Free vibration of single degree of freedom systems

Free vibrations are oscillations about a system’s equilibrium position that occur in the absence of an external excitation. Free vibrations are result of a kinetic energy imparted to the system or of a displacement from the equilibrium position that leads to a difference in potential energy from the system’s equilibrium position.

Let’s consider a model of one degree of freedom system which is shown in Fig 2.1. This system consists of a block of mass m and the spring with stiffens k. When block is displaced a distance a from its equilibrium position, a potential energy ka^2/2 is developed in the spring. When the system is released from equilibrium, the spring force draws the block toward the system’s equilibrium position, with the potential energy being converted to kinetic energy. When the block reaches its equilibrium position the kinetic energy reaches a maximum and the motion continues. The kinetic energy is converted to potential energy until the spring is compressed a distance a. This process of transfer of potential energy to kinetic energy and vice versa is continual in the absence of non-conservative forces.

Remainder:

            Conservative force is a force with the property that the work done in moving a particle between two points is independent of the taken path. For example when you lift a book, the work that you do against gravity in lifting is stored and is available for kinetic energy of the book once you let go. Some conservative forces are: Gravity, Elastic (Hooke’s Law), electric etc.

            Non-conservative force is a force with the property that the work done in moving a particle between two points is dependent on the taken path. For example friction is a non-conservative force. When you move an object the work that you do “against friction” is apparently lost-it is certainly not available to the object as kinetic energy. In general any friction type force, like air resistance is non-conservative force.

            Examples of free vibrations of systems that can be modeled using one degree of freedom include the oscillations of a pendulum about a vertical equilibrium position, the motion of a recoil mechanism of a firearm once it has been fired, and the motion of a vehicle suspension system after the vehicle encounters a pothole.

            Vibrations of the system with one degree of freedom can be described with the second order differential equation. The independent variable is time, while the dependent variable is the chosen generalized coordinate. The chosen generalized coordinate represents the displacement of a particle in the system or an angular displacement and is measured from the system’s equilibrium position.        

            We can derive differential equation by applying one of two methods and these methods are:

1)      The Free Body Diagram Method or

2)      Equivalent System Method

The general solution of the differential equation is a linear combination of two linearly independent solutions. The arbitrary constants, called constants of integration, are uniquely determined on application of two initial conditions. The necessary initial conditions are values of the generalized coordinate and its first time derivative at specified time, usually t = 0.

     The form of the solution of the differential equation depends on system parameters. That means that solution of system with no damping will be different from the solution with viscous damping.