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 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

2.Easy

A 200 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 200 kgk = 40 kN/m

f_n = Hz

3.Medium

A 20 kg mass on a spring with a stiffness of 40 kN/m has a viscous damper giving a damping ratio of 0.35. Find the damped natural frequency f_d of its free vibration in Hz.

m = 20 kgk = 40 kN/mζ = 0.35

f_d = Hz

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 25 mm is followed 4 cycles later by a peak of 5.5 mm. Find the damping ratio ζ.

x₀ = 25 mmx₄ = 5.5 mmn = 4

ζ =

5.Hard

A 40 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 = 5 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 25 kN/mk₂ = 5 kN/m

k_eq = kN/m

6.Hard

A 50 kg mass hangs from spring k₃ = 60 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 60 kN/m and k₂ = 15 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 = 50 kgk₁ = 60 kN/mk₂ = 15 kN/mk₃ = 60 kN/m

f_n = Hz

Answer key

1. ω_n = 17.32 rad/s
2. f_n = 2.251 Hz
3. f_d = 6.667 Hz
4. ζ = 0.06014
5. k_eq = 4.167 kN/m
6. f_n = 4.109 Hz

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