Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engr/Math/Physics 25 AcceleratingPendulum Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Recall 3rd order Transformation

3. A 3rd order Transformation (2)

4. A 3rd order Transformation (3) • Thus the 3-Eqn 1st Order ODE System

5. ODE: LittleOnes out of BigOne V = S = C =

6. ODE: LittleOnes out of BigOne

7. ODE: LittleOnes out of BigOne

8. ODE: LittleOnes out of BigOne      

10. Accelerating Pendulum Problem 9.34 • For an Arbitrary Lateral-Acceleration Function, a(t), the ANGULAR Position, θ, is described by the (nastily) NONlinear 2nd Order, Homogeneous ODE q m W = mg • See next Slide for Eqn Derivation • Solve for θ(t)

12. N-T CoORD Sys Prob 9.34: ΣF = Σma • Use Normal-Tangential CoOrds; θ+ → CCW • Use ΣFT = ΣmaT

13. Prob 9.34: Simplify ODE • Cancel m: • Collect All θ terms on L.H.S. • Next make Two Little Ones out of the Big One • That is, convert the ODE to State Variable Form

14. Convert to State Variable Form • Let: • Thus: • Then the 2nd derivative • Have Created Two 1st Order Eqns

15. SimuLink Solution • The ODE using y in place of θ • Isolate Highest Order Derivative • Double Integrate to find y(t)

16. SimuLink Diagram

17. All Done for Today FoucaultPendulum While our clocks are set by an average 24 hour day for the passage of the Sun from noon to noon, the Earth rotates on its axis in 23 hours 56 minutes and 4.1 seconds with respect to the rest of the universe. From our perspective here on Earth, it appears that the entire universe circles us in this time. It is possible to do some rather simple experiments that demonstrate that it is really the rotation of the Earth that makes this daily motion occur. In 1851 Leon Foucault (1819-1868) was made famous when he devised an experiment with a pendulum that demonstrated the rotation of the Earth.. Inside the dome of the Pantheon of Paris he suspended an iron ball about 1 foot in diameter from a wire more than 200 feet long. The ball could easily swing back and forth more than 12 feet. Just under it he built a circular ring on which he placed a ridge of sand. A pin attached to the ball would scrape sand away each time the ball passed by. The ball was drawn to the side and held in place by a cord until it was absolutely still. The cord was burned to start the pendulum swinging in a perfect plane. Swing after swing the plane of the pendulum turned slowly because the floor of the Pantheon was moving under the pendulum.

18. Prob 9.34 Script File

19. Prob 9.34 Function File

20. Θ with Torsional Damping • The Angular Position, θ, of a linearly accelerating pendulum with a Journal Bearing mount that produces torsional friction-damping can be described by this second-order, non-linear Ordinary Differential Equation (ODE) and Initial Conditions (IC’s) for θ(t): q m W = mg

21. Θ with Torsional Damping • E25_FE_Damped_Pendulum_1104.mdl

22. Θ with Torsional Damping plot(tout,Q, 'k', 'LineWidth', 2), grid, xlabel('t (sec)'), ylabel('\theta (rads)'), title('Accelerating Pendulum Angular Position')