EXACT, SINGLE EQUATION, CLOSED-FORM SOLUTION OF VIBRATING SYSTEMS WITH PIECEWISE LINEAR SPRINGS

A method is proposed to obtain in a single equation the exact, closed-form displacement response of both free undamped vibrations and the steady state of forced damped vibrations of systems with piecewise linear springs, with continuous or discontinuous force–deflection relations. Since the displacement equations are exact the velocity and acceleration responses are obtained by differentiation. The solutions are complete, they apply for any value of time from zero to infinity. The method is designed to be handled in a computer either in conjunction with a Symbolic Mathematics package, like Mathematica or Maple, or without it. However, for certain relatively simple problems the phase-plane solution may be obtained even in a hand-held calculator. Two examples are presented.