Hi August,
To determine the transfer function ( \frac{T}{X} ) from the given system of equations, we need to express ( T ) in terms of ( X ) while eliminating the other variables, ( E1 ) and ( E2 ). Below are the steps to accomplish this using symbolic computation in MATLAB:
- Substitute and Rearrange Equations: Begin by using the provided equations to express ( E1 ) and ( E2 ) in terms of ( X ) and ( T ).
- Solve the System: Solve the system of equations to eliminate ( E1 ) and ( E2), resulting in an expression for ( T ) solely in terms of ( X ).
- Derive the Transfer Function: Rearrange the resulting expression to obtain the transfer function ( \frac{T}{X} ).
Below is the MATLAB code that executes these steps:
syms X T E1 E2 s m k1 k2 k3 c1 I1 r1 r2
% Define the equations
Eqn1 = m*s^2*X + k3*r2*E2 + X*k3 + k2*X + k2*r1*E1 + c1*s*X == 0;
Eqn2 = I1*s^2*E1 - T - k1*r1*r2*E2 + k1*r1^2*E1 + k2*r1*X + k2*r1^2*E1 == 0;
Eqn3 = r2^2*k3*E2 + k3*r2*X + k1*r2^2*E2 - k1*r1*r2*E1 == 0;
% Solve Eqn3 for E2
E2_sol = solve(Eqn3, E2);
% Substitute E2 in Eqn1 and Eqn2
Eqn1_sub = subs(Eqn1, E2, E2_sol);
Eqn2_sub = subs(Eqn2, E2, E2_sol);
% Solve Eqn2_sub for E1
E1_sol = solve(Eqn2_sub, E1);
% Substitute E1 in Eqn1_sub
Eqn1_final = subs(Eqn1_sub, E1, E1_sol);
% Solve for T in terms of X
T_sol = solve(Eqn1_final, T);
% Express T/X as a transfer function
TransferFunction = simplify(T_sol / X);
% Display the transfer function
disp('The transfer function T/X is:');
pretty(TransferFunction)
This code symbolically solves the equations, substitutes the solutions back, and expresses ( T ) in terms of ( X ). The result is a rational function of ( s ), representing the desired transfer function ( \frac{T}{X} ). The simplify function is used to ensure the transfer function is in its simplest form, and the pretty function displays it in a readable format.
To evaluate the transfer function numerically, please assign appropriate numerical values to all constants: ( m, k_1, k_2, k_3, c_1, I_1, r_1) and ( r_2 ).
I hope this helps!
