Damped Free Vibration

12 Key excerpts on "Damped Free Vibration"

• 

  eBook - PDF

  Design Engineer's Reference Guide

  Mathematics, Mechanics, and Thermodynamics

  • Keith L. Richards(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  In practice, the vibrations will gradually reduce in amplitude and die away over a period of time. Figure 6.13 shows the response of a system to varying degrees of damping following an initial displacement ‘A’. The definition of damping is where the free vibration of a system is controlled. A system that is critically damped returns to its stable reference position as quickly as possible without any under- or overshoot. A good example is the machine gun when the recoil mechanism of a gun is designed with critical damping such that the system returns to its firing position in the quickest time without any overshoot. A further example is the automotive suspension system where the vehicle passes over an obstacle and returns the vehicle to its correct ride height as quickly as possible without any undesirable undershoot. 106 Design Engineer’s Reference Guide Damping can be accomplished by a number of methods including 1. Viscous damping 2. Coulomb damping (dry friction) 3. Inertial damping 4. Internal damping The basic nomenclature used in damping is covered in Table 6.1. Table 6.2 tabulates the natural frequencies and coefficients ‘K’ for various modes. 6.3.1 V ISCOUS D AMPING Damping that uses a fluid to provide the damping force is known as viscous damping. Figure 6.14a and b shows two types of viscous dampers. These kinds of dampers are also known as dashpots. In Figure 6.14a, the dashpot uses a fluid (usually oil) and the piston has a number of small orifices passing through the piston.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mechanical Vibrations

  Theory and Application to Structural Dynamics

  • Michel Geradin, Daniel J. Rixen(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Chapter 3 Damped Vibrations of n-Degree-of-Freedom Systems Damping forces have already been introduced briefly in Chapter 1 to underline their non-conservative character.

  Damping is one of the most important aspects to model and predict in structural analysis, as it determines for a major part the performance of a structural system or the amplitude of its vibrations. It is probably also one of the most uncertain and difficult ones to address in vibration analysis and structural dynamics, due to the variety and complexity of its physical origin. Indeed damping can arise from many different sources, such as (Mead 1998):

  • intrinsic damping of the structural material;
  • Coulomb friction and partial-impact energy loss at structural interfaces;
  • energy lost into surrounding air, water or ground and into contained fluids;
  • friction between the structure and mounted equipment, furnishings, payload and people;
  • viscous damping between sliding and lubricated machines surfaces.

  In all generality, the damping forces can be expressed as functions of the structural velocities , the strain rates in the material and the system configuration . The strains being themselves expressed as spatial derivatives of (see Chapter 4, Section 4.1), we may write in all generality:

  3.1

  and this function can be dependent on the history of the behaviour of the system. Inthis chapter, we will make the approximation that the damping is linear in the velocities (viscous damping) and independent of unless otherwise stated. It can thus be described by a dissipation function quadratic in velocities as introduced in Chapter 1, Section 1.3.3.

  Structural damping in itself is a huge subject which could not be covered in a single book chapter. The presentation will focus on the following aspects.

  In Section 3.1 the linear damping model will be described and its essential mathematical properties will be outlined. The classical assumption of modal damping which consists of neglecting the coupling of eigenmodes through damping will be introduced and its validity range discussed. A general method based on modal expansion to construct a proportional damping matrix matching experimentally measured modal damping ratios will be presented.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Vibration

  Fundamentals and Practice, Second Edition

  • Clarence W. de Silva(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  7 Vibration Damping 7.1 Introduction Damping is the phenomenon by which mechanical energy is dissipated (usually con-verted into internal thermal energy) in dynamic systems. Some knowledge of the level of damping in a dynamic system is important in the utilization, analysis, and testing of a system. For example, a device having natural frequencies within the seismic range (i.e., less than 33 Hz) and having relatively low damping could produce damaging motions under resonance conditions when subjected to a seismic disturbance. Also, the device motions could be further magnified by low-frequency support structures and panels having low damping. This illustrates that knowledge of damping in constituent devices, components, and support structure is particularly useful in the design and operation of a complex mechanical system. The nature and the level of component damping should be known in order to develop a dynamic model of the system and its peripherals. Know-ledge of damping in a system is also important in imposing dynamic environmental limitations on the system (i.e., the maximum dynamic excitation the system could with-stand) under in-service conditions. Furthermore, some knowledge of its damping could be useful in order to make design modifications in a system that has failed the accept-ance test. However, the significance of knowledge of damping level in a test object, for the development of test excitation (input), is often overemphasized. Specifically, if the response-spectrum method is used to represent the required excitation in a vibration test, it is not necessary that the damping value used in the development of the required response spectrum specification be equal to the actual damping in the test object. It is only necessary that the damping used in the specified response spectrum be equal to that used in the test-response spectrum.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Applied Solid Dynamics

  • 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Applied Structural and Mechanical Vibrations

  Theory and Methods, Second Edition

  • Paolo L. Gatti(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  This particular condi-tion is called unDamped Free Vibrations and a key point of this phenomenon consists in the fact that the frequency characteristics of the motion depend on the system’s parameters – that is, its mass and elasticity – while the amplitude characteristics depend on the initial conditions. x ( t ) k c m f ( t ) Figure 4.1 Harmonic oscillator. Table 4.1 Analogies between translational and rotational systems Translation Rotation Linear displacement x Angular displacement α Force f Torque M Spring constant k Spring constant k r Damping constant c Damping constant c r Mass m Moment of inertia J Spring law F = k ( x 1 − x 2 ) Spring law M = k r ( α 1 − α 2 ) Damping law F = -c x x ( ) dotnosp dotnosp 1 2 Damping law M c = -r 1 2 ( ) dotnosp dotnosp α α Inertia law F mx = dotnospdotnosp Inertia law M = J dotnospdotnosp α Single degree of freedom systems 121 When some kind of damping is present, on the other hand, energy is lost during the motion and the amplitude of the oscillation decreases with time until it stops completely; this is the case of Damped Free Vibrations . Once again, however, the frequency characteristics of the motion depend on the system’s parameters, and not on the initial conditions that started the motion (no musical instrument could be played in tune if this gen-eral rule did not apply). When, however, damping is ‘sufficiently high’, the system does not vibrate at all but quickly loses its initial energy and simply returns to its equilibrium position without oscillating. We will quantitatively determine the meaning of the term ‘sufficiently high’ in the following sections. 4.2.1 UnDamped Free Vibrations Let us now consider the simple ideal system of Figure 4.2 consisting of a mass m and a massless spring k .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Sound and Vibration

  • Frank Fahy, David Thompson, Frank Fahy, David Thompson(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  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 78 Fundamentals of sound and vibration, second edition dissipates energy, and external force, which provides energy. A state of free vibration is one in which no external forces act, while a forced vibration is one where external forces operate. The external forces might be harmonic (i.e. vary sinusoidally with time), transient (existing only for a finite time), periodic (the forces repeat at regular intervals) or random (unpredictable, such as produced by a rough road or by turbulence). Systems may be the-oretically modelled as being damped or undamped , although all passive physical systems are damped to some extent. We will only consider linear vibrations. For a linear system the principle of superposition holds. In practice, linearity is an idealisation, since all real systems are to some extent nonlinear, but very often they are only very weakly nonlinear, so that their behaviour can be predicted accurately enough by a linear model. A discrete system is one that is made up of a finite number of rigid masses connected by massless springs, while a continuous system is one for which there is a continuous distribution of mass and stiffness throughout the system – under loads, their deformed shapes are continuous in space, like a cantilever beam in bending. 3.1.2 Degrees of freedom The number of degrees of freedom (DOFs) of a system is defined as the number of independent coordinates necessary to describe its motion.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  An Introduction to Mechanical Engineering: Part 2

  • Michael Clifford(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  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. This is because all real structures dissipate energy to a greater or lesser extent. There are many ways this can take place and mechanisms include hysteresis effects in the material, friction between parts, aerodynamic interaction with the surrounding fluid and noise radiated from the surfaces. Most real damping mechanisms are difficult to handle mathematically and we will consider one theoretical damping model, called viscous damping and will only consider discrete dampers. These can be pictured as a piston–cylinder device in which a viscous fluid is displaced from one side of the piston to the other through a constriction such as an orifice. Vehicle shock absorbers are normally based on this idea. In the viscous damping model, we assume that the force in the damper is proportional to the relative velocity between the ends and acts to oppose the imposed motion. The constant of proportionality is called the damping coefficient (normally given the symbol c ) and has units of N/(m/s) [or Ns/m]. Hence the force opposing the motion is c ( x . 2 y . ). Note that dampers do not impose any stiffness on the structure; they only transmit a force if there is relative motion between the ends. If there is no motion, there is no force. In most engineering structures, the level of damping is low. As a result, any discrepancies between the assumed viscous damping model and the actual damping mechanism are generally small, so that the error introduced by our mathematical model is also small.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Engineering Vibration Analysis with Application to Control Systems

  • C. Beards(Author)
  • 1995(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  M , assuming the soil acts similarly in tension and compression is

  Thus the equation of motion becomes Motion is therefore simple harmonic, with frequency An alternative solution can be obtained by considering the energy in the system. In this case, and

  where the loss in potential energy of building weight is given by since for small values of θ. Thus

  Assuming simple harmonic motion, and putting T max = V max , gives

  as before. Note that for stable oscillation, ω > 0, so that

  That is ka 3 b > 12 mgh .

  This expression gives the minimum value of k , the soil stiffness, for stable oscillation of a particular building to occur. If k is less that 12 mgh/a 3 b the building will fall over when disturbed.

  2.2 FREE DAMPED VIBRATION

  All real systems dissipate energy when they vibrate. The energy dissipated is often very small, so that an undamped analysis is sometimes realistic; but when the damping is significant its effect must be included in the analysis, particularly when the amplitude of vibration is required. Energy is dissipated by frictional effects, for example that occurring at the connection between elements, internal friction in deformed members, and windage. It is often difficult to model damping exactly because many mechanisms may be operating in a system. However, each type of damping can be analysed, and since in many dynamic systems one form of damping predominates, a reasonably accurate analysis is usually possible.

  The most common types of damping are viscous, dry friction and hysteretic. Hysteretic damping arises in structural elements due to hysteresis losses in the material. The type and amount of damping in a system has a large effect on the dynamic response levels.

  2.2.1 Vibration with viscous damping

  Viscous damping

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Dynamics of Structures

  • J. Humar(Author)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  Following the presentation of undamped free-vibration response, we discuss the free-vibration response of a damped system. As described in Chapter 2, damping forces may be of several different kinds. The easiest to handle mathemati-cally is viscous damping. Other types of damping are hysteretic damping and Coulomb damping. We discuss the analysis of free damped response under each of the three types of damping resistances mentioned above. 5.2 UNDamped Free Vibration When damping is absent, the equation of motion becomes m ¨ u + ku = 0 (5.2) 176 Dynamics of structures A possible solution of Equation 5.2 is of the form u = Ge λ t (5.3) where G and λ are arbitrary constants to be determined. Substitution of Equation 5.3 in Equation 5.2 gives G λ 2 me λ t + Gke λ t = 0 (5.4) Equation 5.4 will be satisfied provided that G = 0, but as evident from Equation 5.3, this leads to u = 0 and no motion takes place. For motion to take place, G must be nonzero and can be canceled from Equation 5.4. On canceling e λ t as well, we obtain the characteristic equation λ 2 m + k = 0 (5.5a) or λ = ± i ω (5.5b) where ω = k / m . The general solution of Equation 5.2 now becomes u = G 1 e i ω t + G 2 e − i ω t (5.6) By using de Moivre’s theorem, Equation 5.6 can be written in the alternative form u = A cos ω t + B sin ω t (5.7) where A and B are arbitrary constants. To determine the arbitrary constants, we use the specified initial conditions of displacement and velocity at time t = 0. u = u 0 ˙ u = v 0 at t = 0 (5.8) Substitution of the initial conditions gives A = u 0 and B = v 0 /ω , so that Equation 5.7 becomes u = u 0 cos ω t + v 0 ω sin ω t = ρ sin( ω t + φ ) (5.9a) Free-vibration response: Single-degree-of-freedom system 177 where ρ = u 2 0 + v 0 ω 2 (5.9b) tan φ = u 0 ω v 0 Equation 5.9 has been plotted in Figure 5.1c. It shows an oscillatory motion which repeats itself.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Vibration of Continuous Systems

  • Singiresu S. Rao(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  2 Vibration of Discrete Systems: Brief Review 2.1 VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM The number of degrees of freedom of a vibrating system is defined by the minimum number of displacement components required to describe the configuration of the sys- tem during vibration. Each system shown in Fig. 2.1 denotes a single-degree-of-freedom system. The essential features of a vibrating system include: (i) a mass m, producing an inertia force:   ; (ii) a spring of stiffness k, producing a resisting force: kx; and (iii) a damping mechanism that dissipates the energy. If the equivalent viscous damping coefficient is denoted as c, the damping force produced is   . 2.1.1 Free Vibration In the absence of damping, the equation of motion of a single-degree-of-freedom system is given by    +  =  () (2.1) where  () is the force acting on the mass and () is the displacement of the mass m. The free vibration of the system, in the absence of the forcing function  (), is governed by the equation    +  = 0 (2.2) The solution of Eq. (2.2) can be expressed as () =  0 cos    +   0   sin    (2.3) where   is the natural frequency of the system, given by   = √   (2.4)  0 = ( = 0) is the initial displacement and   0 = ( = 0)∕ is the initial velocity of the system. Equation (2.3) can also be expressed as () =  cos(   − ) (2.5) 33 34 Vibration of Discrete Systems: Brief Review k c m x(t) f (t) k a m x(t) f (t) θ k T I 0 θ(t) M t (t) x(t) k f (t) Water tank (mass m) x(t) y(t) k m (a) (d) (e) (b) (c) Figure 2.1 Single-degree-of-freedom systems. or () =  sin(   −  0 ) (2.6) where  = [  2 0 + (   0   ) 2 ] 1∕2 (2.7)  = tan −1   0  0   (2.8)  0 = tan −1  0     0 (2.9) The free vibration response of the system indicated by Eq. (2.5) is shown graphically in Fig. 2.2. The equation of motion for the vibration of a viscously damped system is given by    +    +  =  () (2.10)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Virtual 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)).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Noise and Vibration Analysis for Engineers

  • M. P. Norton, D. G. Karczub(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  Hence, d( T + U ) / d t = − , (1.48) where is power (the negative sign indicates that power is being removed from the system). Power is force × velocity, and the power dissipated from a system with viscous damping is = F v ˙ x = c v ˙ x 2 . (1.49) 1.5.4 Forced vibrations: some general comments So far, only the free vibrations of systems have been discussed. A linear system vibrating under the continuous application of an input excitation is now considered. This is illustrated schematically in Figure 1.13. In general, there can be many input excitations and output responses, together with feedback between some of the inputs and outputs. Some of these problems will be discussed in chapter 5. 22 1 Mechanical vibrations Fig. 1.13. A single input–output linear system. It is useful at this stage to consider the different types of input excitations and output responses that can be encountered in practice. The input or output of a vibration system is generally either a force of some kind, or a displacement, or a velocity, or an accelera-tion. The time histories of the input and output signals can be classified as being either deterministic or random. Deterministic signals can be expressed by explicit mathemat-ical relationships, whereas random signals have to be described in terms of probability statements and statistical averages. Typical examples of deterministic signals are those from electrical motors, rotating machinery and pumps. In these examples, a few spe-cific frequencies generally dominate the signal. Some typical random signals include acoustical pressures generated by turbulence, high speed gas flows in pipeline systems, and the response of a motor vehicle travelling over a rough road surface. Here, the frequency content of the signals is dependent upon statistical parameters.

  Sign up to read

  Learn more about book