Multi-degrees of Freedom

Time response analysis of an n degree of freedom (DOF) system is dependent on n natural frequencies and n mode shapes of the system. An n DOF system is expressed by a set of n ordinary differential equations of the second order. Free Vibrations: Undamped Systems. A system shows its natural behavior when it is force-free and undamped. The time response of the free system is a linear combination of harmonic functions for each coordinate, with the natural frequencies and mode shapes of the system. The coefficients of the harmonic solutions will be determined from the initial conditions. The natural frequencies are solutions of the characteristic equation of the system, and the mode shape corresponding to the natural frequencies, is the solution of the eigenvector equation. Employing the modal matrix U, we can use a linear transformation to change the variables from the generalized coordinates x to principal coordinates p and make the equations decouped. The uncoupled equations can be solved individually. The original solution will be found by combining the principal solutions and the coordinate transformation. Free Vibrations: Damped Systems. To solve free vibrations of multi DOF systems with general damping matrix C, we convert the n second order differential equations of motion to 2n first order differential equations and develop the exponential solutions. A solution of the new first-order equations of motion is exponential function of time, where the exponents are the eigenvalues and relative coefficients are eigenvectors of undamped system. Therefore, the complete solution will be a linear combination of all individual solutions. The unknown coefficients will be calculated from initial conditions. Forced Vibrations. The complete solution of equations of a vibrating system with n degrees of freedom, subjected to given force functions has two parts: a homogenous solution, plus the a particular solution. The homogenous solution is the solution of free vibrations, and the particular solution is the special solutions of the equations associated to the forcing function. To solve the forced vibrations, we convert the n second order differential equations of motion to 2n first order differential equations and develop the exponential solutions.