bajaj562chpt6_2

bajaj562chpt6_2 - Reviewing, what we have discussed so far:...

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1 Reviewing, what we have discussed so far : Generalized coordinates Any number of variables (say, n) sufficient to specify the configuration of the system at each instant to time (need not be the minimum number). In general, let be generalized coordinates. Then, the position vectors are described by 12 , , , n q q q ( , ,.... , ), 1,2,. ..., ii n r r q q q i N
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2 In general, we also have some geometric constraints These are d equations in 3N (scalar) variables. Let be the generalized coordinates or variables i.e., so that the geometric constraints are satisfied. 12 , , , n q q q ( , , , ) 0, 1,2, , . iN i r r t i d  ( , ,.... , , ), 1,2,. ..., ii n r r q q q t i N
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3 If we eliminate all geometric constraints where n – number of generalized coordinates Sometimes, one may not want to solve for all the geometric constraints Then are more than the minimum needed and not all are independent. Now consider the work done by effective forces in any virtual displacement 3. o n n N d . o nn 1 ,..., n qq 1 N ii i W F r
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4 Now, consider the position vector, and its virtual displacement : where virtual displacement in Then, the virtual work is 12 ( , ,..., , ) ii n r r q q q t 1 , 1,. ..., n i i j j j r r q i N q j q . j q 1 1 1 1 N n n N jj i j j i rr W F q F q qq  11 nN i i j j j ji j r W Q q Q F q
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5 Aside : consider the position vector: 1 ( ,. ..., , ) n r r q q t 12 r r r r q q q q t 2 1 n i i rr rdt q dt dt qt  1 n i i i dr dq dt 1 n i i i r rq q Differentiating, we get
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6 Here is generalized force corresponding to the jth generalized coordinate Thus, j Q . j q 11 Nn ii jj ij W F r Q q
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7 Ex 18 : Consider the compound pendulum . Let, x(t) – motion of slider. It is a specified function of time Let be generalized coordinates . Find : generalized forces (corresponding to the generalized coordinates 12 , , QQ , ). O A x(t) y x C B F(t) 1 2
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8 Now, to find the generalized forces , we need to first define the position of C in terms of generalized coordinates : or 1 1 2 [ sin sin( )] C r x L L i 1 1 2 [cos cos( )] Lj 1 1 1 2 1 2 [cos cos( )( )] C r L i 1 1 1 2 1 2 [ sin sin( )( )] 1 1 2 1 [{cos cos( )} {sin C r L i 1 2 1 1 2 sin( )} ] [cos( ) j L i 1 2 2 sin( ) ] j
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9 Aside : first find the velocity to find virtual dispalcement : 1 1 1 2 1 2 [ cos ( )cos( )] c r x L L i  1 1 1 2 1 2 [ sin ( )sin( )] Lj 11 [ cos c r x L 1 2 1 2 ( )] Li 1 1 2 1 2 [ sin ( )]
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10 Now, the force acting at C is: Thus, the virtual work done is : or Note : if the forces are conservative, the generalized forces are : where the potential function is F Fi C W F r 1 1 2 1 [{cos cos( )} W FL 1 2 2 cos( )} ] 1 1 2 2 QQ 1 1 1 2 2 1 2 {cos cos( )} cos( ) Q FL Q FL / con jj Q V q 1 ( ,. .., ) n V q q
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11 The potential function can be found as below: or 1 1 2 1 1 {cos cos( )} V FL Q 1 1 2 1 2 {sin( ) sin( )} ( ) V FL h 1 2 2 2 cos( ) V FL Q 1 2 2 1 sin( ) ( ) V FL h 1 2 1 1 2 1 2 2 1 ( , ) {sin sin( )} ( ) ( ) V FL h h
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12 6.6 Lagrange’s Equations (Important: this derivation is different from the one in the text). The starting point is the D’Alembert’s principle : Recall that there are also d finite and g kinematic constraints to be satisfied by any virtual displacement of the system: 1 ( ) 0 N i i i i i F m r r 
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13 ( finite constraints ) (
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bajaj562chpt6_2 - Reviewing, what we have discussed so far:...

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