TAILIEUCHUNG - Ebook Vibrations (2nd edition): Part 2

(BQ) Part 1 book "Vibrations" has contents: Single-degree-of-freedom systems - subjected to transient excitations, multiple degree-of-freedom systems - governing equations, natural frequencies, and mode shapes, multiple degree of freedom systems - general solution for response and forced oscillations, vibrations of beams. | 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations INTRODUCTION RESPONSE TO IMPULSE EXCITATION RESPONSE TO STEP INPUT RESPONSE TO RAMP INPUT SPECTRAL ENERGY OF THE RESPONSE RESPONSE TO RECTANGULAR PULSE EXCITATION RESPONSE TO HALF-SINE WAVE PULSE IMPACT TESTING SUMMARY EXERCISES INTRODUCTION In Chapter 4, free responses were discussed, and in Chapter 5, responses to harmonic and other periodic excitations were discussed. As illustrated in the last two chapters, a “sudden” change in the state of a system brought about by an initial condition or by a change in the profile of the forcing function results in transients in the response of the system. Here, the initial conditions are assumed to be zero, and the responses to various types of excitations such as impulse excitations, step inputs, ramp inputs, and pulse excitations are considered at length. All of these excitations are characterized by sudden changes 285 CHAPTER 6 Single Degree-of-Freedom Systems Subjected to Transient Excitations 286 in their respective profiles of amplitude with time. When transformed to the frequency domain the responses to such transient excitations can also provide a basis for determining the characteristics of a system. The resulting displacement response, and several design criteria are established based on this information. In the systems considered in this chapter, the inertia element or the base of the system is subjected to a transient forcing. Solution for Response to Transient Excitation When the initial displacement Xo ϭ 0 and the initial velocity Vo ϭ 0, the governing equation for an underdamped system (0 Ͻ z Ͻ 1) with a forcing acting on the mass as shown in Figure is given by Eq. (), which is repeated below. x(t) c f(t) k m t 1 x 1t 2 ϭ mvd Ύe Ϫzvnh sin1vdh2f 1t Ϫ h2dh 0 FIGURE t Spring-mass-damper system subjected to forcing f(t). 1 ϭ mvd Ύe Ϫzvn1tϪh2 sin1vd 3t Ϫ h4 2f

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