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

m = 10 kgk = 20 kN/m

ω_n = rad/s

2.Easy

A 20 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 20 kgk = 10 kN/m

f_n = Hz

3.Medium

A 25 kg mass on a spring with a stiffness of 15 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 = 25 kgk = 15 kN/mζ = 0.7

f_d = Hz

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 3 cycles later by a peak of 8.8 mm. Find the damping ratio ζ.

x₀ = 40 mmx₃ = 8.8 mmn = 3

ζ =

5.Hard

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 12 kN/m and the lower one k₂ = 40 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 10 kgk₁ = 12 kN/mk₂ = 40 kN/m

k_eq = kN/m

6.Hard

A 20 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 50 kN/m and the lower one k₂ = 50 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 20 kgk₁ = 50 kN/mk₂ = 50 kN/m

f_n = Hz

Answer key

1. ω_n = 44.72 rad/s
2. f_n = 3.559 Hz
3. f_d = 2.784 Hz
4. ζ = 0.08007
5. k_eq = 9.231 kN/m
6. f_n = 5.627 Hz

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