1. 8. Solution of Linear Differential Equations: f(t): Input,
u(t): Output
Example 8.1
If f(t)=est then u(t)= H(s)est
Transfer Function:
where H(s) is the transfer function which transforms the input to the output
s3H(s)est+4s2H(s)est+14sH(s)est+20H(s)est=3est+sest
a=[1,4,14,20];roots(a)
(s3+4s2+14s+20)H(s)=3+s
Eigenvalues: -1±3i, -2
Exponential/Harmonic Input:
s= i;
hs=(s+3)/(s^3+4*s^2+14*s+20);
abs(hs), angle(hs)
RESONANCE

2. H(s) Laplace Transform: x(t) y(t) Y(s)=X(s) H(s) Δ(s)=1 1
Final value theorem:
Impulse function
Δ(s)=1
u(t): Step function 1

3. Inverse Laplace transform of H(s)
Impulse Response:
Inverse Laplace transform of H(s)
Eigenvalues:
a=[1,4,14,20];roots(a)
-1±3i, -2
(Partial fraction expansion):
p1=[1,3]; p2=[1,4,14,20]; [r,p,k]=residue(p1,p2)
r(1)= i, r(2)= i, r(3)=0.1
z= i 2*abs(z) phase(z)
ξ= (s=-1±3i), suitable values of Δt=0.099 and t∞=6.283

4. ξ=0. 3162 (s=-1±3i), use Δt=0. 099 and t∞=6
ξ= (s=-1±3i), use Δt=0.099 and t∞=6.283 to plot the response h(t)
clc;clear;
t=0:0.099:6.283;
yt=0.3801*exp(-t).*cos(3*t-1.837)+0.1*exp(-2*t)
plot(t,yt)

5. y(t)  Inverse Laplace Transform of Y(s)
Step Input Response:
y(t)  Inverse Laplace Transform of Y(s)
Singular Points (Eigenvalues) are
-1±3i , -2
and s=0
Partial fraction expansion:
p1=[1,3]; p2=[1,4,14,20,0]; [r,p,k]=residue(p1,p2)
r(1)= i, r(2)= i, r(3)=-0.05, r(4)=0.15
Final value theorem:
yss=0.15

6. clc;clear;
t=0:0.099:6.283;
yt=0.1202*exp(-t).*cos(3*t )-0.05*exp(-2*t)+0.15;
plot(t,yt)

7. Solution due to the Initial Conditions
(Homogeneous Solution):
f=0
t=0 da
p1=[-1.2,-2.3,-9.9]; p2=[1,4,14,20]; [r,p,k]=residue(p1,p2)
r(1)= i, r(2)= i, r(3)=-1.01

9. a) Find the output θ(t) for yA(t)=0.2e-3tcos(17t-1.8)
Problem:
For the mechanical sysem given in the figure, yA is input and θ is output. Take m1=250 kg, m2=350 kg, k=37000 N/m, c=1500 Ns/m and L1=1.2 m. The governing differential equation of the system is given as
m1, L1 θ yA m2
k c
a) Find the output θ(t) for yA(t)=0.2e-3tcos(17t-1.8)
Solution:
yA(t)=est and θ(t)= H(s)est
s=-3+17i ;
h=(1800*s+44400)/(624*s^2+2160*s+53280)
abs(h)
angle(h)