This series of tutorials covers the basic theory of mechanical vibrations, an important subject in mechanical engineering. Establishing the fundamentals of the subject is largely an exercise in applied mathematics, in particular finding and interpreting solutions to homogeneous and non-homogeneous, second order differential equations. For this reason I've included a maths tutorial covering this aspect.

Tutorials in the **Mechanical vibrations** series are as follows.

Mechanical vibrations - introduction and overview

Establishing free body diagrams and equations of motion for spring and mass vibrating system with one degree of freedom for: (a) free vibrations, (b) free vibrations with damping, (c) forced free vibrations, (d) forced free vibrations with damping.

Mechanical vibrations - maths tutorial

Methods for solving ordinary, second order, linear, homogeneous and non-homogeneous equations, specifically: general solutions for homogeneous equations of form **y = e ^{rx}** obtained from the roots of characteristic equation

**ar**; particular solution derived from solving for constant terms of the general solution using two initial conditions; general solution for non-homogeneous equations obtained from sum of the general solution of the complementary homogeneous equation and the particular solution of the non-homogeneous equation obtained by the method of undetermined coefficients.

^{2}+ br + c = 0Deriving the expression for displacement ** x(t) = R.cos(ωt - φ) ** from solutions to the homogeneous differential equation of motion of a free vibrating spring and mass system where **R** is the amplitude and **φ** the phase angle; establishing the angular frequency of a free vibrating system **ω** is the natural frequency **ω _{n}** for all initial conditions of the system.

Deriving an expression for displacement from solutions to the homogeneous differential equation of motion of a free vibrating spring and mass system with damping; definition of dimensionless damping ratio **ζ** and derivation of solutions for ζ > 1, ζ < 1 and ζ = 1 ; worked examples and plots illustrating heavily damped (ζ > 1), critically damped (ζ = 1) and lightly damped (ζ < 1) behaviour; method of estimating value of ζ from experimentally derived plots.

Forced vibrations without damping

Deriving an expression for displacement from solutions to the non-homogeneous differential equation of motion of a free vibrating spring and mass system without damping subjected to a harmonic forcing function **F _{0}cos(ωt)** ; interpretation of the solution as separable transient and steady state responses; plots illustrating amplification factor and phase relationship for steady state response where ω < ω

_{n}and ω > ω

_{n}; illustration of resonance condition where ω ≅ ω

_{n}; development of solution for combined transient and steady state response illustrating the harmonic phenomenon of beating;

Forced vibrations with damping

Deriving expressions for steady state displacement and phase angle using the non-homogeneous differential equation of motion of a free vibrating spring and mass system with damping subjected to a harmonic forcing function **F _{0}cos(ωt)** ; plots illustrating amplification factor and phase relationship for steady state response where ω < ω

_{n}and ω > ω

_{n}; plot illustrating variation of amplification factor with damping ratio

**ζ**.

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