Coordinate Transformation Matrix and Angular Velocity vector formulation. (t) = cos 2 (θt)+ sin 2 (θ t)=1. v (t)=r. = θ( sin (θt) î+ cos(θt) ĵ).

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1 Coorinate Transformation Matri an Angular Velocit vector formulation Consier a vector ating in time, r =r 0 (cos(θ t)î+ sin (θt) ĵ) Define a unit vector, =cos(θ t)î+ sin (θt) ĵ This can be checke for magnitue b e = cos (θt)+ sin (θ t)= In î, ĵ coorinates, all the time-changing occurs on the components, r (t ), r In the coorinate, all the time-changing occurs with the base vector Taking the time erivative to get the velocit: v =r 0 The erivative of the base vector iels: = θ( sin (θt) î+ cos(θt) ĵ) The new base vector, = sin (θt) î+ cos(θt) ĵ is perpenicular to the original base vector,, which can be shown b otting the two vectors, e = cos(θt)sin(θt )+ sin(θt)cos(θ t)=0 Two orthonormal vectors can form a basis for the two imensional space just as well as the î, ĵ pair It is possible to transform between the two bases In matri notation, ] = cos(θ t) sin(θt) sin (θt) cos(θt)] îĵ] Call A ]= cos(θt) sin( θ t) sin (θt) a coorinate transformation matri (CTM) cos(θ t )] What oes A ] T A ] look like? A ] T cos(θt) sin(θ t) A ]= What oes A ] A ] T look like? A ] A ] T = Hence, A ] T = A ] cos(θ t) sin(θt) sin(θ t) cos(θt) ] sin (θt) cos(θt)] = 0 0 ] cos(θt ) sin(θt ) cos(θt) sin (θt) sin (θt) cos(θ t)] sin (θt) cos(θt) ] = 0 0 ] This is generall true for a coorinate transformation matri an will be shown later What oes et ( A ]) look like? et ( A ])=cos (θt)+ sin (θt)= Transforming from right-hane CS to right-hane CS or left-hane CS to left-hane CS will result in a CTM with eterminant equal to one Otherwise, the eterminant will be negative one

2 What are the components of the Coorinate Transformation Matri? The ot prouct between two vectors is A B=( A )( B )cos θ where θ is the angle between the two vectors For unit vectors, e =cos θ because the magnitues are î ĵ k For a general Coorinate Transformation Matri, A ]= ] î ĵ k, it can be seen that the î ĵ k entries in the matri are the cosines of the angles between the vectors themselves Appling the transformation to the original base vectors, î, ĵ an k, gives, ĵ k ĵ k e ]= î î ĵ k ] îĵ k ] This can be verifie b consiering the first row, = î ĵ ] îĵ k ] k =( î )î+ ( ĵ) ĵ+ ( k) k Dot this equalit with î, ĵ, k to see that the relationship is correct Consier A ] A ] T more closel ] î The first entr is a = î ĵ k ĵ]=( î) + ( ĵ) + ( k ) k That's not ver helpful Tr otting =( î )î+ ( ĵ) ĵ+ ( k) k with, =( î )(î )+ ( ĵ)( ĵ e )+ ( k)( k ) Since = an the ot prouct is commutative, it can be seen that the first entr in A ] A ] T is This proceure can be followe with the remaining entries in the CTM to iscover that A ] A ] T = I ] ] î a = î ĵ k ĵ î )( î)+ ( ĵ)( ĵ)+ ( k)( k)= =0 k]=( ] î a = î ĵ k ĵ î)( î )+ ( ĵ)( ĵ)+ ( k)( k)= =0 k]=( An so on

3 The opposite relationship can be shown b noting that îĵ k ] =î ĵ verifie b otting each row with,, an î î e k ĵ e ĵ e k ] ], e which can be Since îĵ k ] îĵ k ] = A ] T = A ]T ] an e e ]= A ] T I ] A ] A ] T = I ], A ] T A ]= I ] ) A ] e ]= T ( A ] A ] T A ] e ]=( T A ]) A ] T e ]= I ] A ] T ] e

4 Cute trick with CTM Since the CTM is a special case of a unitar matri (the general unitar matri has comple elements), A ] A ] T = I ] Taking the time erivative of this relation gives ( A ] A ]T = I ]) Hence, ( A ]) A ]T + A ] ( A ]T )=0] Or, ( A ]) A ]T = A ] ( A ]T )= ( ( A ]) T A ]T ) Defining G ]= ( A ]) A ]T, it can be seen that G ]= G ] T which is the efinition of a skewsmmetric matri Blowing this up, g g g g g g g g g g g g g ]= g g g ], which gives one important piece of information, that g = g =0 g = g =0 g = g =0 Using the cclic permutation, ]= 0 Ω Ω ] gives the matri G Ω 0 Ω Ω Ω 0 g =Ω g =Ω g =Ω to efine Ω, Ω, Ω Defining the angular velocit vector b assembling the components, Ω, Ω, Ω, with the transforme base vectors, Ω= Ω Ω Ω ] ]

5 Return to the problem where r=r ' + r ' + r ' an where r ', r ',r' are invariant with time The components in the original coorinate sstem are ] r ' r ' ]= A r ' r ] r r Then, v= (r ' + r ' + r ' )=r' + r ' r ', r ',r' + r ' b the time invariance of Writing this in matri notation, v =r ' r ' r ' ] The CTM can be use to evaluate Hence, v= r' r ' r ' ] G ] Continuing to reuce, ' (Ω Ω ) v=r r ' (Ω Ω ) r ' (Ω Ω e e ] A ] ]= îĵ k] = A ] A ] T G ] ]= ] 0 Ω Ω Ω 0 Ω ] e ]=r ' r ' r ' ] Ω Ω 0 ] r ' Ω r ' Ω r ' Ω r ' Ω r ' Ω r ' Ω ] e )]= ]= Ω r The point behin this eercise, is that evaluating time erivatives has been converte to evaluating cross proucts, a much less error-prone operation The angular velocit, a concept that will become much more powerful in namics of rigi boies, has also been introuce here, along with a simple metho of calculating it base on the coorinate transformation matri The first steps in setting up a namics problem are to write own all the coorinate sstems involve, to write own all the vectors to be involve, an to calculate the angular velocit for each coorinate sstem It will turn out later that chaining coorinate transformations leas to a simple formulation The angular acceleration,

6 α= Ω = Ω Ω Ω ] Ω Ω Ω ] ]+ ]= Back to the original problem that spawne this asie Ω Ω Ω ] Ω Ω= Ω ]+ Ω Ω ] ] r =r 0 (cos(θt)î +sin (θ t) ĵ)=r 0 The coorinate transformation matri is A]= cos(θ From the CTM, t) sin (θ t) 0 ] sin(θ t) cos(θ t) G ]= (θ t) cos(θ t) 0 t) sin (θt) 0 = θ sin A] A ]T cos(θ t) sin (θ t) 0 sin(θ t) cos(θ t) ]cos(θ ]= 0 θ 0 0] θ Therefore, Ω= θ an v = Ω r = θ r 0 =r 0 θ =r 0 θ( sin (θ t)î +cos(θ t) ĵ)

7 Consier r =r r r ] ] The velocit is v = r = ] r r ] ]+r ] Define the velocit as seen from the ating coorinate sstem as v = r = an substitute the earlier result, r ' r ' r ' ] v = e ]= Ω r ] ] r = v + Ω r The acceleration comes from this equation: v a= = ( v + Ω r )= v, an the velocit is + Ω r + Ω r The first term is v = r r r ] ]+ acceleration as seen in the ating coorinate sstem as noting that ] v, ]= Ω The secon term can be reuce b noting that Ω = Ω Ω Defining α= Ω a = r v ] r = a + Ω v Ω ] Ω Ω ] ]+Ω ] ] Ω Ω, the angular acceleration an noting that ] ] Defining the r ] an ]

8 Ω Ω Ω ], ]= Ω Ω= 0 Ω r = α r The thir term reuces Ω r = Ω ( v + Ω r )= Ω v +Ω (Ω r ) Putting it all together gives the formula for an acceleration epresse in a ating coorinate sstem, a= a + Ω v + α r + Ω v +Ω (Ω r ) Combining terms gives a= a + Ω v + α r +Ω (Ω r )

9 Consier a vector, r = r 0 + r The velocit is v = v 0 + v an the acceleration is a= a 0 + a If r is epresse in a ating coorinate sstem, then the general formulas for velocit an acceleration are: v = v 0 +( v ) + Ω r an a= a 0 +( a ) + Ω ( v ) + α r +Ω (Ω r )