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

m = 2 kgk = 8 kN/m

f_n = Hz

2.Easy

A 250 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 = 250 kgk = 50 kN/m

ω_n = rad/s

3.Medium

A 20 kg mass is supported by a spring with a stiffness of 4 kN/m and a viscous damper with a damping coefficient of 280 N·s/m. Find the damping ratio ζ.

m = 20 kgk = 4 kN/mc = 280 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 2 cycles later by a peak of 5.7 mm. Find the damping ratio ζ.

x₀ = 20 mmx₂ = 5.7 mmn = 2

ζ =

5.Hard

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

m = 50 kgk₁ = 12 kN/mk₂ = 50 kN/m

f_n = Hz

6.Hard

A 100 kg mass hangs from spring k₃ = 100 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 12 kN/m and k₂ = 20 kN/m; the bar stays level. Ignore damping and the masses of the springs and bar. Find the natural frequency f_n of vertical vibration in Hz.

m = 100 kgk₁ = 12 kN/mk₂ = 20 kN/mk₃ = 100 kN/m

f_n = Hz

Answer key

1. f_n = 10.07 Hz
2. ω_n = 14.14 rad/s
3. ζ = 0.495
4. ζ = 0.0994
5. f_n = 2.214 Hz
6. f_n = 2.478 Hz

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