Vibration Analysis Procedure

Many variables of a vibratory system such as the excitations (inputs) and responses (outputs) are time-dependent. The response of the vibrating system generally depends on the initial conditions as well as the external excitations. Most practical vibrating systems are very complex, and it is impossible to consider all the details for mathematical analysis. In the mathematical analysis, we include only the most important features so that we can predict the behaviors of the system under specified input conditions. Often the overall behavior of the system can be determined by considering even a simple model of the complex physical system.
The procedure for analysis can be divided into couple of steps and these steps are:

1. Problem Identification - Modeling the system requires abstraction from the surroundings and the effects of the surroundings. The information to be obtained from the modeling is specified, and the known constants and variable parameters are identified.
2. Assumptions- The next step is the introduction of assumptions into the physical system. This is necessary because in that way we simplify the system. In most cases, it is very hard to analyze a mechanical system if all the effects are included. When assumptions are used, an approximate physical system is modeled. An approximation should only be made if the solution to the resulting approximate problem is easier than the solution to the problem if the assumption were not made and that with the assumption the results of the modeling are accurate enough for the use they are intended. Certain implicit assumptions are used in the modeling of most physical systems. Usually, these assumptions are taken for granted and they are rarely mentioned explicitly. The reason for that is the assumption that you know that. Anyway, I’m going to mention them. These implicit assumptions are:
   1. Physical properties are continuous functions of spatial variables. This continuum assumption implies that a system can be treated as a continuous piece of matter.
   2. The earth is an inertial reference frame, thus allowing the application of Newton’s laws in a reference frame fixed to the earth.
   3. Relativistic effects are ignored.
   4. Gravity is the only external force field and the acceleration due to gravity is 9.81 [m/s^2] or 32.2 [ft/s^2] on the surface of the earth.
   5. All materials are liner, isotropic, and homogeneous
   6. The usual assumptions of mechanics of material apply that is: plane sections remain plane for beams in bending; circular sections under torsional loads do not warp.Explicit assumptions are those specific to a particular problem. An explicit assumption is made to eliminate negligible effects from the analysis or to simplify the problem while retaining appropriate accuracy. An explicit assumption should be verified, if possible, on completion of the modeling.
   ALL SYSTEMS ARE NONLINEAR. Exact mathematical modeling of any physical system leads to nonlinear differential equations which often have no analytical solution. Since the exact solution of linear differential equations can usually be determined, assumptions are often made to linearize the problem. Linearization of the problem often leads to the elimination of nonlinear terms in governing equations or to the approximation of nonlinear terms by linear terms. For example, the governing differential equation that describes oscillations of the pendulum has several nonlinear terms. These terms are:
   1. Geometric nonlinearity – which occurs as a result of the system’s geometry. If the maximum angular displacement of the pendulum bob from its equilibrium position is small enough, the nonlinear term in the differential equation due to the geometric nonlinearity can be approximated by a linear term. As the pendulum oscillates it encounters friction in the form of aerodynamic drag.
   2. As the pendulum oscillates it encounters friction in the form of AERODYNAMIC DRAG. If this DRAG is included in the governing equation in form of the drag force this will lead to a nonlinear term. So in certain cases, it can be neglected.
   When you analyze the results of mathematical modeling, one has to keep in mind that the mathematical model is only an approximation of the true physical system. The reason for that comes from the linearization of the governing equation. So, when you linearize governing equation or to be more precise remove all nonlinear terms from the governing equation the solution would be an approximation of the system. The approximate solution is still good because it will give you the basic perception of the system vibrations. If you need to be more accurate you will include some of the nonlinear terms and then try to find a solution but be careful because sometimes the solution of the nonlinear system isn’t possible or it’s impossible to find the solution. Once you got the solution of the mathematical model you need to check the validity of all assumptions.
3. Basic Laws of Nature - are the physical laws that are applied to all physical system regardless of the material from which the system is constructed. These laws are observable, but cannot be derived from any more fundamental law because they are empirical. The basic laws of nature are: Conservation of mass, conservation of momentum, conservation of energy and the second and third laws of thermodynamics. Conservation of momentum, both linear and angular, is usually the only physical law that is of significance in application to vibrating systems. Application of conservation of mass to vibrations problems is trivial. Applications of the second and the third laws of thermodynamics do not yield any useful information. In the absence of thermal energy, the principle of conservation of energy reduces to the mechanical work-energy principle which is derived from well-known Newton’s laws.
4. Constitutive Equations - provide information about the materials of which a system is made. Different materials behave differently under different conditions. Steel and rubber behave differently because their constitutive equations have different forms. While constitutive equations for steel and aluminum are of the same form, the constants involved in the equations are different. Constitutive equations are used to develop force-displacement relationships for mechanical components that are used in modeling vibrating systems.
5. Geometric Constraints - is often necessary to complete the mathematical modeling of an engineering system. Geometric constraints can be in the form of kinematic relationships between displacement, velocity and acceleration. When application of basic laws of nature and constitutive equations lead to differential equations, the use of geometric constraints is often necessary to formulate the requisite boundary and initial conditions.
6. Mathematical Solution - After isolation of mechanical system and collecting all the necessary data need to build mathematical model you will get governing differential equation. But as I have explained earlier you need to simplify this model in order to develop mathematical solution which will satisfy the governing equation. Exact analytical solutions, when they exist, are preferable to numerical or approximate solutions. Exact solution are available for many linear problems, but for only a few nonlinear problems.
7. Physical Interpretation of Results - After the mathematical solution is complete, the results are formulated. Physical interpretation of mathematical solution is the most important part because it gives us the idea of mechanical system behavior or in this case vibration, the cause of vibrations.

Thanks for reading, please comment. In the next post I will talk about Generalized Coordinates