Multi Degrees of Freedom Systems

Systems with multi degree of freedom (DOF) introduce multi natural frequencies, mode shapes, and mode interaction. This chapter is to study the frequency response of multi DOF systems. The general equations of motion of a linear n DOF vibrating system with a single-frequency harmonic excitation have a steady-state solution made up of a combination of n harmonic functions with the same frequency and different coefficients. The coefficients will be determined by substituting the harmonic solution in the differential equation. The coefficient associated to a generalized coordinate indicates the amplitude of oscillation for the generalized coordinate. The coefficient will be a function of the excitation frequency, and hence, it is called frequency response. Free and undamped vibrations of a system are the basic response of the system which express its natural behavior. Using a set of generalized coordinates, x, a linear undamped-free system is governed by a set of second order differential equations. The natural frequencies of the system are solutions of the characteristic equation of the system, and the mode shape corresponding to each natural frequency is the solution of the eigenvector equation. The eigenvectors are called mode shapes, and they are orthogonal with respect to the mass and stiffness matrices. Employing the orthogonality property of the mode shapes, we can determine the generalized mass and stiffness, associated to each mode shape to decouple the equations. Any type of free, transient, or excited response of a linear vibrating system is dominated by its natural frequencies, mode shapes, and interaction of excitation frequencies. Determination of the natural frequencies and their associated mode shapes is the first step in analysis of a multi DOF vibrating system. There exists at least one mode shape corresponding to each natural frequency. If a matrix [A] has n distinct eigenvalues, then there exist exactly n linearly independent eigenvectors, one associated with each eigenvalue. We can always change the set of generalized coordinates q to a set of principal coordinates p and change the equations of motion to a set of decoupled equations. The square transformation matrix [U] is the modal matrix of the system. Using the modal matrix [U], the natural frequencies of the system can be found easier.