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 10 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 10 kgk = 40 kN/m

f_n = Hz

2.Easy

A 25 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 25 kgk = 25 kN/m

f_n = Hz

3.Medium

A 2 kg mass is supported by a spring with a stiffness of 15 kN/m and a viscous damper with a damping coefficient of 90 N·s/m. Find the damping ratio ζ.

m = 2 kgk = 15 kN/mc = 90 N·s/m

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 2 cycles later by a peak of 5.6 mm. Find the damping ratio ζ.

x₀ = 12 mmx₂ = 5.6 mmn = 2

ζ =

5.Hard

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 60 kN/m and k₂ = 50 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 = 20 kgk₁ = 60 kN/mk₂ = 50 kN/m

f_n = Hz

6.Hard

A 10 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 = 50 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 10 kgk₁ = 10 kN/mk₂ = 50 kN/m

k_eq = kN/m

Answer key

1. f_n = 10.07 Hz
2. f_n = 5.033 Hz
3. ζ = 0.2598
4. ζ = 0.06054
5. f_n = 11.8 Hz
6. k_eq = 8.333 kN/m

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