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 5 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 5 kgk = 12 kN/m

f_n = Hz

2.Easy

A 25 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 25 kgk = 40 kN/m

ω_n = rad/s

3.Medium

A 10 kg mass on a spring with a stiffness of 4 kN/m has a viscous damper giving a damping ratio of 0.6. Find the damped natural frequency f_d of its free vibration in Hz.

m = 10 kgk = 4 kN/mζ = 0.6

f_d = Hz

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed 3 cycles later by a peak of 6.6 mm. Find the damping ratio ζ.

x₀ = 30 mmx₃ = 6.6 mmn = 3

ζ =

5.Hard

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 30 kN/m and the lower one k₂ = 80 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 25 kgk₁ = 30 kN/mk₂ = 80 kN/m

k_eq = kN/m

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 = 12 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 25 kgk₁ = 10 kN/mk₂ = 12 kN/m

k_eq = kN/m

Answer key

1. f_n = 7.797 Hz
2. ω_n = 40 rad/s
3. f_d = 2.546 Hz
4. ζ = 0.08007
5. k_eq = 21.82 kN/m
6. k_eq = 5.455 kN/m

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