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

1. Experiment #4 Dynamic Response of a Vibrating Structure to Sinusoidal Excitation

2. Objectives Perform a standard vibration test to measure the frequency response of a structural system 1. Solve a differential equation describing the motion of a structure with one degree of freedom under sinusoidal excitation 2. Calculate the equivalent viscous damping coefficient (ζ) of a single degree of freedom structure 3.

3. Test Specimen and Test Setup Exciter Load Cell Accelerometer Specimen

4. Part I – Frequency Response Forced Response Free Response Manual Excitation Mechanical Excitation

5. Forced Response Node First Mode – 10.1 Hz First Resonance – displays one node

6. Forced Response Second Mode – 68.8 Hz Nodes Second Resonance – displays two nodes

7. Forced Response Third Mode – 115 Hz Nodes Third Resonance – displays three nodes

8. Forced Response Before 68.6 Hz Resonance 𝑋𝐹0 is small Force Acceleration In phase

9. Forced Response Before 68.6 Hz Resonance 𝑋𝐹0 is small Force Acceleration In phase

10. Forced Response At 68.6 Hz Resonance 𝑋𝐹0 is large Force Acceleration 90° phase shift

11. Forced Response After 68.6 Hz Resonance 𝑋𝐹0 is small Force Acceleration Back in phase

12. Magnitude Ratio vs. Frequency Experimental data indicates that there is a resonance ~ 68 Hz

13. Phase Angle vs. Frequency Experimental data indicates that there is a phase shift of 90° at ~68 Hz

14. Free Response Acceleration Decreasing acceleration and decreasing displacement

15. Part II – Lumped Parameter Model The diagram below describes the motion of our beam: x F = applied force by the exciter X = beam displacement

16. Single DOF Spring-Mass-Damper System The mathematical model we used to describe the motion of the beam was a Single DOF Spring-Mass-Damper System. X(t) = displacement F(t) = applied load m = mass k = spring constant c = damping coefficient

17. Single DOF Spring-Mass-Damper System The lumped parameter model can be modeled by a non-homogeneous differential equation: 𝑚𝑥+𝑐𝑥+𝑘𝑥=𝐹(𝑡) We developed two particular solutions to this DE: 𝜙=tan−1−2𝜁𝜔𝜔𝑛1−𝜔𝜔𝑛2 - Phase angle between forcing function and the displacement of the beam 𝑋𝐹0=1𝑘1−𝜔𝜔𝑛22+2𝜁𝜔𝜔𝑛2 - Magnitude ratio of displacement and applied force

18. Part III – Equivalent Viscous Damping Coefficient (ζ) Three Methods for Finding ζ Half-Power Method Log Decrement Method Best Guess Method

19. Half-Power Method 𝜁𝐻𝑃=𝑓2−𝑓12𝑓𝑛 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). 𝜁𝐻𝑃=69.1142−68.52952∗68.9 𝜻𝑯𝑷=𝟎.𝟎𝟎𝟒𝟐𝟒𝟗

20. Half-Power Method 𝑋𝐹0

21. Log Decrement Method The log decrement method utilizes frequencies at different points along the Free Response result in Part I. 𝛿=1𝑛ln𝑥0𝑥𝑛 - This is the log decrement The log decrement is then used to find the viscous damping coefficient (ζ): 𝜁𝐿𝐷=11+2𝜋𝛿2 𝜁𝐿𝐷=11+2𝜋0.075972 𝜻𝑳𝑫=𝟎.𝟎𝟏𝟐𝟎𝟗

22. Best Guess Method 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. 𝜻𝑩𝑮=𝟎.𝟎𝟏

23. Comparison of HP and LD ζ Values Differential error analysis shows that: 𝜎𝜻𝑯𝑷 =7∗𝜎𝜻𝑳𝑫 Therefore, we conclude that the Log Decrement Method is a much more accurate method of calculating the viscous damping coefficient (ζ).

24. Frequency Response Function Curves - Magnitude Magnitude Ratio vs. Frequency, ω All curves agree as to the location of the resonant frequency The value of ζ affects both the height of the curve and the slope leading up to the resonance 68.6

25. Frequency Response Function Curves – Phase Angle (φ) Phase Shift, φ vs. Frequency, ω All curves indicate that there is a phase shift of ~90° at 68.6 Hz FRF curves don’t correlate well with the experimental phase shift in this region 68.6

27. The magnitude ratio curves produced by our model correlate very well with our experimental data.