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 25 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 = 25 kgk = 100 kN/m

ω_n = rad/s

2.Easy

A 200 kg mass hangs from a spring with a stiffness of 20 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 200 kgk = 20 kN/m

f_n = Hz

3.Medium

A 10 kg mass is supported by a spring with a stiffness of 1 kN/m and a viscous damper with a damping coefficient of 50 N·s/m. Find the damping ratio ζ.

m = 10 kgk = 1 kN/mc = 50 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 4 cycles later by a peak of 9.4 mm. Find the damping ratio ζ.

x₀ = 20 mmx₄ = 9.4 mmn = 4

ζ =

5.Hard

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 40 kN/m and k₂ = 10 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 40 kN/mk₂ = 10 kN/m

k_eq = kN/m

6.Hard

A 50 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 80 kN/m and the lower one k₂ = 5 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 50 kgk₁ = 80 kN/mk₂ = 5 kN/m

f_n = Hz

Answer key

1. ω_n = 63.25 rad/s
2. f_n = 1.592 Hz
3. ζ = 0.25
4. ζ = 0.03003
5. k_eq = 50 kN/m
6. f_n = 1.544 Hz

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