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.

New worksheet Practise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________ Date: ____________

1.Easy

A 25 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 25 kgk = 50 kN/m

ω_n = rad/s

2.Easy

An 8 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 = 8 kgk = 2 kN/m

ω_n = rad/s

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 5 cycles later by a peak of 11.4 mm. Find the damping ratio ζ.

x₀ = 40 mmx₅ = 11.4 mmn = 5

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 4 cycles later by a peak of 4.4 mm. Find the damping ratio ζ.

x₀ = 12 mmx₄ = 4.4 mmn = 4

ζ =

5.Hard

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 100 kN/m and k₂ = 60 kN/m. The mass stays level. Find the equivalent stiffness k_eq of the spring system in kN/m.

m = 40 kgk₁ = 100 kN/mk₂ = 60 kN/m

k_eq = kN/m

6.Hard

A 25 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 = 40 kN/m. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 25 kgk₁ = 5 kN/mk₂ = 40 kN/m

f_n = Hz

Answer key

1. ω_n = 44.72 rad/s
2. ω_n = 15.81 rad/s
3. ζ = 0.03992
4. ζ = 0.03989
5. k_eq = 160 kN/m
6. f_n = 2.122 Hz

Full step-by-step solutions: practise this topic on examproblems.com.