Undamped Free Vibration

11 Key excerpts on "Undamped Free Vibration"

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  Naval Architecture for Marine Engineers

  • W. Muckle(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  This is referred to as the 'frequency' of vibration, so that if Τ is the period and η the frequency then η = Í/T. The vibrations referred to above are called 'free vibrations' and associated with any system there is a definite frequency which is known as the 'natural frequency' of free vibration. Some systems have more than one natural frequency and the vibrating string is a good example of this. Again in some systems the natural frequency depends upon the magnitude of the original displacement, but where one is deahng with small vibrations it is sufficiently accurate to consider that the natural frequency is independent of the amphtude. All known engineering systems are subject to damping forces to a greater or lesser degree; that is to say there are forces such as those due to viscosity which are absorbing energy from the system. In such cases the amphtude of the free vibration will gradually diminish and instead of the type of time displacement curve shown in Figure ILl, that shown in Figure 11 .2 will be obtained. This can easily be demon-strated by the example of the pendulum. The original amphtude will diminish steadily until the pendulum comes to rest. The damping force not only affects the amphtude of vibration but it also has an effect on frequency. However, it is only when the damping force is large relative to the mass of the system that there is any appreciable effect and for all practical purposes the natural frequency of undamped oscillations can be taken as the frequency of the real system with damping. Before deahng with the specific problem of the vibration of a ship it is appropriate to consider some fundamental principles concerning vibrations and for this purpose the simple mass-spring system will be examined. Period Γ Vibration 325 Vibration of undamped mass-spring system Suppose a mass Μ is suspended by means of a spring S as shown in Figure 11.3. The mass of the spring can be ignored as being smah relative to the mass M .

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  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 .

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  Engineering Vibrations

  • William J. Bottega(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Free Vibration of Single Degree of Freedom Systems

  The 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

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  Engineering Vibrations

  • William J. Bottega(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  317 7 Free Vibration of Multi-Degree of Freedom Systems In this chapter we consider the behavior of discrete multi-degree of freedom systems that are free from externally applied dynamic forces. That is, we examine the response of such systems when each mass of the system is displaced and released in a manner that is con-sistent with the constraints imposed on it. We are thus interested in the behavior of the sys-tem when left to move under its own volition. As for the case of single degree of freedom systems, it will be seen that the free vibration response yields fundamental information and parameters that define the inherent dynamical properties of the system. 7.1 THE GENERAL FREE VIBRATION PROBLEM FOR UNDAMPED SYSTEMS AND ITS SOLUTION It was seen in Chapter 6 that the equations that govern discrete multi-degree of freedom systems take the general matrix form of Eq. (6.2). We shall here consider the fundamental class of problems corresponding to undamped systems that are free from applied (external) forces. For this situation, Eq. (6.2) reduces to the form + =  mu ku 0 (7.1) where, for an N degree of freedom system, m and k are the N N × mass and stiffness ma-trices of the system, respectively, and u is the corresponding 1 N × displacement matrix. To solve Eq. (7.1), we parallel the approach taken for solving the corresponding scalar problem for single degree of freedom systems. We thus assume a solution of the form i t e ω = u U (7.2) 318 Engineering Vibrations where U is a column matrix with N , as yet, unknown constants, and ω is an, as yet, un-known constant as well. The column matrix U may be considered to be the spatial distribu-tion of the response while the exponential function is the time dependence. Based on our experience with single degree of freedom systems, we anticipate that the time dependence may be harmonic. We therefore assume solutions of the form of Eq. (7.2).

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  Muckle's Naval Architecture

  • W. Muckle, D A Taylor(Authors)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  However, it is only when the damping force is large relative to the mass of the system that there is any appreciable effect and for all practical purposes the natural frequency of undamped oscillations can be taken as the frequency of the real system with damping. Before dealing with the specific problem of the vibration of a ship it is appropriate to consider some fundamental principles concerning vibrations and for this purpose the simple mass-spring system will be examined. 37 4 VIBRATIO N VIBRATION OF UNDAMPED MASS-SPRING SYSTEM Suppose a mass M is suspended by means of a spring S as shown in Figure 12.3. The mass of the spring can be ignored as being small relative to the mass M. The position of static equilibrium is at A. Let ÷ be the displacement of the mass from this position at some time /. Figure 123 If k is the spring constant then the force generated by the displaceŮ ment ÷ is kx, on the assumption that there is a linear relation between the force and the displacement. The force kx is directed towards A and is the only force acting in the absence of damping, except the constant force due to gravity which can be ignored. Applying Newton's second law, the relation between the mass, the acceleration and the displacement is given by M(d 2 x/dt 2 ) = -kx or d 2 x/dt 2 = ~(k/M)x (12.1) The solution to this differential equation is * = £ s i n ^ -^ + oJ (12.2) where  is an arbitrary constant and would depend upon the initial displacement; ô is a phase angle dependent upon the time at which the displacement is measured. The type of motion represented by equation (12.2) is called 'simple harmonic' and is the type which always exists when there is a linear relation between the displacement of the system and the restoring force generated. The motion is represented by the curve in Figure 12.1 already referred to. VIBRATIO N 37 5 It will be seen that if / is increased by 2ðË/(Ì/1ß) then ÷ will have the same value.

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  Mechanical Vibration

  Theory and Application

  • Haym Benaroya, Mark Nagurka, Seon Mi Han(Authors)
  • 2022(Publication Date)
  • Rutgers University Press

    (Publisher)

  Chapter 2 Single Degree-of-Freedom Undamped Vibration “The simplest model.” We begin our exploration of dynamic systems with models in which one coordinate completely describes the motion of the system. Despite its simplicity, the study of single degree-of-freedom systems is very useful since many of the key principles developed here are applica- ble to all problems of dynamics and vibration. We can explore the behavior of simple models and learn how to derive and solve their governing equations. The concept of oscillation frequency is introduced as the most impor- tant measure of vibratory motion. In this chapter, we learn how to model and solve for the response of theoretical oscillators where there is no loss of energy through damping. In Chapter 3, damp- ing is introduced as are more realistic force models and more complex vibratory environments. An oscillator is generally taken to be a single degree-of-freedom system undergoing repetitive motion. This chapter and the next two study single degree-of- freedom oscillators in increasing complexity. The split has been created for pedagogical reasons. Several ap- plication examples are introduced first to motivate the subsequent reading. 2.1 Motivating Examples Most engineering systems are examples of vibrating sys- tems. Examples include bridges, vehicles, and rotat- ing machinery. Our studies begin with relatively simple models that are single degree-of-freedom idealizations. The student may wonder how it is possible to derive useful results about complex systems with only a sin- gle degree of freedom. Although one coordinate is not enough for a detailed study of a complicated system, it is useful to capture the essential behavior of such systems for preliminary analysis and design. It is also important to note that the idealized model may not have any physi- cal resemblance to the actual system. Rather, the model is a mathematical idealization that reflects key behav- ioral characteristics of the physical system.

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  Engineering Mechanics

  Dynamics

  • Benson H. Tongue, Daniel T. Kawano(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  (9.3) ◆ Analyze system responses under the application of external sinusoidal forces. (9.4) 9.1 UNDAMPED, FREE RESPONSE FOR SINGLE-DEGREE-OF-FREEDOM SYSTEMS Learning Objective: Analyze the undamped, free vibrational response of simple, single-degree-of-freedom systems. The simplest vibratory model is a mass attached to a linear spring (Figure 9.1.1). A linear spring is one for which the deflection is propor- tional to the applied force, f spring = ky, where f spring is the force pulling on the spring, k is the spring constant, and y is the spring’s deflection. The spring has an unstretched length L, and in the absence of gravity (shown in Figure 9.1.1a), the mass will remain motionless a distance L below the upper end of the spring. Imagine that gravity is now included. There exists an equilibrium position at which the force due to gravity (mg) is exactly countered by the force in the spring, as shown in Figure 9.1.1b. −ky k eq + mg = 0 ⇒ y eq = mg k Left undisturbed, the mass will just sit at the lower end of the stretched spring. If we pull it down a little farther (as in Figure 9.1.1c) and then release it, we will see some vibrations. Note in Figure 9.1.1c that a coordi- nate system is put into place with y = 0 at the mass’s static equilibrium posi- tion and with positive y indicating that the mass is below that equilibrium. 566 L k y eq m (a) L k m g y eq L y y = 0 m No gravity (b) (c) g Figure 9.1.1 Spring-mass system Constructing a FBD=IRD for the mass displaced a distance y (Figure 9.1.2) gives us my ¨ = mg − k( y eq + y). Since ky eq = mg, we have my ¨ + ky = 0 (9.1) 9.1 UNDAMPED, FREE RESPONSE FOR SINGLE-DEGREE-OF-FREEDOM SYSTEMS 567 Note that the coefficients of this equation are constant (m and k), it is second-order (two differentiations with respect to time), there is no forcing, and no nonlinearity (terms like x 2 or x ˙ 3 x, for example) is present.

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  Engineering Vibration Analysis with Application to Control Systems

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

    (Publisher)

  Other motions may occur, but they are assumed to be negligible compared to the coordinate considered. A system with one degree of freedom is the simplest case to analyse because only one coordinate is necessary to completely describe the motion of the system. Some real systems can be modelled in this way, either because the excitation of the system is such that the vibration can be described by one coordinate although the system could vibrate in other directions if so excited, or the system really is simple, as for example a clock pendulum. It should also be noted that a one degree of freedom model of a complicated system can often be constructed where the analysis of a particular mode of vibration is to be carried out. To be able to analyse one degree of freedom systems is therefore an essential ability in vibration analysis. Furthermore, many of the techniques developed in single degree of freedom analysis are applicable to more complicated systems. 2.1 FREE UNDAMPED VIBRATION 2.1.1 Translation vibration In the system shown in Fig. 2.1 a body of mass m is free to move along a fixed horizontal surface. A spring of constant stiffness k which is fixed at one end is attached at the other end to the body. Displacing the body to the right (say) from the equilibrium position causes a spring force to the left (a restoring force). Upon release this force gives the body an acceleration to the left. When the body reaches its equilibrium position the spring force is zero, but the body has a velocity which carries it further to the left although it is retarded by the spring force which now acts to the right. When the body is arrested by the spring the spring force is to the right so that the body moves to the right, past its equilibrium position, and hence reaches its initial displaced position. In practice this position will not quite be reached because damping in the system will have dissipated some of the vibrational energy

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  Fundamentals of Structural Dynamics

  • Roy R. Craig, Andrew J. Kurdila(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  c .

  3.5.1 Experimental Determination of the Undamped Natural Frequency

  The undamped natural frequency of a simple SDOF system may be determined from a static-displacement measurement or free-vibration experiment. Example 3.3 illustrates the static-displacement method; Example 3.4 illustrates the free-vibration method.

  Undamped Natural Frequency: Static-Displacement Method

  Example 3.2 Determine the natural frequency of the simple spring–mass system of Fig. 1 by measuring the static deflection.

  Figure 1 Spring–mass SDOF system.

  SOLUTION From

  Eq. 3.4a

  ,

  (1)

  The equilibrium of the mass as it hangs on the spring is expressed by

  (2)

  or

  (3)

  From the force–elongation equation for the spring,

  (4)

  Combining Eqs. 3 and 4 , we obtain

  (5)

  Thus, from Eqs. 1 and 5 , we obtain the following expression for the undamped natural frequency:

  Ans. (6)

  Undamped Natural Frequency: Free-Vibration Method If the damping in the system is small (ζ < 0.2),

  Eq. 3.28a

  shows that

  ωd

  is approximately equal to

  ωn

  Example 3.4 shows how a free-vibration experiment could be used to determine the natural frequency of a lightly damped SDOF system.

  Example 3.3 The natural frequency of a cantilever beam with a lumped mass at its tip (Fig. 1 ) is to be determined dynamically. The mass is deflected by an amount A = 1 in. and released. The ensuing motion, shown in Fig. 2 , indicates that the damping in the system is very small. Compute the natural frequency in radians per second and in hertz. What is the period?

  Figure 1 Cantilever-beam SDOF system.

  Figure 2 Motion of the tip mass

  SOLUTION At point (a)

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  Mechanical Vibration

  Fundamentals with Solved Examples

  • Ivana Kovacic, Dragi Radomirovic(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  3 Free Undamped Vibration of Single‐Degree‐of‐Freedom Systems

  Chapter Outline

  This chapter starts with a brief preview of Lagrange’s equation for free undamped vibration, both the exact one and the approximated one for small oscillations around a stable equilibrium. Fifteen examples are then solved in detail, showing how the equations of motion can be obtained in either of these approaches. Special attention is given to the potential energy (especially when the spring is predeformed) as well as to the way that the kinetic energy can be obtained more easily by considering the position when the system passes through the equilibrium position.

  Chapter Objectives

  • To present Lagrange’s equation of the second kind for free undamped vibration
  • To illustrate a variety of examples in which it yields the equation of motion
  • To demonstrate the importance of the prestress of a spring for its potential energy
  • To describe in detail how the potential energy of a spring can be determined by considering its deformations in two orthogonal directions
  • To demonstrate how the kinetic energy can be obtained more easily by considering the position when the system passes through the equilibrium position

  Theoretical Introduction

  Free oscillations occur when a mechanical system is set in motion with an initial input (initial displacement and/or initial velocity). The corresponding equation of motion can be obtained by using Lagrange’s equation of the second kind. Its form for a one‐degree‐of‐freedom free conservative systems is

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  Structural Dynamics and Probabilistic Analysis for Engineers

  • Giora Maymon(Author)
  • 2008(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  C h a p t e r 1 / Some Basics of the Theory of Vibrations 1.1 A S INGLE D EGREE OF F REEDOM S YSTEM A good physical understanding of the behavior of vibrating structures can be achieved by analyzing the behavior of a single degree of freedom (SDOF) oscillator. The SDOF system is covered by an extremely large number of textbooks (e.g., refs. [1–12]), and is the basis for every academic course in vibration analysis in aerospace, mechanical, and civil engineering schools. It will be discussed briefly in the first chapters to create a common baseline for the analysis of the behavior of the cantilever beam described in most of the examples, and for any other continuous structure. The SDOF system can be excited either by a force (which is a function of time) acting on the mass, or by a forced movement of the support. The first type of excitation is usually called “force excitation” and the latter is called “base excitation.” These two major types of excitations (loadings) are basic to the structural response analyses of both the SDOF and the continuous elastic systems. The classical oscillator contains a point mass m (i.e., all the mass is con-centrated in one point), which is connected to a rigid support through two elements: a linear massless spring with a stiffness k and a viscous damper c (which creates a force proportional to the velocity), or a structural damper h (which creates a force proportional to the displacement and in 90 degrees phase lag behind it). The system can be excited either by a force f acting on the mass or by a base movement x s . The force-excited system is described in Figure 1.1. Note in the figure that two “elements” connect the mass to the 1 2 • Chapter 1 / Some Basics of the Theory of Vibrations k m f ( t ) x ( t ) c F IGURE 1.1 A force-excited SDOF.

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