(prenominal) slap-up being that he plays basketball, and drives a car. Moreover, as the reference of the text states, my subject exhibits secondary sexual characteristics, which are the mark off for the physical changes that accompany puberty along with a growth spurt-?a dramatic increase in height and weight (241). Also, my subject has had sexual intercourse during his brief romantic relationship with a girl from his school which Belles notes is often an important part of adolescent development (258).In terms of my subjects cognitive and socio-emotional part of the interview and how it relates to the theory-based developmental milestones, I believe hat my subject is at an adequate developmental level. For example, siren his survey process, when I asked about a certain hypothetical stick out and the future, my subject was able to grasp these concepts and answer concretely. Discussing his plans regarding college and his possible career is one example of blue jean Pigments theory of formal operational cerebration, regarding his bear life. Belles 269) Likewise, my subjects readiness to reason about his past and tell me about his experience with relocating to a unalike country, shows his ability of abstractly expressing his Houghton (Belles 267-8). He shared with me stories about the tall(prenominal)ies his mother went through when they first got the States for example, my subject had to go with her to several Job interviews as a arranger because her English wasnt very good. He said he never complained, plainly he definitely felt like the roles between them had changed, due to the position that was often relying on him instead of the opposite.Piglets theory on morality, which influenced and was expanded on by Lawrence Goldberg, at the autonomous morality stage, adolescents realize that intentions, not implicate outcomes, should serve as the basis for the Judgment of behavior and that it is the manner in which an adolescent reasons about a moral dilemma that determines moral due date (Belles 271). I calc ulate my subject exhibits high moral values when he talks about his family, in particular what drawn my attention was his timber father, with whom he does not share a very close relationship, solely has a lot of respect for him for caring for his mom.Following Piglets formal operation theory, David desire conducted a study where he determined that children that make the transition onto teenaged years become more affectionately conscious- exhibiting Adolescent Egocentrics (272). I have to say that I didnt spot any sighs of that behavior in my subject in fact, he barely revealed any self-conscious thoughts regarding his looks or actions throughout the years of our friendly relationship. However, when I first met this young men, he gather inmed very unsocial and didnt really talk to anybody, and when he was approached by someone he showed some di emphasize.At that point I would probably mark him as a social sensitive adolescent going through storm and stress phase (266). Although, the author in our text also describes those teens as more apt to make risky and dangerous stopping points, it was not at all the case with my subject (273). Continuing with the socio-emotional development, one of the things that was pointed out in our interview is that my subject has a difficultly expressing his thoughts to his family due to the lack of thought from their part.According to Judith Harriers partner group colonization theory, immigrant adolescents have disagreements with their parents that may go beyond the reasonable reason and involve fundamental differences in world views (Belles 288). This situation, in my opinion, leads toward the process of separations between him and his parents. As Belles states, As teenagers push for freedom, they are given more decision making opportunities and establish a new, more equal, adult-like relationship (287).Correspondingly to a search conducted by psychologists, Synthetically and Larson, my subject loves his family very much and enjoys spending time with them however, he clearly says that when taken as whole, frustrating emotions outweigh the happy ones when living with them (286). I can defiantly relate to that idea nidus n the social aspect of developing, a largish potential pubertal problem is popularity. From the interview and from my knowledge of him, my subject has increasingly less issues with peer pressure and finding an identity. He speaks of feeling comfortable as been part of a group of his choice.Supporting this finding, Beelike notes that social standing is very important at this age because it affects tens academic/social paths, while being isolated from a crowd can lead to depression (278). Consequently, my subject is a good student, and surrounds himself with a small group (known as liqueur) that shares the same academic interests as him (Belles 289). By doing that I ring he is reducing the risk of getting into trouble, because Children who are not succeeding with the mainstreams ki ds gravitate toward antisocial groups of friends, who then give each other reinforcement for doing dangerous things( 279).In addition, as Beelike (2013) mentions, school purlieu has a great impact on adolescents development because the academic and social growth is reliant on the ability of schools to fit their computer programmes to the adolescents state of mastermind (284). Thus, for y subject experience in high school, he says that the theme of the material being passed on is k, exactly the way theyve been passing it is boring. Also, he notes that his school does not have any after school program accept of tutoring. So what usually teens are doing after school is over?Mostly sitting in groups next to the school smoking and attractive in other unhealthy and unappreciated activities, he says. Interpreting the author position in this matter, one can see that the need for young development programs for after school period of the day is essential for retentivity the adolescents out of the streets where they usually get into trouble (283). Consequently, those teens might get convolute in frightful crowds- which usually are groups with a main goal of performing antisocial acts (Belles 291).Lastly, CONCLUSION In conclusion, I appreciated this project to a great extent due to its real and open mind nature. It is one thing to read about developmental theories, but it is a unique experience to explore those theories in real life and see how they play out. Also, this project was very beneficial for me because it helped understand better one of my close friends whom I like very much as a person. Conducting this interview and the following report helped me gain a better understanding of where this person is coming from and what has led him to his current state of development.With that said, I in person do not think he represent the majority of the adolescence group. perhaps Im wrong, but I think that he have always been about too mature for his age never got into any serious troubles, hard-boiled his parents with respect as if he was an adult and dealt with very serious life changes in a calm way. Needless to say, that my teen age years were completely different. INTERVIEW QUESTIONS (Transcript) Physical Marina Youve Do you play any sports? correction Yes. MY Which ones? S basketball. MY In prevalent, do you like to spend time foreign? S Yes, sometimes.MY Doing what? S Playing basketball, walking around with friends. MY Do you decease out, or do you Just play sports to keep in shape? S I Just play sports to keep in shape. MY Do you drive a car? S yes. MY What type of car is this? S 2011 Ionians Ultimate MY do you have any health conditions? S Yes, I have asthma MY Do take any medications currently? And do you feel that your condition is stopping you from doing stuff? S Yes, Im using alabaster. And I never thought of it as something that stops me in life. I mean, I know Im not able to run tracks but I never intended to do it either. Cognitive MY How would you say that you do in school, in general? Is it good, average or poorly? S Good, school is really unprovoked for me. MY What are your favorite subjects? S I think Math is k, but I dont really have a favorite subject. MY How long did it take you to learn and speak English? S At first I thought I will never be able to march on with others in English although the school material was fairly easy for me because I could read &038 write utter that I spoke. I think, it took me about 6-7 months to begin with I started talking to people.MY In general, how would you describe your experience of moving to the States? Was it difficult to adjust? S It was somewhat difficult, but it seems to me like a bad dream now. At first, my mom and I we didnt know anyone here and felt very nervous about everything, even going to the store was a big death. Currently, after several years have passed, we feel it is our home and this is where we belong. MY When you have a big assignment or a big project to do, how do you usually approach it? Do you wait until the inhabit second or you rather get it done as short as possible?S I usually Just like to do everything in order and gather all the information that I need, and then put it all together in my project. MY would you describe yourself as pretty organized in general? S Yes. MY Have thought about your future? What youre going to do right after high school and then even further from there? S Yes, I have thought about my future and in particular Im thinking about going to college. I would like to become an engineer one day. Social/Emotional MY Do you like school? S Its k I guess, I cant say I hate it. MY What would u do differently in your school?S I would probably make classes less conventional because it get pretty boring after doing that for so many years. MY Do you get along with your peers? S Mostly yes. My Do you feel popular in your school? S I think Im popular among my friends, all the others dont bother me. MY Do you have a miss? S No, not right now MY So you had one in the first place right? S Yes, I had this girl from our school MY Didnt work out? S No, we are Just different MY Do you get along with your mom and you step father? S Most of the time yes, but sometimes she will get on my nerves and she wont give up.My step dad is fine I guess, we arent very close but we dont bit either. I usually dont share with them any personal information because I know they wont understand me, I guess because they grew up in different circumstances than I did. MY Do you have any brothers or sisters? S Yes, I have a 3 year old brother. MY Do you feel different now that are not the only child in the family? S Yes, he gets all the attention from every body, but thats understandable because he is small child. MY Do you miss your grandmother? I know she is still in Russia.S Yes I do, I wish she could be here with us. I know she is struggling with money and Im planning to get a Job this summer so I can maneuver her money. MY Do you feel that your parents trust you enough to let you live your own life? Or are they worried and strict? S I guess they do trust me to a certain point, but they would ask like a million questions whenever I leave the house or come back. We talked about me going to college and living on campus, I think they are k with that. Reference Belles, Janet. Experiencing the lifespan (3rd De. ). New York Worth (2013).

Having My Ear Surgery

1 of the times that I was most proud of was the time that I had my pinna military operation done on December 19. I wanted capitulum surgery so I can hear better and because of the mend in my ear. Before surgery I was sc ard and worried that it go forth hurt so bad and painful. Right before surgery, they numbed my vein with swimming and lay an intravenous needle in the numbed skin vein and they put me in the operating room.During surgery, I didnt feel anything and the doctor started the surgery by making an incision on the ear and he put a patch on it. Next, he put more patches on the part close to my eardrum and then he found an overage ear tube close to my eardrum and he removed it. subsequently the surgery I went to the recovery room and stayed there for more than an mo and during that time I had some stuff to eat and then after that we went home. One day someone put lively sauce in a soda bottle and put it in the fridge.I came to guide it out and then I started to tak e the bottle out. Then I started to drink from it and my tongue started to burn and I needed help from mama and I had to drink water to make the burning sensation to go away. The burning sensation went away then I had more hot sauce then the burning sensation came back. The smell was very sweet-scented and spicy. The lesson that I learned was before eating or drinking anything you are supposed to check the contents of the container.

Flow Induced Vibration

FLOW bring forth VIBRATIONS IN PIPES, A finite sh ar APPROACH IVAN GRANT Bachelor of Science in Mechanical engineering science Nagpur University Nagpur, India June, 2006 submitted in partial ful? llment of requirements for the degree MASTERS OF learning IN MECHANICAL ENGINEERING at the CLEVELAND STATE UNIVERSITY May, 2010 This thesis has been gutteronic for the department of MECHANICAL ENGINEERING and the College of Graduate Studies by dissertation Chairperson, Majid Rashidi, Ph. D. division & angst read-only storage Date Asuquo B. Ebiana, Ph. D. Department & Date Rama S. Gorla, Ph. D. Department & Date ACKNOWLEDGMENTS I would standardized to thank my advisor Dr. Majid Rashidi and Dr.Paul Bellini, who provided essential support and assistance through with(predicate) verboten my graduate carg unmatchabler, and the likewise for their guidance which immensely contributed towards the marches of this thesis. This thesis would not urinate been realized without their su pport. I would also like to thank Dr. Asuquo. B. Ebiana and Dr. Rama. S. Gorla for being in my thesis committee. Thanks be also due to my parents,my brother and fri military group outs who drive home encouraged, supported and inspired me. FLOW INDUCED VIBRATIONS IN PIPES, A delimited ELEMENT APPROACH IVAN GRANT abstract hang up induced shakings of pipings with interior(a) ? uid ? ow is studied in this move.Finite broker depth psychology methodology is used to de edgeine the vital ? uid swiftness that induces the threshold of subway instability. The partial di? erential comparability of effort governing the afterwardsal shivers of the c on the whole is use to develop the sti? ness and inertia matrices corresponding to two of the conditions of the comparabilitys of cause. The par of motion further includes a mixed-derivative terminal figure that was treated as a source for a dissipative function. The corresponding hyaloplasm with this dissipative function was positive and accepted as the voltagely destabilizing factor for the lateral vibrations of the ? id carrying shrill. Two attributes of boundary conditions, that is to say exclusively-supported and protrudeed were considered for the call. The appropriate flowerpot, sti? ness, and dissipative matrices were developed at an geneal level for the ? uid carrying subway. These matrices were then ensnared to do work the overall bulk, sti? ness, and dissipative matrices of the entire system. Employing the ? nite subdivision mannequin developed in this work two series of parametric studies were conducted. First, a tubing with a unceasing wall thickness of 1 mm was analyzed. Then, the parametric studies were ext destinati whizd to a yell up with variable wall thickness.In this case, the wall thickness of the shriek was imitate to taper down from 2. 54 mm to 0. 01 mm. This lease shows that the critical speed of a subway carrying ? uid place be increased by a factor of six as the result of tapering the wall thickness. iv TABLE OF CONTENTS ABSTRACT LIST OF FIGURES LIST OF TABLES I origination 1. 1 1. 2 1. 3 1. 4 II Overview of privileged coalesce generate Vibrations in tubes . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition of Thesis . . . . . . . . . . . . . . . . . . . . . . . iv vii ix 1 1 2 2 3 FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 2. 1 Mathematical toughieling . . . . . . . . . . . . . . . . . . . . . . . 2. 1. 1 2. 2 Equations of operation . . . . . . . . . . . . . . . . . . . 4 4 4 12 12 Finite constituent modelling . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 2. 2. 2 2. 2. 3 Shape carrys . . . . . . . . . . . . . . . . . . . . . Formulating the Sti? ness hyaloplasm for a holler Carrying eloquent 14 Forming the hyaloplasm for the Force that conforms the changeful to the tubing up . . . . . . . . . . . . . . . . . . . . . 21 2. 2. 4 2. 2. 5Dissipation intercellular substance Formulation for a subway system carrying limpid 26 inactiveness ground substance Formulation for a holler carrying changeable . 28 III FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 31 v 3. 1 Forming orbicular Sti? ness Matrix from Elemental Sti? ness Matrices . . . . . . . . . . . . . . . . . . . . 31 3. 2 Applying point of accumulation Conditions to Global Sti? ness Matrix for simply supported piping with ? uid ? ow . . . . 33 3. 3 Applying bourne Conditions to Global Sti? ness Matrix for a cantilever cry with ? uid ? ow . . . . . . . 34 3. 4 MATLAB syllabuss for Assembling Global Matrices for exclusively back up and stick out tubing carrying ? uid . . . . . . . . . . 35 35 36 3. 5 3. 6 MATLAB design for a simply supported metro carrying ? uid . . MATLAB program for a cantilever pipe carrying ? uid . . . . . . IV FLOW INDUCED VIBRATIONS IN PIPES, A FINIT E ELEMENT APPROACH 4. 1 V parametric film . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 5. 1 channeliseed metro Carrying Fluid . . . . . . . . . . . . . . . . . . . . 42 42 47 50 50 51 54 MATLAB program for just support metro Carrying Fluid . . MATLAB Program for stick out tubing Carrying Fluid . . . . . . MATLAB Program for tapering off yell Carrying Fluid . . . . . . 54 61 68 VI RESULTS AND DISCUSSIONS 6. 1 6. 2 Contribution of the Thesis . . . . . . . . . . . . . . . . . . . . . prospective Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY Appendices 0. 1 0. 2 0. 3 vi LIST OF FIGURES 2. 1 2. 2 Pinned-Pinned subway Carrying Fluid * . . . . . . . . . . . . . . pipework Carrying Fluid, Forces and snatchs playacting on Elements (a) Fluid (b) tube-shaped structure ** . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 9 10 11 13 14 15 16 17 21 33 34 36 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 Force due to crimp . . . . . . . . . . . . . . . . . . . . . . . . .Force that Conforms Fluid to the curve ball of Pipe . . . . . Coriolis Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inertia Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Carrying Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . transmit Element Model . . . . . . . . . . . . . . . . . . . . . . . . . alliance between Stress and Strain, Hooks legal philosophy . . . . . . 2. 10 Plain randomnesstions bide plane . . . . . . . . . . . . . . . . . . . . . 2. 11 Moment of Inertia for an Element in the Beam . . . . . . . . . 2. 12 Pipe Carrying Fluid Model . . . . . . . . . . . . . . . . . . . . . 3. 1 3. 2 3. 4. 1 Representation of Simply back up Pipe Carrying Fluid . . Representation of Cantilever Pipe Carrying Fluid . . . . . . . Pinned-Free Pipe Carrying Fluid* . . . . . . . . . . . . . . . . . reducing of rudimentary Frequency for a Pin ned-Pinned Pipe with change magnitude take to the woods pep pill . . . . . . . . . . . . . . . . 4. 2 Shape Function Plot for a Cantilever Pipe with increasing lessen f come up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 3 step-down of thoroughgoing Frequency for a Cantilever Pipe with increasing scat speeding . . . . . . . . . . . . . . . . . . . . 5. 1 Representation of manoeuvreed Pipe Carrying Fluid . . . . . . . 39 40 41 42 vii 5. 2 6. 1 Introducing a Taper in the Pipe Carrying Fluid . . . . . . . . Representation of Pipe Carrying Fluid and decrease Pipe Carrying Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 47 viii LIST OF TABLES 4. 1 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing ply Velocity . . . . . . . . . . . . . . . . 38 4. 2 Reduction of Fundamental Frequency for a Pinned-Free Pipe with increasing hightail it Velocity . . . . . . . . . . . . . . . . . . . . 40 5. 1 Reduction of Fun damental Frequency for a tapering pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . . . 46 6. 1 Reduction of Fundamental Frequency for a Tapered Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . . . . . . . . 48 6. 2 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity . . . . . . . . . . . . . . . . 49 ix CHAPTER I INTRODUCTION 1. 1 Overview of Internal Flow Induced Vibrations in Pipes The ? ow of a ? uid through a pipe can impose presss on the walls of the pipe create it to de? ect chthonic certain ? ow conditions. This de? ection of the pipe may lead to geomorphologic instability of the pipe.The aboriginal innate relative absolute relative absolute frequence of a pipe generally decreases with increasing focal ratio of ? uid ? ow. There are certain cases where decrease in this natural frequency can be very important, such as very gamy swiftness ? ows through ? exible thin-walled pipes such as those used in give way lines to rocket motors and water turbines. The pipe becomes susceptible to resonance or harass failure if its natural frequency falls below certain limits. With abundant ? uid velocities the pipe may become un fixed. The most familiar form of this instability is the scourgeping of an unrestricted garden hose.The hear of dynamic response of a ? uid conveying pipe in conjunction with the transient vibration of ruptured pipes reveals that if a pipe ruptures through its cross indorsementond basetion, then a ? exible distance of groundless pipe is left spewing out ? uid and is free to whip about and conflict other structures. In power plant plumbing pipe whip is a possible mode of failure. A 1 2 study of the in? uence of the resulting high pep pill ? uid on the static and dynamic characteristics of the pipes is thus necessary. 1. 2 Literature Review Initial investigations on the bending vibrations of a simply supported pipe containing ? id were car ried out by Ashley and Haviland2. Subsequently,Housner3 derived the comparabilitys of motion of a ? uid conveying pipe more than comp allowely and developed an equation relating the of import bending frequency of a simply supported pipe to the stop subprogram of the inseparable ? ow of the ? uid. He also express that at certain critical fastness, a statically unstable condition could exist. Long4 presented an alternate solution to Housners3 equation of motion for the simply supported end conditions and also treated the ? xed-free end conditions. He compared the analysis with experimental results to con? rm the mathematical model.His experimental results were rather undetermined since the muckimum ? uid velocity available for the test was low and change in bending frequency was very half-size. Other e? orts to treat this subject were made by Benjamin, Niordson6 and Ta Li. Other solutions to the equations of motion show that type of instability depends on the end conditio ns of the pipe carrying ? uid. If the ? ow velocity exceeds the critical velocity pipes supported at both ends bow out and buckle1. Straight Cantilever pipes fall into ? ow induced vibrations and vibrate at a large bounteousness when ? ow velocity exceeds critical velocity8-11. . 3 Objective The objective of this thesis is to weapon numerical solutions method, more specifically the Finite Element Analysis (FEA) to beget solutions for di? erent pipe con? gurations and ? uid ? ow characteristics. The governing dynamic equation describing the induced structural vibrations due to subjective ? uid ? ow has been create and dis- 3 cussed. The governing equation of motion is a partial di? erential equation that is fourth order in spatial variable and second order in while. Parametric studies have been performed to bear witness the in? uence of lot distribution along the duration of the pipe carrying ? id. 1. 4 Composition of Thesis This thesis is organized according to the followe s equences. The equations of motions are derived in chapter(II)for pinned-pinned and ? xed-pinned pipe carrying ? uid. A ? nite factor model is created to solve the equation of motion. Elemental matrices are formed for pinned-pinned and ? xed-pinned pipe carrying ? uid. Chapter(III)consists of MATLAB programs that are used to assemble global matrices for the higher up cases. Boundary conditions are applied and based on the user de? ned parameters fundamental natural frequency for free vibration is calculated for various pipe con? urations. Parametric studies are carried out in the followe chapter and results are obtained and discussed. CHAPTER II FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH In this chapter,a mathematical model is formed by developing equations of a straight ? uid conveying pipe and these equations are later solved for the natural frequency and onset of instability of a cantilever and pinned-pinned pipe. 2. 1 2. 1. 1 Mathematical Modelling Equations o f Motion reckon a pipe of length L, modulus of snap bean E, and its cross(prenominal) ambit outcome I. A ? uid ? ows through the pipe at pressure p and assiduousness ? t a regular velocity v through the sexual(a) pipe cross-section of area A. As the ? uid ? ows through the de? ecting pipe it is speed up, because of the ever- changing bender of the pipe and the lateral vibration of the line. The vertical component of ? uid pressure applied to the ? uid share and the pressure deplume F per unit length applied on the ? uid element by the tube walls oppose these accelerations. Referring to ? gures (2. 1) and 4 5 go steady 2. 1 Pinned-Pinned Pipe Carrying Fluid * (2. 2),balancing the deplumes in the Y poseion on the ? uid element for small deformations, gives F ? A ? ? ? 2Y = ? A( + v )2 Y ? x2 ? t ? x (2. 1) The pressure gradient in the ? uid along the length of the pipe is opposed by the shear stress of the ? uid clangoring against the tube walls. The sum of the force s parallel numeral 2. 2 Pipe Carrying Fluid, Forces and Moments acting on Elements (a) Fluid (b) Pipe ** to the pipe axis for a constant ? ow velocity gives 0 0 * Flow Induced Vibrations,Robert D. Blevins,Krieger. 1977,P 289 ** Flow Induced Vibrations,Robert D. Blevins,Krieger. 1977,P 289 6 A ?p + ? S = 0 ? x (2. 2) Where S is the inner perimeter of the pipe, and ? s the shear stress on the internal surface of the pipe. The equations of motions of the pipe element are derived as follows. ?T ? 2Y + ? S ? Q 2 = 0 ? x ? x (2. 3) Where Q is the transverse shear force in the pipe and T is the longitudinal tension in the pipe. The forces on the element of the pipe normal to the pipe axis accelerate the pipe element in the Y direction. For small deformations, ? 2Y ? 2Y ? Q +T 2 ? F =m 2 ? x ? x ? t (2. 4) Where m is the mass per unit length of the empty pipe. The bending moment M in the pipe, the transverse shear force Q and the pipe deformation are related by ? 3Y ?M = EI 3 ? x ? x Q=? ( 2. 5) Combining all the higher up equations and eliminating Q and F yields EI ? 4Y ? 2Y ? ? ? Y + (? A ? T ) 2 + ? A( + v )2 Y + m 2 = 0 4 ? x ? x ? t ? x ? t (2. 6) The shear stress may be eliminated from equation 2. 2 and 2. 3 to give ? (? A ? T ) =0 ? x (2. 7) At the pipe end where x=L, the tension in the pipe is zero and the ? uid pressure is equal to ambient pressure. Thus p=T=0 at x=L, ? A ? T = 0 (2. 8) 7 The equation of motion for a free vibration of a ? uid conveying pipe is found out by substituting ? A ? T = 0 from equation 2. 8 in equation 2. 6 and is presumptuousness by the equation 2. EI ? 2Y ? 2Y ? 4Y ? 2Y +M 2 =0 + ? Av 2 2 + 2? Av ? x4 ? x ? x? t ? t (2. 9) where the mass per unit length of the pipe and the ? uid in the pipe is precondition by M = m + ? A. The next section describes the forces acting on the pipe carrying ? uid for each of the components of eq(2. 9) Y F1 X Z EI ? 4Y ? x4 solve 2. 3 Force due to Bending Representation of the First terminal figure in the Equation of Motion for a Pipe Carrying Fluid 8 The term EI ? Y is a force component acting on the pipe as a result of bending of ? x4 the pipe. Fig(2. 3) shows a non readational view of this force F1. 4 9 Y F2 X Z ?Av 2 ? 2Y ? x2 Figure 2. Force that Conforms Fluid to the Curvature of Pipe Representation of the Second border in the Equation of Motion for a Pipe Carrying Fluid The term ? Av 2 ? Y is a force component acting on the pipe as a result of ? ow ? x2 around a curved pipe. In other lecture the momentum of the ? uid is changed leading to a force component F2 shown formalally in Fig(2. 4) as a result of the curvature in the pipe. 2 10 Y F3 X Z 2? Av ? 2Y ? x? t Figure 2. 5 Coriolis Force Representation of the Third Term in the Equation of Motion for a Pipe Carrying Fluid ? Y The term 2? Av ? x? t is the force required to dislocate the ? id element as each point 2 in the bridgework rotates with angular velocity. This force is a result of Coriolis E? ect. Fig(2. 5 ) shows a schematic view of this force F3. 11 Y F4 X Z M ? 2Y ? t2 Figure 2. 6 Inertia Force Representation of the Fourth Term in the Equation of Motion for a Pipe Carrying Fluid The term M ? Y is a force component acting on the pipe as a result of Inertia ? t2 of the pipe and the ? uid ? owing through it. Fig(2. 6) shows a schematic view of this force F4. 2 12 2. 2 Finite Element Model Consider a pipeline span that has a transverse de? ection Y(x,t) from its equillibrium position.The length of the pipe is L,modulus of elasticity of the pipe is E,and the area moment of inertia is I. The immersion of the ? uid ? owing through the pipe is ? at pressure p and constant velocity v,through the internal pipe cross section having area A. Flow of the ? uid through the de? ecting pipe is accelerated due to the changing curvature of the pipe and the lateral vibration of the pipeline. From the forward section we have the equation of motion for free vibration of a ? uid convering pipe EI ? 2Y ? 2Y ? 2Y ? 4Y + ? Av 2 2 + 2? Av +M 2 =0 ? x4 ? x ? x? t ? t (2. 10) 2. 2. 1 Shape Functions The essence of the ? ite element method,is to approximate the apart(p) by an scene prone as n w= i=1 Ni ai where Ni are the interpolating design functions prescribed in term of linear independent functions and ai are a set of unknown parameters. We shall now derive the shape functions for a pipe element. 13 Y R R x L2 L L1 X Figure 2. 7 Pipe Carrying Fluid Consider an pipe of length L and allow at point R be at outgo x from the left end. L2=x/L and L1=1-x/L. Forming Shape Functions N 1 = L12 (3 ? 2L1) N 2 = L12 L2L N 3 = L22 (3 ? 2L2) N 4 = ? L1L22 L change the observes of L1 and L2 we get (2. 11) (2. 12) (2. 13) (2. 14) N 1 = (1 ? /l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 15) (2. 16) (2. 17) (2. 18) 14 2. 2. 2 Formulating the Sti? ness Matrix for a Pipe Carrying Fluid ?1 ?2 W1 W2 Figure 2. 8 Beam Element Model For a two dimensional be am element, the chemise matrix in footing of shape functions can be express as ? ? w1 ? ? ? ? ? ?1 ? ? ? W (x) = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 19) where N1, N2, N3 and N4 are the displacement shape functions for the two dimensional beam element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? and at end 2 is given by w2 , ? 2. Consider the point R inside the beam element of length L as shown in ? gure(2. 7) Let the internal make qualification at point R is given by UR . The internal strain talent at point R can be expressed as 1 UR = ? 2 where ? is the stress and is the strain at the point R. (2. 20) 15 ? E 1 ? Figure 2. 9 tattleship between Stress and Strain, Hooks Law Also ? =E Relation between stress and strain for elastic secular, Hooks Law subbing the take to be of ? from equation(2. 21) into equation(2. 20) yields 1 UR = E 2 (2. 21) 2 (2. 22) 16 ???? ???? A1 z B1 w A z B u x Figure 2. 0 Plain sections re main plane Assuming plane sections remain same, = du dx (2. 23) (2. 24) (2. 25) u=z dw dx d2 w =z 2 dx To obtain the internal energy for the whole beam we integrate the internal strain energy at point R over the volume. The internal strain energy for the entire beam is given as UR dv = U vol (2. 26) replace the economic value of from equation(2. 25) into (2. 26) yields U= vol 1 2 E dv 2 (2. 27) tawdriness can be expressed as a product of area and length. dv = dA. dx (2. 28) 17 based on the above equation we now integrate equation (2. 27) over the area and over the length. L U= 0 A 1 2 E dAdx 2 (2. 29) interchange the value of rom equation(2. 25) into equation (2. 28) yields L U= 0 A 1 d2 w E(z 2 )2 dAdx 2 dx (2. 30) Moment of Inertia I for the beam element is given as ?? = ???? ???? dA z Figure 2. 11 Moment of Inertia for an Element in the Beam I= z 2 dA (2. 31) Substituting the value of I from equation(2. 31) into equation(2. 30) yields L U = EI 0 1 d2 w 2 ( ) dx 2 dx2 (2. 32) T he above equation for enumerate internal strain energy can be re indite as L U = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 33) 18 The potential energy of the beam is nothing but the total internal strain energy. Therefore, L ? = EI 0 1 d2 w d2 w ( )( )dx 2 dx2 dx2 (2. 34)If A and B are two matrices then applying matrix station of the transpose, yields (AB)T = B T AT (2. 35) We can express the Potential susceptibility expressed in equation(2. 34) in terms of displacement matrix W(x)equation(2. 19) as, 1 ? = EI 2 From equation (2. 19) we have ? ? w1 ? ? ? ? ? ?1 ? ? ? W = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 ? ? N1 ? ? ? ? ? N 2? ? ? W T = ? ? w1 ? 1 w2 ? 2 ? ? ? N 3? ? ? N4 L (W )T (W )dx 0 (2. 36) (2. 37) (2. 38) Substituting the determine of W and W T from equation(2. 37) and equation(2. 38) in equation(2. 36) yields ? N1 ? ? ? N 2 ? w1 ? 1 w2 ? 2 ? ? ? N 3 ? N4 ? ? ? ? ? ? N1 ? ? ? ? ? w1 ? ? ? ? ?1 ? ? ? ? ? dx (2. 39) ? ? ? w2? ? ? ?2 1 ? = EI 2 L 0 N2 N3 N4 19 where N1, N2, N3 and N4 are the displacement shape functions for the two dimensional beam element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? 1 and at end 2 is given by w2 , ? 2. 1 ? = EI 2 L 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ?? ? w1 ?? ? ?? ? ? ? ? 1 ? ?? ? ? ? ? dx ?? ? ? ?w2? ?? ? ? 2 (2. 40) where ? 2 (N 1 ) ? ? L ? N 2 N 1 ? K = ? 0 ? N 3 N 1 ? ? N4 N1 N1 N2 (N 2 )2 N3 N2 N4 N2N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? N4 N3 ? ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 41) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 42) (2. 43) (2. 44) (2. 45) The element sti? ness matrix for the beam is obtained by substituting the set of shape functions from equations (2. 42) to (2. 45) into equation(2. 41) and desegregation every element in the matrix in equation(2. 40) over the length L. 20 The El ement sti? ness matrix for a beam element ? ? 12 6l ? 12 6l ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? K e = 3 ? ? l ?? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (2. 46) 1 2. 2. 3 Forming the Matrix for the Force that conforms the Fluid to the Pipe A X ? r ? _______________________ x R Y Figure 2. 12 Pipe Carrying Fluid Model B Consider a pipe carrying ? uid and let R be a point at a distance x from a character plane AB as shown in ? gure(2. 12). Due to the ? ow of the ? uid through the pipe a force is introduced into the pipe causing the pipe to curve. This force conforms the ? uid to the pipe at all times. Let W be the transverse de? ection of the pipe and ? be angle made by the pipe due to the ? uid ? ow with the neutral axis. ? and ? tally the unit vectors along the X i j ? nd Y axis and r and ? represent the two unit vectors at point R along the r and ? ? ? axis. At point R,the vectors r and ? can be expressed as ? r = cos?? + sin?? ? i j (2. 47) ? ? = ? sin?? + cos?? i j Expressi on for slope at point R is given by tan? = dW dx (2. 48) (2. 49) 22 Since the pipe undergoes a small de? ection, hence ? is very small. Therefore tan? = ? ie ? = dW dx (2. 51) (2. 50) The displacement of a point R at a distance x from the reference plane can be expressed as ? R = W ? + r? j r We di? erentiate the above equation to get velocity of the ? uid at point R ? ? ? j ? r ? R = W ? + r? + rr ? r = vf ? here vf is the velocity of the ? uid ? ow. Also at time t r ? d? r= ? dt ie r ?? ? d? d? = ?? r= ? d? dt ? Substituting the value of r in equation(2. 53) yields ? ? ?? ? ? j ? r R = W ? + r? + r?? (2. 57) (2. 56) (2. 55) (2. 53) (2. 54) (2. 52) ? Substituting the value of r and ? from equations(2. 47) and (2. 48) into equation(2. 56) ? yields ? ? ? ?j ? R = W ? + rcos?? + sin?? + r? ? sin?? + cos?? i j i j Since ? is small The velocity at point R is expressed as ? ? ? i ? j R = Rx? + Ry ? (2. 59) (2. 58) 23 ? ? i ? j ? ? R = (r ? r?? )? + (W + r? + r? )? ? ? The Y component of velocity R cause the pipe carrying ? id to curve. Therefore, (2. 60) 1 ? ? ? ? T = ? f ARy Ry (2. 61) 2 ? ? where T is the kinetic energy at the point R and Ry is the Y component of velocity,? f is the density of the ? uid,A is the area of cross-section of the pipe. ? ? Substituting the value of Ry from equation(2. 60) yields 1 ? ? ? ? ? ? ? ? ? T = ? f AW 2 + r2 ? 2 + r2 ? 2 + 2W r? + 2W ? r + 2rr?? 2 (2. 62) Substituting the value of r from equation(2. 54) and selecting the ? rst,second and the ? fourth terms yields 1 2 ? ? T = ? f AW 2 + vf ? 2 + 2W vf ? 2 (2. 63) Now substituting the value of ? from equation(2. 51) into equation(2. 3) yields dW 2 dW dW 1 2 dW 2 ) + vf ( ) + 2vf ( )( ) T = ? f A( 2 dt dx dt dx From the above equation we have these two terms 1 2 dW 2 ? f Avf ( ) 2 dx 2? f Avf ( dW dW )( ) dt dx (2. 65) (2. 66) (2. 64) The force acting on the pipe due to the ? uid ? ow can be calculated by combine the expressions in equations (2. 65) and (2. 66) over the length L. 1 2 dW 2 ? f Avf ( ) 2 dx (2. 67) L The expression in equation(2. 67) represents the force that causes the ? uid to conform to the curvature of the pipe. 2? f Avf ( L dW dW )( ) dt dx (2. 68) 24 The expression in equation(2. 68) represents the coriolis force which causes the ? id in the pipe to whip. The equation(2. 67) can be expressed in terms of displacement shape functions derived for the pipe ? =T ? V ? = L 1 2 dW 2 ? f Avf ( ) 2 dx (2. 69) Rearranging the equation 2 ? = ? f Avf L 1 dW dW ( )( ) 2 dx dx (2. 70) For a pipe element, the displacement matrix in terms of shape functions can be expressed as ? ? w1 ? ? ? ? ? ?1 ? ? ? W (x) = N 1 N 2 N 3 N 4 ? ? ? ? ? w2? ? ? ?2 (2. 71) where N1, N2, N3 and N4 are the displacement shape functions pipe element as stated in equations (2. 15) to (2. 18). The displacements and rotations at end 1 is given by w1, ? 1 and at end 2 is given by w2 , ? . Refer to ? gure(2. 8). Substituting the shape functions determined in equations (2. 15) t o (2. 18) ? ? N1 ? ? ? ? ? N 2 ? ? ? ? N1 w1 ? 1 w2 ? 2 ? ? ? N3 ? ? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? N 4 ? ? dx (2. 72) ? ? ? w2? ? ? ?2 L 2 ? = ? f Avf 0 N2 N3 25 L 2 ? = ? f Avf 0 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 )2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 ?? ? w1 ?? ? ?? ? ? ? ? 1 ? ?? ? ? ? ? dx ?? ? ? ?w2? ?? ? ? 2 (2. 73) where (N 1 ) ? ? L ? N 2 N 1 ? ? 0 ? N 3 N 1 ? ? N4 N1 ? 2 N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 ) 2 N1 N4 ? 2 K2 = ? f Avf N4 N3 ? N2 N4 ? ? ? dx ? N3 N4 ? ? 2 (N 4 ) (2. 74) The matrix K2 represents the force that conforms the ? uid to the pipe. Substituting the values of shape functions equations(2. 15) to (2. 18) and integrating it over the length gives us the elemental matrix for the ? 36 3 ? 36 ? ? 4 ? 3 ? Av 2 ? 3 ? K2 e = ? 30l ?? 36 ? 3 36 ? ? 3 ? 1 ? 3 above force. ? 3 ? ? ? 1? ? ? ? ? 3? ? 4 (2. 75) 26 2. 2. 4 Dissipation Matrix Formulation for a Pipe carrying Fluid The superab undance matrix represents the force that causes the ? uid in the pipe to whip creating instability in the system. To formulate this matrix we recall equation (2. 4) and (2. 68) The dissipation function is given by D= L 2? f Avf ( dW dW )( ) dt dx (2. 76) Where L is the length of the pipe element, ? f is the density of the ? uid, A area of cross-section of the pipe, and vf velocity of the ? uid ? ow. Recalling the displacement shape functions mentioned in equations(2. 15) to (2. 18) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 77) (2. 78) (2. 79) (2. 80) The Dissipation Matrix can be expressed in terms of its displacement shape functions as shown in equations(2. 77) to (2. 80). ? ? N1 ? ? ? ? ? N 2 ? L ? ? D = 2? Avf ? N1 N2 N3 N4 w1 ? 1 w2 ? 2 ? ? ? 0 N3 ? ? ? ? N4 (N 1 ) ? ? ? N 2 N 1 ? w1 ? 1 w2 ? 2 ? ? ? N 3 N 1 ? N4 N1 ? 2 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 (2. 81) N1 N2 (N 2 )2 N3 N2 N4 N2 N1 N3 N2 N3 (N 3 ) 2 N4 N3 N1 N4 N2 N4 N3 N4 (N 4 )2 L 2? f Avf 0 ?? ? w1 ?? ? ?? ? ? ? ? 1 ? ?? ? ? ? ? dx ?? ? ? ?w2? ?? ? ? 2 (2. 82) 27 Substituting the values of shape functions from equations(2. 77) to (2. 80) and integrating over the length L yields ? ? ? 30 6 30 ? 6 ? ? ? ? 0 6 ? 1? ?Av ? 6 ? ? De = ? ? 30 ?? 30 ? 6 30 6 ? ? ? ? ? 6 1 ? 6 0 De represents the elemental dissipation matrix. (2. 83) 28 2. 2. 5Inertia Matrix Formulation for a Pipe carrying Fluid Consider an element in the pipe having an area dA, length x, volume dv and mass dm. The density of the pipe is ? and let W represent the transverse displacement of the pipe. The displacement model for the Assuming the displacement model of the element to be W (x, t) = N we (t) (2. 84) where W is the vector of displacements,N is the matrix of shape functions and we is the vector of nodal displacements which is assumed to be a function of time. Let the nodal displacement be expressed as W = weiwt Nodal Velocity can be found by di? erentiating the equation() with time. W = (iw)weiwt (2. 86) (2. 85) Kinetic Energy of a touch can be expressed as a product of mass and the square of velocity 1 T = mv 2 2 (2. 87) Kinetic energy of the element can be found out by integrating equation(2. 87) over the volume. Also,mass can be expressed as the product of density and volume ie dm = ? dv T = v 1 ? 2 ? W dv 2 (2. 88) The volume of the element can be expressed as the product of area and the length. dv = dA. dx (2. 89) Substituting the value of volume dv from equation(2. 89) into equation(2. 88) and integrating over the area and the length yields T = ? w2 2 ? ?W 2 dA. dx A L (2. 90) 29 ?dA = ?A A (2. 91) Substituting the value of A ?dA in equation(2. 90) yields ?? Aw2 2 T = ? W 2 dx L (2. 92) Equation(2. 92) can be written as ?? Aw2 2 T = ? ? W W dx L (2. 93) The Lagrange equations are given by d dt where L=T ? V (2. 95) ? L ? w ? ? ? L ? w = (0) (2. 94) is called the Lagrangian function, T is the kinetic energy, V is the potential e nergy, ? W is the nodal displacement and W is the nodal velocity. The kinetic energy of the element e can be expressed as Te = ?? Aw2 2 ? ? W T W dx L (2. 96) ? and where ? is the density and W is the vector of velocities of element e. The expression for T using the eq(2. 9)to (2. 21) can be written as ? ? N1 ? ? ? ? ? N 2? ? ? w1 ? 1 w2 ? 2 ? ? N 1 N 2 N 3 N 4 ? ? ? N 3? ? ? N4 ? ? w1 ? ? ? ? ? ?1 ? ? ? ? ? dx ? ? ? w2? ? ? ?2 ?? Aw2 T = 2 e (2. 97) L 30 Rewriting the above expression we get ? (N 1)2 ? ? ? N 2N 1 ?? Aw2 ? Te = w1 ? 1 w2 ? 2 ? ? 2 L ? N 3N 1 ? N 4N 1 ?? ? N 1N 2 N 1N 3 N 1N 4 w1 ?? ? ?? ? 2 (N 2) N 2N 3 N 2N 4? ? ? 1 ? ?? ? ? ? ? dx ?? ? N 3N 2 (N 3)2 N 3N 4? ?w2? ?? ? 2 N 4N 2 N 4N 3 (N 4) ? 2 (2. 98) Recalling the shape functions derived in equations(2. 15) to (2. 18) N 1 = (1 ? x/l)2 (1 + 2x/l) N 2 = (1 ? x/l)2 x/l N 3 = (x/l)2 (3 ? 2x/l) N 4 = ? (1 ? x/l)(x/l)2 (2. 9) (2. 100) (2. 101) (2. 102) Substituting the shape functions from eqs(2. 99) to (2. 102) into eq s(2. 98) yields the elemental mass matrix for a pipe. ? ? 156 22l 54 ? 13l ? ? ? ? 2 2? ? 22l 4l 13l ? 3l ? Ml ? M e = ? ? ? 420 ? 54 13l 156 ? 22l? ? ? ? 2 2 ? 13l ? 3l ? 22l 4l (2. 103) CHAPTER III FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 3. 1 Forming Global Sti? ness Matrix from Elemental Sti? ness Matrices Inorder to form a Global Matrix,we start with a 6&2156 null matrix,with its six degrees of freedom being rendition and rotation of each of the leaf nodes. So our Global Sti? ness matrix looks like this ? 0 ? ?0 ? ? ? ?0 =? ? ? 0 ? ? ? 0 ? ? 0 ? 0? ? 0? ? ? ? 0? ? ? 0? ? ? 0? ? ? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 KGlobal (3. 1) 31 32 The two 4&2154 element sti? ness matrices are ? ? 12 6l ? 12 6l ? ? ? ? 4l2 ? 6l 2l2 ? EI ? 6l ? ? e k1 = 3 ? ? l ?? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l ? 12 6l ? 12 6l ? (3. 2) ? ? ? ? 2 2? 4l ? 6l 2l ? EI ? 6l ? e k2 = 3 ? ? l ?? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 6l 2l ? 6l 4l (3. 3) We shall now build the global sti? ness matrix by inserting element 1 ? rst into the global sti? ness matrix. 6l ? 12 6l 0 0? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0? ? ? ? ? ? ? ?? 12 ? 6l 12 ? l 0 0? EI ? ? = 3 ? ? l ? 6l 2 2 2l ? 6l 4l 0 0? ? ? ? ? ? 0 0 0 0 0 0? ? ? ? ? 0 0 0 0 0 0 ? ? KGlobal (3. 4) Inserting element 2 into the global sti? ness matrix ? ? 6l ? 12 6l 0 0 ? ? 12 ? ? ? 6l 4l2 ? 6l 2l2 0 0 ? ? ? ? ? ? ? EI ?? 12 ? 6l (12 + 12) (? 6l + 6l) ? 12 6l ? ? KGlobal = 3 ? ? l ? 6l 2 2 2 2? ? 2l (? 6l + 6l) (4l + 4l ) ? 6l 2l ? ? ? ? ? 0 0 ? 12 ? 6l 12 ? 6l? ? ? ? ? 2 2 0 0 6l 2l ? 6l 4l (3. 5) 33 3. 2 Applying Boundary Conditions to Global Sti? ness Matrix for simply supported pipe with ? uid ? ow When the boundary conditions are applied to a simply supported pipe carrying ? uid, the 6&2156 Global Sti? ess Matrix formulated in eq(3. 5) is modi? ed to a 4&2154 Global Sti? ness Matrix. It is as follows Y 1 2 X L Figure 3. 1 Representation of Simply Supported Pipe Carrying Fluid ? ? 4l2 ?6l 2l2 0 KGlobalS ? ? ? ? EI ?? 6l (12 + 12) (? 6l + 6l) 6l ? ? ? = 3 ? ? l ? 2l2 (? 6l + 6l) (4l2 + 4l2 ) 2l2 ? ? ? ? ? 2 2 0 6l 2l 4l (3. 6) Since the pipe is supported at the two ends the pipe does not de? ect causing its two translational degrees of freedom to go to zero. Hence we end up with the Sti? ness Matrix shown in eq(3. 6) 34 3. 3 Applying Boundary Conditions to Global Sti? ness Matrix for a cantilever pipe with ? id ? ow Y E, I 1 2 X L Figure 3. 2 Representation of Cantilever Pipe Carrying Fluid When the boundary conditions are applied to a Cantilever pipe carrying ? uid, the 6&2156 Global Sti? ness Matrix formulated in eq(3. 5) is modi? ed to a 4&2154 Global Sti? ness Matrix. It is as follows ? (12 + 12) (? 6l + 6l) ? 12 6l ? KGlobalS ? ? ? ? ?(? 6l + 6l) (4l2 + 4l2 ) ? 6l 2l2 ? EI ? ? = 3 ? ? ? l ? ?12 ? 6l 12 ? 6l? ? ? ? 6l 2l2 ? 6l 4l2 (3. 7) Since the pipe is supported at one end the pipe does not de? ect or rotate at that end causing translational and rotational degrees of freedom at that end to go to zero.Hence we end up with the Sti? ness Matrix shown in eq(3. 8) 35 3. 4 MATLAB Programs for Assembling Global Matrices for Simply Supported and Cantilever pipe carrying ? uid In this section,we implement the method discussed in section(3. 1) to (3. 3) to form global matrices from the developed elemental matrices of a straight ? uid conveying pipe and these assembled matrices are later solved for the natural frequency and onset of instability of a cantlilever and simply supported pipe carrying ? uid utilizing MATLAB Programs. Consider a pipe of length L, modulus of elasticity E has ? uid ? wing with a velocity v through its inner cross-section having an outside diameter od,and thickness t1. The expression for critical velocity and natural frequency of the simply supported pipe carrying ? uid is given by wn = ((3. 14)2 /L2 ) vc = (3. 14/L) (E ? I/M ) (3. 8) (3. 9) (E ? I/? A) 3. 5 MATLAB program for a simply supported pipe carrying ? uid The issue of elements,de nsity,length,modulus of elasticity of the pipe,density and velocity of ? uid ? owing through the pipe and the thickness of the pipe can be de? ned by the user. Refer to addendum 1 for the complete MATLAB Program. 36 3. 6MATLAB program for a cantilever pipe carrying ? uid Figure 3. 3 Pinned-Free Pipe Carrying Fluid* The number of elements,density,length,modulus of elasticity of the pipe,density and velocity of ? uid ? owing through the pipe and the thickness of the pipe can be de? ned by the user. The expression for critical velocity and natural frequency of the cantilever pipe carrying ? uid is given by wn = ((1. 875)2 /L2 ) (E ? I/M ) Where, wn = ((an2 )/L2 ) (EI/M )an = 1. 875, 4. 694, 7. 855 vc = (1. 875/L) (E ? I/? A) (3. 11) (3. 10) Refer to Appendix 2 for the complete MATLAB Program. 0 * Flow Induced Vibrations,Robert D.Blevins,Krieger. 1977,P 297 CHAPTER IV FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 4. 1 Parametric Study Parametric study has been carried out in this chapter. The study is carried out on a single span steel pipe with a 0. 01 m (0. 4 in. ) diameter and a . 0001 m (0. 004 in. ) thick wall. The other parameters are tightfistedness of the pipe ? p (Kg/m3 ) 8000 assiduousness of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 human activity of elements n 10 Modulus Elasticity E (Gpa) 207 of MATLAB program for the simply supported pipe with ? uid ? ow is use for these set of parameters with alter ? uid velocity.Results from this study are shown in the form of graphs and tables. The fundamental frequency of vibration and the critical velocity of ? uid for a simply supported pipe 37 38 carrying ? uid are ? n 21. 8582 rad/sec vc 16. 0553 m/sec send back 4. 1 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 Frequency(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 Frequency Ratio(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 709 0. 0180 0 39 Figure 4. 1 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity The fundamental frequency of vibration and the critical velocity of ? uid for a Cantilever pipe carrying ? uid are ? n 7. 7940 rad/sec vc 9. 5872 m/sec 40 Figure 4. 2 Shape Function Plot for a Cantilever Pipe with increasing Flow Velocity Table 4. 2 Reduction of Fundamental Frequency for a Pinned-Free Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 2 4 6 8 9 9. 5872 0 0. 2086 0. 4172 0. 6258 0. 8344 0. 9388 1 Frequency(w) 7. 7940 7. 5968 6. 9807 5. 8549 3. 825 1. 9897 0 Frequency Ratio(w/wn) 1 0. 9747 0. 8957 0. 7512 0. 4981 0. 2553 0 41 Figure 4. 3 Reduction of Fundamental Frequency for a Cantilever Pipe with increasing Flow Velocity CHAPTER V FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH E, I v L Figure 5. 1 Repr esentation of Tapered Pipe Carrying Fluid 5. 1 Tapered Pipe Carrying Fluid Consider a pipe of length L, modulus of elasticity E. A ? uid ? ows through the pipe at a velocity v and density ? through the internal pipe cross-section. As the ? uid ? ows through the de? ecting pipe it is accelerated, because of the changing curvature 42 43 f the pipe and the lateral vibration of the pipeline. The vertical component of ? uid pressure applied to the ? uid element and the pressure force F per unit length applied on the ? uid element by the tube walls oppose these accelerations. The arousal parameters are given by the user. Density of the pipe ? p (Kg/m3 ) 8000 Density of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 Number of elements n 10 Modulus Elasticity E (Gpa) 207 of For these user de? ned values we introduce a taper in the pipe so that the clobber property and the length of the pipe with the taper or without the taper remain the same.This is done by keeping the inner diame ter of the pipe constant and varying the outside diameter. Refer to ? gure (5. 2) The pipe tapers from one end having a thickness x to the other end having a thickness Pipe Carrying Fluid 9. 8mm OD= 10 mm L=2000 mm x mm t =0. 01 mm ID= 9. 8 mm Tapered Pipe Carrying Fluid Figure 5. 2 Introducing a Taper in the Pipe Carrying Fluid of t = 0. 01mm such that the volume of substantive is equal to the volume of material 44 for a pipe with no taper. The thickness x of the tapered pipe is now calculated From ? gure(5. 2) we have Outer Diameter of the pipe with no taper(OD) 10 mm cozy Diameter of the pipe(ID) 9. mm Outer Diameter of thick end of the Tapered pipe (OD1 ) Length of the pipe(L) 2000 mm Thickness of thin end of the taper(t) 0. 01 mm Thickness of thick end of the taper x mm Volume of the pipe without the taper V1 = Volume of the pipe with the taper ? ? L ? 2 V2 = (OD1 ) + (ID + 2t)2 ? (ID2 ) 4 4 3 4 (5. 2) ? (OD2 ? ID2 )L 4 (5. 1) Since the volume of material distribute d over the length of the two pipes is equal We have, V1 = V2 (5. 3) Substituting the value for V1 and V2 from equations(5. 1) and (5. 2) into equation(5. 3) yields ? ? ? L ? 2 (OD2 ? ID2 )L = (OD1 ) + (ID + 2t)2 ? (ID2 ) 4 4 4 3 4 The outer diameter for the thick end of the tapered pipe can be expressed as (5. 4) OD1 = ID + 2x (5. 5) 45 Substituting values of outer diameter(OD),inner diameter(ID),length(L) and thickness(t) into equation (5. 6) yields ? 2 ? ? 2000 ? (10 ? 9. 82 )2000 = (9. 8 + 2x)2 + (9. 8 + 0. 02)2 ? (9. 82 ) 4 4 4 3 4 Solving equation (5. 6) yields (5. 6) x = 2. 24mm (5. 7) Substituting the value of thickness x into equation(5. 5) we get the outer diameter OD1 as OD1 = 14. 268mm (5. 8) Thus, the taper in the pipe varies from a outer diameters of 14. 268 mm to 9. 82 mm. 46The following MATLAB program is utilized to calculate the fundamental natural frequency of vibration for a tapered pipe carrying ? uid. Refer to Appendix 3 for the complete MATLAB program. Res ults obtained from the program are given in table (5. 1) Table 5. 1 Reduction of Fundamental Frequency for a Tapered pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 20 40 60 80 100 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 Frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 Frequency Ratio(w/wn) . 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0The fundamental frequency of vibration and the critical velocity of ? uid for a tapered pipe carrying ? uid obtained from the MATLAB program are ? n 51. 4917 rad/sec vc 103. 3487 m/sec CHAPTER VI RESULTS AND DISCUSSIONS In the present work, we have utilized numerical method techniques to form the basic elemental matrices for the pinned-pinned and pinned-free pipe carrying ? uid. Matlab programs have been developed and utilized to form global matrices from these elemental matrices and fundamental frequency for free vibration has been calculated for various pipe con? gurations and varying ? uid ? ow velocities.Consider a pipe carrying ? uid having the following user de? ned parameters. E, I v L v Figure 6. 1 Representation of Pipe Carrying Fluid and Tapered Pipe Carrying Fluid 47 48 Density of the pipe ? p (Kg/m3 ) 8000 Density of the ? uid ? f (Kg/m3 ) 1000 Length of the pipe L (m) 2 Number of elements n 10 Modulus Elasticity E (Gpa) 207 of Refer to Appendix 1 and Appendix 3 for the complete MATLAB program Parametric study carried out on a pinned-pinned and tapered pipe for the same material of the pipe and subjected to the same conditions reveal that the tapered pipe is more stable than a pinned-pinned pipe.Comparing the following set of tables justi? es the above statement. The fundamental frequency of vibration and the critical velocity of ? uid for a tapered and a pinned-pinned pipe carrying ? uid are ? nt 51. 4917 rad/sec ? np 21. 8582 rad/sec vct 103. 3487 m/sec vcp 16. 0553 m/sec Table 6. 1 Reduction of Fundamental Frequency for a Tapered Pipe with increasin g Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 20 40 60 80 100 103. 3487 0 0. 1935 0. 3870 0. 5806 0. 7741 0. 9676 1 Frequency(w) 40. 8228 40. 083 37. 7783 33. 5980 26. 5798 10. 7122 0 Frequency Ratio(w/wn) 0. 8100 0. 7784 0. 7337 0. 6525 0. 5162 0. 2080 0 9 Table 6. 2 Reduction of Fundamental Frequency for a Pinned-Pinned Pipe with increasing Flow Velocity Velocity of Fluid(v) Velocity Ratio(v/vc) 0 2 4 6 8 10 12 14 16. 0553 0 0. 1246 0. 2491 0. 3737 0. 4983 0. 6228 0. 7474 0. 8720 1 Frequency(w) 21. 8806 21. 5619 20. 5830 18. 8644 16. 2206 12. 1602 3. 7349 0. 3935 0 Frequency Ratio(w/wn) 1 0. 9864 0. 9417 0. 8630 0. 7421 0. 5563 0. 1709 0. 0180 0 The fundamental frequency for vibration and critical velocity for the onset of instability in tapered pipe is approximately three times larger than the pinned-pinned pipe,thus making it more stable. 50 6. 1 Contribution of the Thesis Developed Finite Element Model for vibration analysis of a Pipe Carrying Fluid. Implemented the above developed model to two di? erent pipe con? gurations Simply Supported and Cantilever Pipe Carrying Fluid. Developed MATLAB Programs to solve the Finite Element Models. fit(p) the e? ect of ? uid velocities and density on the vibrations of a thin walled Simply Supported and Cantilever pipe carrying ? uid. The critical velocity and natural frequency of vibrations were determined for the above con? gurations. Study was carried out on a variable wall thickness pipe and the results obtained show that the critical ? id velocity can be increased when the wall thickness is tapered. 6. 2 time to come Scope Turbulence in Two-Phase Fluids In single-phase ? ow,? uctuations are a direct consequence of turbulence developed in ? uid, whereas the situation is clearly more complex in two-phase ? ow since the ? uctuation of the mixture itself is added to the inherent turbulence of each phase. get going the study to a time dependent ? uid velocity ? owing through the pipe. BIBLIOGRAP HY 1 Doods. H. L and H. Runyan E? ects of High-Velocity Fluid Flow in the Bending Vibrations and Static Divergence of a Simply Supported Pipe.National astronautics and Space Administration Report NASA TN D-2870 June(1965). 2 Ashley,H and G. Haviland Bending Vibrations of a Pipe controversy Containing slick Fluid. J. Appl. Mech. 17,229-232(1950). 3 Housner,G. W Bending Vibrations of a Pipe aura Containing Flowing Fluid. J. Appl. Mech. 19,205-208(1952). 4 Long. R. H Experimental and Theoretical Study of transversal Vibration of a tube Containing Flowing Fluid. J. Appl. Mech. 22,65-68(1955). 5 Liu. H. S and C. D. hint Dynamic Response of Pipes Transporting Fluids. J. Eng. for Industry 96,591-596(1974). 6 Niordson,F. I. N Vibrations of a Cylinderical Tube Containing Flowing Fluid. Trans. Roy. Inst. Technol. Stockholm 73(1953). 7 Handelman,G. H A Note on the transverse Vibration of a tube Containing Flowing Fluid. Quarterly of utilise Mathematics 13,326-329(1955). 8 Nemat-Nassar,S. S. N. Prasad and G. Herrmann Destabilizing E? ect on VelocityDependent Forces in Nonconservative Systems. AIAA J. 4,1276-1280(1966). 51 52 9 Naguleswaran,S and C. J. H. Williams Lateral Vibrations of a Pipe transport a Fluid. J. Mech. Eng. Sci. 10,228-238(1968). 10 Herrmann. G and R. W.Bungay On the Stability of Elastic Systems Subjected to Nonconservative Forces. J. Appl. Mech. 31,435-440(1964). 11 Gregory. R. W and M. P. Paidoussis Unstable Oscillations of Tubular Cantilevers Conveying Fluid-I Theory. Proc. Roy. Soc. (London). Ser. A 293,512-527(1966). 12 S. S. Rao The Finite Element Method in Engineering. Pergamon Press Inc. 245294(1982). 13 Michael. R. Hatch Vibration Simulation Using Matlab and Ansys. Chapman and Hall/CRC 349-361,392(2001). 14 Robert D. Blevins Flow Induced Vibrations. Krieger 289,297(1977). Appendices 53 54 0. 1 MATLAB program for Simply Supported Pipe Carrying FluidMATLAB program for Simply Supported Pipe Carrying Fluid. % The f o l l o w i n g MATLAB Progr am c a l c u l a t e s t h e Fundamental % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a % simply supported pipe carrying f l u i d . % I n o r d e r t o perform t h e above t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are used t o c a l c u l a t e % Fundamental N a t u r a l % Frequency w . lc num elements =input ( Input number o f e l e m e n t s f o r beam ) % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e % has t o be d i v i d e d . n=1 num elements +1% Number o f nodes ( n ) i s e q u a l t o number o f %e l e m e n t s p l u s one n o d e l =1 num elements node2 =2 num elements +1 soap nodel=max( n o d e l ) max node2=max( node2 ) max node used=max( max nodel max node2 ) mnu=max no de used k=zeros (2? mnu ) % C r e a t i n g a G l o b a l S t i f f n e s s Matrix o f z e r o s 55 m =zeros (2? nu ) % C r e a t i n g G l o b a l Mass Matrix o f z e r o s x=zeros (2? mnu ) % C r e a t i n g G l o b a l Matrix o f z e r o s % f o r t h e f o r c e t h a t conforms f l u i d % to the curvature of the % pipe d=zeros (2? mnu ) % C r e a t i n g G l o b a l D i s s i p a t i o n Matrix o f z e r o s %( C o r i o l i s character ) t=num elements ? 2 L=2 % T o t a l l e n g t h o f t h e p i p e i n meters l=L/ num elements % Length o f an e l e m e n t t1 =. 0001 od = . 0 1 i d=od? 2? t 1 % t h i c k n e s s o f t h e p i p e i n meter % outer diameter of the pipe % inner diameter of the pipeI=pi ? ( od? 4? i d ? 4)/64 % moment o f i n e r t i a o f t h e p i p e E=207? 10? 9 roh =8000 rohw =1000 % Modulus o f e l a s t i c i t y o f t h e p i p e % Density of the pipe % d e n s i t y o f water ( FLuid ) M =roh ? pi ? ( od? 2? i d ? 2)/4 + rohw? pi ? . 2 5 ? i d ? 2 % mass per u n i t l e n g t h o f % the pipe + f l u i d rohA=rohw? pi ? ( . 2 5 ? i d ? 2 ) l=L/ num elements v=0 % v e l o c i t y o f t h e f l u i d f l o w i n g t h r o u g h t h e p i p e %v =16. 0553 z=rohA/M i=sqrt ( ? 1) wn= ( ( 3 . 1 4 ) ? 2 /L? 2)? sqrt (E? I /M) % N a t u r a l Frequency vc =(3. 14/L)? sqrt (E?I /rohA ) % C r i t i c a l V e l o c i t y 56 % Assembling G l o b a l S t i f f n e s s , D i s s i p a t i o n and I n e r t i a M a t r i c e s for j =1 num elements d o f 1 =2? n o d e l ( j ) ? 1 d o f 2 =2? n o d e l ( j ) d o f 3 =2? node2 ( j ) ? 1 d o f 4 =2? node2 ( j ) % S t i f f n e s s Matrix Assembly k ( dof1 , d o f 1 )=k ( dof1 , d o f 1 )+ (12? E? I / l ? 3 ) k ( dof2 , d o f 1 )=k ( dof2 , d o f 1 )+ (6? E? I / l ? 2 ) k ( dof3 , d o f 1 )=k ( dof3 , d o f 1 )+ (? 12? E? I / l ? 3 ) k ( dof4 , d o f 1 )=k ( dof4 , d o f 1 )+ (6? E? I / l ? 2 ) k ( dof1 , d o f 2 )=k ( dof1 , d o f 2 )+ (6? E?I / l ? 2 ) k ( dof2 , d o f 2 )=k ( do f2 , d o f 2 )+ (4? E? I / l ) k ( dof3 , d o f 2 )=k ( dof3 , d o f 2 )+ (? 6? E? I / l ? 2 ) k ( dof4 , d o f 2 )=k ( dof4 , d o f 2 )+ (2? E? I / l ) k ( dof1 , d o f 3 )=k ( dof1 , d o f 3 )+ (? 12? E? I / l ? 3 ) k ( dof2 , d o f 3 )=k ( dof2 , d o f 3 )+ (? 6? E? I / l ? 2 ) k ( dof3 , d o f 3 )=k ( dof3 , d o f 3 )+ (12? E? I / l ? 3 ) k ( dof4 , d o f 3 )=k ( dof4 , d o f 3 )+ (? 6? E? I / l ? 2 ) k ( dof1 , d o f 4 )=k ( dof1 , d o f 4 )+ (6? E? I / l ? 2 ) k ( dof2 , d o f 4 )=k ( dof2 , d o f 4 )+ (2? E? I / l ) k ( dof3 , d o f 4 )=k ( dof3 , d o f 4 )+ (? ? E? I / l ? 2 ) k ( dof4 , d o f 4 )=k ( dof4 , d o f 4 )+ (4? E? I / l ) % ?????????????????????????????????????????????? 57 % Matrix a s s e m b l y f o r t h e second term i e % f o r t h e f o r c e t h a t conforms % f l u i d to the curvature of the pipe x ( dof1 , d o f 1 )=x ( dof1 , d o f 1 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) x ( dof2 , d o f 1 )=x ( dof2 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) x ( dof3 , d o f 1 )=x ( dof3 , d o f 1 )+ (( ? 36? rohA? v ? 2)/30? l ) x ( dof4 , d o f 1 )=x ( dof4 , d o f 1 )+ ( ( 3 ? rohA? v ? 2)/30? l ) x ( dof1 , d o f 2 )=x ( dof1 , d o f 2 )+ ( ( 3 ? ohA? v ? 2)/30? l ) x ( dof2 , d o f 2 )=x ( dof2 , d o f 2 )+ ( ( 4 ? rohA? v ? 2)/30? l ) x ( dof3 , d o f 2 )=x ( dof3 , d o f 2 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof4 , d o f 2 )=x ( dof4 , d o f 2 )+ (( ? 1? rohA? v ? 2)/30? l ) x ( dof1 , d o f 3 )=x ( dof1 , d o f 3 )+ (( ? 36? rohA? v ? 2)/30? l ) x ( dof2 , d o f 3 )=x ( dof2 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof3 , d o f 3 )=x ( dof3 , d o f 3 )+ ( ( 3 6 ? rohA? v ? 2)/30? l ) x ( dof4 , d o f 3 )=x ( dof4 , d o f 3 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof1 , d o f 4 )=x ( dof1 , d o f 4 )+ ( ( 3 ? rohA? v ? 2)/30? ) x ( dof2 , d o f 4 )=x ( dof2 , d o f 4 )+ (( ? 1? rohA? v ? 2)/30? l ) x ( dof3 , d o f 4 )=x ( dof3 , d o f 4 )+ (( ? 3? rohA? v ? 2)/30? l ) x ( dof4 , d o f 4 )=x ( dof4 , d o f 4 )+ ( ( 4 ? rohA? v ? 2)/30? l ) % ?????????????????????????????????????????????? % D i s s i p a t i o n Matrix Assembly d ( dof1 , d o f 1 )=d ( dof1 , d o f 1 )+ (2? ( ? 30? rohA? v ) / 6 0 ) d ( dof2 , d o f 1 )=d ( dof2 , d o f 1 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 1 )=d ( dof3 , d o f 1 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) 58 d ( dof4 , d o f 1 )=d ( dof4 , d o f 1 )+ (2? ( ? 6? rohA? ) / 6 0 ) d ( dof1 , d o f 2 )=d ( dof1 , d o f 2 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof2 , d o f 2 )=d ( dof2 , d o f 2 )+ ( 2 ? ( 0 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 2 )=d ( dof3 , d o f 2 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof4 , d o f 2 )=d ( dof4 , d o f 2 )+ (2? ( ? 1? rohA? v ) / 6 0 ) d ( dof1 , d o f 3 )=d ( dof1 , d o f 3 )+ (2? ( ? 30? rohA? v ) / 6 0 ) d ( dof2 , d o f 3 )=d ( dof2 , d o f 3 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof3 , d o f 3 )=d ( dof3 , d o f 3 )+ ( 2 ? ( 3 0 ? rohA? v ) / 6 0 ) d ( dof4 , d o f 3 )=d ( dof4 , d o f 3 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) ( dof1 , d o f 4 )=d ( dof1 , d o f 4 )+ ( 2 ? ( 6 ? rohA? v ) / 6 0 ) d ( dof2 , d o f 4 )=d ( dof2 , d o f 4 )+ ( 2 ? ( 1 ? rohA? v ) / 6 0 ) d ( dof3 , d o f 4 )=d ( dof3 , d o f 4 )+ (2? ( ? 6? rohA? v ) / 6 0 ) d ( dof4 , d o f 4 )=d ( dof4 , d o f 4 )+ ( 2 ? ( 0 ? rohA? v ) / 6 0 ) % ???????????????????????????????????????????? % I n e r t i a Matrix Assembly m( dof1 , d o f 1 )=m( dof1 , d o f 1 )+ (156? M? l / 4 2 0 ) m( dof2 , d o f 1 )=m( dof2 , d o f 1 )+ (22? l ? 2? M/ 4 2 0 ) m( dof3 , d o f 1 )=m( dof3 , d o f 1 )+ (54? l ? M/ 4 2 0 ) m( dof4 , d o f 1 )=m( dof4 , d o f 1 )+ (? 3? l ? 2? M/ 4 2 0 ) m( dof1 , d o f 2 )=m( dof1 , d o f 2 )+ (22? l ? 2? M/ 4 2 0 ) m( dof2 , d o f 2 )=m( dof2 , d o f 2 )+ (4? M? l ? 3 / 4 2 0 ) m( dof3 , d o f 2 )=m( dof3 , d o f 2 )+ (13? l ? 2? M/ 4 2 0 ) m( dof4 , d o f 2 )=m( dof4 , d o f 2 )+ (? 3? M? l ? 3 / 4 2 0 ) 59 m( dof1 , d o f 3 )=m( dof1 , d o f 3 )+ (54? M? l / 4 2 0 ) m( dof2 , d o f 3 )=m( dof2 , d o f 3 )+ (13? l ? 2? M/ 4 2 0 ) m( dof3 , d o f 3 )=m( dof3 , d o f 3 )+ (156? l ? M/ 4 2 0 ) m( dof4 , d o f 3 )=m( dof4 , d o f 3 )+ (? 22? l ? 2? M/ 4 2 0 ) m( dof1 , d o f 4 )=m( dof1 , d o f 4 )+ (? 13? l ? 2?M/ 4 2 0 ) m( dof2 , d o f 4 )=m( dof2 , d o f 4 )+ (? 3? M? l ? 3 / 4 2 0 ) m( dof3 , d o f 4 )=m( dof3 , d o f 4 )+ (? 22? l ? 2? M/ 4 2 0 ) m( dof4 , d o f 4 )=m( dof4 , d o f 4 )+ (4? M? l ? 3 / 4 2 0 ) end k ( 1 1 , ) = % A p p l y i n g Boundary c o n d i t i o n s k( ,11)= k ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = k ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = k x(11 ,)= x( ,11)= x ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = x ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = x % G l o b a l Matrix f o r t h e % Force t h a t conforms f l u i d t o p i p e x1=? d(11 ,)= d( ,11)= d ( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = % G l o b a l S t i f f n e s s Matrix 60 d ( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = d d1=(? d ) Kg lobal=k+10? x1 m( 1 1 , ) = m( , 1 1 ) = m( ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) , ) = m( , ( 2 ? mnu? 2 ) ( 2 ? mnu? 2 ) ) = m affectionateness ( t ) zeros ( t ) H=? inv (m) ? ( d1 ) ? inv (m)? Kglobal eye ( t ) zeros ( t ) Evalue=eig (H) % E i g e n v a l u e s v r a t i o=v/ vc % V e l o c i t y Ratio % G l o b a l Mass Matrix % G l o b a l D i s s i p a t i o nMatrix i v 2=imag ( Evalue ) i v 2 1=min( abs ( i v 2 ) ) w1 = ( i v 2 1 ) wn w r a t i o=w1/wn vc % Frequency Ratio % Fundamental N a t u r a l f r e q u e n c y 61 0. 2 MATLAB Program for Cantilever Pipe Carrying Fluid MATLAB Program for Cantilever Pipe Carrying Fluid. % The f o l l o w i n g MATLAB Program c a l c u l a t e s t h e Fundamental % N a t u r a l f r e q u e n c y o f v i b r a t i o n , f r e q u e n c y r a t i o (w/wn) % and v e l o c i t y r a t i o ( v / vc ) , f o r a c a n t i l e v e r p i p e % carrying f l u i d . I n o r d e r t o perform t h e above t a s k t h e program a s s e m b l e s % E l e m e n t a l S t i f f n e s s , D i s s i p a t i o n , and I n e r t i a m a t r i c e s % t o form G l o b a l M a t r i c e s which are used % t o c a l c u l a t e Fundamental N a t u r a l % Frequency w . clc num elements =input ( Input number o f e l e m e n t s f o r Pipe ) % num elements = The u s e r e n t e r s t h e number o f e l e m e n t s % i n which t h e p i p e has t o be d i v i d e d . =1 num elements +1% Number o f nodes ( n ) i s % e q u a l t o number o f e l e m e n t s p l u s one n o d e l =1 num elements % Parameters used i n t h e l o o p s node2 =2 num elements +1 max nodel=max( n o d e l ) max node2=max( node2 ) max node used=max( max nodel max node2 ) mnu=max node used k=zeros (2? mnu ) % C r e a t i n g a G l o b a l S t i f f n e s s Matrix o f z e r o s 62 m =zeros (2? mnu ) % C r e a t i n g G l o b a l Mass Matrix o f z e r o s