# Forced Vibration of a Cantilever Beam with a Lumped Mass at Free End

3.1 Objective of the Experiment

The aim of the experiment is to study the forced vibrations of cantilever beams. The focus is on the fundamental mode of the resonance phenomenon. Data to be collected from the test setup is the force-time and response-time graphs. These signals are processed to obtain the force-frequency, response-frequency and phase-frequency graphs. Finally, calculate the natural frequency, damping ratio and related parameters.

By the use of the actual experimental data virtual experiments are performed to give a feel of actual experiment along with learning of basic concepts while performing the virtual experiments. Then students can do self-evaluations are focal aims of virtual experiments.

3.2 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. Common sources of harmonic excitation are unbalance in rotating machines, forces produced by the reciprocating parts, or the motion of machine itself. 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 causes 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 the vibrating system for a particular natural frequency is known as the mode shape (or 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 the beam.