Mechanical vibrations worksheet with answers

Free printable Mechanical vibrations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

A machine rests on spring mounts, which compress by 25 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.

δ_st = 25 mmg = 9.81 m/s²

f_n = Hz

2.Easy

A 20 kg mass hangs from a spring with a stiffness of 4 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 20 kgk = 4 kN/m

f_n = Hz

3.Medium

An 80 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.3. Find the damped natural frequency f_d of its free vibration in Hz.

m = 80 kgk = 20 kN/mζ = 0.3

f_d = Hz

4.Medium

A 200 kg mass on a spring with a stiffness of 25 kN/m has a viscous damper giving a damping ratio of 0.5. Find the damped natural frequency f_d of its free vibration in Hz.

m = 200 kgk = 25 kN/mζ = 0.5

f_d = Hz

5.Hard

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

m = 20 kgk₁ = 20 kN/mk₂ = 20 kN/m

k_eq = kN/m

6.Hard

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 25 kN/m and k₂ = 50 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 20 kgk₁ = 25 kN/mk₂ = 50 kN/m

k_eq = kN/m

Answer key

1. f_n = 3.153 Hz
2. f_n = 2.251 Hz
3. f_d = 2.401 Hz
4. f_d = 1.541 Hz
5. k_eq = 40 kN/m
6. k_eq = 75 kN/m

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