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

m = 12 kgk = 20 kN/m

f_n = Hz

2.Easy

A 10 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 = 10 kgk = 12 kN/m

ω_n = rad/s

3.Medium

A 20 kg mass on a spring with a stiffness of 200 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 = 20 kgk = 200 kN/mζ = 0.45

f_d = Hz

4.Medium

A 12 kg mass is supported by a spring with a stiffness of 200 kN/m and a viscous damper with a damping coefficient of 150 N·s/m. Find the damping ratio ζ.

m = 12 kgk = 200 kN/mc = 150 N·s/m

ζ =

5.Hard

A 50 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 40 kN/m and k₂ = 5 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 50 kgk₁ = 40 kN/mk₂ = 5 kN/m

k_eq = kN/m

6.Hard

A 5 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 kN/m and the lower one k₂ = 30 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 5 kgk₁ = 5 kN/mk₂ = 30 kN/m

k_eq = kN/m

Answer key

1. f_n = 6.497 Hz
2. ω_n = 34.64 rad/s
3. f_d = 14.21 Hz
4. ζ = 0.04841
5. k_eq = 45 kN/m
6. k_eq = 4.286 kN/m

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