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 50 kg mass hangs from a spring with a stiffness of 200 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 50 kgk = 200 kN/m

ω_n = rad/s

2.Easy

An 8 kg mass hangs from a spring with a stiffness of 2 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 8 kgk = 2 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed 3 cycles later by a peak of 13.7 mm. Find the damping ratio ζ.

x₀ = 20 mmx₃ = 13.7 mmn = 3

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed one cycle later by a peak of 14.6 mm. Find the damping ratio ζ.

x₀ = 20 mmx₁ = 14.6 mmn = 1

ζ =

5.Hard

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 50 kN/m and k₂ = 20 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 20 kgk₁ = 50 kN/mk₂ = 20 kN/m

f_n = Hz

6.Hard

A 100 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 100 kN/m and k₂ = 60 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 100 kgk₁ = 100 kN/mk₂ = 60 kN/m

k_eq = kN/m

Answer key

1. ω_n = 63.25 rad/s
2. f_n = 2.516 Hz
3. ζ = 0.02007
4. ζ = 0.05003
5. f_n = 9.416 Hz
6. k_eq = 160 kN/m

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