1. SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS Example: Mathematical model of a mechanical system is defined as a system of differential equations as follows: where f is input, x 1 are x 2 outputs. At t=0 x 1 =2 and x 2 =-1. a)Find the eigenvalues of the system. b)If f is a step input having magnitude of 3, find x 1 (t). c)If f is a step input having magnitude of 3, find x 2 (t). d)Find the response of x 1 due to the initial conditions. e)Find the response of x 2 due to the initial conditions. f)How do you obtain [sI-A] -1 with MATLAB?

2. Let us obtain the State Variables Form so as to 1 st order derivative terms are left-hand side and non-derivative terms are on the right-hand side. State Variables Form A B D(s) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

3. a) Eigenvalues are roots of the polynomial D(s) or eigenvalues of the matrix A. or b) x 1 (t) due to the forcing General Solution Solution due to the initial conditions Homogeneous Solution Solution due to the input Particular Solution Initial Conditions clc;clear; num=[4.5 67.5]; den=[1 15 -280 0]; [r,p,k]=residue(num,den) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

4. System is instable because of the positive root. c) x 2 (t) due to input clc;clear; num=[6 174]; den=[1 15 -280 0]; [r,p,k]=residue(num,den) Laplace transform of x 2p SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

5. d) x 1 due to the initial conditions. clc;clear; num=[2 -25]; den=[1 15 -280]; [r,p,k]=residue(num,den) e) x 2 due to the initial conditions clc;clear; num=[-1 4]; den=[1 15 -280]; [r,p,k]= residue(num,den) f) [sI-A] -1 with Matlab. clc;clear; syms s; i1=eye(2) A=[-20 15;12 5]; a1=inv(s*i1-A) pretty(a1) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

6. Example: Mathematical model of a system is given below. Where V(t) is input, q 1 (t) and q 2 (t) are outputs. a)Write the equations in the form of state variables. b)Write Matlab code to obtain eigenvalues of the system. c)Write Matlab code to obtain matrix [sI-A] -1. d)Results of (b) and (c) which are obtained by computer are as follows: At t=0and V(t) is a step input having magnitude of 2. Find the Laplace transform of due to the initial conditions. e) Find the Laplace transform of q 1 due to the input. 2 t (s) V 2 (t) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

7. a) State variables are q 1, q 2 and. System of differential equations is arranged so as to 1 st order derivative terms are left- hand side and non-derivative terms are on the right-hand side. b) Matlab code which gives the eigenvalues of the system. A=[-1.5 1.5 0;0 0 1;3.75 -3.75 0]; eig(A) c) Matlab code which produces [sI-A] -1 clc;clear A=[-1.5 1.5 0;0 0 1;3.75 -3.75 0]; syms s; i1=eye(3); sia=inv(s*i1-A); pretty(sia) A B State variables SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

8. d) e) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

9. Example: Write the equation of motion of the mechanical system given below in the State Variables Form. Force applied on the system is F(t)=100 u(t) (a step input having magnitude 100 Newtons) and at t=0 x 0 =0.05 m and dx/dt=0. Find x(t) and v(t). State variables are x and v=dx/dt. m=20 kg c=40 Ns/m k=5000 N/m Matlab program to obtain eigenvalues: >>a=[0 1;-250 -2];eig(a) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

10. Applying Laplace transform and arranging, Solution due to the inputSolution due to the initial conditions SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

11. For x(t) ; clc;clear; num=[0.05 0.1 5]; den=[1 2 250 0]; [r,p,k]=residue(num,den) clc;clear; syms s; A=[0 1;-250 -2]; i1=eye(2); %unit matix with dimension 2x2 siA=s*i1-A; x0=[0.05;0]; %Initial conditions B=[0;0.05]; Fs=100/s; X=inv(siA)*x0+inv(siA)*B*Fs; pretty(X) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

12. Steady-state value (Final value) Initial value, x 0 SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

13. For v(t) clc;clear; num=[-7.5]; den=[1 2 250]; [r,p,k]=residue(num,den) SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

14. Example: Mathematical model of a mechanical system having two degrees of freedom is given below. If F(t) is a step input having magnitude 50 Newtons, find the Laplace transforms of x and θ. R=0.2 m m=10 kg k=2000 N/m c=20 Ns/m State variables SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

15. clc;clear A=[0 0 1 0;0 0 0 1;-400 80 0 0;2000 -600 0 -2]; syms s; eig(A) i1=eye(4); sia=inv(s*i1-A); pretty(sia) If the initial conditions are zero, only the solution due to the input exists. Eigenvalues: System is stable since real parts of all eigenvalues are negative. SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS