Vibrational Dissipative Systems with Two Degrees of Freedom

Forced linear oscillations of dissipative mechanical systems with two degrees of freedom under the action of time-periodic forces are considered. The Lagrange equations are expressed in terms of three positive-definite quadratic forms: kinetic energy, dissipative function, and potential energy. The necessary and sufficient condition for simultaneous reducibility to diagonal forms of symmetric matrices of three real quadratic forms of two variables is formulated and proved. The condition was reduced to the equality of the third-order determinant of the coefficients of quadratic forms to zero. In this case, by linear transformation, the quadratic forms are reduced to the sum of squares and the equations are split into two independent second-order equations. The solution of the system is in a general analytical form. The effectiveness of the method is demonstrated by analyzing the forced oscillations of a double pendulum.