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

m = 100 kgk = 20 kN/m

f_n = Hz

2.Easy

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.

δ_st = 2.5 mmg = 9.81 m/s²

f_n = Hz

3.Medium

A 15 kg mass on a spring with a stiffness of 12 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency f_d of its free vibration in Hz.

m = 15 kgk = 12 kN/mζ = 0.4

f_d = Hz

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 15 mm is followed 3 cycles later by a peak of 3.3 mm. Find the damping ratio ζ.

x₀ = 15 mmx₃ = 3.3 mmn = 3

ζ =

5.Hard

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 25 kN/m and k₂ = 40 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 5 kgk₁ = 25 kN/mk₂ = 40 kN/m

k_eq = kN/m

6.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 40 kN/m and the lower one k₂ = 15 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 40 kN/mk₂ = 15 kN/m

k_eq = kN/m

Answer key

1. f_n = 2.251 Hz
2. f_n = 9.97 Hz
3. f_d = 4.126 Hz
4. ζ = 0.08007
5. k_eq = 65 kN/m
6. k_eq = 10.91 kN/m

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