On forced oscillations of a double mathematical pendulum

For conservative mechanical systems, the method of normal coordinates is known, which uses the theorem on the reduction of two quadratic forms to the sum of squares. In this case, the system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of freedom degrees is defined by three quadratic forms: the kinetic energy of the system and potential energy of the system, and the dissipative Rayleigh function. We study the linear problem of forced oscillations of a double pendulum when the friction coefficients are proportional to the masses. Then all three quadratic forms are reduced to the sum of squares by a single transformation. In normal coordinates the system splits into two independent systems of second order. An analytical solution is constructed in the most general form for arbitrary rod lengths and point masses. A complete analysis of the oscillations in the non-resonant case and in the case of resonances is given. Formulas for the error of the analytical formulas if the proportionality of the friction coefficients and masses is violated are also obtained.