Damped Free Vibration
Damped free vibration refers to the oscillatory motion of a system that gradually decreases in amplitude over time due to the dissipation of energy. This damping effect can be caused by factors such as friction, air resistance, or material properties. In engineering, understanding damped free vibration is crucial for designing structures and machines that can effectively manage and minimize the effects of vibration.
8 Key excerpts on "Damped Free Vibration"
eBook - ePub- James C. Anderson, Farzad Naeim(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
...Chapter 3 Free-Vibration Response of Single-Degree-of-Freedom Systems 3.1 UnDamped Free Vibration The term undamped implies there is no damping or energy dissipation present in the dynamic system, and the term free vibration indicates there is no applied dynamic loading. Therefore, the dynamic system consists of only a mass and a resistance, as shown in Figure 3.1. The motion of the oscillator occurs as a result of the initial conditions that occur at time zero and consist of an initial displacement and/or an initial velocity. Figure 3.1 Undamped single-degree-of-freedom (SDOF) linear oscillator Summing the horizontal forces, including the inertia force, results in the equation 3.1 This equation is a linear homogeneous second-order differential equation with constant coefficients that has a general solution of the form 3.2 Differentiating twice with respect to time, we obtain the following expression for the acceleration: 3.3 Now substituting Equations (3.2) and (3.3) into Equation (3.1) leads to 3.4 Because the exponential term is never zero, the expression in parentheses must be zero. Dividing by m and introducing the notation, we obtain 3.5 which has a solution of the form a = i. Substituting this result into Equation (3.2), we get 3.6 Introducing the Euler equations, e i t = cos t i sin t, we can write the general solution as 3.7 Differentiating Equation (3.7) leads to an equation for the velocity: 3.8 Expressing the constants A and B in terms of the initial conditions at time t = 0 results in 3.9 and the general solution becomes 3.10 Because the applied load is zero, the vibration of the system is initiated by these initial conditions. The solution represents a simple harmonic motion, which is shown graphically in Figure 3.2. The quantity is the circular frequency or angular velocity and is measured in radians per unit of time...
eBook - ePub- Bikash C. Chakraborty, Debdatta Ratna(Authors)
- 2020(Publication Date)
- Elsevier(Publisher)
...The time dependence of response causes a phase shift for the state variable such as strain and is mathematically expressed as a complex quantity. The phase shift represents the extent of damping or loss of energy in a damped oscillatory system. The loss mechanism could be inelastic (dashpot) or viscoelastic damping, magneto-rheological or electro-rheological damping, or active control or shear thinning-type damping. 1.5.1 Expressions for free, undamped vibration An SDOF system of a mass attached to a spring is shown in Fig. 1.3. The rotating or oscillatory machines and many such real-life systems can be modelled as a mass-spring system as shown in the figure. Most metallic objects, such as machines and equipment, have very low inherent losses and the vibrations are approximately undamped. Fig. 1.3 Spring-mass arrangement in an SDOF system: unDamped Free Vibration. The key assumptions for the above system are that the spring has very low mass, and can be neglected and its behaviour is Hookean, that is, the force is linearly proportional to the deflection of the spring. The proportionality constant is termed as Spring Constant, denoted by ‘ k ’. The response of the spring is instantaneous, which means there is no time lag between the force and deformation and also the spring comes to its undeformed state instantly on withdrawal of the force. Therefore, there is no loss of energy due to deformation and retraction cycle. Considering the force balance, the force exerted by the stretching of the spring is balanced by the force due to the acceleration of the mass: m d 2 u dt 2 = − ku (1.4) One solution to the above second-order differential equation can be u = u 0 cos ω n t + v 0 ω n sin ω n t (1.5) where u 0 is the initial displacement, v 0 is the initial velocity, and ω n is the Natural Angular Frequency of the spring-mass system and is given by ω n = k m (1.6) Unit of natural angular frequency (ω n) is radian/s...
eBook - ePub- Jens Holger Rindel(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Chapter 2 Mechanical vibrations 2.1 A SIMPLE MECHANICAL SYSTEM In this chapter, we shall look at one-dimensional vibrations in a simple mechanical system consisting of a mass, a spring and a damping element. Since the vibrations can be fully described in one dimension, this is also called a system with one degree of freedom. First, the use of complex notation is shown for harmonic vibrations as a basis for later applications in acoustic vibrations and sound fields. Next, the vibrations in a resonant mechanical system are dealt with, as a basis for numerous applications in resonant acoustic systems. Finally, the theory for vibration isolation is explained as this has fundamental importance for noise control of machines and equipment in practice. The description follows the terminology and concepts in ISO 2041. Figure 2.1 shows a mechanical system consisting of a mass m placed on an elastic layer on top of a rigid, unmovable surface. The elastic layer has the stiffness k symbolized as a spring and the resistance r symbolized as a dashpot. Some materials have slightly different stiffness for static load and for a dynamic excitation, and thus, we will distinguish between those cases. The acceleration due to gravity means that the system has a static excitation, which leads to a compression of the elastic layer. The static displacement is by definition: x s = m g k s (2.1) where g = 9.81 ms −2 is the acceleration due to gravity and k s is the static stiffness (N/m). In the case of a dynamic excitation of the system by the force F, we have the equation of motion from Newton’s second law: m d 2 x d t 2 + r d x d t + k d x = F (2.2) Figure 2.1 (a) Simple mechanical system. (b) Static deflection due to mass load and gravity. (c) Dynamic excitation by an external force. where x is the displacement from the rest position, t is the time and k d is the dynamic stiffness (N/m). As long as the external force is active, the system is said to display forced vibrations...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...3 Single Degree of Freedom Systems – Free Vibrations The equations of motion derived in Chapter 2 were the free vibration equations of motion for systems. Free vibration equations of motion always have the form (3.1) where represents the effective mass, represents the effective damping, and represents the effective stiffness of the system. Remember that we are considering systems like the independent front suspension described earlier (see Figure 2.2) for the most part and the values of,, and are not as simple as those shown in the standard mass/spring/damper systems that we normally use for illustrating vibratory systems. At the end of Chapter 2, we derived an equation of motion for a system that appeared as (3.2) In this equation, the effective mass is, the effective damping is, and the effective stiffness is. The degree of freedom is rather than but Equations 3.1 and 3.2 will have solutions that are identical in form. The left‐ and right‐hand sides of the equation of motion have different physical meanings. The left‐hand side contains all of the system properties – mass, stiffness, and damping. The right‐hand side indicates what we do to the system – apply forces. At this point the right‐hand side is zero, indicating that we have not applied a force to the system. We call the motions of systems like this free vibrations since the systems are free to do what they want without any external interference. In this chapter, we consider free vibrations of systems, realizing that motion can only ensue if there is a set of initial conditions that starts the motion. 3.1 UnDamped Free Vibrations Setting the effective damping coefficient,, to zero in Equation 3.1 gives the equation of motion for unDamped Free Vibration. It is (3.3) This is a second‐order, linear, ordinary, differential equation and we should be able to recall enough about our differential equations course to solve it...
eBook - ePub- Roy R. Craig, Andrew J. Kurdila(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
...Therefore, this type of member acts like a spring with spring constant (2.16 b) Simply Supported Beam Figure 2.3 d shows a uniform simply supported beam with a transverse midspan load P that causes a deflection δ at the point of application of the load. The relationship between the force P and the deflection δ is given by [2.1] (2.17 a) where E is the modulus of elasticity of the material from which the member is made, I is the moment of inertia of the cross section of the member, and L is the length of the member. Therefore, this type of member acts like a spring with spring constant (2.17 b) 2.2.2 Viscous-Damping Element Just as a spring-type member serves as an energy storage device, there are means by which energy is dissipated from a vibrating structure. These are called damping mechanisms, or simply dampers. Although research studies have proposed numerous ways to describe material damping mathematically, the exact nature of damping in a structure is usually impossible to determine. There are many references that discuss the general topic of damping in vibrating structures (e.g., Refs. [2.2] to [2.4]). The simplest analytical model of damping employed in structural dynamics analyses is the linear viscous dashpot model, which is illustrated in Fig. 2.4. The damping force f D is given by (2.18) and is thus a linear function of the relative velocity between the two ends of the dashpot...
eBook - ePub- Randall F. Barron(Author)
- 2002(Publication Date)
- CRC Press(Publisher)
...9 Vibration Isolation for Noise Control One of the major sources of noise in mechanical equipment is the noise produced by energy radiated from vibrating solid surfaces in the machine. In addition, noise may be produced when vibratory motion or forces are transmitted from the machine to its support structure, through connecting piping, etc. Noise reduction may be achieved by isolating the vibrations of the machine from the connected elements. For the case of noise generated by a vibrating panel on the machine, noise reduction may be achieved by using damping materials on the panel to dissipate the mechanical energy, instead of radiating the energy into the surrounding air. In this chapter, we will consider some of the techniques for vibration isolation for machinery and examine some of the materials used in isolation of vibrations from equipment. There are at least two types of vibration isolation problems that the engineer may be called upon to solve: (a) situations in which one seeks to reduce forces transmitted from the machine to the support structure and (b) situations in which one seeks to reduce the transmission of motion of the support to the machine. Some examples of the first case include reciprocating engines, fans, and gas turbines. An example of the second case is the mounting of electrical equipment in an aircraft or automobile such that motion of the vehicle is not “fed into” the equipment. 9.1 UNDAMPED SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM Many aspects of vibration isolation may be understood by examination of an SDOF system consisting of a mass and a linear spring, as shown in Fig. 9-1. For a more extensive treatment of mechanical vibrations, there are several references available in the literature (Rao, 1986; Tongue, 1996; Thomson and Dahleh, 1998)...
eBook - ePub- Stuart T. Smith(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...For this conservative system, we find, by differentiating with respect to time and noting that the velocity could only be zero at all times in the trivial case, that m q ¨ + λ q = 0 (10.2) Equation 10.2 may be rewritten as the familiar linear, second order harmonic oscillator equation q ¨ + ω n 2 q = 0 (10.3) where ω n = (λ / m) ½ is its natural, or resonant, frequency. At instants of zero velocity, the system holds only potential energy in the form of strain energy and the spring will be at its maximum distortion, so E = 1 2 λ q max 2 (10.4) The maximum amplitude of motion for a given input energy, E, is q max = 2 E / λ (10.5) Equation 10.5 indicates that, for a given input energy and in the absence of significant energy dissipation mechanisms, the amplitude of motion can be reduced only by increasing the stiffness. All practical systems will be excited by internal (motors, gearboxes and crankshafts) and external (adjacent machinery, traffic, people) sources of vibration and by airborne noise. Thus there must be a means of dissipation in every instrument to prevent it slowly absorbing energy and building up oscillations of an unacceptable magnitude. The principle energy dissipation characteristics are viscous, hysteretic and coulomb damping. The first two are most commonly used and also have similar transient characteristics. Thus we analyse only system responses where energy is dissipated by a viscous damper, that is one providing a resisting force directly proportional to the velocity of motion. 10.2 Response of a second order spring/mass/viscous damper system In Figure 10.2 a viscous damper having a dissipative constant b (dimensionally of units N s m −1) has been attached in parallel with the spring of the basic oscillator of Figure 10.1...
eBook - ePubTheory of Nonlinear Structural Analysis
The Force Analogy Method for Earthquake Engineering
- Gang Li, Kevin Wong(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
...9 Application: Structural Vibration Control The structural vibration control began in mechanical engineering in the early 20th century, and subsequently developed in civil engineering. At present, the structural vibration control technique is verified as an effective way to reduce responses of structures under natural hazards such as earthquakes and strong winds. Over the past few years, a number of structural control techniques have been developed and applied in practice, specifically in seismic regions. Structural control systems, in general, fall into four categories: passive, active, semi-active and hybrid control based on whether the power source is necessary. For the active and semi-active control technique, the control force is often determined by control strategies, but the passive control technique is through energy-dissipation devices to dissipate the input energy. In this chapter, the FAM is used to simulate the seismic dynamical analysis of structures with passive and active controlling devices. 9.1 Passive Control Technique The passive control technique is through leading the input energy from environmental loads to special elements or devices to reduce energy-dissipating demand on primary structural members and minimize possible structural damage. These special elements or devices are so-called passive energy dissipation devices (PEDD) due to their energy-dissipation capacity. 9.1.1 Model of Passive Energy-Dissipation Devices The PEDDs are mainly categorized into velocity-dependent dampers and displacement-dependent dampers. (1) Velocity-dependent PEDDs Viscous and viscoelastic dampers are two typical kinds of velocity-dependent PEDDs. The restoring force of the dampers is associated with the inter-story velocity...
