Engineering 25
Chp9
Tutorial:
Prob 9.32
Solution
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Unit Summary
 A summary of the Units of
Measure in this problem:
• y(t) → meters
• dy/dt → meters/second (m/s)
• dy2/(dt)2 → meter/second2
(m/s2)
• m → kg
• k → Newtons/meter (N/m)
• M → Newtons
• K, B → meters
• ωp , ωc → rads/sec (r/s)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
The Physical Situation
f t   10 N sin  p t 
m  3 kg
k k  75N m
m
y
 A FrictionLESS mass-spring
(m-k) system
 SINSOIDAL Forcing
Function, f(t)
• Pull/Push Magnitude, M = 10N
(2.248 lbs)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
The Math Model
 Using Newton’s 2nd Law
(ΣF = ma) find the MassSpring System y(t) ODE
d2y  N
3 kg 2  75  y  10 Nsin  pt 
dt   m 
Or
m y  0 y  ky  M sin  p t 


 With 0th & 1st order I.C.’s

dy
y0  0 m &
 y0  0 m s
dt t 0
 Find y(t) for ωp = 1, 5.1, 10 r/s
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.32 Analytical Soln
% Bruce Mayer, PE * 15Apr12
% ENGR25 * problem 9.32
% file = Prob9_32_Analytical_Soln_1204.m
%* NOTE: Analytical Solutions are NOT Covered in the TextBook
%
% Implements Analytical Soln
%*
y(t) = K*sin(wp*t) - [K*wp/wc]*sin(wc*t)
%* Forcing Fcn: f(t) = 10N*sin(wp*t) = M*sin(wp*t)
%
clear
% clears memory
%
% Eqn Constants
m = 3; % in kg
k = 75; % in N/m
wp =[1 5.1 10]; % in rads/sec
%
% Calc Constants in Soln as a fcn of wp
K = (10./(k-m*wp.^2)) % in m
wc = 5; % rads/sec
B1 = -K.*wp/wc % in m
%
tmax = 20; % solve over adjustable time frame: 20, 100, 500
%* Use 70 sec to determine "Beating" Period
t = linspace(0, tmax, 1000); % 1000 plotting Points pts
tLen = length(t);
%
% Calc soln for 3 wp cases
%
ya = B1(1)*sin(wc*t) + K(1)*sin(wp(1)*t); % wp = 1 rad/sec
yb = B1(2)*sin(wc*t) + K(2)*sin(wp(2)*t); % wp = 5.1 rad/sec
yc = B1(3)*sin(wc*t) + K(3)*sin(wp(3)*t); % wp = 10 rad/sec
%
% Plot one on top of the other using subplot
subplot(3,1,1)
plot(t,ya, 'LineWidth',2), grid, xlabel('t (sec)'),...
ylabel('ya (m)'),title('Case a. wp = 1 r/s')
subplot(3,1,2)
plot(t,yb, 'LineWidth',2), grid, xlabel('t (sec)'),...
ylabel('yb (m)'), title('Case b. wp = 5.1 r/s')
subplot(3,1,3)
plot(t,yc, 'LineWidth',2), grid, xlabel('t (sec)'),...
ylabel('yc (m)'), title('Case b. wp = 10 r/s')
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.29 Analytical Plot
Case a. wp = 1 r/s
0.15
0.1
ya (m)
0.05
0
-0.05
-0.1
-0.15
-0.2
0
2
4
6
8
10
t (sec)
12
14
16
18
20
14
16
18
20
14
16
18
20
Case b. wp = 5.1 r/s
6
4
yb (m)
2
0
-2
-4
-6
0
2
4
6
8
10
t (sec)
12
Case b. wp = 10 r/s
0.15
0.1
yc (m)
0.05
0
-0.05
-0.1
-0.15
-0.2
10
t (sec)
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0
2
4
6
8
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Solve by ODE23
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Solve by ODE23
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.32 Numerical Soln-a
 The Function file for the ODE
Solver Call
function dxdt = dxdt_P9_32(t,z);
% Bruce Mayer, PE * 15Apr12
% ENGR25 * P9.32 ODE soln by Matrix Methods
%
% Receive parameters using GLOBAL constants
global m k wp
%
% the two 1st order eqns inside the dxdt vector [dx1/dt; dx2/dt]
dxdt = [z(2); (10*sin(wp*t) - k*z(1))/m]
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.32 Numerical Soln-a
% Bruce Mayer, PE * 15Apr12
% ENGR25 * problem 9.32
% file = Prob9_32_dxdt_Soln.m
%
% Use with Function File = dxdt_P9_32.m
%
% Implements Numerical Soln
%* y(t) = K*sin(wp*t) - [K*wp/wc]*sin(wc*t)
%** Forcing Fcn: f(t) = 10N*sin(wp*t)
%
clear
% clears memory
%
% Pass Parameters to ODE function dxdt as GLOBAL values
global m k wp
%
% one of the cases below is ACTIVE; comment out others
wp = 1; % case a
%wp = 5.1; % case b
%wp = 10; % case c
m = 3;k = 75; M = 10;
%
% use ode23 solver
%* fcn = dxdt_P9_32.m
%* t interval = 0-20 sec (or more) => t = 0-70 is quite interesting
%* IC's y(0) = 0; dy/dt @ t=0 = 0
%
[t, x] = ode23('dxdt_P9_32', [0, 70], [0, 0]);
f = M*sin(wp*t);
%
% Soln & Forcing Function on Top of Each Other using SubPlot
%
subplot(2,1,1)
plot(t,x(:,1), 'LineWidth',2),xlabel('t (sec)'),ylabel('y(t) (m)'),
title('P8.28 dxdt Form Soln'), grid
subplot(2,1,2)
plot(t,f, 'LineWidth',2),xlabel('t (sec)'),ylabel('f(t) (N)'),
title('Forcing Fcn'), grid
%
disp('showing solution & Forcing-Funtion -- Hit AnyKey to continue')
pause
%
% Soln VALUE & SLOPE on Top of Each Other using SubPlot
%
subplot(2,1,1)
plot(t,x(:,1), 'LineWidth',2),xlabel('t (sec)'),ylabel('y(t) (m)'),
title('Position'), grid
subplot(2,1,2)
plot(t,x(:,2), 'LineWidth',2),xlabel('t (sec)'),ylabel('dy/dt @ t
(m/s)'), title('Slope'), grid
%
Bruce Mayer, PE
Engineering/Math/Physics
25: Computational Methods
16
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Anonymous Function
 Quick Solution Entirely from
Command Window Using an
ANONYMOUS Function
>> m = 3;k = 75; M = 10; wp = 1;
>> dxdt = @(t,z) [z(2); (10*sin(wp*t) - k*z(1))/m]
dxdt =
@(t,z)[z(2);(10*sin(wp*t)-k*z(1))/m]
>> [T, Y] = ode23(dxdt, [0,14], [0, 0]);
>> plot(T,Y, 'LineWidth', 2), grid, legend('y(t)',
'dy/dt = slope')
y(t)
dy/dt = slope
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
2
4
6
8
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10
12
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Anonymous Function
 Plot ONLY(T)
>> >> plot(T,Y(:,1), 'LineWidth', 2), grid,
xlabel('t'), ylabel('y(t)')
0.15
0.1
y(t)
0.05
0
-0.05
-0.1
-0.15
0
2
4
6
8
10
t
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
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14
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.32 Numerical Soln-b
 The Function file for the ODE
Solver Call
function xDot = mksine_P9_32(t,z);
% Bruce Mayer, PE * 15Apr12
% ENGR25 * P 9.32 ODE soln by Matrix Methods
%
% Pass parameters using GLOBAL constants
global m k wp
%
% the matrix elements
A = [0,1;-k/m,0];
B = [0;1/m];
f = 10*sin(wp*t);
xDot=A*z+B*f;
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
P9.32 Numerical Soln-b
% Bruce Mayer, PE * 15Apr12
% ENGR25 * problen 9.32
% file = Prob9_32_Matrix_Soln.m
%
% Use with Function File = mksine_P9_32.m
%
% Implements Numerical Soln
%* y(t) = K*sin(wp*t) - [K*wp/wc]*sin(wc*t)
%* Forcing Fcn: f(t) = 10N*sin(wp*t)
%
clear
% clears memory
%
% Pass Parameters as Global values
global m k wp
%
% one of the cases below is ACTIVE; comment out others
%wp = 1; % case a
wp = 5.1; % case b
%wp = 10; % case c
m = 3;k = 75; M = 10;
%
% use ode23 solver
%* fcn = mksine_P9_32
%* t interval = 0-20 sec
%* IC's y(0) = 0; dy/dy @ t=0 = 0
%
[t, x] = ode23('mksine_P9_32', [0, 20], [0, 0]);
f = M*sin(wp*t);
% Soln & Forcing Function on Top of Each Other using SubPlot
%
subplot(2,1,1)
plot(t,x(:,1), 'LineWidth', 2),xlabel('t (sec)'),ylabel('y(t)
(N)'),...
title('P9.32 Matrix Form Soln'), grid
subplot(2,1,2)
plot(t,f, 'LineWidth', 2),xlabel('t (sec)'),ylabel('f(t) (N)'),...
title('Forcing Fcn'), grid
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Case-a: ωp = 1 rad/sec
P8.28 Matrix Form Soln
0.15
0.1
y(t) (N)
0.05
0
-0.05
-0.1
-0.15
-0.2
0
5
10
t (sec)
15
20
15
20
Forcing Fcn
10
f(t) (N)
5
0
-5
-10
0
5
10
t (sec)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Case-b: ωp = 5.1 rad/sec
P8.28 Matrix Form Soln
6
4
y(t) (N)
2
0
-2
-4
-6
0
5
10
t (sec)
15
20
15
20
Forcing Fcn
10
f(t) (N)
5
0
-5
-10
0
5
10
t (sec)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Case-c: ωp = 10 rad/sec
P8.28 Matrix Form Soln
0.15
0.1
y(t) (N)
0.05
0
-0.05
-0.1
-0.15
-0.2
0
5
10
t (sec)
15
20
15
20
Forcing Fcn
10
f(t) (N)
5
0
-5
-10
0
5
10
t (sec)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
Case-b: 100 sec
P8.28 Matrix Form Soln
8
6
4
y(t) (N)
2
0
-2
-4
-6
-8
0
10
20
30
40
50
t (sec)
60
70
80
 Note the “beating” with a
Period of about 63 sec
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
90
100
SimuLink Solution
 ODE by Newton’s 2nd Law
d y
m 2  ky  M sin  p t 
dt
2
 Solve for Highest Order Term


d y M sin  p t   ky

2
dt
m
2
 Find y by Double Integral


Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PE
 M sin  pt   ky 
y    
dt dt
m


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BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt
SimuLink Model
 P9_32_mk_1104.mdl
 Note Changes in IC’s
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_HW-01_Solution.ppt