ࡱ> !BbRbb cHcJcVcXcZc\c^cbcdcpcrcvcxczccchhhiii@jDjTjj"kJkLkXkZk\k^k`kdkfkrktkvkxkzkkk5mHmH jpEHUj: 5CJUVmH0JmH0J j0JU55>*. when only a few of the system s forces need to be solved forLagrange s Equations s based on vector products. ofeach subject methods The next step is to develop a simple three dimensional model and slowly increase its complexity by adding jointA Brief Synopsis of Kanes Method This paper is the result of my interest in multi-body dynamics and desire to increase my knowledge on the topic. My initial intentions were to teach myself Kanes method (originally called Lagrange form of dAlemberts principle) for developing dynamical equations of motion and then prepare a lecture for class. As I started learning and applying the technique I realized that a small lecture would not be sufficient to teach the method and have time to get to some of the more interesting applications. In this paper I present a proof of Kanes Method using the Principle of Virtual Work, a general procedure for developing equations of motion, and a few applications. Essentially all methods for obtaining equations of motion are equivalent. However, the ease of use of the various methods differs; some are more suited for multibody dynamics than others. The Newton-Euler method is comprehensive in that a complete solution for all the forces and kinematic variables are obtained, but it is inefficient. Applying the Newton-Euler method requires that force and moment balances be applied for each body taking in consideration every interactive and constraint force. Using the Newton-Euler method results in a set of equations that is as numerous as the number of variables. Therefore, the method is inefficient, especially for large multibody systems. The Lagrange technique provides a method for disregarding all interactive and constraint forces that do not perform work. The major disadvantage of this method is the need to differentiate scalar energy functions (kinetic and potential energy). This is not much of a problem for small multibody systems, but becomes an efficiency problem for large multibody systems. Kanes method offers the advantages of both the Newton-Euler and Lagrange methods without the disadvantages. With the use of generalized forces the need for examining interactive and constraint forces between bodies is eliminated. Since Kanes method does not employ the use of energy functions, differentiating is not a problem. The differentiating required to compute velocities and accelerations can obtained through the use of algorithms based on vector products. So Kanes method provides an elegant means to develop the dynamics equations for multibody systems that lends itself to automated numerical computation. (Huston 1990) Derivation of Kanes Equations Using the Principle of Virtual Work Consider an open-chain multibody system with N rigid bodies interconnected subjected to external and constraint forces. The external forces can be transformed into an equivalent force and torque ( EMBED Equation.3  and  EMBED Equation.3 ) passing through  EMBED Equation.3 , the mass center of the body k (k = 1,2N). Similar to the external forces, the constraint forces may be written as  EMBED Equation.3 and  EMBED Equation.3 . Using dAlemberts principle for the force equilibrium of body k, the following is obtained:  EMBED Equation.3  where  EMBED Equation.3 is the inertia force of body k. The concept of virtual work may be described as follows for a system of N particles with 3N degrees of freedom. The systems configuration can be described using qr (r = 1,2,3N) generalized coordinates with force components F1, F2,, F3N applied to the particles along the corresponding generalized coordinates. The virtual work is then defined as:  EMBED Equation.3  Where  EMBED Equation.3  is the resultant force acting on the ith particle and  EMBED Equation.3  is the position vector of the particle in the inertial reference frame.  EMBED Equation.3  is the virtual displacement, which is imaginary in the sense that it is assumed to occur without the passage of time. Now applying the concept of virtual work to our multibody system considering only the work due to the forces on the system we obtain:  EMBED Equation.3  (k = 1,2,,N) The constraints that are commonly encountered are known as workless constraints so  EMBED Equation.3  Which simplifies the virtual work equation to:  EMBED Equation.3  (k = 1,2,,N) or  EMBED Equation.3  (r = 1,2,,3N) * The positions vector may also be written as:  EMBED Equation.3  so  EMBED Equation.3   EMBED Equation.3  Taking the partial derivative of  EMBED Equation.3 with respect to  EMBED Equation.3  yields  EMBED Equation.3  or  EMBED Equation.3  Since the virtual displacement  EMBED Equation.3 is arbitrary without violating the constraints we can write * as:  EMBED Equation.3  where EMBED Equation.3  and  EMBED Equation.3  are the generalized active and inertia forces respectively and are defined as follows:  EMBED Equation.3  and  EMBED Equation.3  In a similar fashion it can be shown using virtual work that the moments can be written as:  EMBED Equation.3  where EMBED Equation.3  and  EMBED Equation.3  are the generalized active and inertia moments respectively and are defined as follows:  EMBED Equation.3  and  EMBED Equation.3  By superposition of the force and moment equations we arrive at Kanes equations:  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  (Adapted from Amirouche 1992) Notation Notation that will be used throughout the rest of the paper:  EMBED Equation.3  - generalized coordinate  EMBED Equation.3  - generalized speed, typically equal to  EMBED Equation.3   EMBED Equation.3  - velocity of P with respect to the Newtonian (inertial) reference frame  EMBED Equation.3  - velocity of the center of mass of body A  EMBED Equation.3  - partial velocity, equal to  EMBED Equation.3  Cn or Sn Cos(n) or Sin(n) General Procedure for using Kanes Method 1) Label important points (important points being defined as all center of mass locations, and locations of applied forces with the exception of conservative constraint forces). 2) Select generalized coordinates (qr) and generalized speeds (ur), then generate expressions for angular velocity and acceleration of all bodies and velocity and acceleration of the important points. 3) Construct a partial velocity table of the form: Generalized Speeds ( ur ) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 1r = 2. . .. . .. . .. . .. . . 4) Fr + Fr* = 0 where the generalized active force, Fr, is defined as:  EMBED Equation.3  and the generalized inertia force, Fr*, is defined as:  EMBED Equation.3  5) Which can then be written in the form:  EMBED Equation.3  Applications In this section of the paper I will step through some of the example problems I did to familiarize myself with Kanes method. I started with simple two-dimensional problems that I could easily confirm the equations of motion using other methods. As I felt more comfortable with Kanes method I moved onto 3D problems involving non-holonomic constraints. Presented here are a few of those problems, which demonstrate the subtleties of Kanes method. Problem 1 The first problem is a spring-mass-pendulum problem with frictionless sliding. This was problem is good for showing the general procedure of Kanes method.     Step 1) Define important points as the center of mass of A and particle P. Step 2) Select generalized coordinates as shown in the figure and generate velocity and acceleration expressions for the important points.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  Step 3) Construct a partial velocity table.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 1 EMBED Equation.3  EMBED Equation.3 r = 20 EMBED Equation.3  Step 4)  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   Step 5) Assemble  EMBED Equation.3   EMBED Equation.3  Problem 2 This problem is useful in that it shows how auxiliary generalized speeds can be introduced to bring constraint forces and torques into evidence. In this case we will introduce u3 to find an expression for Tc (constraint torque about  EMBED Equation.3 ). The joints at O and P are revolute. Body A and B are uniform rods with length 4L and 2L respectively. Body A has two times the mass of body B. Define the direction cosine matrices:  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 C10S1 EMBED Equation.3 010 EMBED Equation.3 -S10C1  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 C10S1 EMBED Equation.3 010 EMBED Equation.3 -S10C1  Step 1) Choose important points: Center of Mass of bodies A and B, and point P. Step 2) Select generalized coordinates as shown in the figure (plus auxiliary generalized coordinate u3) and generate velocity and acceleration expressions for the important points. The prime in the equations below indicates that the specified quantities contain the auxiliary generalized coordinate. Body A  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Point P  EMBED Equation.3  Body B  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 3) Construct a partial velocity table.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 1 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 200 EMBED Equation.3  EMBED Equation.3 r = 3 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  Step 4)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 5) Assemble  EMBED Equation.3   EMBED Equation.3  Problem 3 This non-holonomic system demonstrates the usefulness of choosing the generalized speeds to be something other than the first derivative of the generalized coordinates.  Define the direction cosine matrices:  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 C10S1 EMBED Equation.3 010 EMBED Equation.3 -S10C1  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 001 EMBED Equation.3 S2C20 EMBED Equation.3 -C1S20  Step 1) Choose important points: Center of Mass of bodies C, point  EMBED Equation.3 and EMBED Equation.3 (on body C). Step 2) Select generalized coordinates as shown in the figure and generate velocity and acceleration expressions for the important points.  EMBED Equation.3   EMBED Equation.3  Case A Choose  EMBED Equation.3  (i = 1,2,,5) Imposing the rolling without slip constraint we obtain:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Which gives us constraint relations on  EMBED Equation.3 and EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Pluging in the constraint relations allows us to develop the non-holonomic expression for the velocity of the center of mass of body C:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 3) Construct a partial velocity table.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 1 EMBED Equation.3  EMBED Equation.3 r = 2 EMBED Equation.3  EMBED Equation.3 r = 3 EMBED Equation.3  EMBED Equation.3  Step 4)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 5) Assemble  EMBED Equation.3   EMBED Equation.3  Notice that the inertia matrix, M, is not symmetric. However if the definitions of the generalized speeds are choosen wisely then it is possible to produce a inertia matrix that is symmetric positive definite. Case B - Choose u1, u2, u3 such that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  From which we can derive the expressions for u1, u2, u3:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 3) Construct a partial velocity table.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 r = 1 EMBED Equation.3  EMBED Equation.3 r = 2 EMBED Equation.3  EMBED Equation.3 r = 3 EMBED Equation.3  EMBED Equation.3 Step 4)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 5) Assemble  EMBED Equation.3   EMBED Equation.3  The inertia matrix is now symmetric positive definite. Conclusions Although the applications of Kanes method to multibody systems in this paper has been light, I have shown that is a powerful technique. It offers the advantages of both the Newton-Euler and Lagrange techniques in that it can be comprehensive and efficient. As part of learning Kanes method I have taught myself how to use Autolev, a dynamics software package that uses Kanes method to develop equations of motion and perform simulations. In the future I plan on using Autolev to develop equations of motion for a biped. So far I have successfully built simple models of a biped in the frontal plane to compare to published works. References Amirouche, Farid M.L. Computational Methods in Multibody Dynamics. New Jersey: Prentice Hall, 1992. Anderson, Kurt. Lecture Notes from Multibody Dynamics. Rensselaer Polytechnic Institute. Critchley, James. Personal Interviews. February-May 1999. Huston, Ronald L. Multibody Dynamics. Boston: Butterworth-Heinemann, 1990. Kane, Thomas, and David Levinson. Dynamics: Theory and Applications. New York: McGraw-Hill, 1985.  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F 2" ="mL 2 3()(2u 2 "s 2 c 2 u 12 )+mL 2 ("2u 1 "2u 2 c 2 +s 2 (u 12 +u 22 ))c 2 +s 2 (4u 12 "2u 2 s 2 "u 22 )[] FMicrosoft Equation 3.0 DS Equation Equation.39q4<  .1  0&/i & MathType-`:`SyCompObj1tiveFyfObjInfo7tiveF{OlePres000iveF|2Equation Native Fmbol a!w*wgw a -2 [Symbol {!w*wgw { -2 /] Times New Roman*wgw b -2 .)2 ^&22 %( 2 ")2 442 ( 2 42 $)2 d2 2 (2 63 Times New Roman*wgw | -2 S.2 2 (-12 a*22 )2 2 '12 -%22 !2 2 !22 22 2 2 22 12 2 2 }22 g22 %2 2 12 O 22  2 2  12 "122 3 Times New Roman*wgw c -2 -s2 j,u2 )c 2 H(u2 'u2 t$c 2  u2 Js2 u 2 u2 mL2 c 2 Nu2 uL2  c 2 f s2 $ uL2 TmL2 FFMT Extra }!w*wgw } -2 ,&L2 &L2 d &Symbol d!w*wgw d -2 P++L2 T#+L2 -2 -L2 -L2 > +2 UL2 L2 K2 L2 UtL2 t2 KtL2 tL2 -2 j=L Symbol ~!w*wgw ~ -2 j*L & "Systemw f  - F 3" ="mL 2 3()(2u 1 s 2 c 2 +2u 1 u 2 c 22 )"mL 2 4(4u 12 "2u 2 s 2 "u 22 )+c 2 (2u 1 u 2 c 2 +2u 1 s 2 )[] FMicrosoft Equation 3.0 DS EqN@w<B*FP/FEo7' P56uε]X%O.{8HgVV͎eƧmH~q\QLt 9T99D]yNN;yUy孊r)UީpyV+󎭃vUM-y{6|YiUi^2.:fw2?-\(!+_ly031[pD*1_IyypR" \_+~Za٫gv?mYLD꙯aitz,'\Kg~Izo&8 r MR6R|s)m wKbFR78OG.3fa=o4/)a6:>XyHy~Io,JI`lקVuation Equation.39qH "8mL 2 3"12mL 2 "4mL 2 3s 22 "4mL 2 c 2 0"mL 2 c 2 "mL 2 3"mL 2 0"mOlePres000FEquation Native  _986570462JF"fOlenfo Q:  C  .1   `J& J & MathType-II\I$ I| L IF"I$#S& # c'Symbol !w*wgw -2 B I2 I2 bI2 I2 I2 I2 I2 I2 B !2 !2 b!2 !2 !2 !2 !2 !2  eA-2  8+2  2+2  ++2 `D+2 `<-2 `6+2 `_3+2 `*-2 jE+2 B?+2 j7+2 D3+2 *+2 `f =2 d2 d2 d2  d2 <d2 P2 P2 P2  P2 <P2 B 2 2 b2 2 2 2 2 2 B @2 @2 b@2 @2 @2 @2 @2 @2   -2  &-2 `-2 `-2 `-2 -2  -2 -2 - Times New Roman*wgw -2 k .G22 w G22 k E22 w v@22 w -?12 k =22 w =22 k <22 w 712 w 622 k 122 w 112 k F022 k *22 w *22 w )22 w W(12 y&22 F22 F22 C22 C12 &A22 ?22 ;22 :12 f522 P522 p222 A212 ^022 /22 -22 )22 (22 v'22 G'12 %22 fH12 =G22 D12 =22 <12 E;22 :22 {622 5522 U222 &212 C022 -22 )22 k(22 /'22 %12 =$22 22 12 w .22 k 22 w Q22 k  22 w ) 22 w 22 yC22 22 22  22  22 22 22 22 22 22  22 2Times New Roman*wgw -2  [B162  922  ,162 C P%32 % $262 `G)2 `WB462 `A(2 ` 6)62 `1(2 $362 H)2 8C862 B(2 v>)62 822 5)62 0(2 .462 F.(2 B#362  12  O462 C H32 `062 32 062 42  362 > 42 122 U32  82Times New Roman*wgw -2  AFu2  CmL2  ?u2  o>uL2  =c2  :mL2  M7c 2  3mglc2  0u2  i.mL2  )*c2  (uL2  'u2 % $mL2 `Fu2 `CuL2 `@s2 ` >mL2 `E;c 2 `7mgls2 `y4u2 `1ug2 `/c2 `u.sg2 `,mL2 `;)cg2 ` (sL2 `&ug2 n#mL2 Gcg2 FsL2 Csg 2 h@mgl2 <ug2 ;ug2 :cg2 \9sg2 ^4ug2 h1ug2 /sg2 +mL2 (cg2 'sL2 X&ug2 (%uL2 `"mL2 TL2 `uL2 uL2  sL2  mL2  sL2  8 mL2  pcL2  @sL2 % fmL2 `mL2 nmL2 `" cL2 `mL2 cL2 mL2 sL2  mL2 2mL2 mL Times New Roman*wgw -2 cLMT Extra !w*wgw -2 N &L2 5&L & "Systemwbf"  -L 2 3s 2 c 2 "mL 2 s 2 4mL 2 s 2 1[]2u 1 2u 2 T c []=mL 2 3u 1 u 2 s 2 c 2 +mL 2 (4s 2 (u 12 +u 2 ) 2 +2s 2 c 2 u 1 u 2 )+mgl(8s 1 +s 2 c 1 )mL 2 3u 12 s 2 c 2 "mL 2 s 2 c 2 (u 12 +u 22 )+mgls 1 c 2 "mL 2 s 2 (4u 12 +u 22 )2mL 2 3u 1 u 2 c 22 +16mL 2 u 12 +mglc 2 c 1 +2mL 2 c 22 u 1 u 2 "16mL 2 u 22 []CompObj5FfObjInfoOlePres000~Equation Native ) FMicrosoft Equation 3.0 DS Equation Equation.39qD  .1  &@f & MathType Times New Roman*wgwI 9 -2 `FPy & "System  - X P FMicrosoft Equation 3.0 DS Equation Equation.39q4  .1  _986570430  '  FPE Oleion Native)*,f-h/0 56789@CompObj8GHIJKLMNOPFf`ObjInfogkijlnmporwvyxzOlePres000iveFEquation Native F-_986568882tiveF=P^?fOlenfo3 F&@ & MathType Times New Roman*wgw -2 FPTimes New Roman*wgw -2 m~ & "Systemw/f  -0 2PCompObjfObjInfo.OlePres000vEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q(4<  .1   &  & MathType0 Times New Roman*wgw H -2 C2  B2 f B2 A2 A2 N2 C2 GNSymbol !w*wgw -2  w2 %w2 w2 wMT Extra I!w*wgw I -2  r2 r2 r2 ZrSymbol !w*wgw -2 ' +2 +2 = & "SystemwXf  -@ , N "  C =, N "  A +, A "  B +, B "  C FMicrosoft Equation 3.0 DS Equation Equation.39q_986570896tiveFZ@\fOlenfo3 FCompObjNativefObjInfo3FOlePres000iveFEquation Native F_986571113tiveF{`|fOlenfo3 F    !"#$%&'()*+,-./01234678;>?@ABCDEFGHIJKLMNOPQRSTUWXYZ[^abcdefghijklmnopqrstuvwyz{|}   .1  @ & A & MathType` Times New Roman*wgw" -2 @ 32 @O 32 @12 @Y22 @y22 @*1Times New Roman*wgwC -2 m 2 m"2 Times New Roman*wgw" -2  b2  q2 b2 q2 a2 rqMT Extra !w*wgwC -2  &2 &2 &Symbol" !w*wgw" -2 `+2 h-2 := & "Systemwf  - =2q 1 2a  2 "2q 2 2b  1 +2q 3 2b  3 FMicrosoft Equation 3.0 DS Equation Equation.39q{| > CompObj1tiveF fObjInfo7tiveFOlePres000iveFEquation Native FT.1  @`&  & MathType` Times New Roman*wgw2 [ -2 i2 iTimes New Roman*wgw w -2 q2 .uMT Extra \!w*wgw2 \ -2 mJ&2 mn&Symbol x!w*wgw x -2 = & "Systemwfc  -88 2u i =2q i FMicrosoft Equation 3.0 DS Equation Equation.39q_986570361  '  FGΨ Oleion Native)*,f-h/0 56789@CompObj8GHIJKLMNOPFf`ObjInfogkijlnmporwvyxzOlePres000iveF Equation Native F5_986568989tiveFp(ҚӚfOlenfo3 F9 f .1  &* & MathTypePTimes New Roman*wgwd & -2 02 )2 n 2 L ( Times New Roman*wgw -2 @]22 ~Symbold '!w*wgwd ' -2 =2  -2 J 2 z+2 =`Symbol !w*wgw -2 *Times New Roman*wgwd ( -2  b2  R2 fv2  v Times New Roman*wgw -2 4V C2 4qN2 4;C2 4N2 4P2 4GNSymbold )!w*wgwd ) -2 "wMT Extra !w*wgw -2 r2 r2 *r & "Systemw fB  - , N 2v  2P =, N 2v  C " +, N "  C ("R2b  2 )=0 FMicrosoft Equation 3.0 DS Equation Equation.39q$  .1  @!&!@ & MathType`CompObj1tiveF:fObjInfo7tiveF<OlePres000iveF=Equation Native FV]Times New Roman*wgwh c -2 5 02 7)2 N/2 (2 )2 (2 )]2 N]2 8 (]2 "]2 H]2 H[] Times New Roman*wgw % -2 22 ]22 12 n52 32 4Symbolh d!w*wgwh d -2 =2 -2 2 X+2 $+2  2 Z+2 +2 :=Times New Roman*wgw & -2 b2 R2 b2  R2 n2 u2 n2 u Times New Roman*wgwh e -2 C2 KB2 B2 N2  B2  NSymbol '!w*wgw ' -2 w2 Vw2  wMT Extra f!w*wgwh f -2 Kr2 r2 r & "Systemwf!  -A@ =[u 4 2n  3 +u 5 2n  1 +(, N "  B R2b  2 )]+(, N "  B +, B "  C )("R2b  2 )=0 FMicrosoft Equation 3.0 DS Equation Equation.39q_986568999iveF00Oleion Native F\CompObj2tiveF]fObjInfo6F_OlePres000iveF`Equation Native Fx[_986569011tiveF I)*fOlenfo3 F~ "  .1  &@ & MathType`Times New Roman*wgw . -2 02 N2 2 2 k)2 Nc2 (2 2  Times New Roman*wgw% O -2 12 32 112  52 i32 .42 22 (12 52 `32 %4Symbol /!w*wgw / -2 =2 +2 9+2 C=2  2 -2 0+2 :=Times New Roman*wgw% P -2 ;b2 QR2 u2 n2 Gu2 n2 cu2 b2 3 R2 n2 >u2 n2 Zu Times New Roman*wgw 0 -2 + C2 _ BSymbol% Q!w*wgw% Q -2  wMT Extra 1!w*wgw 1 -2 e r & "Systemwf  -?@ =u 4 2n  3 +u 5 2n  1 "(, B "  C R2b  2 )=u 4 2n  3 +u 5 2n  1 +u 3 R2b  1 =0 FMicrosoft Equation 3.0 DS EqCompObj5FfObjInfoOlePres000Equation Native uation Equation.39q+{b  .1  @'&@' & MathType`Times New Roman*wgw  -2 x&02 z$)2 nb#2 ") 2 cos(2 no2 )o 2 (sin(2 nTs2 nzs2 ni s2 ns2 ns Times New Roman*wgw   -2 #32 "12  12 R12 ^32 12 52 32 42  32  32 (12 52 `32 %4Symbol !w*wgw  -2 R%=2 +2 +2 +2  =2 +2 0+2 :=Times New Roman*wgw   -2 .#n2 n!q2 fn2 q2 R2 u2  n2 u2 Fn2  u2 2 a2 H R2 u2 n2 >u2 n2 Zu & "Systemwmf  -@ =u 4 2n  3 +u 5 2n  1 +u 3 R2a  3 =u 4 2n  3 +u 5 2n  1 +u 3 R(sin(q 1 )2n  1 +cos(q 1 )2n  3 )=0_986571515iveF|RTOleion Native FCompObj2tiveFfObjInfo3F FMicrosoft Equation 3.0 DS Equation Equation.39qW  .1   & & MathTypeP Times New Roman*wgw  -2 OlePres000iveFEquation Native F6_986571533tiveF`u`ufOlenfo3 F4 Times New Roman*wgwI y -2 .u & "Systemw f  -8 u 4 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj5FfObjInfoOlePres000Equation Native 6{  .1  @&` & MathType` Times New Roman*wgw -2 5yTimes New Roman*wgw! -2 .u & "Systemwf  -` u 5 FMicrosoft Equation 3.0 DS Equation Equation.39q { W .1  @ _986569022tiveFӘpZfOlenfo3 FCompObjNativefObjInfo2FOlePres000ive FEquation Native+. Fw@_986569033tiveJMNFPÛśGfOlenfo5tivejkn F&  & MathType`Times New Roman*wgw P -2 2 ) 2 tcos( Times New Roman*wgw -2 12 32 4Times New Roman*wgw Q -2 q2 TR2 u2 .uSymbol !w*wgw -2 = & "Systemwmf  -[@ u 4 =u 3 Rcos(q 1 ) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoOlePres000Equation Native w { W .1  @ &`  & MathType`Times New Roman*wgw N -2 ) 2 dsin( Times New Roman*wgw -2 \12 32 5Times New Roman*wgw O -2 q2 JR2 u2 .uSymbol !w*wgw -2 = & "Systemwf  -[@ u 5 =u 3 Rsin(q 1 ) FMicrosoft Equation 3.0 DS Eq_986572019  '  F󛨑` Oleion Native)*,f-h/0 56789@CompObj8GHIJKLMNOPFf`ObjInfogkijlnmporwvyxzuation Equation.39q ~ .1  &J & MathType`Times New Roman*wgw } -2 )y2 ny2  (y2  y2 cy Times New Roman*wgw OlePres000iveF6Equation Native F_986569470F O" O"fOlenfo  -2 @822 @ 12 @ 52 @32 @4Times New Roman*wgw ~ -2 yb2 R2 Q n2 u2 /n2 u2 v Times New Roman*wgw ! -2 4xB2 4h N2 4C2 4GNSymbol !w*wgw  -2 o2  +2 +2 =`Symbol "!w*wgw " -2 *Symbol !w*wgw -2 +wMT Extra #!w*wgw # -2 r2 6r & "Systemwf;  - , N 2v  C " =u 4 2n  3 +u 5 2n  1 +(, N "  B R2b  2 ) FMicrosoft Equation 3.0 DS Equation Equation.39qd  CompObjfObjInfoOlePres000iveVEquation Native F      !"#$%&'()*+,-.012347:;<=>?@ABCDEFGHIJKLMNOPQSTUVWX[^_`abcdefghijklnopsvwxyz{|}~     XiZ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVW6Y][\^a_`bcdhefgjpklmnorqwstuvx|yz{}~.1   &@ & MathType`Times New Roman*wgw m -2 j)2 NJ2 (2 )2 N2  2  (2 "2  Times New Roman*wgw? 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FZ @CompObj3tiveJMN]`F[ fObjInfo5tivejknF] OlePres000  '  _a^  Equation Native)*,f-h/0 56789i B@_986732040GHIJKLMNOPdF@"cd^_`Oledegkijlnmpor wvyxzk &`! & MathTypeP Times New Roman*wgw -2 @1yTimes New Roman*wgwv - -2 mTimes New Roman*wgw -2 Pb2 fRSymbolv .!w*wgwv . -2 :- & "SystemwHf  -&t "R2b  1 FMicrosoft Equation 3.0 DS Equation Equation.39qd  CompObj00ive cfFl fObjInfoNative+.Fn @OlePres000iveJMNegFo Equation Nativejkn F .1  &@ & MathType`%Symbol !w*wgw -2 P[%Symbol !w*wgw -2 ] Symbol !w*wgw -2 &*Symbol !w*wgw -2 52  +2 -2 != Times New Roman*wgw -2 C2 Cr2 N2 c2 $rTimes New Roman*wgw -2 v2  b2  s2  b2 c2 yg2 m2 FFTimes New Roman*wgw -2 ~2 s)2 Na 2 N] 2 W(2 |~ Times New Roman*wgw -2  32  22  22 2MT Extra !w*wgw -2 r & "Systemwf}  - 2F r ="m c g(c 2 2b  2 +s 2 2b  3 )[]", N 2v  rC" FMicrosoft Equation 3.0 DS Equation Equation.39q\ . .1  `@_986724867tjF U U,Oleion Native  CompObj7ilF fObjInfo OlePres000kmF Equation Native  C_986724989\pFp󧟨z"fOlenfo  & & MathTypePTimes New Roman*wgw4 -2 302 |~ Times New Roman*wgwO F -2  1ySymbol4 !w*wgw4 -2 =Times New Roman*wgwO G -2 FFy & "Systemw=fV  -'w 2F 1 =0 FMicrosoft Equation 3.0 DS Equation Equation.39q k CompObj6tiveorF fObjInfo3F OlePres000iveqs Equation Native F j.1   &( & MathType` Times New Roman*wgww  -2 c22 $2Times New Roman*wgw  -2 |~yTimes New Roman*wgww  - 2 Rgse2 cmy2 FFg Times New Roman*wgw  -2 ~cySymbolw !w*wgww  -2 +=g & "Systemwfh  -Nq 2F 2 =m c Rgs 2 FMicrosoft Equation 3.0 DS Eq_986724968  '  vF ~ Oleion Native)*,f-h/0 56789 @CompObj1GHIJKLMNOPuxF f`ObjInfogkijlnmporwvyxz uation Equation.39q\ . .1  @&( & MathType`Times New Roman*wgw " -2 O02 |~y Times New Roman*wgw Q -2 3OlePres000     wy2  Equation Native)-+,./01 w6y89 C@_986715449GHIJKLMNOP|Fp䟨柨^_`Oledefghijklmnop uvx{ Symbol #!w*wgw # -2 =Times New Roman*wgw R -2 FF & "SystemwfJ  -'rTJ 2F 3 =0CompObj5{~F fObjInfo OlePres000} VEquation Native   FMicrosoft Equation 3.0 DS Equation Equation.39q(q  .1   `$& $ & MathType`-YC'Y"`Symbolh !w*wgwh -2 P#*2 P*2 Y*2 z*Symbol< [!w*wgw< [ -2 ` 2 `.-2 `+2 `2 `2 ` 2 `+2 `M2 `j= Symbolh !w*wgwh -2 j* Times New Roman*wgw< \ -2 #C2 "r2 0!N2 C2 %N2 $C2 `C2 Jr2 KN2 xC2 cN2 C2 C2  C2  N2 C2 C2 $rTimes New Roman*wgwh -2 `"v2 `a2 ` m2 `I2 `8I2 `FFTimes New Roman*wgw< ] -2 d"~2 `? )2 d~2 `(2 `<}2 `~{2 v|~MT Extra !w*wgwh -2 j-"r2 j#r2 jr2 jr2 5v2 (5v2 j r2 j0r2 Vv2 (VvSymbol< ^!w*wgw< ^ -2 ` w2 `8w2 ` w2 `a & "Systemwf  -M ? @ A B C D E F G H I J K L M N O P Q R S T U W X Y Z [ \ ] ^ _ ` a b e h i j k l m n o p q r s t u v w x y z { | } ~  %Symbol %!w*wgw % -2 [%Symbol '!w*wgw ' -2 O]y%Symbol &!w*wgw & -2 \[y%Symbol (!w*wgw ( -2 =]y>Symbol '!w*wgw ' -2 s{y>Symbol )!w*wgw ) -2 >}y Times New Roman*wgw ( -2 @Cy2 @ry2 >Ny2 $ryTimes New Roman*wgw * -2 l<by2 ~:uy2 7cy2 \6uy2 ,5uy2 |2sy2 :1uy2 -by2 +uy2 *uy2 'cy2 &uy2 #by2 !cy2 \ uy2 ,uy2 |sy2 uy2 uy2 uy2 by2 by2 0Iy2 P by2  by2 s Iy2 by2 by2 4Iy2 FFySymbol )!w*wgw ) -2 ?wyMT Extra +!w*wgw + -2 @ry2 :&y2 z1&y2 &&y2 F&ySymbol *!w*wgw * -2 _>y2 d9+y2 4+y2 /+y2 )+y2 %+y2 +y2 -y2 4-y2 y2 +y2 A +y2 -y2 j=y Symbol ,!w*wgw , -2 j*y Times New Roman*wgw + -2 $=3y2 N;3y2 u82y2 372y2 51y2 #32y2 11y2 .2y2 ,2y2 v+3y2 (2y2 l'1y2 2$1y2 g"2y2 ,!3y2 1y2 #2y2 2y2 1y2 2y2 3y2 u3y2 332 232  232  222 c122 K122 11Times New Roman*wgw - -2 N<12 ;)12 0(12 N4.12 p-)12 0&(12 N#12 #)12 (12 N712 N12 N 12 NT 12 N12 N12 |~1 & "Systemw'fZ  -8t 2F r" ="I 11 2b  1 2b  1 +I 22 2b  2 2b  2 +I 33 2b  3 2b  3 []"("2u 2 "u 1 u 2 s 2 +u 1 u 3 c 2 )2b  1 +(2u 1 c 2 +u 3 u 2 )2b  2 +(2u 1 s 2 +u 1 u 2 c 2 +2u 3 )2b  3 []{}", N "  rC FMicrosoft Equation 3.0 DS Equation Equation.39qsI   .1  B&`BA & MathType`CompObj* fObjInfo, OlePres000-  Equation Native V #Symbol, A!w*wgw, A -2 [y#Symbol& !w*wgw& -2 >] Symbol, B!w*wgw, B -2 4A*Symbol& !w*wgw& -2 >2 9-2 4-2 .-2 &-2 +2 +2 +2 +2 B +2 e-2 :- Times New Roman*wgw, C -2 4IAC2 @ Ar2 4v?N2 @ucTimes New Roman*wgw&  -2 S@v2 Y=b2 o<R2 :u2 %8b2 ;7R2 5u2 2b2 2R2 0s2 -/u2 ,b2 o*Rc2 (uc2 'uc2 %bc2 #sc2 !Rc2 / uc2 vbc2 PRc2 uc2 uc2 bc2 7uc2 bc2 uc2  cc2  uc2 [ uc2 sc2 juc2 Rc2 ZmcTimes New Roman*wgw, D -2 X@~2 m=2 mk82 m732 m,2 m_%2 m2 ]2 m2 m2 )2 n[( Times New Roman*wgw&  -2 @>22 4;22 @;22 @822 4622 @632 @322 4l122 @V122 4022 @/12 @;-12 @,22 @)22 @(12 @%32 @$22 @Z#22 4!22 @ 12 @.32 @22 @32 @z12 @I32 @22 @@12 @|32 @ 22 @b 22 @ 12 @S22 @(1MT Extra E!w*wgw, E -2 r@r2 w&2 &2 & & "Systemwf  -. "m c "R[(2u 1 s 2 +u 1 u 2 c 2 +2u 3 )2b  1 +2u 2 2b  3 ]+u 1 u 3 Rc 2 2b  3 +u 12 Rc 2 s 2 2b  3 "u 1 u 2 Rc 2 2b  1 "u 12 s 22 R2b  2 "u 32 R2b  2 "u 22 R2b  2 []", N 2v  rC" FMicrosoft Equation 3.0 DS Equation Equation.39qS9  _986731564F RHIOle c CompObjd fObjInfof OlePres000ive Fg >Equation Native+. F I@_986731570tiveJMNFffGfOlenfo5tivejkn F  .1  4&3( & MathType`Symbol !w*wgw -2 X[SymbolR 4!w*wgwR 4 -2 e ]Times New Roman*wgw -2 83)y2 +22 &(y2 )2 [(y2 P)2 (y2 |~ Times New Roman*wgwR 5 -2 23y2 /22 Y.2y2 -12 c)2y2 0(12 E&2y2 %22 93y2 H22 2y2 12 2y2 12 2y2 332 2y2 J 332 Y 2y2  132 52y2 {222 1yTimes New Roman*wgw -2 1u22 .cy2 ~-u22 F,uy2 (s22 n'uy2 %s22 #Ry2 5"m22 euy2 c22 #uy2 u22 #sy2 u22 sy2 ^I22  uy2 v u22  cy2 ^u22 xcy2 I22 FFy Times New Roman*wgwR 6 -2 T#c2MT Extra !w*wgw -2 2&y2 '&22 &y2 &22 &ySymbolR 7!w*wgwR 7 -2 0+22 ^*+y2  !-22 C+y2 +22 $+y2 T +22 -y2 b=2 Symbol !w*wgw -2 n*y & "Systemwf%  -- 2F 1" ="I 22 c 2 (2u 1 c 2 +u 3 u 2 )+I 33 s 2 (2u 1 s 2 +u 1 u 2 c 2 +2u 3 )[]"m c R 2 s 2 (2u 1 s 2 +2u 1 u 2 c 2 +2u 3 ) FMicrosoft Equation 3.0 DS Equation Equation.39q*$  .1  &CompObj00  '   f ObjInfoNative)*,f-h/056789 @OlePres000GHIJKLMNOPF ^`Equation Nativeijlnmpor wvyxz &`&( & MathType`Times New Roman*wgwq -2 %)y2 (y2 ()y2 A(y2 |~y Times New Roman*wgw n -2 '%22 #22 "22 d"12 22 932 12 22 22 {22 532 12  22  22 _ 12 z22 6112 *21Times New Roman*wgwq -2 {$s2 "#c2 !u2 c2 cu2 'u2 )u2 R2 +m2 c2 _u2 #u2 O s2  u2  u2 u2 I2 FF Times New Roman*wgw o -2 LcSymbolq !w*wgwq -2  -2 -2 -2  +2 {-2 -2 j= Symbol p!w*wgw p -2 p*MT Extra !w*wgwq -2 h&2 & & "Systemw f  -< 2F 2" =I 11 ("2u 2 "u 1 u 2 s 2 +u 1 u 3 c 2 )"m c R 2 (2u 2 "u 1 u 3 c 2 "u 12 c 2 s 2 )_986731589ive FPtOleion Native+. F @CompObj3tiveJMNF fObjInfo5tivejknF  FMicrosoft Equation 3.0 DS Equation Equation.39q) Z .1  @%&%( & MathType`Symbol? !w*wgw? -2 b[Symbol. !w*wgw. OlePres000  '    Equation Native)*,f-h/0 56789 @_986725870GHIJKLMNOPF0Vܷ^_`Oledegkijlnmpor wvyxz  -2 ]Times New Roman*wgw? -2 $)2 22 (2 @)2 (2 |~ Times New Roman*wgw. -2 #32  22 22 H12 22 a12 q22 32 22 O 22  12  22 12 332 #33Times New Roman*wgw? -2 #u2 , c2 u2 u2 s2 u2 ZR2 m2 u2  c2 r u2 6 u2 bs2 u2 I2 FF Times New Roman*wgw. -2 cMT Extra !w*wgw? -2 Q#&2 &2 &2 T&Symbol. !w*wgw. -2 !+2 +2 t-2 +2  +2 -2 j= Symbol? !w*wgw? -2 p* & "Systemwf`  -0f 2F 3" ="I 33 (2u 1 s 2 +u 1 u 2 c 2 +2u 3 )[]"m c R 2 (2u 1 s 2 +2u 1 u 2 c 2 +2u 3 ) FMicrosoft Equation 3.0 DS EqCompObj6tiveF fObjInfo3F OlePres000ive Equation Native F uation Equation.39qJN ' .1  @C&`C & MathTypeSymbol !w*wgw -2 By2 By2 By2 By2 By2 &y2 &y2 &y2                          ! 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FJ @_986726850tiveJMNF4@GfOlenfo5tivejkn FN D " .1  @&0A & MathType` Times New Roman*wgw -2 @32 @^ 32 @ 22 @D 22 @v12 @^1Times New Roman*wgw ? -2 m 2 m 2 mTimes New Roman*wgw -2  b2  u2  b2 mu2 b2 u Times New Roman*wgw @ -2 48C2 4GNSymbol !w*wgw -2 u +2 T+2 t=Symbol A!w*wgw A -2 wMT Extra !w*wgw -2 fr & "Systemwf  -8 , N "  C =u 1 2b  1 +u 2 2b  2 +u 3 2b  3 FMicrosoft Equation 3.0 DS EqCompObj00  '  O f ObjInfoNative)*,f-h/056789Q @OlePres000GHIJKLMNOPFR `Equation Nativeijlnmpor wvyxze uation Equation.39qt : .1  `&PA & MathType` Times New Roman*wgw < -2 @3y2 @ 3y2 @ 1y2 @ 2y2 @2y2 @d1yTimes New Roman*wgw d -2 mK22 m\ 22 2Times New Roman*wgw = -2 b32  q2  b32 q2 a32 q Times New Roman*wgw e -2 48C32 4GN3MT Extra >!w*wgw > -2  &2 &2 &2 frSymbol f!w*wgw f -2  +22 -22 t=2Symbol ?!w*wgw ? -2 w & "Systemwf  -  , N "  C =2q 1 2a  2 "2q 2 2b  1 +2q 3 2b  3 FMicrosoft Equation 3.0 DS Equation Equation.39q<, B .1   (&'A & MathType`_986727088  '  nF ?  Oleion Native)*,f-h/0 56789i @CompObj0GHIJKLMNOPFj f`ObjInfogkijlnmporwvyxzl OlePres000m Equation Native  _986727109F@HJOle   Times New Roman*wgw -2 @{'32 @%32 @"22 @!12 @L22 @h22 @712 @12 @-22 @32 @32 @ 12 @ 22 @; 32 @22 @?22 @22 @*1Times New Roman*wgw{ -2 m ' 2 E&) 2 Z ( 2 m 2 ) 2 ( 2 m 2 ) 2 ( 2 mP 2 ma  2  ) 2 m 2 m 2 ( Times New Roman*wgw -2 &b2 $q2  "s2  q2 b2 c2 q2 Rb2 \q2  b2 q2  b2  q2 b2 Ys2 b2 Dc2 rqMT Extra !w*wgw{ -2 %&2 !&2 &2 &2 &2  &2 &Symbol !w*wgw -2 #+2 ;+2 +2 ~-2 =2 +2  -2 .+2 := & "Systemwf&  -  =2q 1 (c 2 2b  2 +s 2 2b  3 )"2q 2 2b  1 +2q 3 2b  3 =("2q 2 )2b  1 +(2q 1 c 2 )2b  2 +(2q 1 s 2 +2q 3 )2b  3 FMicrosoft Equation 3.0 DS Equation Equation.39qW| > .1   &@ & MathTypeP Times New Roman*wgw -2 CompObj fObjInfo OlePres000 Equation Native  T22 1Times New Roman*wgw -2 q2 .uMT Extra !w*wgw -2 n/&Symbol !w*wgw -2 -2 = & "Systemwf  -8(l u 1 ="2q 2 FMicrosoft Equation 3.0 DS Equation Equation.39qzW F .1   & & MathTypeP_986727160 Fj8l Ole  CompObj fObjInfo OlePres000     2  Equation Native)-+,./01 w6y89 b@_986727192GHIJKLMNOPFxp^_`Oledefghijklmnop uvx{  Times New Roman*wgw -2 -232 12 2Times New Roman*wgw K -2 tc2 Dq2 .uMT Extra !w*wgw -2 m{&Symbol L!w*wgw L -2  = & "Systemw1f  -F,7 u 2 =2q 1 c 2 FMicrosoft Equation 3.0 DS Equation Equation.39q { f CompObj6tiveF fObjInfo3F OlePres000ive Equation Native F |.1  @&` & MathType` Times New Roman*wgw t -2 3y2 2y2 1y2 3yTimes New Roman*wgw # -2 .q2 rs2 6q2 .uMT Extra u!w*wgw u -2 ne&2 nm&Symbol $!w*wgw $ -2 +2 = & "Systemwf  -` 4 u 3 =2q 1 s 2 +2q 3 FMicrosoft Equation 3.0 DS Eq_986728582F@ǡsɡ Oleion Native  CompObj9F fObjInfo uation Equation.39q1<  .1  ,&p,J & MathType`Times New Roman*wgw  -2 +)y2 n*y2 n&y2 #(y2 n( y2 $)y2 ny2 ny2 OlePres000 vEquation Native  _986729759bF:@Ole  ny2 (y2 0y2 n8 y2 ~y Times New Roman*wgw v -2 @+32 @)12 @0'12 @&32 @ 22 @32 @P32 @w22 @522 @f12 @N12 @ 2Times New Roman*wgw  -2 X*b2 ()u2 &b2 >%u2 "R2 b2 R2 b2 u2 b2 ^u2 b2 u2  b2  R2 !v2 v Times New Roman*wgw w -2 4 C2 4N2 4B2 4WN2 4C2 4GNSymbol  !w*wgw  -2 (+2 l$-2 !=2 2 f+2 D+2 +2 =2  2 +2 Z=`Symbol x!w*wgw x -2 *MT Extra !w*wgw  -2 r2 r2 @r2 8rSymbol y!w*wgw y -2 w & "SystemwEf  -L# , N 2v  C " =, N 2v  B +, N "  C R2b  2 ="0 +(u 1 2b  1 +u 2 2b  2 +u 3 2b  3 )R2b  2 =R("u 3 2b  1 +u 1 2b  3 ) FMicrosoft Equation 3.0 DS Equation Equation.39qt : .1  @&$@ & MathType` Times New Roman*wgw) 1 -2 CompObj     2 f ObjInfo('*)-+,./01w6y89 @OlePres000GHIJKLMNOPUVWXY `Equation Nativeijklmnop uvx{                        ! 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