tailieunhanh - Ebook Engineering vibrations: Part 2

(BQ) Part 2 book "Engineering vibrations" has contents: Free Vibration of multi degree of freedom systems, forced vibration of multi degree of freedom systems, dynamics of one-dimensional continua, free vibration of one-dimensional continua, forced vibration of one-dimensional continua. | 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 consistent with the constraints imposed on it. We are thus interested in the behavior of the system when is 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. THE GENERAL FREE VIBRATION PROBLEM 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. . We shall here consider the fundamental class of problems corresponding to undamped systems that are free from applied (external) forces. For this situation, Eq. reduces to the form mu + ku = 0 () where, for an N degree of freedom system, m and k are the N × N mass and stiffness matrices of the system, respectively, and u is the corresponding N × 1 displacement matrix. To solve Eq. (), we parallel the approach taken for solving the correspond341 342 Engineering Vibrations ing scalar problem for single degree of freedom systems. We thus assume a solution of the form () u = Ueiω t where U is a column matrix with N, as yet, unknown constants, and ω is an, as yet, unknown constant as well. The column matrix U may be considered to be the spatial distribution 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. (). If we find harmonic forms that satisfy the governing equations then such forms are, by definition, .