Chapter 2 On the Razi Acceleration

Transcription

1 hapter 2 On the Razi cceleration. Salahuddin M. Harithuddin, Pavel M. Trivailo, and Reza N. Jazar Keywords Razi acceleration Rigid body kinematics Multiple coordinate system kinematics oordinate transormation 2.1 Introduction Jazar presented a revised method o calculating vector derivatives in multiple coordinate rames using the extended variation o the Euler derivative transormation ormula (Jazar 2012). The paper derives the general expression or vector derivatives or a system o three relatively rotating coordinate rames called the mixed derivative transormation ormula. The ormula is demonstrated as ollows: Let,, and be three arbitrary, relatively rotating coordinate rames sharing an origin as shown in Fig Let be a generic vector expressed in the rame,! be the angular velocity o the rame with respect to the rame, and! be the angular velocity o the rame with respect to the rame. The time-derivative o the -vector as seen rom the rame can be expressed as d D d!! (2.1) where the let superscript indicates the rame in which the vector is expressed. The ormula relates the -derivative o a -vector to the -derivative o the same -vector by means o the angular velocity dierences between the three.s.m. Harithuddin P.M. Trivailo ( ) R.N. Jazar School o erospace, Mechanical, and Manuacturing Engineering, RMIT University, Melbourne, VI, ustralia salah.harithuddin@student.rmit.edu.au; pavel.trivailo@rmit.edu.au; reza.jazar@rmit.edu.au Springer International Publishing Switzerland 2015 L. Dai, R.N. Jazar (eds.), Nonlinear pproaches in Engineering pplications, DOI /

2 32.S.M. Harithuddin et al. Fig. 2.1 Three coordinate rames system z z z y y x y x x coordinate rames. The ormula presents a more general expression o the classical derivative transormation ormula between two relatively rotating rames d D d! (2.2) where the rame is assumed to coincide with ( ), thus! D 0. One advantage o the general mixed derivative transormation ormula is that it leaves the need to compute the derivative o d = in its local coordinate rame. n interesting result appears rom the application o a reerence rame system with three coordinate rames. new acceleration term, given the name the Razi acceleration (Jazar 2011), is shown to appear when the derivatives o an acceleration vector are taken rom two dierent coordinate rames. Such dierentiation, which resulting expression is called mixed acceleration, contains the Razi term,.!! / r. To demonstrate the Razi acceleration term, let there be three relatively rotating coordinate rames,, and, shown in Fig ssume a position vector r is expressed in its local rame which is. Taking the irst derivative o r in rom using the vector derivative transormation ormula Eq. (2.2) yields d r D d r! r (2.3) Taking the derivative o d r again, but rom the rame yields: d d r D a

3 2 On the Razi cceleration 33 where D a! v r!! r! v!! r (2.4) a d 2 r v d r d! (2.5) and the let superscripts in (2.4) show that every vector is expressed in the rame. The term!! r is a new term in the acceleration expression called the Razi acceleration. For comparison, let us assume that the rame is not rotating and coincides with the rame, making. Equation (2.4) is now reduced to the classical expression o acceleration between two relatively rotating rames and d 2 2 r D a D a 2! v r!! r (2.6) Equation (2.6) shows that there are three supplementary terms to the absolute acceleration o a point in local rame : The oriolis acceleration 2! v, the tangential acceleration r, and the centripetal acceleration!! r. The dierence between the expressions in Eqs. (2.4) and (2.6) isthe Razi term which has the orm o a vector product between a product o two angular velocities and a position. This shows that adding another rotating coordinate rame and dierentiating a displacement vector rom two dierent rames reveals a new acceleration term. It is shown in the published work o Jazar that the Razi acceleration is a novel inding in classical mechanics. In its current status, however, the Razi acceleration has only ound a limited application in relative kinematics. This paper seeks to reveal the aspect o the Razi acceleration in mechanics o relative rigid body motion speciically in the multiple coordinate system kinematics. We show the inclusion o the new Razi term in the equation o motion o a rigid body described in non-inertial rotating coordinate rames. To do this, we redeine and extend the Euler derivative transormation ormula to be applied in a system with arbitrary number o coordinate rames. We provide an illustrative example o a rigid body in compound rotation motion about a ixed point to show the appearance o the Razi acceleration. We compare the magnitudes o the Razi acceleration with the classical centripetal acceleration to show that in the case o complex rotating motion, the Razi acceleration may appear as a result o rotational inertia.

4 34.S.M. Harithuddin et al. 2.2 Time-Derivative and oordinate Frame The time-derivative o a vector depends on the coordinate rame in which it is dierentiated. In a multiple coordinate rames environment, it is important to know rom which coordinate the derivative vectors, e.g. velocity and acceleration, are calculated and to which coordinate rame it is related. In this section we introduce the derivative transormation ormula and expand it to include more than two reerence rames Euler Derivative Transormation Formula ny vector r in a coordinate rame can be represented by its components using the basis vectors o the rame. Let.OO{ O O k/ be an arbitrary coordinate rame in which r is expressed. The vector is written using the basis vectors o as r D xo{ y O z O k (2.7) For r, a vector unction o time, its components x, y, and z are scalar unctions o time. The time-derivative o the vector r is d r DPxO{ Py O Pz k O x P O{ y P O z P k O (2.8) Since the basis vector is constant with respect to its own rame, the time-derivative o a vector expressed in dierentiated rom its own rame is Pr D d r DPxO{ Py O Pz k O (2.9) This is called a local or simple derivative since the vector derivative is taken rom the coordinate rame in which it is also expressed. The result is a simple derivative o its scalar components. Now consider global-ixed coordinate rame G.OIO JO K/ O and a rotating bodyixed coordinate rame.oo{ O k/ O as shown in Fig Due to the relative motion o the coordinate rames, the basis vectors o one rame are not constant in relation to the other rame. The time-derivative o a vector expressed in dierentiated rom rame G is expressed as G G Pr D d r DPxO{ Py O Pz O k x G d O{ y G d O z G d O k (2.10) Similarly, the time-derivative o a vector expressed in G dierentiated rom rame is expressed as

5 2 On the Razi cceleration 35 Fig. 2.2 inary coordinate rames system a global-ixed rame G and a rotating body rame G J K J k r (t) J î j Î G Pr D d G r D P X O I PY O J PZ O K X d IO Y d JO Z d K O (2.11) n angular velocity vector can be deined in terms o a time-derivative o basis vectors coordinate rame in relation to another coordinate rame. The angular velocity o a rotating rame in coordinate rame G, G!, expressed using the -coordinates, is written as G! D Ok G d O O{ O{ G dk O! O O G d O{ Ok (2.12) nd the angular velocity o a rotating rame G in coordinate rame,! G, expressed using the G-coordinates, is written as G! G D O K! djo IO O I! dko Equations (2.10) and (2.11) can be represented as G G Pr D d r D O J O J! dio O K (2.13) d r G! r (2.14) G Pr D d G G d r D G r G! G G r (2.15) n alternate proo o Eqs. (2.14) and (2.15) is given in ppendix 1.

6 36.S.M. Harithuddin et al. The resulting equations show the method o calculating and transorming the derivative o a vector expressed in one rame to another. This results in the most undamental result o kinematics: Let and be two arbitrary rames which motion is related by the angular velocity!, or every vector expressed in, its time derivative with respect to is given by d D d! (2.16) This ormula is called the Euler derivative transormation ormula. It relates the irst time-derivative o a vector with respect to another coordinate rame by means o angular velocity. Despite its importance and ubiquitous application in classical mechanics, there is no universally accepted name or the Euler derivative transormation ormula in Eq. (2.16). Several terminologies are used: kinematic theorem (Kane et al. 1983), transport theorem (Rao 2006), and transport equation (Kasdin and Paley 2011). These terms, although terminologically correct, are more prevalent in the subject o luid mechanics to reer to entirely dierent concepts. We use Zipel s (Zipel 2000) and Jazar s (Jazar 2011) terminology, which we opine as more descriptive and unique to the subject o derivative kinematics. The name also is a homage to Leonard Euler who irst used the technique o coordinate transormation in his works on rigid body dynamics (Koetsier 2007; Tenenbaum 2004; aruh 1999; Van Der Ha and Shuster 2009) Second Derivative One o the most important applications resulting rom the vector derivative transormation method is the development o inertial accelerations. Inertial accelerations, or sometimes called d lembert accelerations or, misnomerly, ictitious accelerations, are the acceleration terms that appear when the double-derivatives o a position vector are transormed rom a rotating rame to an inertial rame or vice versa. To demonstrate this, let G be an inertial coordinate rame and a non-inertial coordinate rame rotating with respect to G with an angular velocity G!.Usingthe Euler derivative transormation ormula, the irst derivative o a vector r as seen rom G is G d r D G Pr D d r G! r (2.17) Using the ormula again to get the second derivative o the vector r rom G yields

7 2 On the Razi cceleration 37 G d G d G d r D D D D d d 2 d d r G! r r G! r G! 2 r G r 2 G! d r G! r d r G! G! r a G r 2 G! v G! G! r (2.18) Equation (2.18) is the classical expression or the acceleration o a point in a rotating rame as seen rom an inertial rame. The term a is the local acceleration o a point in with respect to the rame itsel regardless o the rotation o in G. The second term G r is called the tangential acceleration as its direction is always tangential to the rotation path. Tangential acceleration term is represented by a vector product o an angular acceleration and a position vector. The third term 2 G! v is an acceleration term traditionally named ater oriolis. The oriolis acceleration term appears rom the two dierent sources: hal o the term comes rom a local dierentiation o G! r, and the other hal is rom the product o angular velocity and the local linear velocity vector. The oriolis acceleration in the classical second derivative between two rames is written as twice the vector product o an angular velocity and a linear velocity. The inal term in the expression G! G! r is called the centripetal acceleration as its direction is towards the center o rotation. The centripetal acceleration has the orm o a vector product o an angular velocity with another vector product o an angular velocity and a position vector Derivative Transormation Formula in Three oordinate Frames The Euler derivative transormation ormula is now extended to include a third coordinate rame. Let,, and be three relatively-rotating coordinate rames sharing a same origin as shown in Fig The irst time-derivative o a -vector in relative to the -rame is written as v D v! r (2.19) nd the irst time derivative o the same -vector in relative to the -rame is written as v D v! r (2.20)

8 38.S.M. Harithuddin et al. Fig. 2.3 Three relatively rotating coordinate rames Z Z Z r y y x y x x ombining Eqs. (2.19) and (2.20) yields v D v!! r (2.21) The resulting equation relates the -derivative o a -vector (2.19) and the - derivative o the same -vector (2.20) by means o the angular velocity dierences between the three coordinate rames. This result also shows that i the relative velocities and angular velocities with respect to one o the three coordinate rames are known, the simple derivative in local rame can be neglected. In another perspective, by introducing an auxiliary coordinate rame, one can orgo altogether the calculation o the vector derivatives in the local coordinate rame. The extended Euler derivative transormation in three coordinate rames can be generalized as P D P!! (2.22) where can be only generic vector. This ormula is called the mixed derivative transormation ormula (Jazar 2012). It can be used to relate vector derivatives in three rames by means o angular velocities dierence between the three coordinate rames Kinematic hain Rule To extend the Euler derivative transormation to account or arbitrary number o moving coordinate rames, we derive the general ormula or the composition o

9 2 On the Razi cceleration 39 3ω 4 4 1ω 2 3 2ω 3 0ω Fig. 2.4 Multiple coordinate rames system with rame 0 as the inertial rame multiple angular velocity vectors. This is useul or, but not limited to, a system with a chain o coordinate rames where the motion o one rame is subject to another. One example o such system is depicted in Fig onsider now n angular velocities representing a set o n.n D 1;2;:::;n/ noninertial rotating coordinate rames in an inertial rame indicated by n D 0. Their relative angular velocities satisy the relation 0 0! n D 0 0! 1 0 1! 2 0 2! 3 n 1 0! n D 0 i 1! i (2.23) i and only i they all are expressed in the same coordinate rame, which in this case rame 0. Using the rule, the composition o the angular velocity 0! n can be represented in any coordinate rame in the set.

10 40.S.M. Harithuddin et al. 0! n D R 0 0 0! n D i 1! i (2.24) Equation (2.24) represents the additional rule o the composition o multiple angular velocity vectors. ny equation o angular velocities that has no indication o the coordinate rame in which they are expressed is incomplete and generally incorrect. 0! n 0! 1 1! 2 2! 3 n 1! n (2.25) ngular acceleration is the irst time-derivative o an angular velocity vector. For an angular velocity vector o with respect to G, or example, the angular acceleration is deined as G D G! D! x O{! y O! z O k (2.26) d G! D P! x O{! P y O P! zk O (2.27) Suppose there are n angular velocity vectors representing a set o n coordinate rames expressed in an arbitrary rame. 0! n D 0! 1 1! 2 2! 3 n 1! n (2.28) To dierentiate the composition o angular velocities, which are expressed in, rom an arbitrary rame g in the set, one has to apply the Euler derivative transormation ormula in Eq. (2.16). g d i 1! i! D g d 0! n g d D 0! 1 1! 2 n 1! n D 0 1 g! 1 0 1! 1 2 g! 2 1! 2 n 1 n g! n n 1! n D i 1 i g! i i 1! i (2.29) where is deined as the dierentiation o! in its own rame d!=. The ormula provides a method to express the irst derivative o a composition o angular velocity vectors.

11 2 On the Razi cceleration 41 s an example, let the number o rotating coordinate rames be n D 3. The angular velocity o rame 3 with respect to the inertial rame 0 is written as a composition o three angular velocities, expressed in rame ! 3 D 3 0! 1 3 1! 2 3 2! 3 (2.30) pplying dierentiation rom rame 0, using the ormula in Eq. (2.29), one obtains 0 d 3 0! 3 D ! 1 3 0! ! 2 3 1! ! 3 3 2! 3 D ! 2 3 1! ! 3 3 2! 3 (2.31) Looking at the third term in the right-hand side, is the rate o magnitudinal change o 3 2! 3 in rame 3, and 3 0! 3 3 2! 3 is the convective rate o change o 3 2! 3 due to its angular velocity o rame 3 with respect to the rame it is dierentiated, 3 0! 3. The second right-hand side term can be interpreted similarly. The total angular acceleration o rame 1 with respect to rame 0 is only as the direction o the angular velocity o rame 1 with respect to rame 0 3 0! 1 does not change in rame rom which it is dierentiated as 3 0! 1 3 0! 1 D 0 Thereore, any equation o angular acceleration that has no indication o in which coordinate rame they are expressed, or without the convective rate o change term np g! i i 1! i is incomplete and generally incorrect. g d 0 n n 1 n (2.32) i 1! i! n 1 n (2.33) 2.3 The Razi cceleration Jazar (2012) showed that the Razi term appears as a relative acceleration due to transorming vector derivative in multiple reerence rames. Here, we show that the Razi term can be exhibited by the same method that the inertial accelerations in a ixed-axis rotation motion are calculated, such as the oriolis and the centripetal acceleration. We examine the system o three nested coordinate rames a bodyixed coordinate rame is rotating in another coordinate rame, which in turn is rotating in a globally ixed, inertial rame, with all coordinate rames sharing a common origin. The position vector o a body point r is expressed in the body rame and its double derivative, hence the acceleration o the body

12 42.S.M. Harithuddin et al. point, is taken rom the inertial rame. ny resulting acceleration term, apart rom the local absolute acceleration a, that appears rom the transormation o the vector derivatives is deined as an inertial acceleration. The derivative transormation between two coordinate rames produces inertial accelerations such as centripetal, oriolis, and tangential acceleration as shown in Eq. (2.18). We show that nested rotations involving more than two coordinate rames produce more inertial accelerations terms, in which one o them is the Razi acceleration Razi cceleration in Mixed cceleration Expression We reproduce the derivation o the Razi acceleration as shown in Jazar (2011, 2012) as a review. onsider three relatively rotating coordinate rames,, and, where the rame is considered as a global-ixed rame, while the rame is rotating in, and a body rame is rotating in. This motion is called compound or nested rotation since the rotation o is inside the rame, which in turn is also rotating inside another rame,. ssume a position vector r is expressed in its local rame. Taking the irst derivative o r rom the -rame using the vector derivative transormation ormula yields d r D d r! r (2.34) Taking the derivative o d r rom the -rame now yields a D d d D D d d r d d r! r r! r! d r! r D a r! v! v!! r!! r (2.35) From the expression, the mixed acceleration expression contains the local acceleration term a; two acceleration terms that have the oriolis orm o a product o an angular velocity and a linear velocity, which are! v and! v; the tangential acceleration term r; and a mixed centripetal acceleration term!! r. The term!! r is the Razi acceleration. Note the dierence between the Razi acceleration and the mixed centripetal acceleration is the order

13 2 On the Razi cceleration 43 o cross product. This indicates that when a coordinate body rame is rotating relative to other two rotating coordinate rame, the mixed double derivative o r contains these inertial acceleration terms: mixed oriolis acceleration, tangential acceleration, mixed centripetal acceleration, and the Razi acceleration. n application example o mixed derivative transormation ormula with this Razi acceleration is illustrated on Jazar (2012, pp ). However, the dynamical interpretation o the mixed acceleration is not clear. In practice, the double derivative o a position vector is usually directly transormed rom the local rame to the inertial rame without undergoing a mid-rame dierentiation. The reason is that a double-dierentiation o a point position in a rotating rame rom the inertial rame gives the additional acceleration terms, which contribute to inertial eect such as centriugal and oriolis orces. In mixed acceleration, in contrast, the dierentiation is taken rom two dierent rames. The mixed acceleration a D d v can be used to complement the acceleration o a point in rom using the mixed derivative transormation ormula in Eq. (2.22). d v D d v!! v (2.36) Next, we show that i there are more than one nested rotation, the Razi acceleration would appear even when the vector derivative is transormed directly rom the local rame to the inertial rame without using the mixed acceleration technique Razi cceleration as an Inertial cceleration The derivation o the Razi acceleration here uses the Euler derivative transormation ormula in conjunction with the kinematic chain rule. It diers rom the mixed acceleration method in such a way that the both derivatives o a body point in a local coordinate rame are taken rom the inertial rame, the same way the oriolis, centripetal, and tangential accelerations are derived. To demonstrate, consider a two coordinate rames system with G as the global-ixed inertial rame and as the rotating body-ixed rame. Since the relative rotation between the two rames can be represented by one angular velocity G!, we can use Eq. (2.16) twice to ind the double derivative o a body-ixed position vector r rom the inertial rame G. Looking at the resulting expression o the acceleration in two rames, shown in Eq. (2.18), one can separate the acceleration terms as ollows

14 44.S.M. Harithuddin et al. a local acceleration in G r tangential acceleration 2 G! v oriolis acceleration G!. G! r/ centripetal acceleration (2.37) Now, consider a slightly complex case o nested rotations. onsider three relatively rotating rames,, and where the -rame is treated as the inertial rame, the -rame is rotating in, and the -rame, which is the body-ixed rame, is rotating in. The motion o a rigid body which is spinning in one or more rotating coordinate rames is called compound rotation or nested rotation. ompound rotation is usually seen in the gyroscopic motion, or the precession and nutation o a rigid body. To express the velocity o a point in as seen rom the inertial rame, the Euler derivative transormation ormula in (2.16) is invoked to transorm the velocity rom the local rame to the inertial rame d r D Pr D Pr! r (2.38) Since, the rame is in, and the rame is in, the angular velocity! can be expanded using the angular velocity addition rule rom Eq. (2.24).! D!! (2.39) The expression o the acceleration o a point in rom the inertial rame can be expressed as d d r D a d! r 2! v!.! r/ (2.40) The angular velocity derivative d! can be expanded using the rule o addition or angular acceleration in (2.29) with n D 2 d! d D!! D!!!! D!! (2.41) The expanded expression o the acceleration o a point in rom the inertial rame now becomes d d r D a r!! r r 2! v!.! r/ (2.42)

15 2 On the Razi cceleration 45 One can now list the acceleration terms expressed in local rame and include the Razi term a local acceleration in r tangential acceleration 2! v oriolis acceleration (2.43)!.! r/ centripetal acceleration!! r Razi acceleration The Razi acceleration!! r comes rom dierentiating the composition o angular velocity vectors rom a dierent rame as shown in Eq. (2.41). Next, we show that or the case o n>2nested rotating rames, Razi acceleration terms appear in the equation o motion. It is important to note that the Razi acceleration can only appear in compound rotation motion. rigid body in a rotation about a ixed axis does not experience an inertial orce caused by the Razi acceleration. This is because a rotation about a ixed axis can always be simpliied to a two coordinate rames system. In such case the Razi acceleration vanishes as there is only one angular velocity, making!! D cceleration Transormation in Multiple oordinate Frames with a ommon Origin Given a set o n non-inertial rotating coordinate rames in nested rotations with i 1! i representing the angular velocity o i-th coordinate rame with respect to the preceding coordinate rame, we present the general expression or the acceleration (the double derivative o a vector) o the n-th coordinate rame as seen rom the inertial coordinate rame 0. 0 d 2 n r D n 2 00 a D nn n a n i 1 i n r 2 n 0! i n n v n 0! i n0! i i 1 n! i n r n 0! i n r! (2.44)

16 46.S.M. Harithuddin et al. where n 0! i 2 n n nn a local acceleration i 1 i n r tangential acceleration n 0! i n nv oriolis acceleration! n 0! i n r centripetal acceleration n0! i i 1 n! i n r Razi acceleration (2.45) The general expressions or acceleration when a position vector in a local rame is dierentiated twice rom the inertial rame are shown in Table 2.1. Examples are given or up to a ive coordinate rames system. The rame is dedicated to the inertial rame and the successive relatively rotating rames are named,, and so on, with the last letter indicates the local coordinate rame. For example, in a system o three coordinate rames, is the inertial rame, is the intermediate rame, and is the local rame. ll vectors are expressed in the local coordinate rame as indicated by the let superscript on each vector. ddition o angular acceleration vectors is simpliied as n P 0 n n n i0 i. The resulting acceleration expression in Table 2.1 shows that i a rigid body moves linearly with respect to an inertial space, its motion can be suiciently described by a linear acceleration vector a. When a rotating reerence rame is introduced and the motion is described in this rame, the additional acceleration terms centripetal, oriolis, and tangential accelerations act as supplementary accelerations to describe the motion in a rotating reerence rame. When another rotating rame is introduced in the existing rotating rame making a three coordinate rames system, the expression adds another acceleration term: the Razi accelerations. s more coordinate rames are introduced in the nested rotating rames system, the acceleration expressions becomes more complex to accommodate the relative motion between the coordinate rames. In general, Eq. (2.44) includes all o the relative accelerations acting on a rigid body in compound motion which are otherwise unobservable with the classical derivative kinematics method and not intuitive to the analyst.

17 2 On the Razi cceleration 47 Table 2.1 Inertial acceleration terms or describing rigid body in multiple rames system No. o rames cceleration terms Enclosed system r a r r D D r D E E r

18 48.S.M. Harithuddin et al. 2.4 Mechanical Interpretation o the Razi cceleration lassical dynamics asserts that there exists a rigid rame o reerence in which Newton s second law o motion is valid. ssuming the mass is constant at all times, the law o motion is written as F D ma (2.46) where m indicates a mass scalar and a represents the acceleration vector. The reerence rame in which this relationship is valid is called an inertial rame. n inertial rame is ixed in space, or at most, moves in relation to ixed points in space with a constant velocity. On the other hand, a non-inertial reerence rame can be deined as a rame which accelerates with respect to the inertially ixed points. The calculation o motion observed rom a non-inertial reerence rame does not ollow Newton s second law due to the relative acceleration between the observer s rame and the inertial rame. Starting rom an inertial rame, the method o derivative kinematics is used to calculate and transorm the vector derivatives to the non-inertial observer. Such transormation produces additional acceleration terms to the Newton s law in Eq. (2.46). For the classical case o one stationary inertial rame and a body rame which rotates about a ixed axis, these additional acceleration terms are generally divided into tangential acceleration, oriolis acceleration, and centripetal acceleration, as shown in the expression in Eq. (2.18). F D m a a tangential a oriolis a centripetal (2.47) Multiplying the acceleration terms with mass gives the expression o the orces that is acting on a point mass in the rotating coordinate rame. In the case o multiple nested coordinate rames, the classical method o transorming derivative between two rames is limiting. This method is suicient or any motion consideration in two-dimensional space, due to the assumption that the direction o a rigid body s angular velocity is ixed with respect to the inertial rame at all time. However, a general ormula or a three-dimensional space is needed to account or the subtle accelerations due to a non-constant angular velocity. The derivative transormation method can be extended to include the eects o moving angular velocities. To do this, the complex rotation caused by the nonconstant angular velocity vector is decomposed into several simpler coordinate rame representation such that each coordinate rame then will have a ixed-axis angular velocity vector. Such technique results on multiple coordinate rames in nested rotations. To analyze such system, we incorporate the extended derivative transormation ormula and the kinematic chain rule, which method reveals new results and application. For the case o three coordinate rames system (two nested rotating rames in an inertial rame), we have seen that the acceleration expression includes

19 2 On the Razi cceleration 49 a new rotational inertia called which is the Razi acceleration. We also have shown the general expression or all our inertial acceleration tangential, oriolis, centripetal, and Razi or an arbitrary number o nested rotating rames. F D m a a tangential a oriolis a centripetal a Razi (2.48) Extending the derivative transormation ormula and incorporating the kinematic chain rule reveal several new results and applications. The Razi acceleration is given a ocus due to its appearance as one o the additional acceleration terms in the system o more than two coordinate rames. We take the simplest case o the Razi acceleration where there are three nested coordinate rames,, and sharing an origin with the body coordinate rame being the innermost rame. Such system can be better illustrated by a lat disc in compound rotation motion, as shown in Fig We ix the coordinate rame in the disc, which is spinning on its own axis with an angular speed P and tilted on a turning shat by degrees. We ix the coordinate rame on the shat, which is turning with respect to the global-ixed rame with an angular speed P. The two angular velocities can be written as! D P O{ and! D P O. The total acceleration acting on a point r on the disc, as seen rom the inertial rame, can be expressed by Eq. (2.42). Decomposing the expression into multiple terms, the Razi acceleration, i expressed in the local body coordinates o,is written as a Razi D!! r (2.49) The cross product o two angular velocities!! is the convective rate o change o! due to the angular velocity o in,!. Thereore, the Razi acceleration can be described as an inertial eect caused by the change o the local angular velocity vector direction. This is separable rom the tangential direction which is due to the change o an angular velocity magnitude, hence producing acceleration in the direction which is tangent to the rotation curve. To examine the direction o action o the Razi acceleration, we visualize the resulting vectors in Fig ssuming the position vector o the point o interest r is constant in. Since the angular velocity o in can be decomposed using the addition rule, i.e.! D!!, the Razi acceleration is written as The total acceleration, thus, can be expressed as d a Razi D!! r (2.50) d r D!! r!.! r/ (2.51) In Fig. 2.6, the total acceleration vector o a body point in is decomposed into the centripetal acceleration vector!.! r/ and the Razi acceleration vector

20 50.S.M. Harithuddin et al. a y y ψ ω x y ω z O θ r(t) x x z z ody coordinate rame spinning in shat coordinate rame. The -rame, in turn, is rotating in inertial rame. b r(t) c y y z O θ z ϕ O x z z x r(t) y -rame rotating in -rame with ^ =θ. -rame rotating in -rame with ^ ω i ω =ϕ. i Fig. 2.5 Describing compound rotation motion using three coordinate rames system!! r. Using the disc to represent the plane o rotation o the -rame, we can see the resultant vector o the Razi acceleration is always in the out-o-plane direction. This can be tested by inding the direction o the cross products in the Razi term using the right-hand rule. part rom the magnitudes o!,!, and r, the magnitude o the Razi acceleration vector depends on the angle between the two angular velocity vectors,. Figures 2.7, 2.8, 2.9, 2.10, 2.11, 2.12 show the magnitudes o Razi

21 2 On the Razi cceleration 51 instantaneous axis o rotation ω ω ω O r = ω ( ω r) acentripetal = ( ω ω ) r arazi Fig. 2.6 Razi acceleration vector a centripetal m/s 2 10 a Razi ψ (deg) Fig. 2.7! D P D 5 rad=s;! D P D 1 rad=s and centripetal acceleration experienced by the body point r in the example. The plots show the eect o to the magnitude o Razi acceleration or dierent combinations o angular velocities. Since most readers are more amiliar with the centripetal acceleration eect and its inluence on bodies under rotation, we compare its magnitude with the magnitude o Razi acceleration.

22 52.S.M. Harithuddin et al a centripetal m/s 2 10 a Razi ψ (deg) Fig. 2.8! D P D 4 rad=s;! D P D 2 rad=s a centripetal m/s 2 10 a Razi ψ (deg) Fig. 2.9! D P D 3 rad=s;! D P D 3 rad=s 2.5 onclusion y revising and extending the classical Euler vector derivative transormation ormula, we update the interpretation o the Razi acceleration and reveal its application in rigid body motion. The Razi acceleration is revealed as one o the inertial eects, apart rom the classical oriolis, centripetal, and tangential accelerations, aecting rigid bodies in compound rotation motion. It is shown to appear as a result o the convective rate o change o a body-rame s angular velocity np in nested rotation. Razi s mathematical expression, n0! i i 1 n! i n r, can only

23 2 On the Razi cceleration a centripetal m/s 2 10 a Razi ψ (deg) Fig. 2.10! D P D 2 rad=s;! D P D 4 rad=s a centripetal m/s 2 10 a Razi ψ (deg) Fig. 2.11! D P D 1 rad=s;! D P D 5 rad=s be discovered by using multiple coordinate rames in analyzing the kinematics o complexly rotating bodies. practical example is illustrated to show the typical direction o action o Razi acceleration. Its magnitude is compared to the more widely acknowledged centripetal acceleration to show that the Razi acceleration can have a signiicant impact to the structures o a body in compound rotation.

24 54.S.M. Harithuddin et al a centripetal m/s ψ (deg) Fig. 2.12! D P D 5 rad=s;! D P D 0 rad=s ppendix 1: lternative Proo o Derivative Transormation Formula ngular velocity can be deined in terms o time-derivative o rotation matrix. onsider the two coordinate rames system as shown in Fig The angular velocity o in relative to G, G! is written as ollows G G! D G PR G R T G! D G R T G PR i expressed in G i expressed in (2.52) To prove Eq. (2.52), we start with the relation between a vector expressed in two arbitrary coordinate rames G and. Finding the derivative o the G-vector rom G, we get G r D G R r (2.53) G G Pr D G PR r D G PR G R T G r D G G! G r (2.54) Here, it is assumed that the vector r is constant in. I it is not constant, we have to include the simple derivative o the vector in and rotate it to G G G Pr D G PR r G R Pr D G G! G r G Pr (2.55) The angular velocities between two arbitrary rames are equal and oppositely directed given that both are expressed in the same rame G! D! G (2.56)

25 2 On the Razi cceleration 55 Using this angular velocity property, rearranging Eq. (2.54) gives G Pr D G G Pr G! G G r (2.57) ppendix 2: Proo o the Kinematic hain Rule or ngular cceleration Vectors To prove Eq. (2.29), we start with a composition o n angular velocity vectors which are all expressed in an arbitrary rame g. g 0! n D g i 1! i D g 0! 1 g 1! 2 g n 1! n D g 1 R 1 0! 1 g 2 R 2 1! 2 g R n.n 1/! n (2.58) We use the technique o dierentiating rom another rame in ppendix 1. Dierentiating the angular velocities rom an arbitrary rame g gives g d g i 1! i D g d g R 1 1 0! 1 g R 2 2 1! 2 g R n n n.n 1/! n D. g 1 R g 1 PR 1 0! 1 /. g 2 R g 2 PR 2 1! 2 /. g n R n.n 1/ n g n PR n.n 1/! n/ D. g 1 R g PR g 1 R1 T g 0! 1/. g 2 R g PR g 2 R2 T g 1! 2/. g n R n.n 1/ n g PR g n Rn T g.n 1/! n/ D. g 0 1 g g! 1 g 0! 1/. g 1 2/ g g! 2 g 1! 2/ g g..n 1/ n g g! n.n 1/! n/ (2.59) The whole expression can be transormed to any arbitrary rame by applying the rotation matrix rom g to, R g.!! g d i 1! i D g d g R g i 1! i D D g d g R g i 1! i! i 1 i g! i i 1! i (2.60)

26 56.S.M. Harithuddin et al. Notations In this article, the ollowing symbols and notations are being used. Lowercase bold letters are used to indicate a vector. Dot accents on a vector indicate that is a timederivative o the vector. r, v (or Pr), and a (or Rr) are the position, velocity, and acceleration vectors! and (or P!) are angular velocity and angular acceleration vectors. apital letter,,, and G are used to denote a reerence rame. In examples where only and G are used, the ormer indicates a rotating, body coordinate rame and the latter indicates the global-ixed, inertial body rame. When a reerence rame is introduced, its origin point and three basis vectors are indicated. For example:.oa 1 a 2 a 3 / G.OXYZ/.oO{ O O k/ apital letters R is reserved or rotation matrix and transormation matrix. The right subscript on a rotation matrix indicates its departure rame, and the let superscript indicates its destination rame. For example: R D rotation matrix rom rame to rame Let superscript is used to denote the coordinate rame in which the vector is expressed. For example, i a vector r is expressed in an arbitrary coordinate rame which has the basis vector.o{; O ; O k/, it is written as r D r 1 O{ r 2 O r 3k O D vector r expressed in rame Let subscript is used to denote the coordinate rame rom which the vector is dierentiated. For example, V Pr D d r D vector r expressed in rame and dierentiated in rame Rr D d d r D vector r expressed in rame and dierentiated twice in rame

27 2 On the Razi cceleration 57 Rr D d d r D vector r expressed in rame and dierentiated in rame and dierentiated again in rame Right subscript is used to denote the coordinate rame to which the vector is reerred, and let subscript is the coordinate rame to which it is related. For example, the angular velocity o rame with respect to rame is written as! D angular velocity vector o with respect to I the coordinate rame in which the angular velocity is expressed is speciied! D angular velocity vector o with respect to expressed in 2.6 Key Symbols a DRr cceleration vector v DPr Velocity vector r Displacement vector ngular acceleration vector! ngular velocity vector F Force vector m Mass scalar n Number o elements in the set o noninertial coordinate rames i ith element in a set ; g Elements in the set o noninertial coordinate rames I; O J; O KO Orthonormal unit vectors o an inertial rame O{; O ; k O Orthonormal unit vectors o a noninertial rame O Origin point o a coordinate rame P ngular speed o body (spin) P ngular speed o body (precession) Nutation angle, angle between spin vector and precession vector R Rotation transormation matrix rbitrary vector r Vector r expressed in -rame Vector Pr expressed in -rame and dierentiated in -rame Pr

28 58.S.M. Harithuddin et al. Pr R Vector Rr expressed in -rame, irst-dierentiated in -rame, and second-dierentiated rom -rame Rotation transormation matrix rom -rame to -rame Reerences aruh H (1999) nalytical dynamics. W/McGraw-Hill, oston Jazar RN (2012) Derivative and coordinate rames Nonlinear Eng 1(1):25 34 Jazar RN (2011) dvanced dynamics: rigid body, multibody, and aerospace applications. Wiley, Hoboken, pp Kane TR, Likins PW, Levinson D (1983) Spacecrat dynamics. McGraw-Hill, New York, pp 1 90 Kasdin NJ, Paley D (2011) Engineering dynamics: a comprehensive introduction. Princeton University Press, Princeton Koetsier T (2007) Euler and kinematics. Stud Hist Philos Math 5: Rao (2006) Dynamics o particles and rigid bodies: a systematic approach. ambridge University Press, ambridge Tenenbaum R (2004) Fundamentals o applied dynamics. Springer, New York Van Der Ha J, Shuster MD (2009) tutorial on vectors and attitude (Focus on Education). ontrol Syst IEEE 29(2): Zipel PH (2000) Modeling and simulation o aerospace vehicle dynamics. I, Virginia

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