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 2 kg mass hangs from a spring with a stiffness of 2 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 2 kgk = 2 kN/m

ω_n = rad/s

2.Easy

A 5 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 5 kgk = 8 kN/m

ω_n = rad/s

3.Medium

A 25 kg mass is supported by a spring with a stiffness of 5 kN/m and a viscous damper with a damping coefficient of 110 N·s/m. Find the damping ratio ζ.

m = 25 kgk = 5 kN/mc = 110 N·s/m

ζ =

4.Medium

A 100 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.5. Find the damped natural frequency f_d of its free vibration in Hz.

m = 100 kgk = 20 kN/mζ = 0.5

f_d = Hz

5.Hard

A 50 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 50 kN/m and the lower one k₂ = 100 kN/m. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 50 kgk₁ = 50 kN/mk₂ = 100 kN/m

k_eq = kN/m

6.Hard

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 30 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 = 20 kgk₁ = 10 kN/mk₂ = 30 kN/m

f_n = Hz

Answer key

1. ω_n = 31.62 rad/s
2. ω_n = 40 rad/s
3. ζ = 0.1556
4. f_d = 1.949 Hz
5. k_eq = 33.33 kN/m
6. f_n = 7.118 Hz

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