How to find transfer function to a second order ODE having a constant term?

and are constants.

Transfer function

[edited by @Sam Chak]

[Aquatris]

What is the specific question? Do you want to know how to take a Laplace Transform?

[PONNADA]

Thanks for your kind response.

Yes sir,I want to find laplace transform, and ultimatley I need the transfer function of the given ODE.

[Aquatris]

So this question in general is not related to Matlab but rather to using Laplace Transform to solve differential equations.

What have you tried so far? Do you know how to take Laplace transform?

[Sam Chak]

Hi @PONNADA,

First things first, make sure to bookmark the Help Center, and then search for the keyword 'Laplace Transform'. You will find an example under this article: Solve Differential Equations of RLC Circuit Using Laplace Transform.

It's always a good idea to start by searching in the Help Center before seeking technical assistance, unless the problem is highly specific and no relevant examples can be found in the documentation.

[Paul]

Is the problem stated correctly? As written, it doesn't appear to be a linear, time-invariant system (unless c1 = 0), in which case there is no transfer function.

[PONNADA]

Sir, thank you for your kind response.

I need the characteristic equation from the transfer function of the said problem. As you said, by removing the constant term from the given equation, and if I find the transfer function, will it affect the characteristic equation?

[Paul]

As far as I can tell, there is no transfer function of the said problem, so there also is not characteristic of that transfer function.

If you set c1 = 0, then the the system is LTI and you can fnd the transfer function of that system and the characteristic equation of that transfer function.

Are you sure that the problem statement is correct as written?

[PONNADA]

Yes Sir, it's a non-deimensional ODE.

Answer 1

[Sam Chak]

Hi @PONNADA

This type of differential equation is commonly encountered in examples involving an ideal undamped mass-spring system subjected to an input force and constant gravitational acceleration. The equation can be rearranged and expressed as a 2-input, 1-output state-space system.

When you transform the state-space system into transfer function form, you'll obtain two transfer functions because the system's response is influenced by two external inputs: one from the manipulatable force and the other from the constant effect of gravity.

You can observe somewhat similar dynamics in a free-falling object:

https://wethestudy.com/mathematics/free-falling-bodies-differential-equations/

c2  = 2;
A   = [0  1; 
     -c2  0];
B   = [0  0; 
       1 -1];
C   = [1, 0];
D   =  0*C*B;

%% State-space system
sys = ss(A,B,C,D, 'StateName', {'Position' 'Velocity'}, ...
    'InputName', {'Force', 'Constant'}, 'OutputName', 'MyOutput')

sys =
 
  A = 
             Position  Velocity
   Position         0         1
   Velocity        -2         0
 
  B = 
                Force  Constant
   Position         0         0
   Velocity         1        -1
 
  C = 
             Position  Velocity
   MyOutput         1         0
 
  D = 
                Force  Constant
   MyOutput         0         0
 
Continuous-time state-space model.

%% Transfer functions
G   = tf(sys)

G =
 
  From input "Force" to output "MyOutput":
     1
  -------
  s^2 + 2
 
  From input "Constant" to output "MyOutput":
    -1
  -------
  s^2 + 2
 
Continuous-time transfer function.

[Sam Chak]

@PONNADA: ... by removing the constant term from the given equation, and if I find the transfer function, will it affect the characteristic equation?

No, it won't affect the characteristic equation. As you can see above, both transfer functions have the same characteristic polynomial, , due to there being only 1 degree of freedom in this single mass-spring system.

By taking the Laplace transform of the differential equation with zero initial conditions, we establish the relationship between the inputs and output.

To determine how the output behaves in response to a given input, we must determine the transfer functions: one from force input to plant output, and the other from constant gravity to output.

[PONNADA]

What a wonderful explanation you provided, sir. Your patience and commitment are evident in the response you provided. Thank you so much. It is really helpful to me.

[Rajiv]

So, what will be total TF from G1 and G2. (i) G1 * G2 or (ii) G1 + G2 (iii) or something else

Answer 2

[Alexander]

Are you looking for something like this:

AnregeAmpl = 1; % mm

fmax = 20; % Hz

f=0.1:0.1:fmax; % Hz

w=2*pi*f; % 1/s

s=ones(size(f))*AnregeAmpl; % mm

c=200000; % N/m

m=20; % kg

d=100; % Ns/m

x=(c+d*j*w)./(c+d*j*w+j*w.*j.*w.*m);

plot(f,abs(x),f,abs(s));grid minor

Replace jw by S and you have your transfere function in S. But maybe your professor is not amused about this. ;=)