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 25 kg mass hangs from a spring with a stiffness of 4 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 25 kgk = 4 kN/m

ω_n = rad/s

2.Easy

A 40 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 40 kgk = 12 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed 3 cycles later by a peak of 2.9 mm. Find the damping ratio ζ.

x₀ = 50 mmx₃ = 2.9 mmn = 3

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed 3 cycles later by a peak of 7.5 mm. Find the damping ratio ζ.

x₀ = 50 mmx₃ = 7.5 mmn = 3

ζ =

5.Hard

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 80 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 = 5 kgk₁ = 15 kN/mk₂ = 80 kN/m

f_n = Hz

6.Hard

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 30 kN/m and k₂ = 100 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 30 kN/mk₂ = 100 kN/m

k_eq = kN/m

Answer key

1. ω_n = 12.65 rad/s
2. ω_n = 17.32 rad/s
3. ζ = 0.1494
4. ζ = 0.1001
5. f_n = 21.94 Hz
6. k_eq = 130 kN/m

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