The analysis of a vibration problem usually involves the following steps:

1] Formulation of the physical problem into a mathematical model : Here, a decision is to be taken as to whether the structure should be treated as a contentious one or can with advantage be considered as a lumped parameter system, as to how many degrees of freedom the model will have and the location of lumped massed etc. also the excitation causing it must be given a mathematically feasible form. it may have to be Fourier analyzed.

2] Writing down the equation of motion : This could be done by first principles of Dynamics. both equilibrium methods and energy methods can be employed, the choice upon the convenience of the resulting analysis.

3] Writing down the solution of the differential equation in step (2) and determining the constants occurring in the solution by using the initial conditions. it may be necessary sometimes to use transforms like Laplace transforms to solve the equations. if it is a multi degree of freedom system, a harmonic solution for each mass may be assumed and natural frequencies determined using matrix methods.

4] Interpretation of the solution obtained in step (3) and its application in design, etc.