1. Example: Obtain the equation of motion of the mechanical system given below. Write the form of the output θ(t) in case of free vibration and determine the proper t and t values in order to plot the response versus time.
M(t)
clc;clear
m=5;
L=2;
k=2000;
c=20;
syms s;
pol=(7/3*m*L^2)*s^2+c*L^2*s+k*L^2
vpa(solve(pol),5) Re
13.065i
-0.857
Img

2. Form of the output θ(t) Re
13.065i
-0.857
Img

3. For complex roots (eigenvalues) - i p
p=-±
Damping ratio Re
13.065i
-0.857
Img
For complex roots (eigenvalues)
For p= i
(Suitable time increment)
(Time to reach the steady-state response)

4. (Free vibration) m
x(t)
(Characteristic polinom)
Roots are repeated if
Roots are real and different if
Roots are complex conjugate if  

5. MODELING MECHANICAL SYSTEMS USING ANSYS
Example:
ωn=18.71 rad/s=2.98 Hz 2 m
node k
k=35000 N/m
node
m=100 kg 1
Run notepad.exe and write the code given below
/prep7
et,1,mass21
/solu
To run the program using ANSYS Input window
/input,e01,txt
et,2,combin14
d,all,ux,0
r,1,0,100
d,all,uz,0
r,2,35000,0
d,1,uy,0
n,1,0,0
antype,2
n,2,0,0.25
modopt,lanb,1
type,1
solve finish
real,1
General Postproc > Results Summary
e,2
type,2
real,2
e,1,2
eplot

6. Example: Model the mechanical system using ANSYS and find the damped and andamped natural frequencies.
Using Matlab
clc;clear
m=5;
L=2;
k=2000;
c=20;
syms s;
pol=(7/3*m*L^2)*s^2+c*L^2*s+k*L^2
vpa(solve(pol/(2*pi)),5)

8. Undamped natural frequencies Damped natural frequencies
(rad/s)
(Hz)
Undamped natural frequencies
Damped natural
frequencies
clc;clear;
m1=5;
m2=3;
k1=1000;
k2=2000;
c1=5;
c2=5;
M=[m1,0;0 m2];
C=[c1+c2,-c2;-c2,c2];
K=[k1+k2,-k2;-k2,k2];
syms s;
sist=M*s^2+C*s+K;
p=solve(det(sist));
display ('Eigenvalues are')
vpa(p,5)
(Hz)

9. ANSYS Model
Constants used in the model
Declerations for element types
Definition of Real Constants
Creation of nodes
Creation of elements
Preprocessing
Solution

10. Example: Spring supported beam model
ANSYS APDL (Ansys Parametric Design Language) CODE