RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1

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1 RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the point P is lso fee to move in ny configution of the spce (figue ). Fo the moment, P is consideed without mss, nd only the kinemtic spect will be nlyzed. his helps to intoduce the vey impotnt concept of the ngul velocity vecto. If the point P is moving in the efeence system S, u (t) is the vecto tht descibes its position t the time t in this system. P is lso moving espect to the mobile efeence system S nd q (t) is the vecto tht descibes its position in this system. Now, it s possible to expess the position of P by vecto eqution whee the eltion between the two systems ppes. In fct it s ( ) ( t ) R ( t ) ( t ) u = +, () t q whee (t) is the position of the cente O in the system S nd R (t) is the ottion mtix tht tnsltes vecto fom the vecto bse of S to the bse of S. P R q u S S O O Figue Befoe to expess velocity nd cceletion of the point P, it s necessy to conside some impotnt popeties of the ottion mtix. o give moe simple fomultion, in the following sections time dependence will be not lwys explicitly indicted.

2 PROPERIES OF HE ROAION MARIX Fo the popety of the ottion mtix to be othogonl, it is R R = I. () When this eqution is deived with espect to time we obtin d ( R R ) = R R + R R 0 =. () Now, it s possible to define skew symmetic mtix Γ with the following popety: Γ ( R R ) = R R = R R = Γ = ; (4) nd the components of this mtix e 0 Γ = R R = 0. (5) 0 he mtix Γ defines the ngul velocity of the system S with efeence to the system S. he poduct of the mtix Γ with vecto v is equl to the vecto poduct between new vecto nd the vecto v. In fct it s possible to expess this esult s Γv = v. (6) is the ngul velocity vecto nd =. (7) Fo the second deivtive of the eqution () we obtin by simple opetions

3 d ( R ) = + = + ( ) ( R RR RR RR RR RR ). (8) It s impotnt to explicte, fom the (8), the tem R R, tht it will let us to clculte the expession fo the cceletion of the point P in the globl efeence system. his tem is d R R = ( R R ) ( R R ) ( RR ) (9) nd fo the eqution (5) it s possible to wite R RΓ d = Γ ( ) + (0) nd fo the sme gument of the (6), if v is vecto defined in the globl efeence system, we hve the following vecto opetion: d. () ( RR ) vγ= Γ( v ) + = v + v ( )

4 VELOCIY AND ACCELERAION By diffeentiting the fundmentl eqution () with espect to time, we clculte velocity nd cceletion of the point P. Fo the velocity of the point in the inetil efeence system we hve ( t ) = ( t ) + ( t ) ( t ) + ( t ) ( t ) u R q R q. () It s moe inteesting to wite this eqution using the following expession whee thee velocity vectos compe: v = v + v. () v is the bsolute velocity, i.e., the velocity of the point P mesued with efeence to the globl efeence system nd it coesponds to the vecto ( t ) v = u ; (4) v is the eltive velocity of the point P, i.e. the velocity of the point mesued in the mobile efeence system, but expessed using the vesos of the globl efeence. In fct, using the ottion mtix between the two efeence systems, we hve ( t ) ( t ) v = R q ; (5) v is the dg velocity, tht is the velocity of the point P s if it ws fixed to the mobile efeence system nd moving with it: ( t ) ( t ) ( t ) v = + R q. (6) It s inteesting to wite down the dg velocity s v = v + ( P O ) O. (7) Such fomul expesses the motion lw fo igid body when it s fixed with the mobile efeence system. Fo the equtions () nd () we obtin 4

5 = + R q = + R R (u ) = + (u ). (8) v Diffeentiting the eqution () with espect to time, fo the cceletion of the point P, we hve ( + R q ) + R q R u = +. (9) q We cn wite this eqution s sum of thee cceletions: = + +. (0) c is the bsolute cceletion, i.e., the cceletion of the point P mesued with efeence to the globl efeence system nd it coesponds to the vecto ( t ) = u ; () is the dg cceletion, nd fo the eqution (0) it s possible to wite down = + R q = + R R ( u ) + [ ( u )] ( u ) = + Γ( u ) + Γ ( u ), () = +, () ( P O ) + [ ( P )] = + O O ; (4) is the eltive cceletion, i.e., the cceletion of the point P efeed to the mobile efeence system but expessed by the vesos of the inetil: = R q ; (5) c is the Coiolis cceletion defined in the globl efeence system s ( ) ( ) = R q = R R R q = Γv = v. (6) c 5