Steady-state solutions of certain types of differential equations by piecewise linearization

Two methods are presented for finding steady-state solutions of differential equations of any order governing certain systems that are acted upon by a harmonic force and have one nonlinear element with hysteresis represented by piecewise linearization. Both methods need the solution of a set of linear algebraic equations. In the first method, the unknowns are the Fourier coefficients of the steady-state solution, while in the second method, the unknowns are the values of the derivatives of the steady-state solution at a break point of the piecewise linearized characteristic. In both methods, the unknowns have to be calculated for different values of the time angle at the break point, yielding different corresponding values of the amplitude of the forcing term. The required solution is that consistent with the given amplitude of this forcing term. In the first method, the parameters involved in the multiple-input describing functions of the nonlinear element are unified by normalization. Comparison of the two methods is given, and the advantages of the piecewise linearization of characteristics is discussed.