## Introduction: Procedure for Vibration Analysis

Like many engineering problems, the analysis of a vibration problem is typically carried out in a series of logical steps.

### Mathematical Modeling

Most real vibrational systems are hopelessly complex and it would be impossible to consider all of the details of the problem. As a result, we try to simplify the problem as much as possible while retaining all of the important and relevant features. For example, we often represent solid bodies as being rigid (they’re not), consider springs as linear and massless (they’re not), ignore damping, etc. The purpose is to make the system as simple as possible to analyze while still retaining all of the important features of the original problem. Often we start with an overly simple model to understand the basics of a problem, and then add complexities as required to more accurately represent the quantities of interest. A good rule of thumb is to use the simplest possible model which adequately captures the behaviour of interest.

All mechanical vibrational systems contain at a minimum a means to store potential energy (a spring) and a means to store kinetic energy (a mass). A real system will also have some means of dissipating energy (friction, viscous damper, etc.).

### Derivation of Governing Equations

Once the mathematical model is available, we use it to derive the governing equations of motion. Typically this involves drawing Free Body Diagrams and Mass-Acceleration Diagrams (FBD/MAD) of various components of the system and applying Newton’s Laws. However, there are other principles that can be used to obtain the desired equations which may apply in certain cases:

- Conservation of energy,
- Influence coefficients,
- D’Alembert’s Principle and Lagrange’s Equations,
- Many others.

For discrete systems we usually obtain second order ordinary differential equations. For continuous systems we generally have partial differential equations. These equation may be linear or nonlinear depending on the model used. If they are nonlinear we may choose to linearize them (by limiting the amplitudes of motion to be small for example). This is another approximation introduced into the analysis.

### Solution of Governing Equations

Once the equations of motion have been obtained, we must solve them to find the response of the system. Depending on the type of equation there are many solutions methods possible:

- Standard solutions procedure for differential equations (generally only applicable for linear equations),
- Laplace transforms,
- Matrix methods (modal analysis),
- Numerical solutions (finite element method).

For nonlinear problems, typically numerical solutions are used. All of these solutions will generally depend on initial conditions.

### Interpretation of Results

Once the equations have been solved the last (and most important) step is to interpret the results in the context of the real physical situation. It is important to be clear about what the goal of the original analysis was and also about the effects of all of the simplifying assumptions and approximations that were necessarily made.