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 25 kg mass hangs from a spring with a stiffness of 15 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 25 kgk = 15 kN/m

f_n = Hz

2.Easy

A 200 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 200 kgk = 100 kN/m

ω_n = rad/s

3.Medium

An 8 kg mass is supported by a spring with a stiffness of 200 kN/m and a viscous damper with a damping coefficient of 1260 N·s/m. Find the damping ratio ζ.

m = 8 kgk = 200 kN/mc = 1260 N·s/m

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 10 mm is followed 5 cycles later by a peak of 5.3 mm. Find the damping ratio ζ.

x₀ = 10 mmx₅ = 5.3 mmn = 5

ζ =

5.Hard

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 30 kN/m and the lower one k₂ = 25 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 10 kgk₁ = 30 kN/mk₂ = 25 kN/m

k_eq = kN/m

6.Hard

A 40 kg mass hangs from spring k₃ = 20 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 = 60 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 = 50 kN/mk₂ = 60 kN/mk₃ = 20 kN/m

f_n = Hz

Answer key

1. f_n = 3.898 Hz
2. ω_n = 22.36 rad/s
3. ζ = 0.4981
4. ζ = 0.0202
5. k_eq = 13.64 kN/m
6. f_n = 3.274 Hz

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