ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engr/Math/Physics 25 Accelerating Pendulum

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Recall 3 rd order Transformation

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods A 3 rd order Transformation (2)

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods A 3 rd order Transformation (3)  Thus the 3-Eqn 1 st Order ODE System

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods ODE: LittleOnes out of BigOne V =S =C =

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods ODE: LittleOnes out of BigOne

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods ODE: LittleOnes out of BigOne

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods ODE: LittleOnes out of BigOne      

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Problem 9.34  Accelerating Pendulum  For an Arbitrary Lateral- Acceleration Function, a(t), the ANGULAR Position, θ, is described by the (nastily) NONlinear 2 nd Order, Homogeneous ODE See next Slide for Eqn Derivation  Solve for θ(t)  m W = mg

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Prob 9.34: ΣF = Σma N-T CoORD Sys  Use Normal-Tangential CoOrds; θ+ → CCW  Use ΣF T = Σma T

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Prob 9.34: Simplify ODE

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Convert to State Variable Form

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods SimuLink Solution  The ODE using y in place of θ  Isolate Highest Order Derivative  Double Integrate to find y(t)

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods SimuLink Diagram

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today Foucault Pendulum 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 ( ) 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.

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Prob 9.34 Script File

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Prob 9.34 Function File

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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):  m W = mg L = 1.6 metersD = 0.07 meters/secg = 9.8 meters/sec 2 n = 0.40 meters/sec 3 b = −3.0 meters/sec 2

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Θ with Torsional Damping  E25_FE_Damped_Pendulum_1104.mdl

ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Θ with Torsional Damping