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 200 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 200 kgk = 150 kN/m

ω_n = rad/s

2.Easy

An 80 kg mass hangs from a spring with a stiffness of 15 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 80 kgk = 15 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 10 mm is followed 3 cycles later by a peak of 1.5 mm. Find the damping ratio ζ.

x₀ = 10 mmx₃ = 1.5 mmn = 3

ζ =

4.Medium

A 250 kg mass on a spring with a stiffness of 100 kN/m has a viscous damper giving a damping ratio of 0.3. Find the damped natural frequency f_d of its free vibration in Hz.

m = 250 kgk = 100 kN/mζ = 0.3

f_d = Hz

5.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 15 kN/m and the lower one k₂ = 10 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 15 kN/mk₂ = 10 kN/m

k_eq = kN/m

6.Hard

A 25 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 80 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 = 25 kgk₁ = 80 kN/mk₂ = 100 kN/m

k_eq = kN/m

Answer key

1. ω_n = 27.39 rad/s
2. f_n = 2.179 Hz
3. ζ = 0.1001
4. f_d = 3.036 Hz
5. k_eq = 6 kN/m
6. k_eq = 180 kN/m

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