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

An 8 kg mass hangs from a spring with a stiffness of 1 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 8 kgk = 1 kN/m

ω_n = rad/s

2.Easy

A 250 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 250 kgk = 100 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 5 cycles later by a peak of 4 mm. Find the damping ratio ζ.

x₀ = 50 mmx₅ = 4 mmn = 5

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed one cycle later by a peak of 9.3 mm. Find the damping ratio ζ.

x₀ = 12 mmx₁ = 9.3 mmn = 1

ζ =

5.Hard

A 5 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 40 kN/m and the lower one k₂ = 100 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 5 kgk₁ = 40 kN/mk₂ = 100 kN/m

f_n = Hz

6.Hard

A 20 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 20 kN/m and the lower one k₂ = 80 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 20 kgk₁ = 20 kN/mk₂ = 80 kN/m

k_eq = kN/m

Answer key

1. ω_n = 11.18 rad/s
2. ω_n = 20 rad/s
3. ζ = 0.08014
4. ζ = 0.04053
5. f_n = 12.03 Hz
6. k_eq = 16 kN/m

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