Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

[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

[email protected]

Engr/Math/Physics 25

Accelerating

Pendulum



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



A 3rd order Transformation (2)

ydtydxy

dtdyxyx 2

2

321

dtdxyx

xdtdxyx

xdtdxyx

33

32

2

21

1

5ln731973

OR 5ln731973

1233

txxxx

tyyyy



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



ODE: LittleOnes out of BigOneV =

S =

C =



ODE: LittleOnes out of BigOne











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



L

mW = mg

ta

L

mW = mg

ta



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



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



Convert to State Variable Form Let: Thus:

Then the 2nd derivative

Have Created Two 1st Order Eqns



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



SimuLink Diagram



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.



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')



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)];



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



Θ with Torsional Damping

E25_FE_Damped_Pendulum_1104.mdl



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')

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

Documents

pe engineeringmathphysics

computational methods

computational methodsproblem

computational methodsdemo

computational methodsrecall

computational methodsat

computational methodsconvert

computational methodssee