Genres: Science Engineering. This book serves as an introduction to the subject of vibration engineering at the undergraduate level. The style of the prior editions has been retained, with the theory, computational aspects, and applications of vibration presented in as simple a manner as possible. As in the previous editions, computer techniques of analysis are emphasized.
Expanded explanations of the fundamentals are given, emphasizing physical significance and interpretation that build upon previous experiences in undergraduate mechanics.
Numerous examples and problems are used to illustrate principles and concepts. The book is ideal for undergraduate students, researchers, and practicing engineers who are interested in developing a more thorough understanding of essential concepts in vibration analysis of mechanical systems.
Presents a clear connection between continuous beam models and finite degree of freedom models; Includes MATLAB code to support numerical examples that are integrated into the text narrative; Uses mathematics to support vibrations theory and emphasizes the practical significance of the results. Download Textbook Of Mechanical Vibrations books , This comprehensive and accessible book, now in its second edition, covers both mathematical and physical aspects of the theory of mechanical vibrations.
This edition includes a new chapter on the analysis of nonlinear vibrations. The text examines the models and tools used in studying mechanical vibrations and the techniques employed for the development of solutions from a practical perspective to explain linear and nonlinear vibrations.
To enable practical understanding of the subject, numerous solved and unsolved problems involving a wide range of practical situations are incorporated in each chapter. This text is designed for use by the undergraduate and postgraduate students of mechanical engineering.
Download Mechanical Vibrations 2nd Edition books , Written specifically for the students of Mechanical Engineering, "Mechanical Vibrations" is a succinctly written textbook. Without being verbose, the textbook delves into all concepts related to the subject and deals with them in a laconic manner. Concepts such as Freedom Systems, Vibration Measurement and Transient Vibrations have been treated well for the student to get profounder knowledge in the subject.
Download Mechanical Vibrations books , Aiming at undergraduate and postgraduate students of mechanical engineering, the book has been written with a long teaching experience of the author. Lucid and beyond traditional writing style makes the text different from other books. In this text, every effort has been taken to make the subject easy and interesting.
The concepts have been explained in such a manner that students do not require any prerequisite knowledge. The text amalgamated with real-world examples help students adhere to the book and learn the concepts on their own. Throughout the book, engaging and thought-provoking approach has been followed.
Chapter 5 shows how to obtain the response of single-degree-of-freedom systems to non-harmonic forcing functions. The Fourier series representation of a periodic function as a series of sines and cosines enables us to use the results of Chapter 4 to obtain the response. Then the Laplace transform method is introduced. This method provides a means of obtaining the response of a linear system, of any order, for most of the commonly found forcing functions.
The final section shows how to use Simulink to obtain the general forced response. Many practical vibration applications require a model having more than one degree of freedom in order to describe the important features of the system response. One of these, the root locus plot, was developed for control system design and has useful applications in vibration, but has been ignored in the vibration literature. In addition, MATLAB can be used not only to solve the differential equations numerically, but also to perform some of the algebra required to obtain closed form solutions.
Chapter 7 considers how to design systems to eliminate or at least reduce the effects of unwanted vibration. In order to determine how much the vibration should be reduced, we need to know what levels of vibration are harmful, or at least disagreeable. To reduce vibration, it is often important to understand the vibration source, and these two topics are discussed in the beginning of the chapter. The chapter then treats the design of vibration isolators, which consist of a stiffness element and perhaps a damping element, and which are placed between the vibration source and the surrounding environment.
The chapter also treats the design of vibration absorbers. The chapter concludes with a discussion of active vibration control, which uses a power source such as a hydraulic cylinder or an electric motor to provide forces needed to counteract the forces producing the unwanted vibration. For such systems it is more convenient to use matrix representation of the equations of motion and matrix methods to do the analysis. This is the topic of Chapter 8, which begins by showing how to represent the equations of motion in compact matrix form.
Then systematic procedures for analyzing the modal response are developed. Besides providing a compact form for representing the equations of motion and performing the analysis, matrix methods also form the basis for several powerful MATLAB and Simulink functions that provide useful tools for modal analysis and simulation.
These are described at the end of the chapter. In such cases we must resort to using measurements of the system response. Chapter 10 treats the vibration of systems that cannot be described adequately with lumped-parameter models consisting of ordinary differential equations. The chapter begins by considering how to model the simplest distributed system, a cable or string under tension. Torsional and longitudinal vibration of rods are also described by such a model. The partial differential equation model — the wave equation — is second-order, and we introduce two methods for solving such an equation.
The second solution method, separation of variables, is also useful for solving the fourth-order model that describes beam vibration. This model is also derived in the chapter. Examples are then provided to show how to use MATLAB to solve the resulting transcendental equations for the natural frequencies. In Chapter 11 we introduce the finite element method, which provides a more accurate system description than that used to develop a lumped-parameter model, but that for complex systems is easier to solve than partial differential equations.
The method is particularly useful for irregular geometries, such as bars that have variable cross sections, and for systems such as trusses that are made up of several bars. The text has three appendices. Appendix B is an introduction to numerical methods for solving differential equations, and it focuses on the Runge-Kutta family of algorithms. Appendix C contains physical property data for common materials. Typical Syllabi The first seven chapters constitute a basic course in mechanical vibration.
The remaining four chapters can be used to provide coverage of additional topics at the discretion of the instructor. Some examples of such coverage are as follows. Cover Chapter 8 for treatment of matrix methods and modal analysis. Cover Chapter 9 for an introduction to experimental methods, including spectral analysis and the Fast Fourier transform. Cover Chapter 10 for distributed parameter models.
Cover both Chapters 10 and 11 for finite element analysis. Chapters 8 through 11 can be covered in any order, with the exception that Chapter 10 should be covered before Chapter This feedback significantly influenced the final result. The author wishes to express his appreciation to Joe Hayton for this support, and to the rest of the editorial staff for their expert help during the production process.
The author is grateful to the following reviewers for providing especially useful comments in this project. The Department of Mechanical Engineering and Applied Mechanics at URI has always encouraged teaching excellence via textbook writing, and the author wishes to acknowledge particularly the support of the department chair, Professor Arun Shukla.
These are the spring element, which produces a restoring force or moment as a function of the displacement of the mass element, and the damping element, which produces a restoring force or moment as a function of the velocity of the mass element.
In this chapter we establish the basic principles for developing mathematical models of these elements and apply these principles to several commonly found examples.
To be useful, numerical values eventually must be assigned to the parameters of a mathematical model of a spring or a damping element, in order to make predictions about the behavior of the physical device. Therefore, this chapter includes two sections that show how to obtain numerical parameter values from data by using the least-squares method.
If you are such a reader, however, you are strongly encouraged to consider learning MATLAB, because it will be very useful to you in the future. Other examples are a guitar string, vibratory finishers, vibratory conveyors, and vibratory sieves for sorting objects by size. Perhaps the premier example of bad vibration is the building motion caused by an earthquake. In an engineering sense, good vibration is vibration that is useful, and bad vibration is vibration that causes discomfort or damage.
In this text you will learn how to design systems that make use of vibration and how to design systems that reduce or protect against vibration. The material is a necessary prerequisite to related but more specialized courses in finite element analysis, acoustics, modal analysis, active vibration control, and fatigue failure.
Inherent in the study of vibration is oscillation. Electrical circuits can have oscillatory voltages or currents, but these are not called vibratory systems, and their study is not called vibration.
Air pressure oscillation is called sound, but the study of sound is called acoustics, not vibration. The term vibration is usually used to describe the motion of mechanical objects that oscillate or have the potential to oscillate. For example, if we place a pendulum in molasses and give it a push, the pendulum will not oscillate because the fluid is so sticky, but the pendulum itself has the potential to oscillate under the influence of gravity—were it not for the molasses. Oscillation, or vibration, of a mechanical object is caused by a force or moment that tries to return the object to an equilibrium, or rest, position.
Such a force is called a restoring force or moment. Restoring forces and moments are usually functions of displacement, such as the gravitational moment acting on the pendulum. A common theme in mechanical vibration is the interplay between restoring forces and frictional forces, which are constant and oppose motion, or between restoring forces and damping forces, which also oppose motion but are velocity dependent like the force exerted on the pendulum by the molasses.
The study of mechanical vibration has a long tradition. We will not present a detailed history of the subject but merely mention several of the most important investigators. Pythagoras, who was born in B. Galileo, born in A. Of course, Newton, born in A. Many famous mathematicians and physicists contributed to our understanding of vibration. One of the most famous names in vibration is Rayleigh, who in published his classic work, The Theory of Sound. Many illuminating stories concerning the application of vibration theory to practical problems are given by Den Hartog [Den Hartog, ].
Knowledge of vibration is important for the modern engineer. One reason is that so many engineering devices contain or are powered by engines or motors. These often produce oscillatory forces that cause mechanical vibration which can result in unwanted noise, uncomfortable motion, or structural failure. Examples where the resulting vibration can have serious consequences include coolant pumps in submar- ines because of the noise they generate , fuel pumps in aircraft and rocket engines, helicopter rotors, turbines, and electrical generating machinery.
Often the motion of the object itself produces vibration. Examples are air turbulence acting on an aircraft and road forces acting on a vehicle suspension. The environment can produce vibration, as with earthquakes or wind forces acting on a structure.
These gradients can cause structural vibration that interferes with the operation of the instruments, as was the case early on with the Hubble space telescope until the problem was corrected. In fact, satellite structures are particularly susceptible to vibration because they must be light in weight for their size.
The parameter v is called the radian frequency. It is the frequency of oscillation of the function expressed as radians per unit time, for example, as radians per second. The related frequency is cycles per unit time and is often denoted by f. When expressed as cycles per second, the SI unit for frequency is the hertz, abbreviated as Hz.
Thus 1 Hz is one cycle per second. The sine function A sin vt is plotted in Figure 1. The amplitude A is shown in the figure. The period P is the time between two adjacent peaks and is thus the time required for the oscillation to repeat. Thus the velocity is zero when the displacement and acceleration are at a minimum or a maximum. The acceleration is at a minimum when the displacement is at a maximum. These functions are plotted in Figure 1. It occurs when the force acting on the mass is a restoring force that is proportional to the displacement.
To find A and f, given B and C, compare Equations 1. This function is illustrated in Figure 1. This function is shown in Figure 1. The oscillation amplitudes of Equation 1. A displacement represented by Equation 1. The exponential is not always a decaying exponential.
In this case the oscillation amplitude grows indefinitely with time. Table 1. Its solution is given in Table 1. Sometimes the quadratic roots are complex numbers, and in this case it is often more convenient to express the quadratic factor as shown in the table.
We will often need to work with algebraic expressions containing complex numbers. The properties of complex numbers that we will find useful are listed in Table 1. Note that a complex number z may be represented in several ways, two of which are based on Figure 1. TABLE 1. However, in order to make quantitative statements based on the resulting models, a set of units must be employed. A common system of units in business and industry in English-speaking countries has been the foot-pound-second FPS system.
The FPS system is also known as the U. The FPS system is a gravitational system. The pound is selected as the unit of force and the foot and second as units of length and time, respectively. Therefore, the unit of energy in this system is the foot pound ft-lb. Another energy unit in common use for historical reasons is the British thermal unit Btu. The relationship between the two is given in Table 1.
Power is the rate of change of energy with time, and a common unit is horsepower. The meter and the second are selected as the length and time units, and the kilogram is chosen as the mass unit.
The derived force unit is called the newton. However, mechanical systems are often made up of rigid bodies connected with elastic elements. Sometimes the element is not intended to be elastic but deforms anyway because it is subjected to large forces or torques.
This can be the case with a drive shaft that transmits the motor torque to the driven object. Un- intended deformation may also occur in the boom and cables of a crane lifting a heavy load. The most familiar spring is probably the helical-coil spring, such as those used in vehicle suspensions and those found in retractable pens and pencils.
The purpose of the spring in both applications is to provide a restoring force. However, considerably more engineering analysis is required for the vehicle spring application because the spring can cause undesirable motion of the wheel and chassis, such as vibration.
Because the pen motion is constrained and cannot vibrate, we do not need as sophisticated an analysis to see if the spring will work. Many engineering applications involving elastic elements, however, do not contain coil springs but rather involve the deformation of beams, cables, rods, and other mechanical members.
In this section we develop the basic elastic properties of many of these common elements, which are called spring elements. The free length is the length of the spring when no tensile or compressive forces are applied to it.
The free length is L. The greater the deflection compression or extension , the greater the restoring force. Some references, particularly in the automotive industry, use the term spring rate or stiffness instead of spring constant.
We must decide whether extension is represented by positive or negative values of x. This choice depends on the particular application. TensileTest of a Rod A plot of the data for a tension test on a rod is given in Figure 1. This experiment could have been repeated using compressive instead of tensile force.
Note that the larger the deformation, the greater the error that results from using the linear model. Formulas for Spring Constants Table 1. Some of the expressions in the table contain the constants E and G, which are the modulus of elasticity and the shear modulus of elasticity of the material.
The formula for the spring constant of a coil spring is derived in references on machine design. Other mechanical elements that have elasticity, such as beams, rods, and rubber mounts, are usually represented pictorially as a coil spring. An exception is the air spring, which is used in pneumatic suspension systems. Its spring constant must be derived from the thermodynamic relations for a gas. Analytical Determination of the Spring Constant In much engineering design work we do not have the elements available for testing, and thus we must be able to calculate their spring constant from the geometry and material properties.
To do this we can use results from the study of the strength of materials. The following examples show how this is accomplished. The rod length is L and its area is A. Thus a steel rod 0. Beams can have a variety of shapes and can be supported in a number of ways. The beam geometry, beam material, and the method of support determine its spring constant. This illustrates the effect of the support arrangement on the spring constant. A leaf spring in a vehicle suspension is shown in Figure 1.
Leaf springs are constructed by strapping together several beams. The value of the total spring constant depends not only on the spring constants of the individual beams but also on the how they are strapped together, the method of attachment to the axle and chassis, and whether any material to reduce friction has been placed between the layers.
Some formulas are available in the automotive literature. See, for example, [Bosch, ]. Torsional Spring Elements Table 1. This is an example of a torsional spring, which resists with an opposing torque when twisted. This is analogous to the free length position of a translational spring. Although the same symbol k is used for the spring constant, note that the units of the torsional and translational spring constants are not the same. For a hollow cylindrical shaft, Figure 1.
Thus there are two spring constants for a solid rod or a hollow shaft, a translational constant and a torsional constant. Figure 1. Fernand Porsche in the s. As the ground motion pushes the wheel up, the torsion bar twists and resists the motion.
A coil spring can also be designed for axial or torsional loading. Thus there are two spring constants for coil springs: a translational constant, which is given in Table 1. When two springs are connected side-by-side, as in Figure 1.
The symbols for springs connected end-to-end look like the symbols for electrical resistors similarly connected. Such resistors are said to be in series, and therefore springs connected end-to-end are sometimes said to be in series. However, the equivalent electrical resistance is the sum of the individual resistances, whereas series springs obey the reciprocal rule of Equation 1.
This similarity in appearance of the symbols often leads people to mistakenly add the spring constants of springs connected end-to- end, just as series resistances are added.
Springs connected side-by-side are sometimes called parallel springs, and their spring constants should be added. According to this usage, then, parallel springs have the same deflection; series springs experience the same force or torque. As we will see in later chapters, solution of vibration models to predict system behavior requires solution of differential equations.
We will see that differential equations based on linear models of the forces and moments are much easier to solve than ones based on nonlinear models. We therefore try to obtain a linear model whenever possible. Sometimes the use of a linear model results in a loss of accuracy, and the engineer must weigh this disadvantage with advantages gained by using a linear model.
If the model is nonlinear, we can obtain a linear model that is an accurate approximation over a limited range of the independent variable. The next example illustrates this technique, which is called linearization. Consider the pendulum shown Linearization of in Figure 1. If the spring in Figure 1.
The cosine function requires more care. This does not pose any difficulties because the potential energy expression does not appear in the differential equation of motion. It appears only in the integrated form, which is stated as conservation of mechanical energy.
Spring Constant Assume that the resulting motion is small enough to be only horizontal, and determine the of a Lever System expression for the equivalent spring constant that relates the applied force f to the resulting displacement x.
Solution From the triangle shown in Figure 1. Thus the free-body diagram is as shown in Figure 1. For static equilibrium, the net moment about point O must be zero. Thus the original system is equivalent to the system shown in Figure 1.
That is, the force f will cause the same displacement x in both systems. Note that although these two springs appear to be connected side-by-side, they are not in parallel, because they do not have the same deflection. Thus their equivalent spring constant is not given by the sum, 2k. The linear approximation may also be developed with an analytical approach based on the Taylor series. If Rn approaches zero for large n, the expansion is called the Taylor series.
This is a linear relation. These cases are shown in Figure 1. A hard spring is used to limit excessively large deflections. The spring stiffness k is the slope of the force-deflection curve and is constant for the linear spring element. For a hard spring, its slope and thus its stiffness increase with deflection.
The stiffness of a soft spring decreases with deflection. We therefore try to obtain a linear spring model whenever possible because the resulting equation of motion is much easier to solve. The following example illustrates how a linear model can be obtained from a nonlinear description. Best Mechanical Vibration Books Collection. Mechanical Vibration. Mechanical Vibrations. Mechanical Vibrations, SI Edition. Mechanical VIbrations. Structural Dynamics and Vibration in Practice.
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