Bajaj562chpt6_1 - CHAPTER 6 LAGRANGES EQUATIONS(Analytical Mechanics 1 Ex 1 Consider a particle moving on a fixed horizontal surface z Let r P be

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1 CHAPTER 6 LAGRANGE’S EQUATIONS (Analytical Mechanics)
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2 Ex. 1 : Consider a particle moving on a fixed horizontal surface . Let, be the position and F be the total force on the particle. The FBD is: The equation of motion is mr F r r t p  ( , , ) P r O z y x P m (x,y,0) F 1 N F 1 -mgk f
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3 In component form, the equation of motion is Also, motion is restricted to xy plane z = 0 - equation of constraint It is a geometric restriction on where the particle can go in the 3-D space. Clearly, there is a constraint reaction (force) that needs to be included in the total force F . ( , , , , , , ) Px mx F x y z x y z t     ( , , , , , , ) Py my F x y z x y z t    ( , , , , , , ) Pz mz F x y z x y z t
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4 Ex. 2: Consider a particle moving on a surface. Now, the motion is confined to a prespecified surface (e.g. a roller coaster ). The surface is defined by the relation: f (x, y, z) - c = 0 - equation of constraint. The equation of motion will again be the same. Surface f(x,y,z)=c z m y x O
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5 The constraints in the two examples are geometric or configuration constraints . They could be independent of time t, or could depend explicitly on it. For an N particle system, if the positions of particles are given by , the constraint can be written as: This is an equation of a finite or geometric or holonomic constraint. 12 ( , , , , ) 0 N f r r r t 1 2 3 , , ,..... r r r
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6 Ex. 3: Double pendulum: it consists of two particles and two massless rigid rods The masses are Number of coordinates required is 4 - used to define the configuration There are certain constraints on motion : m x y 1 1 1 :( , ) m x y 2 2 2 :( , ) 12 ( 0: planar motion) zz 2 2 2 2 2 2 1 1 1 2 2 1 2 1 ( ), ( ) ( ) l x y l x x y y x y O l 1 l 2 m 1 m 2 (x 2 ,y 2 ) (x 1 ,y 1 )
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7 2 equations of constraint (they are holonomic, geometric, finite etc.) Degrees-of-freedom : the number of independent coordinates needed to completely specify the configuration of the system (4 - 2) = 2. One could perhaps find another set of two coordinates (variables) that are independent: e.g., the two angles with the vertical. Then, there are no constraints on 1 2 , , 1 2 , ,
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8 Ex. 4 : A dumbbell moving in space one possible specification of position is: - these are 6 variables or coordinates, and there is one constraint 1 1 1 1 2 2 2 2 : , , ; : , , m x y z m x y z 2 2 1 2 2 1 2 2 1 2 ( ) ( ) ( ) x x y y z z Z X Y O m 1 C m 2
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9 degrees-of-freedom of the system 6 - 1 = 5 another possible specification for the configuration of the system: Location of center of mass C: ; and orientation of the rod : ( , ). These are independent no constraint relation for these variables. ( , , ) x y z c c c Z X Y O m 1 C m 2
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10 generalized coordinates - any number of variables needed to completely specify the configuration of a system. e.g., for the dumbbell in space motion: there are two sets of generalized coordinates Important : some sets consist of independent coordinates (no constraints) where as others are not independent.
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This note was uploaded on 12/27/2011 for the course ME 562 taught by Professor Bajaj during the Fall '10 term at Purdue University-West Lafayette.

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Bajaj562chpt6_1 - CHAPTER 6 LAGRANGES EQUATIONS(Analytical Mechanics 1 Ex 1 Consider a particle moving on a fixed horizontal surface z Let r P be

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