Free Vibration of a Cantilever Beam(Continuous System)

4.1 Objective of the experiment

To obtain natural frequencies of a cantilever beam up to the second mode, experimentally; and to observe the response of the system subjected to a small initial disturbance and virtualization of the experiment. This virtual experiment is based on a theme that the actual experimental measured vibration data are used.

4.2 Basic Definitions

Free vibration takes place when a system oscillates under the action of forces inherent in the system itself due to initial disturbance, and when the externally applied forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamical system, established by its mass and stiffness distribution.

In case of continuous system the properties of the system are the function of spatial coordinates. The system has infinite number of degrees of freedom and infinite number of natural frequencies.

In actual practice there is always some damping (e.g., the internal molecular friction, viscous damping, aerodynamical damping, etc.) present in the system which causes the gradual dissipation of vibration energy, and it results gradual decay of amplitude of the free vibration. Damping has very little effect on natural frequency of the system, and hence, the calculations for natural frequencies are generally made on the basis of no damping. Damping is of great importance in limiting the amplitude of oscillation at resonance.

The relative displacement configuration of the vibrating system for a particular natural frequency is known as the eigen function in continuous system. The mode shape corresponding to lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental (or the first) mode. The displacements at some points may be zero. These points are known as nodes. Generally n th mode has ( n -1) nodes (excluding end points). The mode shape changes for different boundary conditions of a beam.