1. MAE 82 – Engineering Mathematics
Class Notes 6: Second Order Differential Equation – Non Homogeneous – Force Vibration
MAE 82 – Engineering Mathematics

2. Tacoma Narrow (WA)
Video - Galloping Gertie - The Collapse of the Tacoma Narrows Bridge

3. Dynamic Balance
Video - The vibrations resulting from the rotor destructs this Chinook helicopter

4. Force Vibrations – Introduction
impulse
step
ramp
sin

5. Force Vibrations – Introduction – Frequency Response – Bode Plot

6. Force Vibrations – Introduction Frequency Analysis – Fourier Series

7. Force Vibrations – Effect of Gravity

8. Force Vibrations – Classification Damped Oscillation – Undamped Oscillation

9. Force Vibrations Undamped Oscillation – Case 1

10. Force Vibrations Undamped Oscillation – Case 1
Using the undermined coefficient method
Solve for A
Resulted in

11. Force Vibrations Undamped Oscillation – Case 1
Given the solution
Using the Initial Conditions
Resulted in

12. Force Vibrations Undamped Oscillation – Case 1
Free response(Depends on I.C.)
Force response(Depends on the input)

13. Force Vibrations Undamped Oscillation – Case 1
Examining the factor of the force Response
Static Deflection Factor
Magnitude Factor
Free response(Depends on I.C.)
Force response(Depends on the input)

14. Force Vibrations Undamped Oscillation – Case 1

15. Force Vibrations Undamped Oscillation – Case 1

16. Force Vibrations Undamped Oscillation – Case 1

17. Force Vibrations Undamped Oscillation – Case 1

18. Force Vibrations Undamped Oscillation – Case 1

19. Force Vibrations Undamped Oscillation – Case 3
Undamped Oscillator (case 3)
The solution is similar to the case when
Same as

20. Force Vibrations Undamped Oscillation – Case 3

21. Force Vibrations Undamped Oscillation – Case 2 (Resonance)
Undamped Oscillator

22. Force Vibrations Undamped Oscillation – Case 2 (Resonance)IC
Forced

23. Force Vibrations Undamped Oscillation – Case 2 (Resonance)

24. Glass
Video - Wine glass resonance in slow motion

25. Force Vibrations – Damped Oscillations

26. Force Vibrations – Damped Oscillations
Substitute into the differential equation
Coefficient of

27. Force Vibrations – Damped Oscillations
Convert the solution to the form of a single trigonometric function with a phase shift
where

28. Force Vibrations – Damped Oscillations

29. Force Vibrations – Damped Oscillations
Factor out of the denominator

30. Force Vibrations – Damped Oscillations

31. Single DOF - Resonance
Video - SDOF Resonance Vibration Test