Microsoft Word - 4%5B1%5D.2.Free Vibration1_08jan10
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1 EXPERIMENT 2 FREE VIBRATIONS OF A CANTILEVER BEAM (SINGLE DEGREE OF FREEDOM SYSTEM) 2.1 Objective of the Experiment To experimentally obtain the fundamental natural frequency and the damping ratio of a cantilever beam. This virtual experiment is based on a theme that only the actual e xperimental measured vibration data are used to design the virtual experiments. 2.2 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 steady-state 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 actual practice, there is always some damping (e.g., the internal molecular friction, viscous damping, aerodynamical damping, etc.) present in the system which cause the gradual dissipation of vibration energy, and it result 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. 2.3 Theoretical Natural Frequency for A Cantilever Beam A cantilever beam with rectangular cross-section is shown in Figure 2.1(a), the free transverse (bending) vibration can be generated by giving an initial displacement at the free end of the beam. Figure 2.1(b) shows a cantilever beam under the free vibration. When a system is subjected to free vibration and the system is considered as a discrete system in which the beam is considered as mass-less and the whole mass is concentrated at the free end of the beam. The governing equation of motion for such single-DOF system is give as
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