What is the difference between linear and non linear vibration?

Answer 1

In general all systems are nonlinear but we simplify this nonlinear vibration to linear ones so that we can get approximate results. Approximate results are still good results in many cases. For example when you analyze the vibrations of the simple pendulum for small vibrations you don't need to include aerodynamic drag which is a nonlinear in its nature. By neglecting the nonlinear parts we can derive the second order differential equations which describes the motion of the system in this case gives linear vibration of simple pendulum. Another good example would be an examination of system which consists of block of mass m, spring with stiffness k and viscous damper with damping coefficient c and let's say that the block of mass m is in contact with the surface. Now the spring stiffness and the viscous damping are in reality nonlinear but for small vibration we assume they are linear. The bloc of mass m is in contact with the surface so that means that between the block and the surface is a friction. So if we analyze this system with nonlinear terms we would need to include the nonlinear stiffness, nonlinear damping coefficient and nonlinear friction. These would result in the time consuming calculation and in the end the results would little more precise than the approximation.

In nonlinear analysis we attack the differential equation which describes the motion of nonlinear system with small parameter and with this we expand the solution. This method is called perturbation method. To solve nonlinear systems you need to use specific perturbation method and these methods are: Straightforward expansion, domain perturbation, multiple scale analysis etc. For more information check my site Linear Vibration.