Mechanical vibrations worksheet with answers

A 40 kg mass hangs from a spring with a stiffness of 15 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

A machine rests on spring mounts, which compress by 2.5 mm under its weight. Treat it as an undamped single-degree-of-freedom system and use g = 9.81 m/s². Find the natural frequency f_n of vertical vibration in Hz.

A 100 kg mass is supported by a spring with a stiffness of 100 kN/m and a viscous damper with a damping coefficient of 1260 N·s/m. Find the damping ratio ζ.

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 4 cycles later by a peak of 1.9 mm. Find the damping ratio ζ.

A 25 kg mass hangs from spring k₃ = 25 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 15 kN/m and k₂ = 30 kN/m; the bar stays level. Ignore damping and the masses of the springs and bar. Find the natural frequency f_n of vertical vibration in Hz.

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 50 kN/m and k₂ = 80 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.