Technology & Engineering
Free Vibration
Free vibration refers to the natural oscillation of a system in the absence of external forces or disturbances. This phenomenon occurs when a system is displaced from its equilibrium position and then released, causing it to oscillate at its natural frequency. Free vibration is a fundamental concept in engineering and is important for understanding the behavior of structures and mechanical systems.
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10 Key excerpts on "Free Vibration"
eBook - PDF- Alok Sinha(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND Free Vibration The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development and the performance of many modern engi-neering products: automobiles, jet engines, rockets, bridges, electric motors, electric generators, and so on. Whenever a mechanical sys-tem contains storage elements for kinetic and potential energies, there will be vibration. The vibration of a mechanical system is a contin-ual exchange between kinetic and potential energies. The vibration level is reduced by the presence of energy dissipation elements in the system. The problem of vibration is further accentuated because of the presence of time-varying external excitations, for example, the problem of resonance in a rotating machine, which is caused by the inevitable presence of rotor unbalance. There are many situations where the vibration is caused by internal excitation, which is depen-dent on the level of vibration. This type of vibration is known as self-excited oscillations, for example, the failure of the Tacoma suspension bridge (Billah and Scanlan, 1991) and the fluttering of an aircraft wing. This course deals with the characterization and the computation of the response of a mechanical system caused by time-varying excitations, which can be independent of or dependent on vibratory response. In general, the vibration level of a component of a machine has to be decreased to increase its useful life. As a result, the course also 1 2 Vibration of Mechanical Systems examines the methods used to reduce vibratory response. Further, this course also develops an input/output description of a dynamic system, which is useful for the design of a feedback control system in a future course in the curriculum.
eBook - PDF- Frank Fahy, David Thompson, Frank Fahy, David Thompson(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
77 Chapter 3 Fundamentals of vibration Brian Mace 3.1 INTRODUCTION Vibration of engineering structures is commonplace: for example, cars vibrate due to forces produced by the engine, or by road roughness; air-craft vibrate due to aerodynamic loads and turbulent boundary layers; and buildings vibrate due to forces produced by wind loading and earthquakes. We need to be able to model, analyse and design structures so that their vibrational behaviour is acceptable, and this requires us to understand the fundamentals behind the physics of vibration. Vibration is usually, but not always, a small-amplitude oscillatory motion about a static equilibrium position. Its effects are often unwanted: large displacements and stresses may occur, especially due to resonance, and these may lead to fatigue, breakage, wear or improper operation; structural vibration produces sound which radiates into the surrounding air (indeed, sound is also a form of vibration); vibration can produce physical discom-fort, motion sickness or physiological effects such as vibration white finger; instabilities can occur that lead to growing vibrations that often end in failure, such as flutter and galloping. However, not all vibration is undesir-able: music is one example. This chapter concerns the fundamental aspects of vibration. Emphasis will be placed on relatively simple systems (with one or two degrees of free-dom – see next section) and the fundamentals of their physical behaviour. In practice, vibration analysis often involves a finite element (FE) model of which the number of elements can be very large. Although this might be complicated, the underlying physics is the same – natural frequencies, reso-nance, modes and so on – and hence, understanding of the fundamentals is essential. 3.1.1 Some terminology and definitions The basic properties of a vibrating system are inertia (mass) and stiffness, which store kinetic and potential energy respectively, damping, which
eBook - PDFDesign Engineer's Reference Guide
Mathematics, Mechanics, and Thermodynamics
- Keith L. Richards(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
89 Mechanical Vibrations 6.1 INTRODUCTION Mechanical vibrations are defined as oscillations in mechanical dynamic systems and are the motions of a particle or a body or system of connected bodies that have been displaced from a posi-tion of equilibrium. The majority of vibrations are undesirable in machines or structures as they can result in increased stresses, causing increased wear such as fretting and increased bearing loads. Mechanical fatigue can also result from vibrations and rotating machine parts including aero engine parts and will need careful balancing in order to prevent any damage resulting from vibrations. Although most vibration problems are undesirable, such as the Tacoma Narrow Bridge failure in the United States in the 1940, and innumerable airframe failures resulting from vibration-induced fatigue, some mechanical systems such as the Beal free-piston Stirling engine rely on the vibration characteristics of the system to function correctly. In the mining and quarrying industries, these rely on sifting different sized particles using vibrating screed beds. In the manufacturing industry, vibra-tion conveyors are used to convey components from one machining process to another. Vibrations can be classified into four basic categories: 1. Free 2. Forced 3. Self-excited 4. Random Free Vibration of a system is vibration that occurs in the absence of any external force. External force acting on a system will cause forced vibrations; in this instance, the exciting force is continuously supplying energy to the system. These types of vibrations may be either determin-istic or random (see Figure 6.1). Self-excited vibrations are periodic and deterministic oscillations. Under certain conditions, the equilibrium state becomes unstable and any disturbances will cause the perturbations to grow until some effect limits any further growth.
eBook - PDF- D. G. Gorman, W. Kennedy(Authors)
- 2017(Publication Date)
- Butterworth-Heinemann(Publisher)
9 Vibration of a single degree of freedom system 9.1 Introduction In mechanical engineering there are many examples of cyclic or periodic motion, i.e. motion which is repeated in equal intervals of time. Perhaps the simplest example is the pendulum, which when displaced from its equilibrium position and subsequently released will perform oscillatory motion of a given frequency and time interval; the restoring effect in such a device is, of course, gravity. If, however, a mechanical system which possesses both mass and elasticity is deflected from its equilibrium position and then released, the ensuing oscillatory motion is referred to as vibration. The frequency of the resulting motion will correspond to one of the so-called natural frequencies of the system; these are the frequencies the system will adopt when influenced only by the local parameters of mass and stiffness, i.e. in the absence of external effects. Associated with each of these natural frequencies is a normal mode shape which depicts the manner of movement of the system as it performs periodic motion. The amplitude of the oscillatory motion will, for all practical systems, diminish with time due to the dissipation of kinetic energy resulting from molecular friction within the material providing the elasticity, or alternatively by some form of externally applied damping. In later chapters the more complex vibratory motion associated with mechanical systems which consists of distributed mass and elasticity, as well as those having many concentrated masses, will be examined in detail. In this chapter, however, we shall confine our analysis to the simplest configuration, namely, the single degree of freedom system which requires only a single coordinate to define the displacement of the system at any instant; this single coordinate may be the line displacement in a rectilinear system or the rotational displacement in an angular system.
eBook - ePub- William J. Bottega(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Free Vibration of Single Degree of Freedom SystemsThe most fundamental system germane to the study of vibrations is the single degree of freedom system. By definition (see Section 1.1 ), a single degree of freedom system is one for which only a single independent coordinate is needed to describe the motion of the system completely. It was seen in Chapter 1 that, under appropriate circumstances, many complex systems may be adequately represented by an equivalent single degree of freedom system. Further, it will be shown in later chapters that, under a certain type of transformation, the motion of discrete multi-degree of freedom systems and continuous systems can be decomposed into the motion of a series of independent single degree of freedom systems. Thus, the behavior of single degree of freedom systems is of interest in this context as well as in its own right. In the next few chapters the behavior of these fundamental systems will be studied and basic concepts of vibration will be introduced.2.1 Free Vibration of Undamped Systems
The oscillatory motion of a mechanical system may be generally characterized as one of two types, Free Vibration or forced vibration. Vibratory motions that occur without the action of external dynamic forces are referred to as Free Vibrations, while those resulting from dynamic external forces are referred to as forced vibrations. In this chapter we shall study Free Vibrations of single degree of freedom systems. We first establish the general form of the equation of motion and its associated solution.2.1.1 Governing Equation and System Response
It was seen in Chapter 1 that many mechanical systems can be represented as equivalent single degree of freedom systems and, in particular, as equivalent mass-spring systems. Let us therefore consider the system comprised of a mass m attached to a linear spring of stiffness k that is fixed at one end, as shown in Figure 2.1 . Let the mass be constrained so as to move over a horizontal frictionless surface, and let the coordinate x measure the position of the mass with respect to its rest configuration, as indicated. Thus, x = 0 corresponds to the configuration of the horizontally oriented system when the spring is unstretched. We wish to determine the motion of the mass as a function of time, given the displacement and velocity of the mass at the instant it is released. If we let the parameter t represent time, we will know the motion of a given system if we know x(t) for all times of interest. To accomplish this, we must first derive the equation of motion that governs the given system. This is expedited by examination of the dynamic free-body diagram (DFBD) for the system, also known as the kinetic diagram, depicted in Figure 2.2 . In that figure, the applied force acting on the system (the cause) is shown on the left-hand side of the figure, and the inertia force (the response) is shown on the right-hand side of the figure. The kinetic diagram is simply a pictorial representation of Newton’s Second Law of Motion and, as was seen in Section 1.5 , greatly aids in the proper derivation of the governing equations for complex systems. In Figure 2.2 , and throughout this text, we employ the notation that superposed dots imply (total) differentiation with respect to time (i.e., x ≡ dx/dt
- M. P. Norton, D. G. Karczub(Authors)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
A finite system undergoing Free Vibrations will vibrate in one or more of a series of specific patterns: for instance, consider the elementary case of a stretched string which is struck at a chosen point. Each of these specific vibration patterns is called a mode shape and it vibrates at a constant frequency, which is called a natural frequency. These natural frequencies are properties of the finite system itself and are related to its mass and stiffness (inertia and elasticity). It is interesting to note that if a system were infinite it would be able to vibrate freely at any frequency (this point is relevant to the propagation of sound waves). Forced vibrations, on the other hand, take place under the excitation of external forces. These excitation forces may be classified as being (i) harmonic, (ii) periodic, (iii) non-periodic (pulse or transient), or (iv) stochastic (random). Forced vibrations occur at the excitation frequencies, and it is important to note that these frequencies are arbitrary and therefore independent of the natural frequencies of the system. The phenomenon of resonance is encountered when a natural frequency of the system coincides with one of the exciting frequencies. The concepts of natural frequencies, modes of vibration, forced vibrations and resonance will be dealt with later on in this chapter, both from an elastic continuum viewpoint and from a macroscopic viewpoint. The concept of damping is also very important in the study of noise and vibration. Energy within a system is dissipated by friction, heat losses and other resistances, and any damped Free Vibration will therefore diminish with time. Steady-state forced vibrations can be maintained at a specific vibrational amplitude because the required energy is supplied by some external excitation force. At resonance, it is only the damping within a system which limits vibrational amplitudes.
eBook - PDFVirtual Experiments in Mechanical Vibrations
Structural Dynamics and Signal Processing
- Bin Tang, Michael J. Brennan(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
9 2 Fundamentals of Vibration 2.1 Introduction A vibrating system can be characterised in both the time and frequency domain. The quantities used to characterise the system can be obtained theoretically or experimentally, and are used extensively in this book. This chapter is devoted to deriving these quantities for a simple mechanical system. Thorough knowledge of such a system is essential for the deeper understanding of mechanical vibrations in general. Further, an understanding of the dynamics of a vibrating system in terms of its physical properties is extremely helpful in the interpretation of experimental data. No previous knowledge of vibrations is assumed in this chapter, as all the results are derived from first principles, requiring only a basic understanding of mechanics. 2.2 Basic Concepts – Mass, Stiffness, and Damping There are three fundamental physical properties of a vibrating system. They are mass, stiffness, and damping. Although they tend to exist in a distributed form in the real world, for an initial study of vibration it is convenient to represent them in lumped parameter form using idealised elements as shown in Figure 2.1. Note that only translational linear elements are considered for simplicity, rather than rotational and/or nonlinear elements, which also exist in the real world. The interested reader is referred to more-in-depth texts on linear and nonlinear vibration, such as Tse et al. (1978), Inman (2007), Worden and Tomlinson (2001), Thomsen (2003), Kovacic and Brennan (2011), and Rao (2016). The stiffness element is represented by a linear, massless spring with stiffness k, which has units of N/m. It is shown in Figure 2.1ai. The equations relating the forces at each end of the spring to the corresponding displacements are given by f 1 (t) = k(x 1 (t) − x 2 (t)) (2.1a) and f 2 (t) = k(x 2 (t) − x 1 (t)).
eBook - ePub- Juan Carlos A. Jauregui Correa, Alejandro A. Lozano Guzman(Authors)
- 2020(Publication Date)
- Academic Press(Publisher)
Chapter OneFundamentals of mechanical vibrations
Abstract
The basis for most conditioning monitoring systems is the analysis of vibration signals. Thus, a fundamental way to understand the causes and effects of vibrations is the study of the oscillatory movements and the interactions among the different components of a machine or a set of machines. These oscillatory movements, in the context of design and machinery analysis, are known as mechanical vibrations, or more generically vibrations. The essence of conditioning monitoring is the analysis of the relationship between the input signal (the source of vibration) and the output response (the output signal), and the evolution of the dynamic behavior of the machine. In most cases, the machinery can be considered a linear system, although there are some particular cases that will be analyzed in other chapters, and the output signal will be a linear response of the excitation forces. Even though a machine is a complex system composed of a large number of mechanical elements, its dynamic response can be represented as a simple lumped-mass system.Keywords
Mechanical vibrations; Forced vibrations; Free Vibrations; Multiple degrees of freedom; Continuous systemsGeneral considerations
The basis for most conditioning monitoring systems is the analysis of vibration signals. Thus, a fundamental part to understand the causes and effects of vibrations is the study of oscillatory movements and the interactions among the different components of a machine or a set of machines. These oscillatory movements, in the context of design and machinery analysis, are known as mechanical vibrations or, more generically, vibrations. The essence of conditioning monitoring is the analysis of the relationship between the input signal (the source of vibration) and the output response (the output signal), and the evolution of the dynamic behavior of the machine. In most cases, the machinery can be considered as a lineal system, although there are some particular cases that will be analyzed in other chapters, and the output signal will be a linear response of the excitation forces. Even though a machine is a complex system composed of a large number of mechanical elements, its dynamic response can be represented as a simple lumped-mass system.
eBook - PDFVibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications
Nonlinear Dynamic Effects, General Approach, Applications
- Iliya I Blekhman(Author)
- 2000(Publication Date)
- World Scientific(Publisher)
It is only since the beginning of this century that a period of a rapid development of vibrational technology has started. °By vibration we mean here mechanical oscillations whose period is much shorter than that at which the motion of the system is being considered, and whose swing is far smaller than the characteristic size of the system. 4 Chapter 1 Introduction Without that technology, a number of most important industries, such as min-ing, the processing of natural resources, chemical technology, metallurgy, the manufacture of building materials, and the erecting of various constructions would have been absolutely unthinkable. 6 The diversity of trends in using vibration is reflected in the titles of sec-tions of the book [114]: Vibration shifts, Vibration transforms (in Vibrorhe-ology), Vibration separates and classifies, Vibration intensifies the pro-cesses and the treatment of workpieces, Vibration consolidates - vibration destroys, Vibration unites (self-synchronization of unbalanced rotors), Vi-bration maintains rotation - vibration retards rotation, Vibration cancels vibration - vibration intensifies vibration (the generalized principle of auto-balancing), Vibration helps in measuring - vibration hinders measuring, Vibration cures - vibration causes diseases. The title of a popular book by Goncharevich is also remarkable: Vibration as a nonstandard way [209]. The use of vibration made it possible to literally revolutionize many indus-tries, providing a great technical and economical effect. Potentialities, however, have not yet been exhausted. The application of vibrational technology seems to be most promising in the future. Apart from the book [114], there is an extensive literature in Russian (mono-graphs and reference books) devoted to general and special problems of the use of vibration in technology (see, e.g., [46, 84, 145, 192, 207, 208, 209, 215, 226, 259, 301, 302, 306, 315, 323, 364, 400, 402, 420, 417, 440, 443, 445, 454, 475, 550, 159, 227]).
No longer available |Learn more- Michael Clifford(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
As well as solving static problems, most commercial finite element codes can also solve vibration problems by calculating a mass matrix in addition to the stiffness matrix. This can be used to set up and solve an eigenvalue problem in the form: ([ K ] 2 v 2 [ M ]){ X } 5 {0} This can be solved to give the natural frequencies and mode shapes for the structure. Learning summary By the end of this section you should have learnt: 3 a systematic approach for setting up the equations of motion for single-degree-of-freedom and lumped mass–spring systems using the following three steps Step 1 convert the physical structure into a dynamic mass–spring model Step 2 draw free-body diagram(s) Step 3 apply the appropriate form(s) of Newton’s second law of motion to give the equation(s) of motion for the system; 3 to obtain the natural frequency of single-degree-of-freedom systems from the coefficients in the equation of motion; 3 the equations of motion for lumped mass–spring systems and how to obtain the natural frequencies and the corresponding mode shapes; 3 to set up the boundary condition equations for shaft/beam vibration problems and obtain the frequency equation and an expression for the mode shapes. Structural vibration 405 6.3 Response of damped single- degree-of-freedom systems Introduction So far, we have looked at Free Vibration of systems and at their natural frequencies and mode shapes. This section will analyse the response of single-degree-of-freedom systems to external excitation. This takes the form either of applied forces and/or moments or of imposed displacement on part of the system. We will also introduce the effects of damping. Damping is the phenomenon that dissipates energy in a structure. If there was no damping and a structure was set vibrating and then left, the mathematics would suggest that it would carry on vibrating forever. This, of course, is impossible and the structure would stop vibrating sooner or later.
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