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 12 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 12 kgk = 10 kN/m

ω_n = rad/s

2.Easy

A 10 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 = 10 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 30 mm is followed 3 cycles later by a peak of 4.5 mm. Find the damping ratio ζ.

x₀ = 30 mmx₃ = 4.5 mmn = 3

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed 5 cycles later by a peak of 8.5 mm. Find the damping ratio ζ.

x₀ = 30 mmx₅ = 8.5 mmn = 5

ζ =

5.Hard

A 50 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 = 25 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 50 kgk₁ = 20 kN/mk₂ = 25 kN/m

k_eq = kN/m

6.Hard

A 40 kg mass hangs from spring k₃ = 30 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 25 kN/m and k₂ = 5 kN/m; the bar stays level. Ignore damping and the masses of the springs and bar. Find the natural frequency f_n of vertical vibration in Hz.

m = 40 kgk₁ = 25 kN/mk₂ = 5 kN/mk₃ = 30 kN/m

f_n = Hz

Answer key

1. ω_n = 28.87 rad/s
2. ω_n = 100 rad/s
3. ζ = 0.1001
4. ζ = 0.04011
5. k_eq = 11.11 kN/m
6. f_n = 3.082 Hz

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