Mechanical vibrations worksheet with answers

A 40 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

A machine rests on spring mounts, which compress by 8 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 15 kg mass is supported by a spring with a stiffness of 40 kN/m and a viscous damper with a damping coefficient of 120 N·s/m. Find the damping ratio ζ.

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed 3 cycles later by a peak of 9.4 mm. Find the damping ratio ζ.

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 15 kN/m and the lower one k₂ = 80 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

A 25 kg mass hangs from spring k₃ = 30 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 12 kN/m and k₂ = 60 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.