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 10 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 100 kgk = 10 kN/m

ω_n = rad/s

2.Easy

A machine rests on spring mounts, which compress by 4 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 = 4 mmg = 9.81 m/s²

f_n = Hz

3.Medium

A 12 kg mass is supported by a spring with a stiffness of 12 kN/m and a viscous damper with a damping coefficient of 40 N·s/m. Find the damping ratio ζ.

m = 12 kgk = 12 kN/mc = 40 N·s/m

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 10 mm is followed 2 cycles later by a peak of 7.8 mm. Find the damping ratio ζ.

x₀ = 10 mmx₂ = 7.8 mmn = 2

ζ =

5.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 25 kN/m and the lower one k₂ = 10 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 25 kN/mk₂ = 10 kN/m

k_eq = kN/m

6.Hard

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 50 kN/m and k₂ = 30 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 10 kgk₁ = 50 kN/mk₂ = 30 kN/m

f_n = Hz

Answer key

1. ω_n = 10 rad/s
2. f_n = 7.882 Hz
3. ζ = 0.0527
4. ζ = 0.01977
5. k_eq = 7.143 kN/m
6. f_n = 14.24 Hz

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