Acta Mech DOI 10.1007/s00707-016-1611-8

O R I G I NA L PA P E R

Te Wen Tu

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

Received: 12 September 2015 / Revised: 20 November 2015 © Springer-Verlag Wien 2016

Abstract This paper proposes that Kane’s equations for a simple nonholonomic system are the first-order form of generalized speeds. When the first-order form of Kane’s equations is put in matrix form, the element of the mass matrix is identical to the inertia coefficient. Because the orthogonal set of partial velocities will decouple the first-order equations, one can use the orthogonal criterion to generate efficient equations of motion. With the presented first-order form, Kane’s equations are different from Maggi’s or Gibbs–Appell’s equations. Moreover, in order to clarify the relationship between Kane’s approach and classical approaches, we start from Kane’s equations and introduce kinetic energy or acceleration energy functions to derive Lagrange’s or Gibbs–Appell’s forms of Kane’s equations for the system. We found that the Lagrange’s forms of Kane’s equations can be used to solve nonholonomic systems without introducing Lagrangian multipliers. At last, the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations are, respectively, used to depict the derivation of first-order equations of motion for a rolling coin.

List of symbols Scalar ai (i = 1, 2, 3) A Asr , Bs ci C, C ∗ Fi g G G˜ Ii K r , K r∗ , Kˆ r∗ CK

r,

C K ∗ , C Kˆ ∗ r r

mi m r s , C (m r s ) M

Components of acceleration Function used in Lagrange’s and Gibbs–Appell’s forms of Kane’s equations Functions of qs (s = 1, . . . , n) and t Abbreviation of cos qi A rigid body and its mass center Components of force FC Local gravitational acceleration Gibbs function or acceleration energy function Equal to G minus A Components of principal moment of inertia r th generalized active force, r th generalized inertia force, and part of the r th generalized inertia forces for particles system r th generalized active force, r th generalized inertia force, and part of the r th generalized inertia forces for rigid body C Mass of a particle Inertia coefficients for a particles system and for rigid body C Mass of a rigid body

T. W. Tu (B) Department of Mechanical Engineering, Air Force Institute of Technology, Kaohsiung 820, Taiwan, ROC E-mail: [email protected]; [email protected] Tel.:+886-7-6256040

T. W. Tu

n N p Pi Pr qi (i = 1, . . . , n) R si S t T Ti ur vi (i = 1, 2, 3) v¯i (i = 1, 2, 3) Wsr , X s , Yr s , Z r

Number of generalized coordinates Newtonian reference frame Degree of freedom A typical particle Functions used in Lagrange’s form of Kane’s equations Generalized coordinates Radius of a rolling coin Abbreviation of sin qi A nonholonomic system Time Kinetic energy function Components of force TC r th generalized speed Velocity components of a rolling coin Velocity components of mass center of a rolling coin Functions of qs (s = 1, . . . , n) and t

Vectors ai ∗ aC bi ci ∗ FC , FC Fi I ni (i = x, y, z) ri ∗ TC , TC ∗ vC vi ∗ ∗ v˜ rC , v˜ tC Pi Pi v˜ r , v˜ t

Acceleration of Pi Acceleration of mass center of rigid body C ∗ Constitute the base vectors of a reference frame Constitute the base vectors of a rigid body Resultant active force and inertia force in rigid body C Resultant of all contact forces and/or distance forces Central inertia dyadic of a rigid body The Newtonian unit vector used in the rolling coin Position vector Resultant active torque and inertia torque whose line of action passes through C ∗ Velocity of mass center of rigid body C Velocity of particle Pi Vector functions of qi (i = 1, . . . , n) and t for a rigid body C Vector functions of qi (i = 1, . . . , n) and t for a particle Pi

Greek symbols αC αi (i = 1, 2, 3) ν ωC ωi (i = 1, 2, 3) ˜C ˜ rC , ω ω t

Angular acceleration of a rigid body C Components of angular acceleration Number of particles Angular velocity of rigid body C Components of angular velocity Angular velocities of qi (i = 1, . . . , n) and t for rigid body C

Subscript i, r, s, t

Indices

1 Introduction High-speed digital computers (even for personnel computers) have made it possible to numerically solve many real-time multi-body systems, such as spacecraft, railway vehicles, roboters, and mechanisms. When we deal with sufficiently complicated systems by using computers, however, the tasks of formulating and structuring equations of motion are not as simple as we expected. Therefore, it is important to adopt an efficient and powerful approach for generating equations of motion to both reduce computer storage and shorten execution time simultaneously.

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

In general, several methods can be utilized to solve multi-body dynamics systems, such as Newton–Euler’s, Lagrange’s, Gibbs–Appell’s, and Kane’s. Nevertheless, Kane’s method [1–10] is the most advantageous: It automatically eliminates workless constraint forces, permits the use of a varied class of dependent variables, and applies to both holonomic and nonholonomic systems. In 1985, the special issue in science journal [11] claimed that Kane was ‘the man who mastered motion.’ Also, Kane’s method always led to first-order equations, which are easily integrated numerically that makes it easier for analysis. Multi-body dynamicists (such as Huston [12]) preferred Kane’s method to develop equations of motion for complex systems. Tracking the naming process of Kane’s method, we found that professor Kane was lack of confidence in his method. Before 1968 [4], Kane called his equations ‘Lagrange’s form of d’Alembert’s principle.’ In 1980, Kane and Levinson [5] compared Kane’s method with the other classical methods to derive the equations of motion for complex spacecraft, and Kane himself claimed his method as ‘last method.’ Owing to the interrogation by his student, Levinson, Kane finally named his equations ‘Kane’s dynamical equations’ [9,10] (for brevity, we call them Kane’s equations). Although Kane’s method is more systematic and requires less labor, two major points still need to be further studied. The first one is that we can obtain the first-order equations of motion at the last minute, but what is the explicit first-order form of Kane’s equations? Kane presented that the so-called Kane’s equations were the vanishing forms of generalized active forces plus generalized inertia forces; hence, dynamicists cannot directly perceive the explicit form of Kane’s equations with generalized speeds [9,10]. In 1995, Loduha and Ravani [13] proposed a congruency transformation method to obtain first-order decoupled equations of motion for a holonomic system. They also claimed that they needed to incorporate an orthogonal complement with the congruency transformation to generate the first-order decoupled equations for a nonholonomic system. Nevertheless, because the first-order equations of Kane’s equations had never been found, researchers were not confident of using Kane’s method to solve real-time complex systems. The other one concerns the close relationship between Kane’s approach and the classical approaches, such as Gibbs–Appell’s and Maggi’s. Kane and Levinson [5] compared Kane’s method with the Gibbs–Appell’s method and stated that Gibbs–Appell’s method was more labor-intensive than Kane’s method when it was taken to solve the same problem. However, Kane did not elucidate the relationship between his method and Gibbs–Appell’s. Chen [14] and Desloge [15] claimed that Kane’s method was fundamentally equivalent to the Gibbs–Appell’s method. Later, in a companion study, Desloge [16] also stated that Kane’s equations were an intermediate set of equations arising in the derivation of the Gibbs–Appell’s equations from Newton’s equations. In addition, Borri et al. [17] also advocated the equivalence of Kane’s and Maggi’s equations in 1990. However, either Kane’s rebuttal [18] or Levinson’s rebuttal [19] was too weak to defend. Kane and Levinson were focused on that Kane’s method was free of the principle of virtual work and required less labor to derive the equations of motion. Later, based on Jourdain’s variational equation, Wang and Pao [20] deduced a minimal set of general equations of motion for simple nonholonomic system. They found the equation was the same as that derived by Kane in 1961 [2] but established by Appell in 1903. Hence, they named it as Appell’s equation. Following this line of reasoning, some researches abandoned Kane’s equations. For example, Ginsberg only included the Gibbs–Appell’s equations in his textbook [21], and Chen [22] used the Maggi’s equations to solve the equations of motion of mechanical systems under servo constraints. Previous research has shown that Kane’s equations are either the equivalence of Maggi’s equations or the intermediate forms of Gibbs–Appell’s equations as derived from d’Alembert’s principle. This paper generates all required first-order equations of Kane’s equations for a simple nonholonomic system. This includes the first-order equations of generalized speeds and Kane’s equations together with first-order equations between generalized coordinates and generalized speeds. With these equations, the beginner can clearly understand the merit of Kane’s method in computer implementation. Then, we try a reverse approach that begins from Kane’s equations and then introduces kinetic energy and acceleration energy functions to yield four alternative forms, two with Lagrange’s forms and two with Gibbs–Appell’s forms. Finally, it is to show how we can use the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations to solve the classical problem—a rolling coin.

2 Kane’s equations In consistence with the following derivation, Kane’s equations are simply shown below, and the notations used in Kane’s textbook [10] are directly taken. Consider a system S consisting of ν particles (say P1 , P2 , . . . , Pν ) with p ( p < n) degrees of freedom in a Newtonian reference frame N (Fig. 1). Let q1 , . . . , qn be n generalized

T. W. Tu

n3

P2

Fi

Pν

P1

Pi

ri

N

S

mi a i

n2

n1 Fig. 1 A dynamic system having ν particles

coordinates characterizing the configuration of S in N , and generalized speeds u r (r = 1, . . . , n) are defined as the linear combination of q˙s (s = 1, . . . , n), i.e., ur =

n 

Yr s q˙s + Z r

(r = 1, . . . , n)

(1)

s=1

where Yr s and Z r are functions of q1 , . . . , qn , and the time t. These functions must be chosen such that Eq. (1) can be uniquely solved as q˙s =

n 

Wsr u r + X s

(s = 1, . . . , n)

(2)

r =1

where Wsr and X s are certain functions of q1 , . . . , qn , and t. If the (n − p) motion constraint equations of system B can be written as us =

p 

Asr u r + Bs

(s = p + 1, . . . , n),

(3)

r =1

then S is called a simple nonholonomic system by Kane [10]. Note that Asr and Bs are functions of q1 , . . . , qn , and t. When no motion constraint equations exist (i. e., p = n), the system turns to stand for a holonomic system.

2.1 For a particles system The velocity vi in N of a typical particle Pi belonging to S (Fig. 1) can be uniquely expressed as vi =

p 

v˜ rPi u r + v˜ tPi

(i = 1, . . . , ν)

(4)

r =1

where the vectors v˜ rPi and v˜ tPi are functions of q1 , . . . , qn , and t. v˜ rPi is called the r th nonholonomic partial velocity of Pi in N [10], and the tilde notations denote the vector quantities for the nonholonomic system. If Fi denotes the resultant of all contact forces and/or distance forces acting on Pi of S, and ai is the acceleration of Pi in N , then, from d’Alembert’s principle, we have Fi − m i ai = 0

(i = 1, . . . , ν)

(5)

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

where m i is the mass of Pi . Dot multiplication of Eq. (5) with v˜ rPi and subsequent summation yields ν 

v˜ rPi · Fi +

i=1

ν 

v˜ rPi · (−m i ai ) = 0

(r = 1, . . . , p)

(6)

i=1

or K r + K r∗ = 0

(r = 1, . . . , p).

(7)

Equations (7) are called ‘Kane’s dynamical equations’ [9,10] for the simple nonholonomic system S. Here, the generalized active force K r and the generalized inertia force K r∗ , respectively, are defined as ν 

v˜ rPi · Fi

(r = 1, . . . , p),

(8)

v˜ rPi · (−m i ai )

(r = 1, . . . , p),

(9)

Kr =

i=1

K r∗ =

ν  i=1

Many researches regarded Eqs. (7) as Maggi’s equations or the relevant form of Gibbs–Appell’s equations. 2.2 For a rigid body system Now, we consider a typical particle Pi of rigid body C (formed by ν particles, say P1 , P2 , . . . , Pν ) belonging to a simple nonholonomic system S possessing p degrees of freedom in a Newtonian reference frame N ∗ (Fig. 2a). The angular velocity of C, ωC , and the velocity of mass center C ∗ , vC , can be expressed as ωC = ∗

vC =

p  r =1 p 

˜C ˜ rC u r + ω ω t , ∗

v˜ rC u r + v˜ tC

(10)

∗

(11)

r =1 ∗

∗

∗

˜C ˜ rC and v˜ rC are called r th partial ˜ rC , ω ˜ rC , and v˜ tC are functions of q1 , . . . , qn , and t. ω where the vectors ω t ,v angular velocity and r th partial velocity of C, respectively. Using the kinematic equations, the velocity vi and the acceleration ai of Pi are given as ∗

ai = a

n3

C

c2 C

*

n3

(a)

(i = 1, . . . , ν)

S

N

(13) C

ωC

FC

ri

C*

Pi mi a i

αC

(12)

TC

c1

n2

n1

C

Fi

Pν

P1 c3

N

(i = 1, . . . , ν),

+ α × ri + ω × (ω × ri ) C

C

ωC P2

C∗

vi = vC + ωC × ri

Pi S

αC

n2

n1

Fig. 2 A rigid body system. a A rigid body. b The equivalent force system

(b)

T. W. Tu ∗

where ri is the position vector from C ∗ to Pi ; αC and aC are the angular acceleration of C and the acceleration of C ∗ . If a set of contact and/or distance forces acting on C is equivalent to a couple of torque TC along with a force FC whose line of action passes through C ∗ (Fig. 2b), then the generalized active forces and generalized inertia forces for the rigid body [10] are written as ∗

C

˜ rC · TC K r = v˜ rC · FC + ω

C

K r∗

=

∗ v˜ rC

C∗

·F

˜ rC +ω

(r = 1, . . . , p),

C∗

·T

(r = 1, . . . , p)

∗

(14) (15)

∗

where the resultant inertia force FC and resultant inertia torque TC , whose line of action passes through C ∗ , are given by ∗

∗

FC = −MaC , C∗

T

(16)

= −α · I − ωC × I · ωC . C

(17)

Here, M is the mass of C and I is the central inertia dyadic of C. Substituting Eqs. (14) and (15) into Eq. (7) yields Kane’s equations for the rigid body as K r + C K r∗ = 0

C

(r = 1, . . . , p).

(18)

3 The complete first-order form of Kane’s equations For a beginner to Kane’s method, he or she cannot understand directly what the merits of Kane’s equations (Eqs. (7)) are in computer implementation. In this Section, we will derive the first-order form of Kane’s equations to show why Kane’s method is more suitable for numerical analysis. 3.1 The relationship between time derivative of generalized speeds and Kane’s equations First, considering the particles system and differentiating Eq. (4) with respect to time t in N , the acceleration of Pi of S is obtained as ai =

p 

(v˙˜ rPi u r + v˜ rPi u˙ r ) + v˙˜ tPi

(i = 1, . . . , ν).

(19)

r =1

Substitution from Eq. (19) into Eq. (6) gives p 

m r s u˙ s = Kˆ r∗ + K r

(r = 1, . . . , p).

(20)

s=1

Here, m r s is called an inertia coefficient of a particle which can be found in Kane’s textbook [10] and is defined as mr s =

ν 

m i (˜vrPi · v˜ sPi )

(r, s = 1, . . . , p).

(21)

i=1

Also, Kˆ r∗ (r = 1, . . . , p) represent the part of the r th generalized inertia forces and are defined as Kˆ r∗ = −

ν  i=1

 p

m i vrPi ·

v˙˜ sPi u s + v˙˜ tPi



(r = 1, . . . , p).

(22)

s=1

Next, consider a rigid body system. Substituting Eqs. (10) and (11) into Eq. (12) leads to vi =

p  ∗ ∗ ˜ rC × ri )u r + (˜vtC + ω ˜C (˜vrC + ω t × ri ) r =1

(i = 1, . . . , ν).

(23)

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

Comparing Eq. (23) with Eq. (4) will give ∗

˜ rC × ri v˜ rPi = v˜ rC + ω v˜ tPi

=

∗ v˜ tC

˜C +ω t

(r = 1, . . . , p),

× ri .

(24) (25)

Substituting Eqs. (24) and (25) into Eq. (22), the part of the r th generalized inertia forces for the rigid body system is derived as C

Kˆ r∗ = −

p 

∗

∗

∗

∗

˙˜ C )u s − (M v˜ C · v˙˜ C + ω ˙˜ C ) − ω ˜ rC · I · ω ˜ rC · I · ω ˜ rC · ωC × I · (ωC − ω ˜C (M v˜ rC · v˙˜ sC + ω t ). s t r t

s=1

(26) Therefore, for a rigid body system, Eq. (6) can be written as p 

C

(m r s )u˙ s = C Kˆ r∗ + C K r

(r = 1, . . . , p)

(27)

s=1

where C (m r s ) is called inertia coefficient of the rigid body and is derived from Eq. (21) by using Eq. (24) as C

∗

∗

˜ rC · I · ω ˜C (m r s ) = M v˜ rC · v˜ sC + ω s

(r, s = 1, . . . , p).

(28)

Now, Eqs. (20) and (27) can be rewritten in matrix form as ˙ p×1 = {F} p×1 . [M] p× p {u}

(29)

Observing the above equation, we clearly understand that Kane’s equations are the first-order form of generalized speeds. However, in Kane’s textbooks, such as Eq. (21) in page 282 of [9], we can find the equation X u˙ = Y , but the explicit elements of X and Y are not found.

3.2 The dynamic decoupling From Eqs. (20) and (21), the decoupled condition for a particles system is v˜ rPi · v˜ sPi = 0

r  = s.

(30)

If either v˜ rPi or v˜ sPi is a null vector, or they are orthogonal to each other, Eq. (30) can be satisfied automatically. Now, if dynamic decoupling occurs, the form of equations of motion for the particles system is u˙ r =

1 ˆ∗ ( K + Kr ) m rr r

(r = 1, . . . , p).

(31)

Likewise, for a rigid body system, the decoupled conditions are ∗

∗

∗

∗

v˜ rC · v˜ sC = 0

r = s

(32)

r  = s.

(33)

and ωrC · ωC s =0

When dynamic decoupling occurs, the form of the equations of motion for the rigid body system is u˙ r =

1 C (m

rr )

(C Kˆ r∗ + C K r )

(r = 1, . . . , p).

(34)

T. W. Tu

3.3 The first-order form between generalized coordinates and generalized speeds Upon substitution of Eq. (3) into Eq. (2) one obtains the equations ⎛ ⎛ ⎞ ⎞ p n n    ⎝Wsr + q˙s = Wsk Akr ⎠ u r + ⎝ X s + Wsk Bk ⎠ r =1

k= p+1

(s = 1, . . . , n).

(35)

k= p+1

After p generalized speeds u r are solved out, we can employ this equation to solve n generalized coordinates qi for the nonholonomic system. Therefore, Eq. (20) or Eq. (27) plus Eq. (35) will construct the complete ( p + n) Kane’s equations with first-order form for a particles system or for a rigid body system, respectively.

4 The energy forms of Kane’s equations In order to let classical dynamicists understand Kane’s method quickly, we try a reversal approach. That is to say, stemming from Kane’s equations and introducing kinetic energy or acceleration energy functions, we can obtain four alternative forms of Kane’s equations. From Eq. (4), the partial velocity v˜ rPi of the particle Pi of S in N can be expressed as v˜ rPi =

∂vi ∂u r

(r = 1, . . . , p).

(36)

Likewise, from Eq. (19), the vector v˜ rPi of the particle Pi in N can be written as v˜ rPi =

∂ai ∂ u˙ r

(r = 1, . . . , p).

(37)

4.1 Alternative forms of generalized active forces Substituting into Eq. (8) from Eq. (36), one obtains the generalized active force as Kr =

ν  ∂vi · Fi ∂u r

(r = 1, . . . , p).

(38)

i=1

Similarly, if one substitutes Eq. (37) into Eq. (8), this yields the other form as ν  ∂ai ∂ A Kr = · Fi = ∂ u˙ r ∂ u˙ r

(r = 1, . . . , p)

(39)

i=1

where the function A is defined as A=

ν 

ai · Fi .

(40)

i=1

4.2 Alternative forms of generalized inertia forces Upon substitution of Eq. (36) into Eq. (9), one obtains the generalized inertia force K r∗ =



ν  ∂vi d ∂T · (−m i ai ) = − + Pr , ∂u r dt ∂u r i=1

(r = 1, . . . , p)

(41)

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

where the kinetic energy function T and the functions Pr (r = 1, . . . , p) of S are defined, respectively, as ν

1 m i vi · vi , 2 i=1

ν  d ∂ vi Pr = m i vi · dt ∂ u r T =

(42) (r = 1, . . . , p).

(43)

i=1

Similarly, if one substitutes Eq. (37) into Eq. (9), this yields K r∗ =

ν  ∂ai ∂G · (−m i ai ) = − ∂ u˙ r ∂ u˙ r

(r = 1, . . . , p)

(44)

i=1

where the Gibbs function (or the acceleration energy function) G is defined as ν

G=

1 m i ai · ai . 2

(45)

i=1

4.3 Alternative forms of Kane’s equations After combining these alternative forms of generalized active forces (Eqs. (38) and (39)), with those of generalized inertia forces (Eqs. (41)), we can develop two Lagrange’s forms of Kane’s equations for the nonholonomic system S having p degrees of freedom as follows:

d ∂T − Pr = K r dt ∂u r

d ∂T ∂A − Pr = dt ∂u r ∂ u˙ r

(r = 1, . . . , p), (r = 1, . . . , p).

(46) (47)

We call Eqs. (46) and (47) Lagrange’s form I and Lagrange’s form II of Kane’s equations, respectively. Applying the same approach to the other form of generalized inertia forces (Eq. (44)), we can get the Gibbs–Appell’s forms of Kane’s equations as follows: ∂G = Kr (r = 1, . . . , p), ∂ u˙ r ∂A ∂ G˜ ∂G − =0 or =0 ∂ u˙ r ∂ u˙ r ∂ u˙ r

(48) (r = 1, . . . , p)

(49)

where the function G˜ is defined as G˜ = G − A.

(50)

We call Eqs. (48) and (49) Gibbs–Appell’s form I and Gibbs–Appell’s form II of Kane’s equations, respectively. Note that Eq. (49) is equal to that presented by Desloge [24], but both derivative procedures are different.

5 Explicit forms of functions C K r , A, T, Pr , G for a rigid body When we consider a typical particle Pi of rigid body C belonging to a simple nonholonomic system S possessing p degrees of freedom in a Newtonian reference frame N , the explicit forms of functions C K , A, T, P , G for the rigid body can be derived. r r

T. W. Tu

5.1 Vector forms Firstly, the generalized active functions C K r (r = 1, . . . , p) and the function A can be proven as ∗

C

∂vC ∂ωC Kr = · FC + · TC ∂u r ∂u r ∗ A = aC · FC + αC · TC .

(r = 1, . . . , p),

(51) (52)

At the same time, the kinetic energy function T , the functions Pr (r = 1, . . . , p), and Gibbs function G can be expressed as 1 1 ∗ ∗ T = MvC · vC + ωC · I · ωC , (53) 2 2 

∗ d ∂ωC ∂ vC C∗ d C Pr = Mv · +ω ·I· (r = 1, . . . , p), (54) dt ∂ u r dt ∂ u r 1 ∗ ∗ (MaC · aC + αC · I · αC + 2αC · ωC × I · ωC ). 2 For brevity, formal proofs of Eqs. (53)–(55) are omitted. G=

(55)

5.2 Scalar forms Let c1 , c2 , c3 form a right-handed set of mutually perpendicular unit vectors, each parallel to a central principal axis of C, but not necessarily fixed in C (Fig. 2a), and αi , ωi , Ii , ai , vi , Fi , Ti be defined as 

αi = αC · ci , 

ai = a

C∗

· ci ,



ωi = ωC · ci , 

vi = v

C∗

· ci ,



Ii = ci · I · ci 

Fi = FC · ci ,

(i = 1, 2, 3), 

Ti = TC · ci

(56) (i = 1, 2, 3).

(57)

Substituting into Eqs. (51)–(53) and Eq. (55) from Eqs. (56)–(57), one can thus generate the explicit forms of functions C K r , A, T , and G as follows: ∂v1 ∂v2 ∂v3 ∂ω1 ∂ω2 ∂ω3 C K r = F1 + F2 + F3 + T1 + T2 + T3 (r = 1, . . . , p), (58) ∂u r ∂u r ∂u r ∂u r ∂u r ∂u r (59) A = F1 a1 + F2 a2 + F3 a3 + T1 α1 + T2 α2 + T3 α3 , 1 1 2 2 2 2 2 2 (60) T = M(v1 + v2 + v3 ) + (I1 ω1 + I2 ω2 + I3 ω3 ), 2 2 1 G = [M(a12 + a22 + a32 ) + I1 α12 + I2 α22 + I3 α32 ] − ω2 ω3 α1 (I2 − I3 ) 2 − ω3 ω1 α2 (I3 − I1 ) − ω1 ω2 α3 (I1 − I2 ). (61) Moreover, if one performs the time derivative of unit vectors c1 , c2 , c3 , one can get dc1 (62) = ωC × c1 = ω3 c2 − ω2 c3 , dt dc2 (63) = ωC × c2 = −ω3 c1 + ω1 c3 , dt dc3 (64) = ωC × c3 = ω2 c1 − ω1 c2 . dt With mathematical operations in Eq. (54), one can get the explicit forms of functions Pr (r = 1, . . . , p) as follows:







 d ∂v1 d ∂v2 ∂v3 ∂v2 ∂v1 ∂v3 Pr = Mv1 − ω3 − ω1 + ω2 + Mv2 + ω3 dt ∂u r ∂u r ∂u r dt ∂u r ∂u r ∂u r







 d ∂v3 d ∂ω1 ∂v2 ∂v1 ∂ω3 ∂ω2 − ω2 − ω3 +Mv3 + ω1 + I 1 ω1 + ω2 dt ∂u r ∂u r ∂u r dt ∂u r ∂u r ∂u r







 d ∂ω2 d ∂ω3 ∂ω1 ∂ω3 ∂ω2 ∂ω1 − ω1 − ω2 + ω3 + I 3 ω3 + ω1 . (65) +I2 ω2 dt ∂u r ∂u r ∂u r dt ∂u r ∂u r ∂u r

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

z

N q2

nz

q4

nx

b2

b3

O

R

q1

y C

q3

Cˆ

q5

ny

b1

H

x Fig. 3 A rolling coin Table 1 The directional cosines between frames N and B b1 b2 b3

nx

ny

nz

c1 s1 s2 s1 c2

s1 c1 s2 c1 c2

0 c2 s2

6 Illustrative example To demonstrate how we can use the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms to derive the equations of motion, the classical problem of a rolling coin [10] as depicted in Fig. 3 is considered. In this figure, a circular disk C of radius R in contact with a horizontal plane H that is fixed in a Newtonian reference 

frame N is shown. Mutually perpendicular unit vectors nx , n y , and nz = nx × n y are fixed in N . In addition, b1 , b2 , and b3 form a dextral set of orthogonal unit vectors (called reference frame B). Here, b1 is parallel to the tangent to the rim of C at the contact point Cˆ between N and C, b2 is parallel to the line connecting Cˆ to C ∗ (the mass center of C), and b3 is normal to the plane of C. ˆ constitute a set of n = 5 The angles q1 , q2 , q3 together with q4 and q5 , the x- and y- coordinates of C, ˆ one can get two motion generalized coordinates. If one assumes no slipping condition is taking place at C, constraint equations. As a consequence, one has a simple nonholonomic system having three degrees of freedom, i.e., p = 3. Moreover, the directional cosines between unit vectors b1 , b2 , b3 and nx , n y , nz are listed in Table 1, where ci (i = 1, 2) and si (i = 1, 2) stand for cos qi and sin qi (i = 1, 2), respectively. Firstly, if the generalized speeds u i (i = 1, 2, 3) are chosen as u i = N ωC · bi

(i = 1, 2, 3),

(66)

then the angular velocity of C in N can be expressed as N

ωC = u 1 b1 + u 2 b2 + u 3 b3 .

(67)

Note that the relationship between generalized speeds and time derivative of generalized coordinates is given by u 1 = −q˙2 ,

u 2 = c2 q˙1 ,

u 3 = s2 q˙1 + q˙3 .

(68)

From Eq. (10), one can get the partial angular velocities as ˜C ω 1 = b1 ,

˜C ω 2 = b2 ,

˜C ω 3 = b3

(69)

T. W. Tu

˜C and ω t = 0. To facilitate the derivation efficiency, we can obtain N

ω B = N ωC + C ω B = u 1 b1 + u 2 b2 + u 2 tan q2 b3 .

(70)

Next, the velocity of mass center C ∗ of the coin C is N C∗

v

=

N

d (q4 nx + q5 n y + Rb2 ) = v¯1 b1 + v¯2 b2 + v¯3 b3 dt

(71)

where v¯i (i = 1, 2, 3) are given by v¯1 = −Ru 2 tan q2 + u 4 c1 + u 5 s1 , v¯2 = (−u 4 s1 + u 5 c1 )s2 , v¯3 = Ru 1 + (u 4 s1 − u 5 c1 )c2 ,

(72)

and u 4 and u 5 are defined as u 4 = q˙4 ,

u 5 = q˙5 .

(73)

Employing the pure rolling condition of C yields N Cˆ

∗

v = N vC + N ωC × (−Rb2 ) = 0.

(74)

Thereafter, one obtains the motion constraint equations u 4 = (u 2 tan q2 − u 3 )Rc1 ,

u 5 = (u 2 tan q2 − u 3 )Rs1 .

(75)

With the aid of these equations, one can rewrite Eq. (71) as N C∗

v

= −Ru 3 b1 + Ru 1 b3

(76)

from which one arrives at the partial velocities ∗

v˜ 1C = Rb3 ,

∗

v˜ 2C = 0,

∗

v˜ 3C = −Rb1 ,

(77)

∗

and v˜ tC = 0. It is worthwhile to note that partial angular velocities and partial velocities satisfy the orthogonal conditions from Eq. (69) and Eq. (77). Hence, the equations of motion will be dynamic decoupling. The set of gravitational forces exerted on C is equivalent to a force FC whose line of action passes through the mass center C ∗ of C. Explicitly, force FC and torque TC are given as FC = −Mg(c2 b2 + s2 b3 ),

TC = 0

(78)

where M is the mass of C and g is the local gravitational acceleration. Now, the generalized active forces r (r = 1, 2, 3) for the rolling coin are

CK

C

K 1 = −Mg Rs2 ,

C

K 2 = 0,

C

K 3 = 0.

(79)

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

6.1 The first-order form of Kane’s equations The first-order form of Kane’s equations for the rolling coin is C

(m rr )u˙ r = C Kˆ r∗ + C K r

(r = 1, 2, 3).

(80)

Upon substitution of Eqs. (69) and (77) to Eq. (28), one obtains the inertia coefficients as C

(m 11 ) =

5 M R2, 4

C

(m 22 ) =

1 M R2, 4

C

(m 33 ) =

3 M R2 2

(81)

where I1 = I2 = 41 M R 2 and I3 = 21 M R 2 have been used. In order to compute C Kˆ r∗ (r = 1, 2, 3), we perform ∗ ∗ ˜ rC (r = 1, 2, 3), and ω ˜C the time derivatives of v˜ rC (r = 1, 2, 3), v˜ tC , ω t as follows: ∗ v˙˜ 1C = Ru 2 b1 − Ru 1 b2 ,

∗ v˙˜ 2C = 0,

∗ v˙˜ 3C = −Ru 2 tan q2 b2 + Ru 2 b3 ,

∗ v˙˜ tC = 0,

(82)

and ˙˜ C = −u 2 tan q2 b1 + u 1 b3 , ω 2

˙˜ C = u 2 tan q2 b2 − u 2 b3 , ω 1

˙˜ C = u 2 b1 − u 1 b2 , ω 3

˙˜ C = 0. (83) ω t

It follows from Eqs. (26), (69), (77), (82), and (83) that 3 1 Kˆ 1∗ = − M R 2 u 2 u 3 + M R 2 u 22 tan q2 , 2 4 C ˆ∗ K3 = M R2u1u2. C

C

M R2 M R2 Kˆ 2∗ = − u 1 u 2 tan q2 + u1u3, 4 2 (84)

Substituting Eqs. (79), (81), and (84) into Eq. (80), we obtain the decoupled first-order equations of motion of the rolling coin as (m 11 )u˙ 1 = C Kˆ 1∗ + C K 1 , C (m 22 )u˙ 2 = C Kˆ 2∗ + C K 2 , C

C

(m 33 )u˙ 3 = C Kˆ 3∗ + C K 3 ,

u˙ 1 = −1.2u 2 u 3 + 0.2u 22 tan q2 − 0.8gs2 /R,

(85)

u˙ 2 = 2u 1 u 3 − u 1 u 2 tan q2 , 2 u˙ 3 = u 1 u 2 . 3

(86) (87)

6.2 Lagrange’s forms of Kane’s equations Differentiating Eqs. (67) and (76) with respect to the generalized speeds u r (r = 1, 2, 3), respectively, we obtain ∂ N ωC = b1 , ∂u 1

∂ N ωC = b2 , ∂u 2

∂ N ωC = b3 ∂u 3

(88)

and ∗

∂ N vC = Rb3 , ∂u 1

∗

∂ N vC = 0, ∂u 2

∗

∂ N vC = −Rb1 . ∂u 3

(89)

Employing Eqs. (62)–(64) as well as Eq. (70), one can get N

db1 = u 2 tan q2 b2 − u 2 b3 , dt

N

db2 = −u 2 tan q2 b1 + u 1 b3 , dt

N

db3 = u 2 b1 − u 1 b2 . dt

Differentiating Eqs. (88) and (89) with respect to time t and using Eq. (90) one can derive N C

N C

∂ ω ∂ ω N d N d = u 2 tan q2 b2 − u 2 b3 , = −u 2 tan q2 b1 + u 1 b3 , dt ∂u 1 dt ∂u 2 N C

∂ ω N d = u 2 b1 − u 1 b2 dt ∂u 3

(90)

(91)

T. W. Tu

and N

d dt

N

d dt

 

∂ N vC ∂u 1

∗

∂ N vC ∂u 3

∗

= R(u 2 b1 − u 1 b2 ),

N



d dt



∂ N vC ∂u 2

∗

= 0,

= −Ru 2 (tan q2 b2 − b3 ).

(92)

Employing Eqs. (53) and (54), the kinetic energy function T and the functions Pr (r = 1, 2, 3) are 1 1 M R 2 (u 21 + u 23 ) + (I1 u 21 + I2 u 22 + I3 u 23 ), 2 2 3 1 1 1 2 P2 = M R 2 u 1 u 3 − M R 2 u 1 u 2 tan q2 , P1 = − M R u 2 u 3 + M R 2 u 22 tan q2 , 2 4 2 4 P3 = M R 2 u 1 u 2 . T =

(93)

(94)

Next, the angular acceleration of C in N may be expressed as α =

N C

B

d N C  N B N C ω + ω × ω = α1 b1 + α2 b2 + α3 b3 , dt

(95)

and αi (i = 1, 2, 3) are given by α1 = u˙ 1 + u 2 (u 3 − u 2 tan q2 ),

α2 = u˙ 2 − u 1 (u 3 − u 2 tan q2 ),

α3 = u˙ 3 .

(96)

The acceleration of C ∗ in N now is N C∗

a

=

N

d N C∗  N B N C∗ + ω × v = a 1 b1 + a 2 b2 + a 3 b3 v dt

(97)

where ai (i = 1, 2, 3) are given by a1 = R(u 1 u 2 − u˙ 3 ),

a2 = −R(u 2 u 3 tan q2 + u 21 ),

a3 = R(u˙ 1 + u 2 u 3 ).

(98)

With the aid of Eqs. (52), (78), (95), and (97), one obtains A = −Mg R[s2 (u˙ 1 + u 2 u 3 ) − c2 (u 2 u 3 tan q2 + u 21 )].

(99)

Finally, by Eq. (46) or Eq. (47), the same results of Eqs. (85)–(87) are derived.

6.3 Gibbs–Appell’s forms of Kane’s equations Substituting Eqs. (67), (95), and (97) into Eq. (55), the Gibbs function G is G=

1 [M(a12 + a22 + a32 ) + I1 α12 + I2 α22 + I3 α32 ] − u 2 u 3 α1 (I2 − I3 ) 2 − u 3 u 1 α2 (I3 − I1 ) − u 1 u 2 α3 (I1 − I2 ).

Employing Eqs. (48) or (49), we generate the same equations as Eqs. (85)–(87).

(100)

First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

6.4 The complete first-order form of equations of motion The first-order equations of motion for the rolling coin can be put in matrix form as ⎫ ⎡ ⎤⎧ ⎫ ⎧ 1 0 0 ⎨ u˙ 1 ⎬ ⎨ −1.2u 2 u 3 + 0.2u 22 tan q2 − 0.8gs2 /R ⎬ ⎣ 0 1 0 ⎦ u˙ 2 = 2u 1 u 3 − u 1 u 2 tan q2 . ⎭ 2 0 0 1 ⎩ u˙ 3 ⎭ ⎩ u u 1 2 3

(101)

When the initial conditions of generalized speeds are specified, we can obtain the generalized speeds numerically. In addition, from Eq. (68), we can obtain the time derivative of the generalized coordinates qi (i = 1, 2, 3) as q˙1 = u 2 sec q2 ,

q˙2 = −u 1 ,

q˙3 = u 3 − u 2 tan q2 .

(102)

Combining Eq. (102) with Eq. (73), the relationship between the time derivative of generalized coordinates and generalized speeds in matrix form is ⎫ ⎤⎧ ⎫ ⎧ ⎡ 1 0 0 0 0 ⎪ q˙1 ⎪ ⎪ u 2 sec q2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ q˙2 ⎬ ⎪ ⎪ ⎨ ⎪ ⎬ ⎢0 1 0 0 0⎥⎨ −u 1 ⎥ ⎢ (103) ⎢ 0 0 1 0 0 ⎥ q˙3 = u 3 − u 2 tan q2 . ⎪ ⎣0 0 0 1 0⎦⎪ ⎪ q˙4 ⎪ ⎪ ⎪ ⎪ ⎪ u 4 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 0 0 0 1 q˙5 u5 When the initial conditions of the generalized coordinates are specified in Eq. (103), the generalized coordinates can be solved by numerical integration.

7 Discussion As depicted above, we have generated the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations for the simple nonholonomic system. In the following, we will discuss further three aspects: What Kane’s equations are, how to choose the generalized speeds, and what the merit of alternative forms of Kane’s equations is.

7.1 Kane’s method and Kane’s equations The so-called ‘Kane’s method’ appeared in 1973 by Passerello and Huston [22]. Later, many multi-body dynamicists used the terminology to represent their dynamic equations derived from Kane’s equations. By contrast, the so-called Kane’s equations, Eqs. (7), are always attributed to Maggi’s equations or the intermediate forms of Gibbs–Appell’s equations as derived from d’Alembert’s principle. However, if one renames Kane’s equations the first-order form derived in the paper, i.e., Eqs. (20) or (27), no one can say Kane’s equations are equivalent to Maggi’s or Gibbs–Appell’s equations.

7.2 The choice of generalized speeds Due to the importance of generalized speeds in Kane’s equations, we tracked the development of generalized speeds in Kane’s method. In 1959 [1], Kane did not present the concept of generalized speeds. Later, Kane set n u r =  i=1 q˙i in the paper of Ref. [2]. In the paper of 1965 [3] and in the textbook of 1968 [4], Kane took n u r = s=1 Ar s q˙s and regarded u r as the linear combination of q˙1 , . . . , q˙n . Because the role of generalized speeds in Kane’s equations is very important, the way to choose the generalized speeds to derive the efficient equations of motion is the major concern by many researchers. In 1983, Kane [6,7] claimed if the generalized speeds are chosen optimally, one can obtain the efficient equations, but he did not show how to choose them. Later, Mitiguy and Kane [25] proposed three guidelines for choosing generalized speeds for complicated multi-body systems in 1996. This work found that as long as the orthogonal conditions of partial velocities are satisfied, one can generate the first-order decoupling equations of motion.

T. W. Tu

Therefore, the orthogonal condition on partial velocities could be taken as the requirement on choosing the generalized speeds to yield the efficient equations of motion. For example, if the generalized speeds u i (i = 1, 2, 3) in the example of the rolling coin now are chosen as u i = q˙i

(i = 1, 2, 3),

(104)

then the angular velocity of C in N can be expressed as N

ωC = u 1 (c2 b2 + s2 b3 ) − u 2 b2 + u 3 b3 .

(105)

With the aid of Eq. (10), one can get the partial angular velocities as ˜C ω 1 = c2 b2 + s2 b3 ,

˜C ω 2 = −b2 ,

˜C ω 3 = b3 ,

(106)

˜C and ω t = 0. Notice that partial angular velocities do not satisfy the orthogonal conditions to one another. Hence, using the set of general speeds in Eq. (104), one will derive the coupling equations of motion. 7.3 The alternative forms of Kane’s equations The objective for the derivation of four alternative forms of Kane’s equations is to construct the relationship between Kane’s approach and the classical approaches. The two Lagrange’s forms of Kane’s equations can be used to solve both holonomic and nonholonomic systems without introducing Lagrangian multipliers. Lagrange’s form I does not calculate acceleration, whereas the other forms need so. Also, Lagrange’s form I is more general and simpler than Passerello and Huston’s form [23]. Because the four alternative forms need to calculate such functions as C K r , A, T, Pr , G, we recommend that the dynamicists can harness the first-order form of Kane’s equations, Eqs. (20) or (27), to derive the equations of motion for dynamics systems. 8 Conclusions This paper has developed the first-order form of Kane’s equations for a simple nonholonomic system to demonstrate the reason why Kane’s method is powerful in numerical analysis. We suggest that the complete first-order equations could be named as the new form of Kane’s equations to distinguish from Maggi’s or Gibbs– Appell’s equations. Also, stemming from Kane’s equations and introducing kinetic energy and acceleration energy functions, four alternative forms of Kane’s equations, two Lagrange’s forms and two Gibbs–Appell’s forms, are proposed. By the derived forms, the classical dynamicists can quickly get familiar with Kane’s equations. Based on the first-order form, Lagrange’s forms, and Gibbs–Appell’s forms of Kane’s equations, this study wishes to wide spread Kane’s method for dynamicists in solving complex dynamics systems. Acknowledgments This research was supported by the National Science Council of Taiwan, the Republic of China, under Grant NSC 87-2212-E-344-001 and is gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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First-order form, Lagrange’s form, and Gibbs–Appell’s form of Kane’s equations

14. Chen, B.: Analytical Dynamics. Taiwan Advanced Education Inc., Taipei (1988). (in Chinese) 15. Desloge, E.A.: A comparison of Kane’s equation of motion and the Gibbs–Appell equations of motion. Am. J. Phys. 54, 470–472 (1986) 16. Desloge, E.A.: Relationship between Kane’s equations and the Gibbs–Appell equations. J. Guid. Control Dyn. 10, 120–122 (1987) 17. Borri, M., Bottasso, C., Mantegazza, P.: Equivalence of Kane’s and Maggi’s equations. Meccanica 25, 272–274 (1990) 18. Kane, T.R.: Rebuttal to ‘A comparison of Kane’s equation of motion and the Gibbs–Appell equations of motion’. Am. J. Phys. 54, 472 (1986) 19. Levinson, D.A.: Rebuttal to ‘Equivalence of Kane’s and Maggi’s equations’. Meccanica 27, 63–64 (1992) 20. Wang, L.S., Pao, Y.H.: Jourdain’s variational equation and Appell’s equations of motion for nonholonomic dynamical systems. Am. J. Phys. 71, 72–82 (2003) 21. Ginsberg, J.H.: Advanced Engineering Dynamics. Harper and Row Publishers Inc., New York (1988) 22. Chen, Y.H.: Equations of motion of mechanical systems under servo constraints: the Maggi approach. Mechatronics 18, 208–217 (2008) 23. Passerello, C.E., Huston, R.L.: Another look at nonholonomic systems. ASME J. Appl. Mech. 40, 101–104 (1973) 24. Desloge, E.A.: Efficacy of the Gibbs–Appell method for generating equations of motion for complex systems. J. Guid. 12, 114–116 (1989) 25. Mitiguy, P.C., Kane, T.R.: Motion variables leading to efficient equations of motion. Int. J. Robot. Res. 15, 522–532 (1996)