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

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

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

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

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

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)

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)

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

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