MATLAB: Problem with Response function in a mass-spring-damper System

massspringdampersystemsvibration analysis

Hello!

So I am attempting to determine the response of an under-damped system (There is a spring, Mass and damper in the system). I have an error in my response function, x(t) that I cant seem to figure out. The error is as follows:

Error using * Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches the number of rows in the second matrix. To perform elementwise multiplication, use '.*'

Here is my Response Function:

And here is My command Window with the error:

%%Damped Response
c = 8;    Dampinging Coefficient
m = 2;    Mass
k = 16    Spring Constant
x0 = 1    X
xd0 = -1  X Dot
wn = sqrt(k/m)     Natural Frequency
zeta = c/2/m/wn;
B = x0;
A = (xd0 + zeta*wn*x0)/wn/sqrt(1-zeta^2);
C = sqrt(A^2+B^2);
x = DampedResponse(zeta,wn,A,B,t);
figure; plot(t,x,t,C*exp(-zeta*wn*t),'--r',t,-C*exp(-zeta*wn*t),'--r'); grid on;
if true
  % code
end
Response Function For my System:
function x = DampedResponse(zeta,wn,A,B,t)
wd = wn*sqrt(1-zeta^2);
x = exp(-zeta*wn*t)*(A*sin(wd*t)+B*cos(wd*t));
end

Best Answer

You forgot to include ‘t’. When I provided it, the problem turned out to be solved by using element-wise multiplication in this assignment in ‘DampedResponse’:

x = exp(-zeta*wn*t).*(A*sin(wd*t)+B*cos(wd*t));
                   ↑ ← INSERT DOT-OPERATOR HERE

Related Solutions

MATLAB: Plot all four functions

It only plots the first one because your Zeta is always 1 (c/cc = 1). Moreover, even if you changed Zeta to be other values, it would overwrite your graph since you do not have "hold on" for the figure.

instead of using if statements, why dont you define 4 different functions like this:

clear,clc
%% Givens
m=2; %kg
k=200; %N/m
x0=0.05; %meters
x_dot=2; %m/s
%% Solution Part A
Wn= (k/m)^0.5;
c= (2*m*Wn);
Cc= (2*m*Wn);
%% Solution Part B
Zeta=1;
C1=x0;
C2=x_dot+Wn*x0;
x_1 =  @(t) (C1+C2*t)*exp(-Wn*t);
Zeta=0;
X= (((x0^2)*(Wn^2)+(x_dot^2)+(2*x0*x_dot*Zeta*Wn))^0.5)/(((1-(Zeta^2))^0.5)*Wn);
phi= atan((x_dot+(Zeta*Wn*x0))/(x0*Wn*((1-(Zeta^2))^0.5)));
Wd= ((1-(Zeta^2))^0.5)*Wn;
x_0 = @(t) X*(exp(-Zeta*Wn*t))* cos((Wd*t)-phi);
Zeta=0.5;
X= (((x0^2)*(Wn^2)+(x_dot^2)+(2*x0*x_dot*Zeta*Wn))^0.5)/(((1-(Zeta^2))^0.5)*Wn);
phi= atan((x_dot+(Zeta*Wn*x0))/(x0*Wn*((1-(Zeta^2))^0.5)));
Wd= ((1-(Zeta^2))^0.5)*Wn;
x_05 = @(t) X*(exp(-Zeta*Wn*t))* cos((Wd*t)-phi);
Zeta = 2;
C1=x0;
C2=x_dot+Wn*x0;
x_2= @(t) C1*exp(-Zeta+sqrt(Zeta^2-1)*Wn*t)+C2*exp(-Zeta-sqrt(Zeta^2-1)*Wn*t);
figure(1)
fplot(x_0,[0 1],'b')
hold on
fplot(x_05,[0 1],'r')
fplot(x_1,[0 1],'g')
fplot(x_2,[0 1],'k')
hold off
legend('Zeta = 0','Zeta = 0.5','Zeta = 1','Zeta = 2')
axis([0 1 -0.5 0.5])
%% Solution Part C

This way you can plot them all.

MATLAB: Won’t this plot

You can make the privious three plot in single description using for loop, define "a" in array. #Recomended. The error arises because of non equal length of t and v, as v result from diff. Hence I have ignored the last element of t to make them both equal. You can opt out the first one also, as per your requirements.

m=20; %Mass
k=30; %Spring constant
c=0.9; %Damping constant






A=1;
a=0;
g=c/(2*m); %Damping coefficient






wn=sqrt(k/m); %Natural frequency






wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency






t=linspace(0,40,1000); %Time value







x=A*exp(-g*t).*cos(wd*t-a); %x(t)







subplot(4,3,1),plot(t,x)
title('Question 1')
xlabel('Time')
ylabel('Displacement (m)');
grid on;
c=0.9; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,20,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,1),plot(t,x)
title('Question 2')
xlabel('Time (s)')
ylabel('Displacement (m)')
grid on;
c=0; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,40,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,2),plot(t,x)
title('Question 3')
xlabel('Time (s)')
ylabel('Displacement (m)')
grid on;
c=1; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,40,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,3),plot(t,x)
 
c=5; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,40,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,4),plot(t,x)
 
c=15; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,40,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,5),plot(t,x)
 
c=50; %Damping constant
A=1;
a=0;
g= c/(2*m); %Damping coefficient
wn=sqrt(k/m); %Natural frequency
wd=sqrt(((wn)^2)-(g)^2); %Damped natural frequency
t=linspace(0,40,1000); %Time value
x=A*exp(-g*t).*cos(wd*t-a); %x(t)
subplot(4,2,6),plot(t,x)
legend('0','1','5','15','50')
t=linspace(0,40,1000); %Time value
x=A*exp(-g.*t).*cos(wd.*t-a); %x(t)
v=diff(x)./diff(t); %Velocity
subplot(4,2,7),plot(t(1:end-1),v);

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