Free Vibration Of A Cantilever Objective

Free quivering Of A stick out ObjectiveThe purpose of this experiment is to determine the raw(a) frequency of a jut disperse study both undamped and damped let off vibration motion of a protrude institutionalise.Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium. In general, there are two types of vibration, free and forced. Free vibration is maintained by gravitational or elastic restoring force. Forced vibration is caused by an external periodic or intermittent force applied to the system. Both of these types of vibration may be e real damped or undamped. Undamped vibrations can continue indefinitely because frictional effects are neglected in the analysis. Basically, if a system that is subjected to an initial disturbance and is left to vacillate on its own, the subsequent vibration is known as the free vibration. Vibrations without damping would result in a continuous vibration of the particular oscillatory b ody. As a matter of fact, it will produce a displacement-time interpret of such nature as shown in the following figure. This graph is commonly referred to as the simple charitable motion. http//upload.wikimedia.org/wikipedia/commons/4/44/Simple_harmonic_motion.pngFigure 1 Simple Harmonic Motion (Displacement-Time Graph)However, in reality, just as with many other scientific theories, this is out of the question because friction and other forces are present both internally and externally. As the system is subjected to these forces, this phenomena is called damping. The principle effect of damping is to reduce the amplitude of an oscillation, non to change its frequency. So, the graph of the amplitude of a normal damped oscillation might look like the followinghttp//www.efunda.com/formulae/vibrations/sdof_images/SDOF_UnderDamped_Response.gifFigure 2 Graph of Damped Oscillation (Displacement-Time Graph)Apparatus and Materials1. Cantilever burn apparatus-Modulus of elasticity of aluminium(E) 70GPa-Dimension of the cantilever beam 927mm (L) - 19.09mm (W) - 6.35mm (H)-Mass of the cantilever beam 292.59g Mass of the silencer 122 g2. Strain gauge3. Strain registrar4. Viscous damperExperimental ProceduresFigure 3 Experiment Setup without Viscous DamperFigure 4 Experiment Setup with Viscous DamperThe computer and the railway line recorder were switched on.The assay recorder application software was started by double clicking on the DC104REng shortcut icon on the computer desktop.The experiment setup was shown in Figure 3. The operation of the strain recorder and the recorder application software were referred to the operational manual.The viscous damper was removed if it was attached to the beam.The beam was held and displaced by, ymax, -20mm, -15mm, -10m, -5mm, 0, 5mm, 10mm, 15mm,and 20mm. The strain recorder reading for each displacement value from the Numerical Monitor screen of the application software was recorded manually.The relationship of the di splacement (of the free closing curtain of the beam) and the strain recorder reading was obtained by plotting an appropriate graph using a spreadsheet.The beam is displaced by 30mm and the beam is left to vibrate on its own. The strain recorder reading was recorded by clicking on the Play and Stop button.The recorded file was retrieved by clicking on the Read USB button.The graph of the beam displacement versus time, t was plotted.The experiment was repeated by using beam displacement of 50mm.The viscous damper was connected as shown in Figure 3. Steps 7 and 10 were repeated by using beam displacement of 30mm and 50mm respectively.a) Theoretical CalculationsAs given in the experimentModulus of elasticity of aluminium,E = 70 GPaLength of the cantilever beam, L = 0.927mWidth of the cantilever beam,b = 0.019mThickness of the cantilever beam, h = 0.006mMass of the cantilever beam, mcantilever = 0.293 kgMass of the damper, mdamper = 0.122 kgb) Experimental Results and CalculationsFree V ibration of Cantilever channelize at 30mm DisplacementNatural Circular Frequency of Beam with Viscous Damper,The free vibration of the theoretical earthy frequency of the cantilever beam in this experiment is 5.75Hz while the experimental natural frequency of the cantilever beam is 6.25Hz for amplitude of 30mm and 6.25Hz for amplitude of 50mm.The viscously damped vibration of the theoretical natural frequency of the cantilever beam is 3.45Hz and the experimental natural frequency of the cantilever beam for amplitude of 30mm and 50mm are 3.57Hz and 3.57Hz.Percentage error, % x light speed%Free Vibration of Cantilever Beam at 30mm DisplacementPercentage error, %= x 100%=8.70%Free Vibration of Cantilever Beam at 50mm DisplacementPercentage error, %= x 100%=8.70%Viscously Damped Vibration of Cantilever Beam at 30mm DisplacementPercentage error, %= x 100%=3.48%Viscously Damped Vibration of Cantilever Beam at 50mm DisplacementPercentage error, %= x 100%=3.48%In this experiment, it is c alculated that the percentage error for the free vibration is both 8.70% for 30mm and 50mm. For the viciously damped vibration, the percentage error for the 30mm and 50mm were both 3.48%. The results of the experiment were slightly inaccurate. This may be caused by the external force which is the air resistance as the beam oscillates. Another factor is caused by parallax error which occurred during the measuring of displacement before the beam was released as our eye level was not perpendicular to the scale of the metre rule. When the beam is released, it slightly hit the bottom of the container which decreases the original force released drastically which affects the amplitude of oscillation. Furthermore, the Modulus of Elasticity(Youngs Modulus) was given, which might not be accurate. All these source of error may affect the results of the experiment to be inaccurate.The results of the experiment can be improved by measuring the Modulus of Elasticity. Using a deeper container woul d also avoid the damper to hit the bottom of the container. Furthermore, the experiment can be done in a senselessness box to avoid air resistance. Eye level should be adjusted until it is perpendicular to the metre rule scale. This steps can increase the accuracy of the results.The damped period, damped natural frequency and the damping ratio of the system of free vibration is.When the amplitude is 30mm or 50mm for both cases, they have the same damped period, damped natural frequency and damped ratio. The percentage error for the free vibration of 30mm and 50mm were both 8.70% while for the viciously damped of 30mm and 50mm were both 3.48%. This indicates that the difference in amplitudes do not affect the frequency of the oscillation. From the general equation of frequency , where c=speed of wave= wavelengthThis formula proves that amplitude or displacement does not affect the frequency of the oscillation.If the strain gauge is mounted on the other end of the cantilever beam, th e results would be not accurate as the gauge is very sensitive to changes. At the other end, there is not much difference in the change of length which affects the strain. In Figure 3, the strain can be detected more easily as the change in length is very obvious. This is because when the free end is free to vibrates, the vibration will be strong but the compression and tension that results on the surface of pressure sensor is not that strong.Conclusion The theoretical natural frequency , for this experiment is 5.75Hz for free vibration cantilever. The value of both 30mm and 50mm frequency obtained were 6.25Hz which have percentage error of 8.70%. Whereas, the theoretical damped natural frequency , is 3.45 Hz. The value of both 30mm and 50mm frequency obtained were 3.57Hz which have percentage error of 3.48%.Furthermore, the results proves that the displacement does not affect the vibration(frequency) of the oscillation of the cantilever beam.