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 100 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 100 kgk = 50 kN/m

f_n = Hz

2.Easy

A 15 kg mass hangs from a spring with a stiffness of 4 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 15 kgk = 4 kN/m

ω_n = rad/s

3.Medium

A 500 kg mass is supported by a spring with a stiffness of 100 kN/m and a viscous damper with a damping coefficient of 5660 N·s/m. Find the damping ratio ζ.

m = 500 kgk = 100 kN/mc = 5660 N·s/m

ζ =

4.Medium

A 40 kg mass on a spring with a stiffness of 5 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency f_d of its free vibration in Hz.

m = 40 kgk = 5 kN/mζ = 0.4

f_d = Hz

5.Hard

A 50 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 40 kN/m and the lower one k₂ = 50 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 50 kgk₁ = 40 kN/mk₂ = 50 kN/m

f_n = Hz

6.Hard

A 5 kg mass hangs from spring k₃ = 5 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 30 kN/m and k₂ = 100 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 = 5 kgk₁ = 30 kN/mk₂ = 100 kN/mk₃ = 5 kN/m

f_n = Hz

Answer key

1. f_n = 3.559 Hz
2. ω_n = 16.33 rad/s
3. ζ = 0.4002
4. f_d = 1.631 Hz
5. f_n = 3.355 Hz
6. f_n = 4.939 Hz

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