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 80 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 2 kgk = 80 kN/m

f_n = Hz

2.Easy

A 250 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 250 kgk = 100 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 5 cycles later by a peak of 21.3 mm. Find the damping ratio ζ.

x₀ = 40 mmx₅ = 21.3 mmn = 5

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 5 cycles later by a peak of 4.7 mm. Find the damping ratio ζ.

x₀ = 12 mmx₅ = 4.7 mmn = 5

ζ =

5.Hard

A 100 kg mass hangs from spring k₃ = 12 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 5 kN/m and k₂ = 5 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 = 5 kN/mk₂ = 5 kN/mk₃ = 12 kN/m

f_n = Hz

6.Hard

A 20 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 = 25 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 20 kgk₁ = 15 kN/mk₂ = 25 kN/m

k_eq = kN/m

Answer key

1. f_n = 31.83 Hz
2. f_n = 3.183 Hz
3. ζ = 0.02005
4. ζ = 0.02982
5. f_n = 1.175 Hz
6. k_eq = 9.375 kN/m

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