Chapter 2: Introduction to Damping in Free and Forced Vibrations

Transcription

1 Chapter 2: Introduction to Daping in Free and Forced Vibrations This chapter ainly deals with the effect of daping in two conditions like free and forced excitation of echanical systes. Daping plays an iportant role in dying out the vibration aplitude effectively by absorbing the excitation energy. Exaple -1.2 [1]: A achine can ove in a vertical degree-offreedo only. It is ounted elastically to a rigid foundation. Assue that the achine can be regarded as a point ass and that the isolator is an ideal spring, the spring rate of which is k. What is the ounted resonance frequency of the achine in the following cases: a) = 1 kg, (i) k = 1 kn/. (ii) k = 1 kn/. (iii) k = 1 MN/. b) = 1 kg, (i) k = 1 kn/. (ii) k = 1 kn/. (iii) k = 1 MN/. Solution-1:, and f / 2 3 a) (i) f Hz. 3 (ii) f Hz. 6 (iii) f Hz. 3 b) (i) f Hz. 3 (ii) f Hz. 6 (iii) f Hz.

2 Proble: 1.3 A achine ounted on vibration isolators is odeled as a single degree-of-freedo syste. The relevant paraeters are estiated to be as follows: ass = 37 kg, spring rate k= 2 x 15 N/, daping constant =.2 per second. Calculate the natural frequency of the ounted achine and the displaceent aplitude of the achine, if it is excited at that frequency by a force with peak aplitude of 1 N. Solution: The paraeters are the ass = 37 kg, spring rate k = 2.15 N/ and daping constant =.2 s The eigen-frequency is: rad/s Eigen-frequency: Let: x( t) A ei t f 2 F( t) A 3.7 Hz. ( ) e i t So the odulus is: which at A : Introduction to daping: Daping is a phenoenon by which echanical energy is dissipated (usually converted as theral energy) in dynaic systes. Vibrating systes can encounter daping in various ways like

3 Interolecular friction Sliding friction Fluid resistance Three priary echaniss of daping are as: Internal daping of aterial Structural daping at joints and interface Fluid daping through fluid -structure interactions Two types of external dapers can be added to a echanical syste to iprove its energy dissipation characteristics: Active dapers require external source of power Passive dapers Does not required MATERIAL (Internal) Daping Internal daping originates fro energy dissipation associated with: icrostructure defects (grain boundaries & ipurities), thero elastic effects (caused by local teperature gradients) eddy-current effects (ferroagnetic aterials), dislocation otion in etals, etc. Types of Internal daping: Viscoelastic daping Hysteretic daping Daping estiation of any syste is the ost difficult process in any vibration analysis. The daping is generally coplex and generally for echanical systes it is so sall to copute.

4 Fig 1.6 Spring Mass Daper syste There are three cases of interest. The discussion about these three cases is as follows: A. Under-daped Vibrations: This case occurs if the paraeters of the syste are such that (< ξ <1). In this case the discriinate, ω ξ 1 becoes negative, and the roots of equation (1.11) becoes coplex. Thus, the solution of eqn.(1.11) yields as follows. Or, x(t) = e (A cos ω t + Bsin ω t) (1.12) x(t) = Ce sin(ω t + ϕ) Where, A, B, C and applying the initial conditions. are constant, their values ay be deterined by

5 k + ( + ) + ( + )) + ( ) X(t g g Fig 1.6(a) Free body diagra (FBD) At t = t, x = x, v = v =, = + = ( + ) + ( ) = tan + The underdaped response has the for as shown in Figure 1.7. It depicts that the aplitude of the vibrations are decaying with tie. Fig. 1.7 Response of an underdaped syste

6 B. Over-daped Vibrations:. This case occurs if the paraeters of the syste are such that > 1. In this condition the discriinate, ω 1 becoes negative, and the roots of equation (1.11) becoe negative real nubers. Thus the solution of eqn.(1.11) yields as follows. ( ) = + (1.13) Where A and B are constant and can be deterined applying the initial conditions as given follows. = = The response of an overdaped syste is shown in Fig It ay be seen that an over-daped syste does not oscillate, but rather returns to its rest position exponentially. The over daping affects the syste response as shown in Fig. 1.9.

7 Fig. 1.8 Response of an overdaped syste Fig. 1.9 Effect of Over-daping C. Critically daped: When the value of daping coefficient becoes 1. It is known as critically daped syste. In this condition discriinate, ω 1 becoes zero, and the roots of equation (1.11) becoe negative repeated real nubers. Thus the solution of eqn. (1.11) yields as follows. ( ) = [( + ) + ] (1.14)

8 The response of a critically daped syste is shown in Fig Fig. 1.1 Response of a critically daped syste Fig Overall response of a echanical syste

9 FORCED RESPONSE The preceding analysis considers the vibration of a coponent or structure as a result of soe initial disturbance (i.e., v and x ). In this section, the vibration of a spring ass daper syste subjected to an external force is exained. This external force ay be of the for step function, ipulse function, haronic or rap function. k + ( + ) + ( + )) + ( ) F(t) X(t Mg+ F(t) g (a) (b) (c) Fig 1.12 (a) Scheatic of a forced spring ass daper syste, (b) free body diagra of the syste in part (a), (c) free body diagra due to static condition. In ost of the situation the forcing function F(t), is periodic and having the following haronic for. F(t) = F sinωt where, F is the aplitude of the applied force and ω is the frequency of the applied force, soeties called driving frequency. Fro fig. 1.12, the equation of otion of a forced syste ay be expressed as follows.

10 x + cx + kx = F(t) = F sinωt (1.17) The solution of this equation ay be deterined as; x = Hoogeneous solution (copleen func) + Particular solution Fig Coplete solutaion (Transient (CF) and periodic (PI) solution) The hoogeneous solution can be easily obtained as the atheatical approach discussed for the solution of equation Hoogeneous solution and particular solution are usually referred to as the transient response and the steady state response sequentially. Physically, it is to assue that the steady state response will follow the forcing function. Hence, it is tepting to assue that the particular solution has the for ( ) = sin( ) (1.18) Where, X = steady state aplitude θ = phase shift at steady state Fundaentals of Sound and Vibrations by KTH Sweden [1], this book is used under IITR-KTH MOU for course developent.