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 5 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 8 kgk = 5 kN/m

f_n = Hz

2.Easy

A 25 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 25 kgk = 50 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 4 cycles later by a peak of 0.9 mm. Find the damping ratio ζ.

x₀ = 40 mmx₄ = 0.9 mmn = 4

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed one cycle later by a peak of 21.9 mm. Find the damping ratio ζ.

x₀ = 30 mmx₁ = 21.9 mmn = 1

ζ =

5.Hard

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 25 kN/m and the lower one k₂ = 40 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 10 kgk₁ = 25 kN/mk₂ = 40 kN/m

f_n = Hz

6.Hard

A 50 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 = 100 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 50 kgk₁ = 10 kN/mk₂ = 100 kN/m

f_n = Hz

Answer key

1. f_n = 3.979 Hz
2. ω_n = 44.72 rad/s
3. ζ = 0.1493
4. ζ = 0.05003
5. f_n = 6.243 Hz
6. f_n = 2.146 Hz

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