Procedure

(a) Experimental Set-up A physical model of a cantilever held in a fixture that rests on a table of a shaker is seen on the screen. The values of amplitude and frequency of vibration of the shaker table can be entered. The shaker is made to vibrate harmonically over a frequency range with suitable step size of frequency. It vibrates at the lowest frequency for some time, jumps to next frequency obtained through the step-size of frequency and vibrates at that frequency for some time and jumps to next frequency, and so on. It continues to do so till it reaches the highest frequency of the range provided. Thus a sine-sweep is carried out over the frequency range. As the cantilever along with the fixture is mounted on the shaker table, vibration of shaker table provides excitation to fixed end of the cantilever, i.e. support-excitation to the cantilever. An accelerometer is shown mounted at suitable place on the cantilever. Output of the accelerometer goes to vibration analyser displaying displacement (twice integrated from acceleration) versus real time. The frequency of excitation, in Hz, amplitude of vibration of the point where the accelerometer is attached and its phase difference with respect to the excitation are displayed in the window. The whole assembly, cantilever, exciter, accelerometer can be seen vibrating. The frequency of excitation is swept through a suitable range and a plot of amplitude and phase difference of vibration of free end versus excitation-frequency can be seen plotted. Observation of the plot are made and analyzed. Results are tallied with relevant numerical calculations.

(b) Procedure:

1. Enter the dimensions of the cantilever in metre - The length, and width and thickness of cross section of the cantilever.
2. Press the button calculate which will calculate the area and moment of inertia of the cross section of the cantilever.
3. Select the material of the cantilever. The density and Young's Modulus of the material will be displayed immediately.
4. Press the button 'Vibrate'. The shaker table will start vibrating along with the fixture and the cantilever.
5. An accelerometer is shown attached to the cantilever at the free-end of the cantilever. This is connected to vibration analyser. Thus the magnitude of vibration of the free end can be measured. A plot of magnitude of vibration of the free end of the cantilever (i.e. the response) and the frequency of vibration of shaker table (i.e. the excitation) is obtained. Observations of the plot are made and analysed.

(c) Observation and analysis:

1. Enter the dimensions of the cantilever in metre - The length, and width and thickness of cross section of the cantilever. Observe the plot x v/s ω. Note that it has number of peaks. These peaks correspond to resonances, i.e. when the frequency of excitation matches with one of the natural frequencies of the cantilever. Thus the excitation frequencies corresponding to the peaks are natural frequencies of vibration of the cantilever. The lowest one is the fundamental frequency.

(d) Do This

Compare the values of frequencies of resonance with those obtained from analytical formulae from books. Ponder on differences in the values, if any. Remember that formulae are derived from idealization of actual systems.