Vibration of Continuous Systems

Broad, up-to-date coverage of advanced vibration analysis by the market-leading author Successful vibration analysis of continuous structural elements and systems requires a knowledge of material mechanics, structural mechanics, ordinary and partial differential equations, matrix methods, variational calculus, and integral equations. Fortunately, leading author Singiresu Rao has created Vibration of Continuous Systems, a new book that provides engineers, researchers, and students with everything they need to know about analytical methods of vibration analysis of continuous structural systems. Featuring coverage of strings, bars, shafts, beams, circular rings and curved beams, membranes, plates, and shells-as well as an introduction to the propagation of elastic waves in structures and solid bodies-Vibration of Continuous Systems presents: \* Methodical and comprehensive coverage of the vibration of different types of structural elements \* The exact analytical and approximate analytical methods of analysis \* Fundamental concepts in a straightforward manner, complete with illustrative examples With chapters that are independent and self-contained, Vibration of Continuous Systems is the perfect book that works as a one-semester course, self-study tool, and convenient reference.Content: Chapter 1 Introduction: Basic Concepts and Terminology (pages 1–32): Chapter 2 Vibration of Discrete Systems: Brief Review (pages 33–67): Chapter 3 Derivation of Equations: Equilibrium Approach (pages 68–84): Chapter 4 Derivation of Equations: Variational Approach (pages 85–122): Chapter 5 Derivation of Equations: Integral Equation Approach (pages 123–150): Chapter 6 Solution Procedure: Eigenvalue and Modal Analysis Approach (pages 151–173): Chapter 7 Solution Procedure: Integral Transform Methods (pages 174–204): Chapter 8 Transverse Vibration of Strings (pages 205–233): Chapter 9 Longitudinal Vibration of Bars (pages 234–270): Chapter 10 Torsional Vibration of Shafts (pages 271–316): Chapter 11 Transverse Vibration of Beams (pages 317–392): Chapter 12 Vibration of Circular Rings and Curved Beams (pages 393–419): Chapter 13 Vibration of Membranes (pages 420–456): Chapter 14 Transverse Vibration of Plates (pages 457–540): Chapter 15 Vibration of Shells (pages 541–606): Chapter 16 Elastic Wave Propagation (pages 607–646): Chapter 17 Approximate Analytical Methods (pages 647–699): Excerpt CHAPTER 1

Introduction: Basic Concepts and Terminology

1.1 CONCEPT OF VIBRATION

Any repetitive motion is called vibration or oscillation. The motion of a guitar string, motion felt by passengers in an automobile traveling over a bumpy road, swaying of tall buildings due to wind or earthquake, and motion of an airplane in turbulence are typical examples of vibration. The theory of vibration deals with the study of oscillatory motion of bodies and the associated forces. The oscillatory motion shown in Fig. 1.1(a) is called harmonic motion and is denoted as

x(t) = X cos ωt (1.1)

where X is called the amplitude of motion, ω is the frequency of motion, and t is the time. The motion shown in Fig. 1.1(b) is called periodic motion, and that shown in Fig. 1.1(c) is called nonperiodic or transient motion. The motion indicated in Fig. 1.1(d) is random or long-duration nonperiodic vibration.

The phenomenon of vibration involves an alternating interchange of potential energy to kinetic energy and kinetic energy to potential energy. Hence, any vibrating system must have a component that stores potential energy and a component that stores kinetic energy. The components storing potential and kinetic energies are called a spring or elastic element and a mass or inertia element, respectively. The elastic element stores potential energy and gives it up to the inertia element as kinetic energy, and vice versa, in each cycle of motion. The repetitive motion associated with vibration can be explained through the motion of a mass on a smooth surface, as shown in Fig. 1.2. The mass is connected to a linear spring and is assumed to be in equilibrium or rest at position 1. Let the mass m be given an initial displacement to position 2 and released with zero velocity. At position 2, the spring is in a maximum elongated condition, and hence the potential or strain energy of the spring is a maximum and the kinetic energy of the mass will be zero since the initial velocity is assumed to be zero. Because of the tendency of the spring to return to its unstretched condition, there will be a force that causes the mass m to move to the left. The velocity of the mass will gradually increase as it moves from position 2 to position 1. At position 1, the potential energy of the spring is zero because the deformation of the spring is zero. However, the kinetic energy and hence the velocity of the mass will be maximum at position 1 because of conservation of energy (assuming no dissipation of energy due to damping or friction). Since the velocity is maximum at position 1, the mass will continue to move to the left, but against the resisting force due to compression of the spring. As the mass moves from position 1 to the left, its velocity will gradually decrease until it reaches a value of zero at position 3. At position 3 the velocity and hence the kinetic energy of the mass will be zero and the deflection (compression) and hence the potential energy of the spring will be maximum. Again, because of the tendency of the spring to return to its uncompressed condition, there will be a force that causes the mass m to move to the right from position 3. The velocity of the mass will increase gradually as it moves from position 3 to position 1. At position 1, all of the potential energy of the spring has been converted to the kinetic energy of the mass, and hence the velocity of the mass will be maximum. Thus, the mass continues to move to the right against increasing spring resistance until it reaches position 2 with zero velocity. This completes one cycle of motion of the mass, and the process repeats; thus, the mass will have oscillatory motion.

The initial excitation to a vibrating system can be in the form of initial displacement and/or initial velocity of the mass element(s). This amounts to imparting potential and/or kinetic energy to the system. The initial excitation sets the system into oscillatory motion, which can be called free vibration. During free vibration, there will be exchange between potential and kinetic energies. If the system is conservative, the sum of potential energy and kinetic energy will be a constant at any instant. Thus, the system continues to vibrate forever, at least in theory. In practice, there will be some damping or friction due to the surrounding medium (e.g., air), which will cause loss of some energy during motion. This causes the total energy of the system to diminish continuously until it reaches a value of zero, at which point the motion stops. If the system is given only an initial excitation, the resulting oscillatory motion eventually will come to rest for all practical systems, and hence the initial excitation is called transient excitation and the resulting motion is called transient motion. If the vibration of the system is to be maintained in a steady state, an external source must replace continuously the energy dissipated due to damping.

1.2 IMPORTANCE OF VIBRATION

Any body having mass and elasticity is capable of oscillatory motion. In fact, most human activities, including hearing, seeing, talking, walking, and breathing, also involve oscillatory motion. Hearing involves vibration of the eardrum, seeing is associated with the vibratory motion of light waves, talking requires oscillations of the laryng (tongue), walking involves oscillatory motion of legs and hands, and breathing is based on the periodic motion of lungs. In engineering, an understanding of the vibratory behavior of mechanical and structural systems is important for the safe design, construction, and operation of a variety of machines and structures.

The failure of most mechanical and structural elements and systems can be associated with vibration. For example, the blade and disk failures in steam and gas turbines and structural failures in aircraft are usually associated with vibration and the resulting fatigue. Vibration in machines leads to rapid wear of parts such as gears and bearings, loosening of fasteners such as nuts and bolts, poor surface finish during metal cutting, and excessive noise. Excessive vibration in machines causes not only the failure of components and systems but also annoyance to humans. For example, imbalance in diesel engines can cause ground waves powerful enough to create a nuisance in urban areas. Supersonic aircraft create sonic booms that shatter doors and windows. Several spectacular failures of bridges, buildings, and dams are associated with wind-induced vibration, as well as oscillatory ground motion during earthquakes.

In some engineering applications, vibrations serve a useful purpose. For example, in vibratory conveyors, sieves, hoppers, compactors, dentist drills, electric toothbrushes, washing machines, clocks, electric massaging units, pile drivers, vibratory testing of materials, vibratory finishing processes, and materials processing operations such as casting and forging, vibration is used to improve the efficiency and quality of the process.

1.3 ORIGINS AND DEVELOPMENTS IN MECHANICS AND VIBRATION

The earliest human interest in the study of vibration can be traced to the time when the first musical instruments, probably whistles or drums, were discovered. Since that time, people have applied ingenuity and critical investigation to study the phenomenon of vibration and its relation to sound. Although certain very definite rules were observed in the art of music, even in ancient times, they can hardly be called science. The ancient Egyptians used advanced engineering concepts such as the use of dovetailed cramps and dowels in the stone joints of major structures such as the pyramids during the third and second millennia B.C.

As far back as 4000 B.C., music was highly developed and well appreciated in China, India, Japan, and perhaps Egypt. Drawings of stringed instruments such as harps appeared on the walls of Egyptian tombs as early as 3000 B.C. The British Museum also has a nanga, a primitive stringed instrument from 155 B.C. The present system of music is considered to have arisen in ancient Greece.

The scientific method of dealing with nature and the use of logical proofs for abstract propositions began in the time of Thales of Miletos (640–546 B.C.), who introduced the term electricity after discovering the electrical properties of yellow amber. The first person to investigate the scientific basis of musical sounds is considered to be the Greek mathematician and philosopher Pythagoras (582–507 B.C.). Pythagoras established the Pythagorean school, the first institute of higher education and scientific research. Pythagoras conducted experiments on vibrating strings using an apparatus called the monochord. Pythagoras found that if two strings of identical properties but different lengths are subject to the same tension, the shorter string produces a higher note, and in particular, if the length of the shorter string is one-half that of the longer string, the shorter string produces a note an octave above the other. The concept of pitch was known by the time of Pythagoras; however, the relation between the pitch and the frequency of a sounding string was not known at that time. Only in the sixteenth century, around the time of Galileo, did the relation between pitch and frequency become understood.

Daedalus is considered to have invented the pendulum in the middle of the second millennium B.C. One initial application of the pendulum as a timing device was made by Aristophanes (450–388 B.C.). Aristotle wrote a book on sound and music around 350 B.C. and documents his observations in statements such as "the voice is sweeter than the sound of instruments" and "the sound of the flute is sweeter than that of the lyre." Aristotle recognized the vectorial character of forces and introduced the concept of vectorial addition of forces. In addition, he studied the laws of motion, similar to those of Newton. Aristoxenus, who was a musician and a student of Aristotle, wrote a three-volume book called Elements of Harmony. These books are considered the oldest books available on the subject of music. Alexander of Afrodisias introduced the ideas of potential and kinetic energies and the concept of conservation of energy. In about 300 B.C., in addition to his contributions to geometry, Euclid gave a brief description of music in a treatise called Introduction to Harmonics. However, he did not discuss the physical nature of sound in the book. Euclid was distinguished for his teaching ability, and his greatest work, the Elements, has seen numerous editions and remains one of the most influential books of mathematics of all time. Archimedes (287–212 B.C.) is called by some scholars the father of mathematical physics. He developed the rules of statics. In his On Floating Bodies, Archimedes developed major rules of fluid pressure on a variety of shapes and on buoyancy.

China experienced many deadly earthquakes in ancient times. Zhang Heng, a historian and astronomer of the second century A.D., invented the world's first seismograph to measure earthquakes in A.D. 132. This seismograph was a bronze vessel in the form of a wine jar, with an arrangement consisting of pendulums surrounded by a group of eight lever mechanisms pointing in eight directions. Eight dragon figures, with a bronze ball in the mouth of each, were arranged outside the jar. An earthquake in any direction would tilt the pendulum in that direction, which would cause the release of the bronze ball in that direction. This instrument enabled monitoring personnel to know the direction, time of occurrence, and perhaps, the magnitude of the earthquake.

The foundations of modern philosophy and science were laid during the sixteenth century; in fact, the seventeenth century is called the century of genius by many. Galileo (1564–1642) laid the foundations for modern experimental science through his measurements on a simple pendulum and vibrating strings. During one of his trips to the church in Pisa, the swinging movements of a lamp caught Galileo's attention. He measured the period of the pendulum movements of the lamp with his pulse and was amazed to find that the time period was not influenced by the amplitude of swings. Subsequently, Galileo conducted more experiments on the simple pendulum and published his findings in Discourses Concerning Two New Sciences in 1638. In this work, he discussed the relationship between the length and the frequency of vibration of a simple pendulum, as well as the idea of sympathetic vibrations or resonance.

Although the writings of Galileo indicate that he understood the interdependence of the parameters—length, tension, density and frequency of transverse vibration—of a string, they did not offer an analytical treatment of the problem. Marinus Mersenne (1588–1648), a mathematician and theologian from France, described the correct behavior of the vibration of strings in 1636 in his book Harmonicorum Liber. For the first time, by knowing (measuring) the frequency of vibration of a long string, Mersenne was able to predict the frequency of vibration of a shorter string having the same density and tension. He is considered to be the first person to discover the laws of vibrating strings. The truth was that Galileo was the first person to conduct experimental studies on vibrating strings; however, publication of his work was prohibited until 1638, by order of the Inquisitor of Rome. Although Galileo studied the pendulum extensively and discussed the isochronism of the pendulum, Christian Huygens (1629–1695) was the person who developed the pendulum clock, the first accurate device developed for measuring time. He observed deviation from isochronism due to the nonlinearity of the pendulum, and investigated various designs to improve the accuracy of the pendulum clock.

The works of Galileo contributed to a substantially increased level of experimental work among many scientists and paved the way to the establishment of several professional organizations, such as the Academia Naturae in Naples in 1560, Academia dei Lincei in Rome in 1606, Royal Society in London in 1662, the French Academy of Sciences in 1766, and the Berlin Academy of Science in 1770.

The relation between the pitch and frequency of vibration of a taut string was investigated further by Robert Hooke (1635–1703) and Joseph Sauveur (1653–1716). The phenomenon of mode shapes during the vibration of stretched strings, involving no motion at certain points and violent motion at intermediate points, was observed independently by Sauveur in France (1653–1716) and John Wallis in England (1616–1703). Sauveur called points with no motion nodes and points with violent motion, loops. Also, he observed that vibrations involving nodes and loops had higher frequencies than those involving no nodes. After observing that the values of the higher frequencies were integral multiples of the frequency of simple vibration with no nodes, Sauveur termed the frequency of simple vibration the fundamental frequency and the higher frequencies, the harmonics. In addition, he found that the vibration of a stretched string can contain several harmonics simultaneously. The phenomenon of beats was also observed by Sauveur when two organ pipes, having slightly different pitches, were sounded together. He also tried to compute the frequency of vibration of a taut string from the measured sag of its middle point. Sauveur introduced the word acoustics for the time for the science of sound.

Isaac Newton (1642–1727) studied at Trinity College, Cambridge and later became professor of mathematics at Cambridge and president of the Royal Society of London. In 1687 he published the most admired scientific treatise of all time, Philosophia Naturalis Principia Mathematica. Although the laws of motion were already known in one form or other, the development of differential calculus by Newton and Leibnitz made the laws applicable to a variety of problems in mechanics and physics. Leonhard Euler (1707–1783) laid the groundwork for the calculus of variations. He popularized the use of free-body diagrams in mechanics and introduced several notations, including e = 2.71828 ..., f(x), [summation], and i = [square root of -1]. In fact, many people believe that the current techniques of formulating and solving mechanics problems are due more to Euler than to any other person in the history of mechanics. Using the concept of inertia force, Jean D'Alembert (1717–1783) reduced the problem of dynamics to a problem in statics. Joseph Lagrange (1736–1813) developed the variational principles for deriving the equations of motion and introduced the concept of generalized coordinates. He introduced Lagrange equations as a powerful tool for formulating the equations of motion for lumped-parameter systems. Charles Coulomb (1736–1806) studied the torsional oscillations both theoretically and experimentally. In addition, he derived the relation between electric force and charge.
(Continues...) Excerpted from Vibration of Continuous Systems by Singiresu S. Rao. Copyright © 2007 by John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons.
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Broad, up-to-date coverage of advanced vibration analysis by the market-leading author

Successful vibration analysis of continuous structural elements and systems requires a knowledge of material mechanics, structural mechanics, ordinary and partial differential equations, matrix methods, variational calculus, and integral equations. Fortunately, leading author Singiresu Rao has created Vibration of Continuous Systems, a new book that provides engineers, researchers, and students with everything they need to know about analytical methods of vibration analysis of continuous structural systems.

Featuring coverage of strings, bars, shafts, beams, circular rings and curved beams, membranes, plates, and shells—as well as an introduction to the propagation of elastic waves in structures and solid bodies—Vibration of Continuous Systems presents:

• Methodical and comprehensive coverage of the vibration of different types of structural elements
• The exact analytical and approximate analytical methods of analysis
• Fundamental concepts in a straightforward manner, complete with illustrative examples

With chapters that are independent and self-contained, Vibration of Continuous Systems is the perfect book that works as a one-semester course, self-study tool, and convenient reference.