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 15 kg mass hangs from a spring with a stiffness of 80 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 15 kgk = 80 kN/m

ω_n = rad/s

2.Easy

A 15 kg mass hangs from a spring with a stiffness of 5 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 15 kgk = 5 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 3 cycles later by a peak of 18.8 mm. Find the damping ratio ζ.

x₀ = 40 mmx₃ = 18.8 mmn = 3

ζ =

4.Medium

A 12 kg mass on a spring with a stiffness of 25 kN/m has a viscous damper giving a damping ratio of 0.45. Find the damped natural frequency f_d of its free vibration in Hz.

m = 12 kgk = 25 kN/mζ = 0.45

f_d = Hz

5.Hard

A 5 kg mass hangs from spring k₃ = 40 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 20 kN/m and k₂ = 15 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 = 5 kgk₁ = 20 kN/mk₂ = 15 kN/mk₃ = 40 kN/m

f_n = Hz

6.Hard

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 20 kN/m and k₂ = 60 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 20 kN/mk₂ = 60 kN/m

k_eq = kN/m

Answer key

1. ω_n = 73.03 rad/s
2. ω_n = 18.26 rad/s
3. ζ = 0.04002
4. f_d = 6.487 Hz
5. f_n = 9.725 Hz
6. k_eq = 80 kN/m

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