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

m = 2 kgk = 10 kN/m

f_n = Hz

2.Easy

A 2 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 2 kgk = 25 kN/m

ω_n = rad/s

3.Medium

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

x₀ = 10 mmx₁ = 7.8 mmn = 1

ζ =

4.Medium

A 5 kg mass on a spring with a stiffness of 40 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency f_d of its free vibration in Hz.

m = 5 kgk = 40 kN/mζ = 0.4

f_d = Hz

5.Hard

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 30 kN/m and k₂ = 10 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 10 kgk₁ = 30 kN/mk₂ = 10 kN/m

k_eq = kN/m

6.Hard

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 80 kN/m and k₂ = 10 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 = 80 kN/mk₂ = 10 kN/m

f_n = Hz

Answer key

1. f_n = 11.25 Hz
2. ω_n = 111.8 rad/s
3. ζ = 0.03951
4. f_d = 13.05 Hz
5. k_eq = 40 kN/m
6. f_n = 10.68 Hz

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