# Forced Vibration of a Cantilever Beam(Continuous System)

5.1 Objective of the Experiment

The aim of the experiment is to analyze the forced vibrations of a cantilever beam considering it as a continuous system, the phenomena of resonances, the phase of the vibration signal; and also to obtain the fundamental natural frequency and damping ratio of the system, and compare the results with theoretically calculated values.

The basic aim of these experiments is to provide a feel of actual experiments along with learning of basics while performing the virtual experiments.

5.2 Basic Definitions

Forced vibration: When a dynamic system is subjected to a steady-state harmonic excitation, it is forced to vibrate at the same frequency as that of the excitation. The harmonic excitation can be given in many ways like with constant frequency and variable frequency or a swept-sine frequency, in which the frequency changes from the initial to final values of frequencies with a given time-rate (i.e., ramp).

If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered and dangerously large oscillations may result, which results in failure of major structures, i.e., bridges, buildings, or airplane wings etc. At the point of resonance, the displacement of the system is maximum.

Thus, calculation of natural frequencies is of major importance in the study of vibrations. Because of friction & other resistances, vibrating systems are subjected to damping to some degree due to dissipation of energy. 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 a continuous system for a natural frequency is known as the eigen function of that system. For every natural frequency, there would be a corresponding eigen function. The mode shape corresponding to lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental (or the first) mode shape. The displacements at some points may be zero. These points are known as nodes. Generally for higher modes the number of nodes increases. In case of beam, mode shape changes with boundary conditions.

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