Mechanical Vibrations
Springs, damping and resonance.
Key idea
Springs, damping and resonance.
In depth
A mass on a spring obeys m x′′ + c x′ + k x = F(t). With no damping, it oscillates at ω = √(k/m). Damping gives under-, critical or over-damped behaviour; forcing near ω gives resonance.
Worked example
m = 1, k = 4, c = 0: x = A cos 2t + B sin 2t, period π.
Formula
Real-world uses
- Designing bridges, buildings and washing machines to avoid dangerous resonance
Common mistake
Ignoring damping when it is significant; it decides if oscillations die out.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Mechanical Vibrations?
Springs, damping and resonance. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Mechanical Vibrations?
The key formula is mx″ + cx′ + kx = F(t).
Where is Mechanical Vibrations used in real life?
Real-world examples: Designing bridges, buildings and washing machines to avoid dangerous resonance.
What should I learn before Mechanical Vibrations?
Make sure you are comfortable with Variation of Parameters, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Mechanical Vibrations?
Ignoring damping when it is significant; it decides if oscillations die out.
How can I practise Mechanical Vibrations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Mechanical Vibrations?
Move on to RLC Circuits next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.