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 10 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 = 10 kgk = 50 kN/m

ω_n = rad/s

2.Easy

A 50 kg mass hangs from a spring with a stiffness of 20 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 50 kgk = 20 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed 5 cycles later by a peak of 2.1 mm. Find the damping ratio ζ.

x₀ = 50 mmx₅ = 2.1 mmn = 5

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed one cycle later by a peak of 7.2 mm. Find the damping ratio ζ.

x₀ = 12 mmx₁ = 7.2 mmn = 1

ζ =

5.Hard

A 40 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 = 40 kN/m and k₂ = 50 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 = 40 kgk₁ = 40 kN/mk₂ = 50 kN/mk₃ = 100 kN/m

f_n = Hz

6.Hard

A 25 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 12 kN/m and k₂ = 40 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 25 kgk₁ = 12 kN/mk₂ = 40 kN/m

f_n = Hz

Answer key

1. ω_n = 70.71 rad/s
2. ω_n = 20 rad/s
3. ζ = 0.1004
4. ζ = 0.08103
5. f_n = 5.477 Hz
6. f_n = 7.259 Hz

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