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

m = 150 kgk = 200 kN/m

f_n = Hz

2.Easy

A 150 kg mass hangs from a spring with a stiffness of 20 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 150 kgk = 20 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 15 mm is followed 2 cycles later by a peak of 9.1 mm. Find the damping ratio ζ.

x₀ = 15 mmx₂ = 9.1 mmn = 2

ζ =

4.Medium

A 100 kg mass on a spring with a stiffness of 15 kN/m has a viscous damper giving a damping ratio of 0.35. Find the damped natural frequency f_d of its free vibration in Hz.

m = 100 kgk = 15 kN/mζ = 0.35

f_d = Hz

5.Hard

A 100 kg mass hangs from spring k₃ = 60 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 50 kN/m and k₂ = 80 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 = 50 kN/mk₂ = 80 kN/mk₃ = 60 kN/m

f_n = Hz

6.Hard

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 100 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 10 kgk₁ = 10 kN/mk₂ = 100 kN/m

k_eq = kN/m

Answer key

1. f_n = 5.812 Hz
2. f_n = 1.838 Hz
3. ζ = 0.03974
4. f_d = 1.826 Hz
5. f_n = 3.225 Hz
6. k_eq = 110 kN/m

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