Chemical Engineering - Apllied Mechanics

Kinematics of material points

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Note: In the following, vectors are written in boldface. Contrary to the common notation used in mathematics, a generic vector AB, with origin in A and endpoint (tip) in B, will be denoted ( B – A ) instead of ( A – B ). In fact, if x A, y A, z A are the coordinates of the point A and x B, y B, zB are the coordinates of the point B, the vector AB can be expressed as: ( ) ( ) ( ) ( ) B A B A BA BA xx y y zz =−= − + − + − AB i jk = = = 1. KINEMATICS OF A MATERIAL POINT 1.1 Absolute reference systems Let us consider an absolute reference system represented by a set of Cartesian axes, x-y (two-dimensional system, 2D) or x-y-z (three-dimensional system, 3D). Let us denote O the origin of the axes. Being the reference system absolute the axes are fixed, that is the origin O does not move and the axes do not rotate. For the sake of simplicity, first let us consider a two-dimensional case study. The positive versus of the axes x and y is indicated by the versors (unit or basis vectors) i and j. Since the axes of the reference system do not rotate, the versors i and j are constant. Therefore, their derivatives with respect to time are null: 0 t d d =i = 0 t d d =j = (1.1.1)= In the case of a=point=that moves in a 3D space the positive versus of the axes=x-y-z is indicated by the versors i , j and k. 1.1.1 Absolute position, velocity and acceleration of a material point Let us consider a material point that moves in the x-y plane determined by an absolute reference system of Cartesian axes. The loci curve of the time-varying position of the point P(x,y) is the trajectory, s, which can be a general curve or a straight line (Figure 1.1.1). In general, the trajectory can be expressed by means of a function y = f (x), or by z = f (x, y), in the case of a point that moves in a 3D space. Being the reference system fixed, the above mentioned trajectory s is called absolute trajectory of the material point, expressed with respect to an absolute system of Cartesian coordinates, whose origin is O. For a given time t the point position is defined by the vector () −PO , which can be expressed as follows,= using the Euler�s notation:= ()() ieϑ −− = PO PO = () ieϑ ρ −= PO = (1.1.2)= where () −PO , or= ρ, is the magnitude of the vector () −PO , while θ is the vector phase. Since the position of P varies with time, the two parameters ρ and ϑ =are timeϑdependent functions. That is, ()t ρρ= = and ()t ϑϑ= .= = = = Figure 1.1 .1 Absolute trajectory , s, and position vector of a particle Figure 1 .1.2 Absolute finite displacement of a particle = The position vector () −PO =can be=also=expressed as:= ( )= + xy −PO i j = that is:= ( )= ()+ () xt yt −PO i j = (1.1.3)= Since this position vector is defined with respect to fixed reference axes, x-y, the vector () −PO =represents= the absolute position of P. If we consider two instants, t 1 and t 2, the point position changes from P 1 to P 2 and the finite displacement of P is defined by the vector 21 () − PP =(Figure=1.1.2). That is by the difference between the vectors= 2 () − PO =and 1 ()− PO .= 2 1 2 121 2 1 ( )=( ) ( )= ( )+ ( )xx y y − −−− − − P P P O PO i j = (1.1.4)= The instantaneous velocity=v of the point can be determined by deriving the position vector () −PO =with= respect to time:= ( ) () ( ) = = + =+d dx d y xy dt dt dt− v PO i j ij =(1.1.5)= Since=the vector= ()− PO =is=the absolute position of the=point, its derivative gives the corresponding=absolute velocity. The two terms xand y, are the components of the velocity=v along the directions x and y. The magnitude of v is given by 22 v= + xy =while the phase α can be expressed as: -1-1-1 = tan tan = tan dy dy dt dy dx dt dx dx α     =          = (1.1.6)= Therefore, the phase α of the velocity vector v is the angle of the straight line that is tangent to the trajectory in the considered point P. Then, the absolute velocity vector v is always tangent, in P, to the absolute trajectory (Figure 1.1.4). Figur e 1. 1.3 Absolute position of a particle Figure 1 .1.4 Absolute velocity of a particle = The same vector v can be also expressed in this form: =v iieeαα= vv = (1.1.7)= where the magnitude of vector v is v = v(t) and the phase α is a time-dependent function: α = α (t). Let us denote t the unity vector that defines the direction and versus of the velocity vector v at time t. That is: () =1 iiiteeeααα= = t . Therefore, in general, the unity vector t depends on time, t: = () t tt . If the= instantaneous value of the phase, α, of the unity vector changes, the corresponding angular speed, α=or= ω, is given by: d dt αω α = =  . Then, the eq.( 1.1.7) can be written as:= =v vt .= The eq.(1.1.5) can be=also=rewritten as:= () ( ) ( () ) == it d d te dt dt ϑ ρ − v PO = or= iie ieϑϑ ρ ρϑ= +v   =(1 .1.8)= The imaginary index i, that is not a vector, is equivalent to the vector expression: ( 2)i ie ϑπ+ ≅= j.= Therefore, the eq.(1.1.8) can be=rewritten as:= ( 2) iieeϑϑπ ρ ρϑ + = +v   (1 .1.9) The two terms of this equation are vectors having amplitudes ρ=and ρϑ , respectively, while the corresponding phases are= ϑ =and ( 2)ϑπ + . That is, these two vectors are orthogonal and they=are the components of the velocity vector v along the directions given by the angles ϑ =and ( 2)ϑπ + .= That is, the direction of the absolute velocity vector=v is t, as it is tangent to the absolute trajectory. However, the same vector v can be decomposed into two orthogonal vectors directed as the axes x-y, or directed as ϑ = and ( 2)ϑπ + .= = Then, the absolute acceleration, a, of the material point can be obtained by evaluating the derivative of the velocity vector v, with respect to time. That is: () ( + ) () ( ) = =+ =+ 22 22 d d x y dx d yxy dt dt dt dtv a   ij =i j i j =(1.1.10)= The eq.(1.1.10) gives the projections=of the acceleration a onto the axes x, y. Contrary to the velocity v, the acceleration vector a does not have a pre-established direction, with the exception of some particular cases like that of a rectilinear trajectory. Usually, it is worthwhile to project the vector a onto the couple of orthogonal directions that are normal and tangential to the absolute trajectory. These directions are identified by the versors n and t, respectively. The components of the acceleration a projected onto the normal and tangential directions can be obtained as follows: nt =a a+ ant = (1.1.11)= Let us consider a short arc=of the absolute trajectory evaluated=about=the current position of the material= point. In general=this trajectory is curvilinear.= The terms a n=and a t=are positive or negative scalar quantities. Then, the eq.(1.1.10) can be rewritten as:= ( ) (v )( ) (v) ( ) = = = + = +v d d ddd d dt dt dt dt dt dt v v av tt t tt = (1.1.12)= It is important to consider that=t is a time-dependent unity vector: = () t tt =as the direction of t may change time by time. Therefore, the derivative of the unity vector t with respect to time must be evaluated. For a given position () −PO =the centre of curvature, C, and the radius of curvature *ρ =(also called= osculatory radius)=of the=trajectory can be evaluated=(Figure=1.1.5). These two geometrical parameters can= vary with time=depending on the shape of the trajectory. For a rectilinear trajectory the radius of curvature is infinite while a circular trajectory has a constant=radius= *ρ .= = = = Figure 1. 1.5 =Trajectory c entre of curvature == Figure 1. 1.6 Changes in the tangent versor t = Now, let us consider the very small – e.g. infinitesimal – displacement ds of the point that occurs in an infinitesimal time interval 21 =( - ) dt t t . Therefore, the=point=position changes from P 1=to P 2. During the time interval= dt =the inclination angle of the radius of curvature= *ρ =is subjected to an infinitesimal change equal to=d ϕ. The angular speed with which the segment ()CO− =rotates is given by:= =d dt ϕ ω =(1.1.13)= Owing to the infinitesimal length of the considered time interval also the length of the trajectory that the= point=has travelled=is so small that it can be approximated with a very short arc of a circumference=having a= radius= *ρ , equal to the radius=of curvature of the trajectory. That is, during the infinitesimal time interval d t the radius of curvature *ρ =can be=assumed to be=constant. Figure=1.1.6=shows the two unity vectors t 1 and t 2, that are tangent to the trajectory in P 1 and P 2. For the sake of clarity, the dimension of the infinitesimal angle d ϕ has been exaggerated in Figure 1.1.6. Figure 1. 1.7 Changes in the tangent versor t = If the origin of the unity vector t 1 is shifted in P 2 it is possible to point out the small vector dt that represents the difference between the two unity vectors t 1 and t 2. That is: 21 =( ) d − t tt . The magnitude of the vector dt is =1 dd ϕ t .=The vector=dt represents the change of the unity vector t 1 caused by the change of curvature of the trajectory when the position of P moves from P 1 to P 2. Note that, as t 1 is a unity vector, only its phase can change. Being the interval d t infinitesimal, the direction of the vector dt coincides with that of the unity vector n, that is normal to the trajectory in P, whereas the versus of dt is opposite to that of the unity vector n. Since the infinitesimal rotation d ϕ occurs in the infinitesimal interval d t, the derivative ()d dt t =is given by:= () ( )dd dt dt ϕ = − t n= that is:= () d dt ω = − t n =(1.1.14)= With regards to this, be careful=not to confuse the versor t, which is a vector, with the quantity t (e.g. time). As said above, the radius of curvature *ρ =is constant during=the infinitesimal time interval d t. Therefore, the small displacements d s can be expressed as: * ds d ρϕ= = (1.1.15)= Hence, the magnitude, v, of the velocity vector=v, can be written as: ** v d dt ϕ ρ ρω= = = (1.1.16)= Then, the eq.(1.1.12) can be rewritten as: 2 *2 * (v) ( ) (v)(v)(v) v = +vvd dd d d dt dt dtdtdt ωρω ρ =−− − a t t tn=t n=tn = (1.1.17)= Therefore, by considering the eq.(1.1.17),=the two components a n=and a t=of the=absolute acceleration a, directed in accordance with the unity vectors n and t are: 2 *2 n * v a= ρω ρ −− = = t (v) a= d dt = (1.1.18)= That is the acceleration vector=a can be expressed by means of two components, a t and a n, which are oriented along the directions tangential and normal, respectively, to the trajectory, evaluated for the current position of the point. The two acceleration vectors a t and a n are called the tangential and normal components of the acceleration a. That is, the two vectors a t and a n are tangential and normal to the absolute trajectory of the material point. In the case of a rectilinear trajectory, at least close to the current position P, the radius of curvature is infinite and the normal component a n of the acceleration nullifies. Moreover, in accordance with the eq.(1.1.18), also in the case of a constant magnitude of the velocity v, the acceleration a can be not null in all the cases for which the angular speed ω is not null and the trajectory is not a straight line. The normal component, a n, of the acceleration a is also called centripetal. The versus of this acceleration component is always towards the centre of curvature of the trajectory. For a given value of the velocity v, the lower is the radius of curvature ρ the higher is the centripetal acceleration a n. In general, both components a t and a n are not null, though their instantaneous value can nullifies depending on the shape of the trajectory and the characteristics of the motion of the material point. As said above, the velocity vector v is always tangent to the trajectory, while the acceleration vector a can be always projected onto the tangential and normal direction to the trajectory, at any time t. Therefore, the acceleration a can be written as: =+xy a  ij = or:= 2 *2 * (v)(v) v =dd dtdt ρω ρ−− at n= t n = (1.1.19)= The acceleration=a can also be obtained by deriving the eq.(1.1.7) with respect to time. That is: ( )2 = i i ii dd ei e e e dtdt απ α αα αα + +=+ vv av v  = (1.1.20)= with= ωα = and * = ρω v . Therefore the eq.( 1.1.20) can be rewritten as:= ( )2 = i i* d ee dt απ α ρ ωω + + v a = (1.1.21)= The sign of the angular speed ω can be positive (counter-clockwise) or negative (clockwise). This determines the sign of the last term of the eq.(1.1.21). The same kinematic parameters v and a can also be determined evaluating the first order and second order derivatives of the absolute position vector () −PO . Be careful of the fact that in the following equations the term= ρ is the magnitude of the position vector () −PO =(not a radius of curvature). That is:= ( 2) ii eeϑϑπ ρ ρϑ + = + v  =( 1.1.22)= and= () ( ) (() )()( ) == 22 itii 22 d d te d d e i e dtdt dt dt ϑϑϑ ρρ ρθ −+ = = v a   PO (1.1.23) That is: = ii i i i ieieieiie ϑϑ ϑ ϑ ϑ ρ ρϑ ρϑ ρϑ ρϑ ϑ + +++ = a        = ( 2)( 2) 2 ( ) 2 ii i i ee e ϑϑπϑπϑπ ρ ρϑ ρϑ ρϑ ++ + + ++    =( 1.1.24)= The sum of the two velocity vectors on the right side of eq.(1.1.22) is always tangential to the trajectory of the material point. The eq.(1.1.24) shows that the acceleration vector=a can be decomposed into four vectors that are mutually perpendicular. The phases of these vectors are: ϑ , ( 2)ϑπ+ , ()ϑπ+ . θhese=acceleration= components are usually different from the above mentioned components=a t and a n (see eq.(1.1.19)). However, their sum gives the same acceleration vector expressed by the eq.(1.1.21). The theoretical background above illustrated, mainly for a point that moves along a planar trajectory, can easily be extended to a more general case in which the trajectory is defined in a 3D space. 1.2 Centripetal acceleration It is important to emphasise that the normal component a n of the absolute acceleration a is always centripetal. That is its versus is always directed towards the centre of curvature of the trajectory. With regard to this, it is important to take into account the sign of the angular speed ω with which the versor t, tangent in P to the trajectory, rotates. For instance, the material point shown in Figure 1.2.1 moves leftward on the trajectory s. The velocity of the material point can be written as: =v iieeαα= vv (1.2.1) while the acceleration can be expressed as: ( ) ( ) 22 =v ii i iii dd d ei e e e e e dtdtdt απαπ α ααα αα α ++ +=+ =+ vv v av v   (1.2.2) Owing to the trajectory curvature, the derivative of the angle α with respect to time, α, which coincides with the angular speed ω, is positive, as its versus is counter-clockwise. Therefore, the angle 2 απ+ indicates the versus of the centripetal component, a n, of the acceleration a. Conversely, the material point shown in Figure 1.2.2 moves rightward on the trajectory s. In this case the derivative of the angle α with respect to time, α, which coincides with the angular speed ω, is negative, as its versus is clockwise. That is: ( ) ( ) 232 =v ii i iii dd d ei e e e e e dtdtdt απαπ α ααα αα α ++ +=+ =+ vv v av v   (1.2.3) Therefore, the angle 32 απ+ indicates the versus of the centripetal component, a n, of the acceleration a. Anyhow, also in this case the component a n of the acceleration a is centripetal. Figure 1 .2.1 Particle moving leftward Figure 1. 2.2 Particle moving rightward 1.3 Examples Some very simple examples of the evaluation of the kinematic parameters of a material point are shown below. At first a rectilinear trajectory has been considered. In this case the radius of curvature of the trajectory is ∞ . Therefore, the component=a t of the absolute acceleration is null. = vv iieeαα= = vv t= (1.3.1)= Then, a circular trajectory has been considered as second case study. In this case the radius of curvature of= the trajectory is constant. Therefore, the absolute velocity=v and acceleration a of the material point can be evaluated as: ( 2)ie ϑπ ρϑ + =v  = ( 2) 2 ( ) = iiee ϑ πϑπ ρϑ ρϑ ++ + a  =( 1.3.2)= = = = Figure 1. 3.1 Particle mo ving along a rectilinear trajectory = Figure 1. 3.2 Particle moving along a circular trajectory = 1.4 Mobile reference systems In the previous Section the basic kinematic parameters of a material point have been defined with respect to an absolute Cartesian reference system, the axes of which do not rotate while the origin O is fixed. However, the kinematic parameters of a material point can also be evaluated using a mobile reference system, that is a Cartesian system of axes (x-y, x-y-z) whose origin can move in the 3D space while the single axes can rotate. Let us consider a set of absolute Cartesian coordinates, x’-y’, whose origin, O’, is fixed while the axes do not rotate. Then, let us consider a pair of Cartesian coordinates, x-y, whose origin is O. The point O can move in the plane x’-y’, while the axes of this Cartesian reference system can rotate about the z axis (Figure 1.4.1). In the end, let us denote s the absolute trajectory of a material point, defined with respect to the absolute reference system x’-y’. At time t, the direction of the x axis, evaluated with respect to the horizontal axis x’, is given by the time-dependent angle ()tα =(Figure=1.4.1).= = = = Figure 1.4.1 Position vector with respect to a m obile reference sys tem =x-y Figure 1.4. 2 Velocity vector with respect to a m obile reference system x-y = The position of the origin O of the relative Cartesian system ( x-y ), defined with respect to the absolute Cartesian system ( x’-y’ ), is given by the vector () − ' OO . The instantaneous velocity, v O, and acceleration, a O, (Figure 1.4.2) of the origin O can be expressed as: ' O () = d dt − OO v = ' O O ( ) () = 2 2 dd dt dt − = OO v a = (1.4.1)= The instantaneous values of the angular speed and angular acceleration with which the axes x-y rotate about the z axis are given by the two vectors ω =and ω , respectively. In accordance with=common conventions, the scalar=value of the angular speed= ω =is positive if the rotation about=the z axis is counter-clockwise. The same standard is used to define the positive versus of the angular acceleration ω .= Then, the magnitude of the=angular speed and angular acceleration of the mobile reference system,=about=the z axis, are given by: [] () =dt dtα ω = [] [] () () = 2 2d t dtdt dt αω ω =  = (1.4.2)= The general motion of the mobile reference system is=roto-translational. However, pure translational motions ( 0 ωω = =  ) or=a pure=rotation about the z axis ( OOva0 = = ) can occur.= If the=x-y axes rotates, the unity vectors i and j are not constant terms as their orientation is time-dependent. Therefore, when the position vector is differentiated with respect to time, the derivatives d dti and d dtj may give not null terms. In fact, the unity vectors i and j can be expressed as: ieα = i = ( 2)ie απ+ =j = (1.4.3)= By differentiating these expressions it is possible to obtain: ( 2) () i ii d de ie e dt dt α α απ αω ω + = = = =  i j= ( 2) ( ) () ii ii d d e d ie= ii e e dt dt dt απα αα αω ω + = = =−=− j i =(1.4.5 )= = The position vector () − ' PO =of the=material point=can be expressed as:= () ()=()()=()+xy − −+− −+ ''' PO OO PO OO i j = (1.4.6)= where the vector= () − ' OO =is the position vector of the origin of the mobile reference system with respect to the absolute=reference system.= 1.4.1 Instantaneous motion of the mobile reference system The motion of a mobile reference system (reference frame) can be: - a pure translation. In this case the axes of the reference frame do not rotate, but the trajectory described by the origin, O, of the axes can be rectilinear or curvilinear; - a pure rotation about an axis passing through the origin O of the reference frame; - the super-position of a translation and a rotation (roto-translation). Let us consider a planar motion and a Cartesian reference system whose axes are x-y. If the origin O of the mobile reference system is fixed, that is if the vector (O – O’) is constant, the instantaneous motion of the mobile reference system is a pure rotation about a fixed axis passing through point O. In this case the mobile reference system is called rotating. Conversely, if the axes x-y do not rotate, while the vector (O – O’) is not constant, the instantaneous motion of the mobile reference system is a pure translation. In this case the mobile reference system is called translating. In the end, if the axes x-y rotate and vector (O – O’) is time- variant, the mobile reference system is called roto-translating. It is important to remark that the curve described by the origin O of a translating reference system can be both a straight line and a curvilinear trajectory. The basic characteristic of such a mobile reference system is only given by the fact that the axes x-y do not rotate at any time. Figure 1.4.3 shows some examples of translating reference systems. Figure 1.4.4 shows an example of a rotating reference system: the origin O of the axes x-y is fixed, in this case. In the end, Figure 1.4.5 shows an example of a roto-translating reference system. Figure 1.4.3 Translational reference system: the axes do not rotate= Figure 1.4.4 Rotational reference system:= the origin O does not move = = Figure 1.4.5 Roto -trans lational reference system: the origin O moves and the x-y axes rotate For an observer, who is moving together with the mobile reference system, the material point describes a relative trajectory that is different from the absolute one. Besides, if the material point P is virtually joined to the mobile reference system and an infinitesimal motion – compatible with the actual instantaneous motion of the reference system – is assigned to it, the material point describes a very short arc of trajectory – infinitesimal – whose shape depends on the type of the instantaneous motion of the reference system, i.e.: - an infinitesimal translation if the reference system is a translating coordinate system; - an infinitesimal rotation if the reference system is a purely rotating coordinate system; - an infinitesimal roto-translation if the reference system is a roto-translating coordinate system; In the following, this trajectory is called drag trajectory (traiettoria di trascinamento). Therefore, for a material point, an absolute trajectory, a relative trajectory and a drag trajectory can be defined whenever a mobile reference system is used for a kinematical analysis. Be aware of the fact that, in general, the relative and drag trajectories must be determined at any time t. 1.4.2 Velocity The absolute velocity v of the point P can be obtained by evaluating the derivative, with respect to time, of the eq.(1.4.6). That is: ()() '' O ( ) ( )( ) ddd xy x y dt dt dt ωω − −− = = + =++ + −PO OO PO vv ij j i = (1.4.7)= That is (by denoting with ∧bc the crossϑproduct between the vectors b and c): ( ) ( ) OrOr ()( ) xy = ++ ∧ − = ++ ∧ +vvvvv POi j ωω = (1.4.8)= This equation can be rewritten as:= ( ) ( ) 12 rOrOr tt rt ()( ) xy  =+ +∧− =+ +∧ + =+ + =+   vvvvvv vv vv ωω POi j = (1.4.9)= The term=v r is the relative velocity of P, that is the velocity evaluated with respect to the mobile reference system x-y. This velocity vector is tangent to the relative trajectory of P, which is determined by the loci described by the time-varying position P(x,y), defined with respect to the mobile reference system. ( ) r xy = + v ij = (1.4.10)= The second term, v t, of the eq.(1.4.9) is the drag velocity. The vector v t is the component of the absolute velocity v that is caused by the instantaneous motion of the mobile reference system that has been considered for the kinematic analysis. In general, the velocity v t is given by two terms that depend on the translational motion and the rotation of the reference system. In fact: ( ) 12 tOt t () xy =+∧ + = + vvv v ω ij = (1.4.11)= The drag velocity v t is tangent to the drag trajectory. Therefore, every velocity vector – absolute, relative or drag – is always tangent to the corresponding trajectory. In the case of a purely translating reference system the velocity v t is only given by the term v O of the origin of the axes x-y. In fact, in this case the term ω is null. Conversely, in the case of a purely rotating reference system the velocity v t is only given by the term: t () xy =∧+ v ω ij (1.4.12) where the term ω, which is a vector orthogonal to the plane x-y, is the instantaneous angular speed with which the axes of the mobile reference system rotate. In the general case of a roto-translating reference system both terms of eq.(1.4.11) can be not null. The absolute trajectory of the point P can be obtained taking into account the relative trajectory and the drag trajectory. The components of the velocity vector v t2, in the directions x-y-z, can be obtained with an easily technique evaluating the determinant of the following 3 × 3 square matrix: ( ) ( ) ( ) ( ) 2t PO z y zx y x x yz ωωω ω ω ω ω ω ω =∧− = = − − − + − v ω x yz y z x z x y i jk i jk (1.4.13) where ωx, ωy, and ωz are the angular velocity with which the mobile reference system rotates about the x, y and z axes, respectively. In the example shown in Figures 1.4.1 and 1.4.2 the angular velocities ωx and ωy are null as the motion of the material point P occurs in the x’-y’ plane. 1.4.3 Acceleration The absolute acceleration a of the point can be obtained by evaluating the derivative, with respect to time, of the eq.(1.4.9). That is: '' O ( ) ( ) ( ) () ( ) ( )22 2 2 22d d d d dx y d x ydt dt dt dt dtdt ωω − −− + − = = +=+ +PO OO PO v a ij j i (1.4.13) Substituting the expressions (1.4.5) into the eq.(1.4.13) we obtain: 22 O ( )( )()( )( ) xy xy x y xy xy ωω ω ω ωω ωω =+++−+−− +−+− aa      ij ji i j ji ji 22 O ()2()( )()xy xy x+y xy ωω ω ω ωω =++ + − −+ − aa     ij ji i j ji [ ] [ ] 22 O ()2()( )() x yx+ y x+ y x+ y ωω =+ + + ∧ −+∧aa    ij ij i j ij ωω (1.4.14 ) Then, the eq.(1.4.14) can be rewritten as: [] 2 Or ( )2( )() ()xy ω =+ + + ∧ − −+ ∧− aav   i jPO PO ωω ( ) [ ] Or ( )2( ) () () xy =+ + + ∧ + ∧ ∧− + ∧ −  aav   i jPO PO ω ωωω (1.4.15 ) r tC =++aa a a (1.4.16) where: r ()xy = + a  ij (1.4.17) ( ) [ ] { } { } 12n 2 t1 2 tOt t t t t () () =+ ∧ ∧− + ∧ − = + + = +  aaa a a a a  ωωω PO PO (1.4.18) Corr 2( ) = ∧ av ω (1.4.19) The term a O represents the instantaneous acceleration of the origin, O, of the mobile reference system evaluated with respect of the absolute reference system. The term a r of the eq.(1.4.16) is the relative acceleration of the material point evaluated with respect to the mobile reference system. This acceleration vector can be projected onto a couple of orthogonal directions that are normal and tangential to the relative trajectory s r of the point P. These two terms can be denoted a rn and a rt, respectively. The term a t of the eq.(1.4.16) is the drag acceleration of the material point. In the case of a roto-translating reference system the two acceleration vectors 2nta =and 2tta =are normal and tangent=to the component of the drag trajectory associated with the rotation only of the mobile coordinate system x-y-z. The acceleration vector 2tta can be obtained evaluating the determinant of the following 3 × 3 square matrix: ( ) ( ) ( ) ( ) 2tt PO z y zx y x x yz ωωω ω ω ω ω ω ω =∧− = = − − − + − a         ω x yz y z x z x y i jk i jk (1.4.20) where ωx, ωy, and ωz are the angular acceleration with which the mobile reference system rotates about the x, y and z axes, respectively. In the example shown in Figures 1.4.1 and 1.4.2 the angular acceleration ωx and ωy are null as the motion of the material point P occurs in the x’-y’ plane. The acceleration vector 2nta can be also expressed as: ( ) ( ) ( ) 2nt PO PO x+ y ωω22 =∧ ∧− = − =  a ωω ij (1.4.21) This vector is the centripetal component of the acceleration 2ta . It is always directed from the point P to the centre of curvature of the component of the drag trajectory associated with the rotation only of the mobile reference system. In the end, the term a Cor of the eq.( 1.4.16) is the Coriolis’ acceleration vector. This term depends on the instantaneous values of both the angular speed with which the axes of the mobile reference system rotate and the relative velocity v r. The acceleration a Cor can be obtained evaluating the determinant of the following 3 × 3 square matrix: Corr rr r 2( ) 2 vvvx yz ωωω = ∧= av ω x yz i jk (1.4.22) Owing to the cross-product rules, this vector is orthogonal to the relative velocity vector v r. As said above, each velocity vector is tangent to the respective trajectory (absolute, relative, drag). Conversely, the direction of the acceleration vector is not necessarily tangent to the respective trajectory. However, it can be always decomposed into two vectors, normal and tangent, respectively, to the corresponding trajectory (absolute, relative, drag). In the case of a purely translating reference system the drag acceleration a t is only given by the term a O of the origin of the axes x-y. Besides, also the Coriolis’ acceleration a Cor is null. In fact, in this case, the terms ω and ω are null. Therefore, in this case the expression of the absolute acceleration a is given by: rO = +aa a (1.4.23) Conversely, in the case of a purely rotating reference system, the acceleration a O is null while the drag acceleration a t is only given by the terms: ( ) [ ] rCor () () =+ ∧ ∧− + ∧ − +  aaa  ωωω PO PO (1.4.24) The approach above described allows one to evaluate the absolute velocity and acceleration of a material point P that moves along a 2D trajectory. Other methods, here not described for the sake of brevity, are available in literature to perform this task as well as to determine the kinematic parameters of a material point that moves in a 3D space.