Bruce Mayer, PE Regsitered Electrical & Mechanical Engineer BMayer@ChabotCollege
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
description
Transcript of Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx1
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Accelerating
Pendulum
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx2
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Recall 3rd order Transformation
5ln731973
OR 5ln731973 2
2
3
3
tyyyy
tydtdy
dtyd
dtyd
27
37
47
:sIC' and
72
27
ydtyd
ydtdy
y
t
t
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx3
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
A 3rd order Transformation (2)
ydtydxy
dtdyxyx 2
2
321
dtdxyx
xdtdxyx
xdtdxyx
33
32
2
21
1
5ln731973
OR 5ln731973
1233
txxxx
tyyyy
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx4
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
A 3rd order Transformation (3) Thus the 3-Eqn 1st Order ODE System
319735ln73
2
1
12333
322
211
xxxtxdtdx
xxdtdx
xxdtdx
3-IC277
2-IC377
1-IC477
3
2
1
xy
xy
xy
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx5
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
ODE: LittleOnes out of BigOneV =
S =
C =
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx6
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
ODE: LittleOnes out of BigOne
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx7
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
ODE: LittleOnes out of BigOne
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx8
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
ODE: LittleOnes out of BigOne
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx9
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx10
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 2nd Order, Homogeneous ODE
0cossin
tagL• See next Slide for Eqn Derivation
Solve for θ(t)
L
m W = mg
ta
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx11
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
L
mW = mg
ta
L
mW = mg
ta
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx12
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Prob 9.34: ΣF = Σma
N-T CoORD Sys
n
T
LdtdLs
dtsd
LdtdL
dtdsLdds
2
2
2
2
Use Normal-Tangential CoOrds; θ+ → CCW
cos
sin
,, taaLa
WF
TbaseTS
T
cossin taLmmg
L
mW = mg
ta
L
mW = mg
ta
sinW
costa
Ldds
Use ΣFT = ΣmaT
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx13
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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 FormL
mW = mg
ta
L
mW = mg
ta
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx14
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Convert to State Variable Form Let: Thus:
Then the 2nd derivative
Have Created Two 1st Order Eqns
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx15
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)
0cossin2
2
ytaygdtydL
L
ygytadtyd sincos2
2
dtdtL
ygytay
sincos
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx16
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
SimuLink Diagram
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx17
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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.
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx18
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Prob 9.34 Script File
% Bruce Mayer, PE * 05Nov11% ENGR25 * problem 9.34% file = Demo_Prob9_34.m% %This script file calls FUNCTION pendacc%clear % clears memoryglobal m b; % globalize accel calc constants% Acceleration, a(t) = m*t + b% ask user for max time; suggest starting at 25tmax = input('tmax = '); %%set the case consts, and IC's y(0) & dy(0)/dt%=> remove the leading "%" to toggle between casesm = 0, b = 5, y0 = [0.5 0]; % case-a%m = 0, b = 5, y0 = [3 0]; % case-b%m = 0.5, b = 0, y0 = [3 0]; % case-c% m = 0.4, b = -4, y0 = [1.7 2.3]; % case-d => EXTRA%%Call the ode45 routine with the above data inputs[t,x]=ode45('pendacc', [0, tmax], y0);%%PLot theta(t)subplot(1,1,1)plot(t,x(:,1)), xlabel('t (sec)'), ylabel('theta (rads)'),...
title('P9.34 - Accelerating Pendulum'), grid;disp('Plotting ONLY theta - Hit Any Key to continue')pause%Plot the FIRST column of the solution “matrix” %giving x1 or y.subplot(2,1,1)plot(t,x(:,1)), xlabel('t (sec)'), ylabel('theta (rads)'),...
title('P9.34 - Accelerating Pendulum'), grid;%Plot the SECOND column of the solution “matrix” %giving x2 or dy/dt.subplot(2,1,2)plot(t,x(:,2)), xlabel('t (sec)'), ylabel('dtheta/dt (r/s)'), grid;disp('Plotting Both theta and dtheta/dt; hit any key to continue')
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx19
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Prob 9.34 Function Filefunction dxdt = pendacc(t_val,z);% Bruce Mayer, PE * 05Nov05% ENGR25 * Prob 8-30% %This is the function that makes up the system %of differential equations solved by ode45%% the Vector z contains yk & [dy/dt]k%%Globalize the Constants used to calc the Accelglobal m b% set the physical constantsL = 1; % in mg = 9.81; % in m/sq-Sec%%DEBUG § => remove semicolons to reveal t_val & zt_val; z;%% Calc the Cauchy (State) valuesdxdt(1)= z(2); % at t=0, dxdt(1) = dy(0)/dtdxdt(2)= ((m*t_val + b)*cos(z(1)) - g*sin(z(1)))/L;% at t = 0, dxdt(2) =((m*t_val + b)*cos(y(0)) - g*sin(y(0)))/L; %% make the dxdt into a COLUMN vectordxdt = [dxdt(1); dxdt(2)];
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx20
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):
L
m W = mg
taD
0cossin btngDL
rads 8.20 secrads 9.100
tdt
d
L = 1.6 meters D = 0.07 meters/sec g = 9.8 meters/sec2
n = 0.40 meters/sec3 b = −3.0 meters/sec2
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx21
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Θ with Torsional Damping
E25_FE_Damped_Pendulum_1104.mdl
[email protected] • ENGR-25_Tutorial_P9-34_Accelerating_Pendulum.pptx22
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Θ with Torsional Damping
0 10 20 30 40 50 60 70 80 90 100-3
-2
-1
0
1
2
3
t (sec)
(ra
ds)
Accelerating Pendulum Angular Position
plot(tout,Q, 'k', 'LineWidth', 2), grid, xlabel('t (sec)'), ylabel('\theta (rads)'), title('Accelerating Pendulum Angular Position')