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 20 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 20 kgk = 8 kN/m

ω_n = rad/s

2.Easy

A 2 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 2 kgk = 100 kN/m

f_n = Hz

3.Medium

An 80 kg mass is supported by a spring with a stiffness of 8 kN/m and a viscous damper with a damping coefficient of 960 N·s/m. Find the damping ratio ζ.

m = 80 kgk = 8 kN/mc = 960 N·s/m

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed 3 cycles later by a peak of 20.6 mm. Find the damping ratio ζ.

x₀ = 30 mmx₃ = 20.6 mmn = 3

ζ =

5.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 10 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 = 40 kgk₁ = 10 kN/mk₂ = 12 kN/m

f_n = Hz

6.Hard

A 100 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 30 kN/m and the lower one k₂ = 15 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 100 kgk₁ = 30 kN/mk₂ = 15 kN/m

f_n = Hz

Answer key

1. ω_n = 20 rad/s
2. f_n = 35.59 Hz
3. ζ = 0.6
4. ζ = 0.01994
5. f_n = 1.859 Hz
6. f_n = 1.592 Hz

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