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

m = 8 kgk = 80 kN/m

f_n = Hz

2.Easy

A 12 kg mass hangs from a spring with a stiffness of 200 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 12 kgk = 200 kN/m

f_n = Hz

3.Medium

A 5 kg mass is supported by a spring with a stiffness of 50 kN/m and a viscous damper with a damping coefficient of 300 N·s/m. Find the damping ratio ζ.

m = 5 kgk = 50 kN/mc = 300 N·s/m

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed 5 cycles later by a peak of 1.6 mm. Find the damping ratio ζ.

x₀ = 20 mmx₅ = 1.6 mmn = 5

ζ =

5.Hard

A 100 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 12 kN/m and the lower one k₂ = 80 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 100 kgk₁ = 12 kN/mk₂ = 80 kN/m

f_n = Hz

6.Hard

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 60 kN/m and k₂ = 20 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 5 kgk₁ = 60 kN/mk₂ = 20 kN/m

k_eq = kN/m

Answer key

1. f_n = 15.92 Hz
2. f_n = 20.55 Hz
3. ζ = 0.3
4. ζ = 0.08014
5. f_n = 1.626 Hz
6. k_eq = 80 kN/m

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