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 50 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 50 kgk = 12 kN/m

ω_n = rad/s

2.Easy

A machine rests on spring mounts, which compress by 25 mm under its weight. Treat it as an undamped single-degree-of-freedom system and use g = 9.81 m/s². Find the natural frequency f_n of vertical vibration in Hz.

δ_st = 25 mmg = 9.81 m/s²

f_n = Hz

3.Medium

An 80 kg mass is supported by a spring with a stiffness of 200 kN/m and a viscous damper with a damping coefficient of 2400 N·s/m. Find the damping ratio ζ.

m = 80 kgk = 200 kN/mc = 2400 N·s/m

ζ =

4.Medium

A 50 kg mass on a spring with a stiffness of 80 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 = 50 kgk = 80 kN/mζ = 0.45

f_d = Hz

5.Hard

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 25 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 10 kN/mk₂ = 25 kN/m

k_eq = kN/m

6.Hard

A 40 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 12 kN/m and the lower one k₂ = 50 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 12 kN/mk₂ = 50 kN/m

k_eq = kN/m

Answer key

1. ω_n = 15.49 rad/s
2. f_n = 3.153 Hz
3. ζ = 0.3
4. f_d = 5.685 Hz
5. k_eq = 35 kN/m
6. k_eq = 9.677 kN/m

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