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

m = 50 kgk = 100 kN/m

f_n = Hz

2.Easy

A machine rests on spring mounts, which compress by 10 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 = 10 mmg = 9.81 m/s²

f_n = Hz

3.Medium

A 10 kg mass on a spring with a stiffness of 10 kN/m has a viscous damper giving a damping ratio of 0.7. Find the damped natural frequency f_d of its free vibration in Hz.

m = 10 kgk = 10 kN/mζ = 0.7

f_d = Hz

4.Medium

A 50 kg mass is supported by a spring with a stiffness of 150 kN/m and a viscous damper with a damping coefficient of 2190 N·s/m. Find the damping ratio ζ.

m = 50 kgk = 150 kN/mc = 2190 N·s/m

ζ =

5.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 kN/m and the lower one k₂ = 100 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 40 kgk₁ = 5 kN/mk₂ = 100 kN/m

f_n = Hz

6.Hard

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 10 kN/m and the lower one k₂ = 20 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 25 kgk₁ = 10 kN/mk₂ = 20 kN/m

k_eq = kN/m

Answer key

1. f_n = 7.118 Hz
2. f_n = 4.985 Hz
3. f_d = 3.594 Hz
4. ζ = 0.3998
5. f_n = 1.737 Hz
6. k_eq = 6.667 kN/m

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