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 80 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 = 80 kgk = 50 kN/m

ω_n = rad/s

2.Easy

A 15 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 15 kgk = 40 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 one cycle later by a peak of 29.2 mm. Find the damping ratio ζ.

x₀ = 40 mmx₁ = 29.2 mmn = 1

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 25 mm is followed 3 cycles later by a peak of 8.1 mm. Find the damping ratio ζ.

x₀ = 25 mmx₃ = 8.1 mmn = 3

ζ =

5.Hard

A 20 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 60 kN/m and the lower one k₂ = 12 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 20 kgk₁ = 60 kN/mk₂ = 12 kN/m

f_n = Hz

6.Hard

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 50 kN/m and k₂ = 100 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 5 kgk₁ = 50 kN/mk₂ = 100 kN/m

k_eq = kN/m

Answer key

1. ω_n = 25 rad/s
2. ω_n = 51.64 rad/s
3. ζ = 0.05003
4. ζ = 0.05968
5. f_n = 3.559 Hz
6. k_eq = 150 kN/m

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