The linkage consists of two slender bars Bar OC has a mass o

The linkage consists of two slender bars. Bar OC has a mass of m and bar AC has a mass of 2m. The slider at B has negligible mass and friction. The force P is applied such that it is always perpendicular to bar AC. The spring has a constant k and an up stretched length of.l_0. Determine the equation of motion of the system using a) Lagrange\'s Equations; b) Newton- Euler equations.

Solution

solution;

1)lagranges method means by energy method as follows

kinetic energy of link ca about b=.5*I1*w^2

kinetic energy of link ao about o is=-.5*I2*w^2

w is same for botn link

spring potential energy=.5*k*x^2

mass kinetic energy=.5*3m*x\'^2

on adding all and on differentiating we get

putting m\'\'=b*x\'\'

and m\'=b*x\'

3mx\'\'+kx+(I1-I2/b^2)x\'\'=0

this is equation of motion

4)by newton euler method,

DOF=3*n-2*p1=3*4-2*4=4

hence systewill have four equation then we have to merge them to one

for x direction we have

3mx\'\'+kx-P2sinm=0

3mg=P2cosm

-I1m\'\'+P2b+Fcob=0

-Fcob+I2m\'\'=0

from three .four equation we get

P2=(I2-I1/b)m\'\'

on putting and

m\'\'*b=x\'\'

m\'*b=x\'

so finaly we get

3mx\'\'+kx+(I1-I2/b^2)x\'\'sinm=0

on neglecting angle sinm=1, we get

3mx\'\'+kx+(I1-I2/b^2)x\'\'=0

7)in this way we get both way we get same eqaution of motion