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

m = 40 kgk = 12 kN/m

f_n = Hz

2.Easy

A 40 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 40 kgk = 10 kN/m

f_n = Hz

3.Medium

A 2 kg mass on a spring with a stiffness of 40 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 = 2 kgk = 40 kN/mζ = 0.5

f_d = Hz

4.Medium

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

x₀ = 10 mmx₁ = 5.3 mmn = 1

ζ =

5.Hard

A 100 kg mass hangs from spring k₃ = 60 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 5 kN/m and k₂ = 50 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 = 100 kgk₁ = 5 kN/mk₂ = 50 kN/mk₃ = 60 kN/m

f_n = Hz

6.Hard

A 50 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 80 kN/m and k₂ = 100 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 = 50 kgk₁ = 80 kN/mk₂ = 100 kN/m

f_n = Hz

Answer key

1. f_n = 2.757 Hz
2. f_n = 2.516 Hz
3. f_d = 19.49 Hz
4. ζ = 0.1005
5. f_n = 2.696 Hz
6. f_n = 9.549 Hz

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