Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 50

The simple mass‐spring‐damper system excited by a harmonic force was discussed in the preceding sections assuming a single mass which could move in one axis only. This single‐degree‐of‐freedom system idealization is reasonable when the mass is fairly rigid, the springs are lightweight and its motion can be described by means of one variable. For simple systems vibrating at low frequencies, it is also often possible to represent continuous systems with discrete or lumped parameter models. However, real systems have more than just one degree of freedom and, consequently, more than one natural frequency of vibration. For example, systems with more than one mass or systems in which a mass has considerable translation or rotation in more than one direction need to be modeled as multi‐degree of freedom systems. In multi‐degree of freedom systems, we have to consider the relationship between the motions of the various masses, i.e. their relative motion.

The general form of the equation that governs the forced vibration of an n‐degree‐of‐freedom linear system with viscous damping can be written in matrix form as

(2.22)

where [M] is the n × n mass matrix, [R] is the n × n damping matrix (that incorporates viscous damping terms in the matrix formulation), [K] is the n × n stiffness matrix, q is the n‐dimensional column vector of time‐dependent displacements, and f(t) is the n‐dimensional column vector of dynamic forces that act on the system. Therefore, the system governed by Eq. (2.22) exhibits motion which is governed by a set of n simultaneously second‐order differential equations. These equations can be derived using either Newton's laws for free body diagrams or energy methods. In particular, it can be shown that the mass and stiffness matrix are symmetric. This fact is assured if energy methods are used to derive the differential equations. However, symmetric mass and stiffness matrices can also be obtained after algebraic manipulation of the equations. In general, damping matrices are not symmetric unless the system is proportionally damped, i.e. the damping matrix is a linear combination of the mass matrix and stiffness matrix.

The algebraic complexity of the solution grows exponentially with the number of degrees of freedom and the general solution of Eq. (2.22) can be difficult to obtain for systems with a large number of degrees of freedom. Therefore, approximate and numerical approaches are often required to obtain the vibration properties and system response of a multi‐degree of freedom system.