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 150 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 = 150 kgk = 50 kN/m

ω_n = rad/s

2.Easy

A 25 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 25 kgk = 8 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 2 cycles later by a peak of 7.3 mm. Find the damping ratio ζ.

x₀ = 12 mmx₂ = 7.3 mmn = 2

ζ =

4.Medium

A 20 kg mass on a spring with a stiffness of 2 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency f_d of its free vibration in Hz.

m = 20 kgk = 2 kN/mζ = 0.4

f_d = Hz

5.Hard

A 50 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 = 20 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 50 kgk₁ = 60 kN/mk₂ = 20 kN/m

f_n = Hz

6.Hard

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 12 kN/m and the lower one k₂ = 30 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 10 kgk₁ = 12 kN/mk₂ = 30 kN/m

f_n = Hz

Answer key

1. ω_n = 18.26 rad/s
2. f_n = 2.847 Hz
3. ζ = 0.03952
4. f_d = 1.459 Hz
5. f_n = 2.757 Hz
6. f_n = 4.66 Hz

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