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Dynamic Response Of A Vibrating Structure To Sinusoidal Excitation

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This is a presentation that I gave to the engineering department as part of a semester-long lab course.

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• Second derivative of sin is –sin
• Dynamic Response Of A Vibrating Structure To Sinusoidal Excitation

1. 1. Experiment #4<br />Dynamic Response of a Vibrating Structure to Sinusoidal Excitation<br />
2. 2. Objectives<br />Perform a standard vibration test to measure the frequency response of a structural system <br /> 1.<br />Solve a differential equation describing the motion of a structure with one degree of freedom under sinusoidal excitation<br />2.<br />Calculate the equivalent viscous damping coefficient (ζ) of a single degree of freedom structure<br /> 3.<br />
3. 3. Test Specimen and Test Setup<br />Exciter<br />Load Cell<br />Accelerometer<br />Specimen<br />
4. 4. Part I – Frequency Response<br />Forced Response<br />Free Response<br />Manual Excitation<br />Mechanical Excitation<br />
5. 5. Forced Response<br />Node<br />First Mode – 10.1 Hz<br />First Resonance – displays one node<br />
6. 6. Forced Response<br />Second Mode – 68.8 Hz<br />Nodes<br />Second Resonance – displays two nodes<br />
7. 7. Forced Response<br />Third Mode – 115 Hz<br />Nodes<br />Third Resonance – displays three nodes<br />
8. 8. Forced Response<br />Before 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />In phase<br />
9. 9. Forced Response<br />Before 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />In phase<br />
10. 10. Forced Response<br />At 68.6 Hz Resonance <br />𝑋𝐹0 is large<br /> <br />Force<br />Acceleration<br />90°<br />phase shift<br />
11. 11. Forced Response<br />After 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />Back in phase<br />
12. 12. Magnitude Ratio vs. Frequency<br />Experimental data indicates that there is a resonance ~ 68 Hz<br />
13. 13. Phase Angle vs. Frequency<br />Experimental data indicates that there is a phase shift of 90° at ~68 Hz<br />
14. 14. Free Response<br />Acceleration<br />Decreasing acceleration and decreasing displacement<br />
15. 15. Part II – Lumped Parameter Model <br />The diagram below describes the motion of our beam: <br />x<br />F = applied force by the exciter<br />X = beam displacement<br />
16. 16. Single DOF Spring-Mass-Damper System<br />The mathematical model we used to describe the motion of the beam was a Single DOF Spring-Mass-Damper System. <br />X(t) = displacement<br />F(t) = applied load <br />m = mass<br />k = spring constant<br />c = damping coefficient<br />
17. 17. Single DOF Spring-Mass-Damper System<br />The lumped parameter model can be modeled by a non-homogeneous differential equation:<br />𝑚𝑥+𝑐𝑥+𝑘𝑥=𝐹(𝑡)<br /> <br />We developed two particular solutions to this DE:<br />𝜙=tan−1−2𝜁𝜔𝜔𝑛1−𝜔𝜔𝑛2<br /> <br />- Phase angle between forcing function and the displacement of the beam<br />𝑋𝐹0=1𝑘1−𝜔𝜔𝑛22+2𝜁𝜔𝜔𝑛2<br /> <br />- Magnitude ratio of displacement and applied force<br />
18. 18. Part III – Equivalent Viscous Damping Coefficient (ζ)<br />Three Methods for Finding ζ<br />Half-Power Method<br />Log Decrement Method<br />Best Guess Method<br />
19. 19. Half-Power Method<br />𝜁𝐻𝑃=𝑓2−𝑓12𝑓𝑛<br /> <br />The half-power method utilizes frequencies on either side of the natural frequency along with the natural frequency to approximate the viscous damping coefficient (ζ). <br />𝜁𝐻𝑃=69.1142−68.52952∗68.9<br /> <br />𝜻𝑯𝑷=𝟎.𝟎𝟎𝟒𝟐𝟒𝟗<br /> <br />
20. 20. Half-Power Method<br />𝑋𝐹0<br /> <br />
21. 21. Log Decrement Method<br />The log decrement method utilizes frequencies at different points along the Free Response result in Part I.<br />𝛿=1𝑛ln𝑥0𝑥𝑛<br /> <br />- This is the log decrement<br />The log decrement is then used to find the viscous damping coefficient (ζ): <br />𝜁𝐿𝐷=11+2𝜋𝛿2<br /> <br />𝜁𝐿𝐷=11+2𝜋0.075972<br /> <br />𝜻𝑳𝑫=𝟎.𝟎𝟏𝟐𝟎𝟗<br /> <br />
22. 22. Best Guess Method<br />The Best Guess Method involved simply picking a value for ζ and then plotting the theoretical curves alongside the experimental data. The correct value of ζ is found when the theoretical curves match the experimental data.<br />𝜻𝑩𝑮=𝟎.𝟎𝟏<br /> <br />
23. 23. Comparison of HP and LD ζ Values<br />Differential error analysis shows that:<br />𝜎𝜻𝑯𝑷 =7∗𝜎𝜻𝑳𝑫 <br />Therefore, we conclude that the Log Decrement Method is a much more accurate method of calculating the viscous damping coefficient (ζ).<br /> <br />
24. 24. Frequency Response Function Curves - Magnitude<br />Magnitude Ratio vs. Frequency, ω<br />All curves agree as to the location of the resonant frequency<br />The value of ζ affects both the height of the curve and the slope leading up to the resonance<br />68.6 <br />
25. 25. Frequency Response Function Curves – Phase Angle (φ)<br />Phase Shift, φ vs. Frequency, ω<br />All curves indicate that there is a phase shift of ~90° at 68.6 Hz<br />FRF curves don’t correlate well with the experimental phase shift in this region<br />68.6 <br />
26. 26. Conclusions<br /><ul><li> We successfully performed experiments to find both the forced and free response of our test specimen.
27. 27. The magnitude ratio curves produced by our model correlate very well with our experimental data.
28. 28. Our theoretical phase angle curves fit the experimental data well up until the resonance, and show the expected phase shift.</li></li></ul><li>Recommendations<br /><ul><li>I recommend that there be a more permanent test setup for the frequency response experiments in such a way that it can’t be altered between lab sessions.</li>