Exact and approximate analytical solutions of oscillator with piecewise linear asymmetrical damping

The objective of this work is to present exact and approximate analytical solutions of a 1-DOF oscillator with asymmetrical damping under harmonic excitation, and to explore the role of asymmetrical damping in the response of such oscillator. The response of the case with overdamped expansion and underdamped compression is shown in details to illustrate the effect of the asymmetry ratio (β) on the transition time, the mean position and energy dissipation. The solutions for all other cases are also shown, indicating similar effects. Higher asymmetry ratio (β>1) provides lower peak acceleration in high frequency, while (β<1) results in lower acceleration in low frequency. As it is shown that asymmetrical damping can be advantageous over symmetrical systems regarding vibration isolation and energy transfer in certain frequency ranges, the solutions shown here can aid the design of such asymmetrical systems.