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- 1. Experiment #4<br />Dynamic Response of a Vibrating Structure to Sinusoidal Excitation<br />
- 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. Test Specimen and Test Setup<br />Exciter<br />Load Cell<br />Accelerometer<br />Specimen<br />
- 4. Part I – Frequency Response<br />Forced Response<br />Free Response<br />Manual Excitation<br />Mechanical Excitation<br />
- 5. Forced Response<br />Node<br />First Mode – 10.1 Hz<br />First Resonance – displays one node<br />
- 6. Forced Response<br />Second Mode – 68.8 Hz<br />Nodes<br />Second Resonance – displays two nodes<br />
- 7. Forced Response<br />Third Mode – 115 Hz<br />Nodes<br />Third Resonance – displays three nodes<br />
- 8. Forced Response<br />Before 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />In phase<br />
- 9. Forced Response<br />Before 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />In phase<br />
- 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. Forced Response<br />After 68.6 Hz Resonance <br />𝑋𝐹0 is small<br /> <br />Force<br />Acceleration<br />Back in phase<br />
- 12. Magnitude Ratio vs. Frequency<br />Experimental data indicates that there is a resonance ~ 68 Hz<br />
- 13. Phase Angle vs. Frequency<br />Experimental data indicates that there is a phase shift of 90° at ~68 Hz<br />
- 14. Free Response<br />Acceleration<br />Decreasing acceleration and decreasing displacement<br />
- 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. 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. 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. 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. 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. Half-Power Method<br />𝑋𝐹0<br /> <br />
- 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. 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. 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. 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. 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. Conclusions<br /><ul><li> We successfully performed experiments to find both the forced and free response of our test specimen.
- 27. The magnitude ratio curves produced by our model correlate very well with our experimental data.
- 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>

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