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 40 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 40 kgk = 10 kN/m

ω_n = rad/s

2.Easy

An 80 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 80 kgk = 25 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 15 mm is followed one cycle later by a peak of 8 mm. Find the damping ratio ζ.

x₀ = 15 mmx₁ = 8 mmn = 1

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed one cycle later by a peak of 11.6 mm. Find the damping ratio ζ.

x₀ = 30 mmx₁ = 11.6 mmn = 1

ζ =

5.Hard

A 50 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 100 kN/m and the lower one k₂ = 80 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 50 kgk₁ = 100 kN/mk₂ = 80 kN/m

k_eq = kN/m

6.Hard

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 5 kN/m and k₂ = 80 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 = 5 kgk₁ = 5 kN/mk₂ = 80 kN/m

f_n = Hz

Answer key

1. ω_n = 15.81 rad/s
2. ω_n = 17.68 rad/s
3. ζ = 0.09955
4. ζ = 0.1495
5. k_eq = 44.44 kN/m
6. f_n = 20.75 Hz

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