1. Newton-Raphson Example 4:
Solutions of system of nonlinear equations:
Newton-Raphson Example 4:
The kinematic equations for a Four-Bar mechanism can be written as (5th semester, Mechanisms Course)
L2=0.15 m L3=0.45 m L4=0.28 m s1=0.2 m s1 L2 L3 L4 θ2 θ3 θ4 3 4 2 1
Where link 2 is the input member. How do you calculate θ3 and θ4 when θ2=120°.
-0.075
0.13

2. Following changes are made in the computer program.
Solutions of system of nonlinear equations:
Following changes are made in the computer program.
clc, clear
x=[0.5 1] ; err=[ ];
niter1=10;niter2=50;
err=transpose(abs(err));
for n=1:niter2 x
%Error Equations
a(1,1)=-0.45*sin(x(1));a(1,2)=0.28*sin(x(2));
a(2,1)=0.45*cos(x(1));a(2,2)=-0.28*cos(x(2));
b(1)=-(0.45*cos(x(1))-0.28*cos(x(2))-0.275);
b(2)=-( *sin(x(1))-0.28*sin(x(2))); %
bb=transpose(b);eps=inv(a)*bb;x=x+transpose(eps);
if n>niter1
if abs(eps)<err
break
else
display ('Roots are not found')
end
(Initial angle values must be given in RADIAN)
ANSWER:
θ3=0.216 rad (12.37°)
θ4=0.942 rad (53.97°)
Alternative solution with MATLAB
clc;clear
[x,y]=solve('0.45*cos(x)-0.28*cos(y)=0.275',' *sin(x)-0.28*sin(y)=0');
vpa(x,6)
vpa(y,6)

3. Newton-Raphson Example 5:
Solutions of system of nonlinear equations:
Newton-Raphson Example 5:
Kinematic equations for a crank mechanism are given below (5th semester Mechanisms Course)
L2=0.15 m L3=0.6 m θ3 L2 L3 θ2 s
Where link 2 (crank) is the input member. How dou you calculate θ3 and s with computer when θ2=60°.
0.075
0.1299

4. Following changes are made in the computer program.
Solutions of system of nonlinear equations:
Following changes are made in the computer program.
clc, clear
x=[-1 0.8] ; err=[ ];
niter1=10;niter2=50;
err=transpose(abs(err));
for n=1:niter2 x
%Error Equations
a(1,1)=-0.6*sin(x(1));a(1,2)=-1;
a(2,1)=0.6*cos(x(1));a(2,2)=0;
b(1)=-( *cos(x(1))-x(2));
b(2)=-( *sin(x(1))); %
bb=transpose(b);eps=inv(a)*bb;x=x+transpose(eps);
if n>niter1
if abs(eps)<err
break
else
display ('Roots are not found')
end
ANSWER:
θ3= rad (-12.5°)
s= m
Alternative solution with MATLAB
clc;clear
[x,y]=solve(' *cos(x)-y=0',' *sin(x)=0');
vpa(x,6)
vpa(y,6)

5. Newton-Raphson Example 6:
Solutions of system of nonlinear equations:
Newton-Raphson Example 6:
The time-dependent locations of two cars denoted by A and B
are given as
At which time t, two cars meet?

6. Newton-Raphson Example 6:
Solutions of system of nonlinear equations:
Newton-Raphson Example 6:
clc;clear
t=solve('t^3-t^2-4*t+3=0');
vpa(t,6)
Alternative Solutions with MATLAB
clc, clear
x=1;err=0.001;
niter=20; %
for n=1:niter
f=x^3-x^2-4*x+3;
df=3*x^2-2*x-4;
eps=-f/df;
x =x+eps;
if abs(f)<err
break
end
display('Answer is='),x
Using roots command in MATLAB
a=[ ]; roots(a)
ANSWER
t=0.713 s
t=2.198 s

7. Graph Plotting Example 7:
From a vibration measurement on a machine, the damping ratio and undamped vibration frequency are calculated as 0.36 and 24 Hz, respectively. Vibration magnitude is 1.2 and phase angle is -42o. Write the MATLAB code to plot the graph of the vibration signal.
Given:
z=0.36
ω0=24*2*π (rad/s)
A=1.2
Φ=-42*π/180 (rad)=-0.73 rad
ω0= rad/s ω -σ α

8. yt=1.2*exp(-54.3*t).*cos(140.7*t+0.73); plot(t,yt) xlabel(‘Time (s)');
Graph Plotting:
clc;clear
t=0:0.002:0.1155;
yt=1.2*exp(-54.3*t).*cos(140.7*t+0.73);
plot(t,yt)
xlabel(‘Time (s)');
ylabel(‘Displacement (mm)');