Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

Подождите немного. Документ загружается.

MECHANICAL SYSTEMS, CLASSICAL MODELS

442

quasi-co-ordinates are the latter ones) do vanish, if we take into account the relations

(22.2.31); we assume thus that these last

m quasi-co-ordinates are linked to the

generalized velocities by the relations

−+ −+

, 1,2,...,

smk smkj

aqk mπ,

(22.2.32)

where the matrix

−+

⎡⎤

⎣⎦

,smkj

is of rank m , the first −sm quasi-co-ordinates (which,

in particular, can be generalized co-ordinates) being connected to the generalized

velocities by the arbitrary linear relations

==−, 1,2,...,

kkj

qk s mπα ,

(22.2.33)

where the matrix

⎡⎤

⎣⎦

α is of rank −sm. In concordance with the relations (22.2.32),

we can write

−+ −+

δ= δ=

, 1,2,...,

smk smkj

aqk mπ ,

(22.2.32')

and, taking into account the constraint relations (22.2.31), it results that these variations

vanish

−+

δ==0, 1,2,...,

smk

kmπ .

(22.2.32'')

In this case, the last

m terms vanish in Lagrange’s equations in quasi-co-ordinates

(22.2.22), written in the form of the principle of virtual work

⎡⎛ ⎞ ⎤

∂∂ ∂

−+ −δ=

⎜⎟

⎢⎥

∂∂ ∂

⎣⎝ ⎠ ⎦

** *

kj k

jj i

TT T

γπ π

ππ π





;

(22.2.34)

taking into account that the first variations δ

π are independent, the equation (22.2.34)

leads to the system of equations (which does not contain the constraint generalized

forces, as in case of Lagrange’s equations with multipliers, e.g., in case of the equations

(22.2.22'))

−

⎛⎞∂∂ ∂

−+ = = −

⎜⎟

∂∂ ∂

⎝⎠

∑

** *

, 1,2,...,

kj k

jj i

TT T

Qj sm

γπ

ππ π





(22.2.35)

with the notations

∂

⎛⎞

=− = = −

⎜⎟

∂∂

⎝⎠

, 1,2,..., , , 1,2,...,

il im

kj ml lj

isjksm

γββ .

(22.2.35')

We notice that the matrix

⎡⎤

⎣⎦

β is the inverse of the matrix

[

]

α .

Dynamics of Non-holonomic Mechanical Systems

443

The system of

−sm equations (22.2.35), together with the m constraint relations

(22.2.31) and with the

−sm linear relations (22.2.33), form the −2sm equations of

motion for the non-holonomic mechanical system in quasi-co-ordinates. Associating a

corresponding number of initial conditions, we can determine the

−2sm unknown

functions of time

−

12 1 2

, ,..., , , ,...,

ssm

qq qππ π 

. We mention that the kinetic energy

is, in general, a function of all the

s quasi-velocities

π , the coefficients

γ depending

on the generalized co-ordinates

, ,...,

qq q.

22.2.3 Applications

In what follows, we give firstly some simple applications: the motion of the skate

and the motion of a circular disc on a plane; we study then the motion of a sphere on a

horizontal plane and the motion of a carriage with two or four wheels

22.2.3.1 Motion of a Skate

The motion of a rigid skate on the ice plane has been considered in Sec. 13.2.2.6 (see

Fig. 3.16). Let

,xx be the co-ordinates of the middle of the skate (modelled as a segment

of a line of length

2l ) with respect to a system of axes

Ox x in the ice plane and let θ be

the angle made by the skate with the

Ox -axis. The constraint relation is of the form

−=

tan d d 0xxθ

;

(22.2.36)

it is non-holonomic (and scleronomic), as it has been shown in Chap. 3, Subsec. 2.2.6,

by means of the Theorem 3.2.2 of Frobenius. As a matter of fact, this property can be

put in evidence on another way too. Indeed, assuming that the constraint is a geometric

one, we can find a relation of the form

(,,) 0fx x θ , the differential consequence of

which is

∂∂ +∂∂ +∂∂ =

11 22

( / )d ( / )d ( / )d 0fxx fx x fθθ ; taking into account

(22.2.36), it results

∂∂ ∂

⎛⎞

++=

⎜⎟

∂∂ ∂

⎝⎠

tan d d 0

ff f

θθ

The displacements

dx (as well as

dx ) and dθ are arbitrary, so that

∂∂ ∂

+==

∂∂ ∂

tan 0, 0

ff f

;

hence,

f does not depend on θ , while the first relation, where θ is arbitrary, leads to

∂∂=

/0fx and ∂∂=

/0fx . In this case, the function f depends of no one of the

three arguments, so that it cannot represent a finite connection between

,xx and

The number of the generalized co-ordinates cannot be made less than three, but these

ones cannot have a free variation, the mechanical system being thus non-holonomic

MECHANICAL SYSTEMS, CLASSICAL MODELS

444

with two degrees of freedom. Another demonstration of the non-holonomy of the

considered mechanical system starts from the fact that, for a position (

,xx

) of the

middle of the skate, the relation

(,,) 0fx x θ leads to a certain value

θ of the

angle

θ , which is not true, as we will show. Thus, if we start from the position (

,xx)

with the value

θ and if we reach the position (

,xx) on two different arcs of curve,

i.e.

211

()xfx and

[

]

=∈

21111

(), ,xfxx xx, respectively, then one sees that we

reach the final position with two different angles

θ and

θ , ≠

θθ, arbitrarily taken;

the functions

f and

f must satisfy the conditions

′′

=== =

′′

=== =

00 0

11 211 211 11 1

00 0

21 221 221 21 2

() ,() ,()tan,()tan,

( ) , ( ) , ( ) tan , ( ) tan .

fx x fx x fx fx

θθ

The angle

θ can be equal to

θ or to

θ ; but we have supposed that ≠

θθ. One can

find easily pairs of functions

f and

f which satisfy the above conditions, e.g.

()

−

=+− − =

−

21 11 11 11 1 1

tan tan

() () (),()fx fx xx xxgxgx

θθ

where the functions

[

]

11 21 11 1 11

(), (), (), ,fx fx gx x x x∈ are continuous; the

arbitrary function

()gx must satisfy the mentioned condition. The problem is thus

reduced to the construction of a function

()fx which must satisfy the conditions

imposed to it. Because the condition (22.2.36) maintains its form after a translation of

vector

−−

1122

(, )xxxx

, we can assume, for the sake of simplicity, that

0xx and we take

()

[]

()

[]

()

⎛⎞

=+−

⎜⎟

⎝⎠

⎛⎞

−+− +

⎜⎟

⎝⎠

=−

11 1 1 2

11 2 1

121

() tan tan 2

tan 2 tan 3 tan ,

() tan tan ;

fx x x

xxx

θθ

θθ θ

θθ

hence, the constraint relation (22.2.36) is a kinematic one.

Using the generalized co-ordinates

===

11223

,,qxqxqθ

in the space

Λ , the

kinetic energy is given by

()

=++

*222

12 3

TMqq Iq

 ,

(22.2.37)

Dynamics of Non-holonomic Mechanical Systems

445

where

M is the mass of the skate and =

/3IMl is the moment of inertia of it with

respect to its middle. Obviously, the generalized forces vanish (

0, 1,2, 3

Qj== );

comparing the constraint relation (22.2.36) with (22.2.14), we notice that

= 1m ,

== =−=

10 11 12 13

0, tan , 1, 0aa a aθ , the constraint generalized forces being

==−=

1112 3

,,0RaR Rλλ (we denoted =

λλ). Lagrange’s equations with

multipliers (22.2.13) will be of the form

==−=

123

tan , , 0

Mq Mq qλθ λ   ;

(22.2.38)

we associate the initial conditions (at the moment

= 0t )

=== = = =

123 1 2 3

(0) (0) (0) 0, (0) , (0) 0, (0)qqq qvq q ω.

(22.2.38')

The third equation (22.2.38) leads to

() ()tqt tθω.

(22.2.39)

Eliminating the generalized co-ordinate

q and the multiplier λ between the relations

(22.2.36) and the first two equations (22.2.38), we obtain

tan 0qq tωω  ,

wherefrom, by integration,

() () sin

xt qt tω

;

(22.2.39')

we get then

()

==−

() () 1 cos

xt qt tω

(22.2.39'')

Lagrange’s multiplier being given by

=−

00 0

cosMv tλωω.

(22.2.39''')

We notice thus that the middle of the skate has a uniform motion, of velocity

v , on

the circle

()

+− =

xxRR,

(22.2.40)

of radius

/Rvω .

An analogous study can be made for the motion of a ski.

We can introduce also the quasi-co-ordinates

123

,,πππ by the relations

=+=δ=δ+δ

=− + δ =− δ + δ

=δ=δ

112112

212212

dcosdsind d, cos sin,

dsindcosd, sin cos,

dd, ,

xxvt xx

xx xx

πθ θ πθ θ

πθ θπθ θ

πθπθ

(22.2.41)

MECHANICAL SYSTEMS, CLASSICAL MODELS

446

wherefrom

=−δ=δ−δ

=+ δ=δ+δ

=δ=δ

112112

212212

d cos d sin d , cos sin ,

d sin d cos d , sin cos ,

dd, .

θπ θπ θπ θπ

θπθπ

(22.2.41')

If we observe that

d d , , d d ,

iijjiijjj j

kk jkk

qqq qπα πα βπ βπ=δ=δ= δ=δ,

=,, 1,2,3ijk , then we obtain the matrices

[]

⎡⎤⎡⎤

−

⎢⎥⎢⎥

=− =

⎡⎤

⎢⎥⎢⎥

⎣⎦

⎢⎥⎢⎥

⎣⎦⎣⎦

cos sin 0 cos sin 0

sin cos 0 , sin cos 0

001 001

θθ θ θ

αθθβθθ

;

thus, using the formulae (22.1.23), it results

11 22

23 32 31 13

1, 1γγ γγ=− =− =− =−

, the

other 23 components

γ vanishing. The transposition relations (22.1.23') become

11 11 22 2332

22 11 22 3113

d d cos (d d ) sin (d d ) (d d ),

ddsin(dd)cos(dd)(d d),

dddd.

xx xx

ππ θ θ ππππ

ππ θθ

δ−δ = δ−δ + δ−δ − δ− δ

δ −δ =− δ −δ + δ −δ − δ − δ

δ−δ =δ−δ

(22.2.42)

Lagrange’s equations in quasi-co-ordinates (22.2.22') become (with

= 0

γ )

⎛⎞

∂∂∂

−− =+

⎜⎟

∂∂∂

⎝⎠

⎛⎞∂∂∂

−+ =+

⎜⎟

∂∂∂

⎝⎠

⎛⎞∂∂∂ ∂

−+ − =+

⎜⎟

∂∂∂ ∂

⎝⎠

***

311

112

***

322

221

*** *

1233

332 1

TTT

TTT T

πππ

ππ

πππ π











Taking into account (22.2.41'), the kinetic energy (22.2.37) will be given by

()

=++

*222

12 3

TM Iππ π .

(22.2.37')

The generalized forces being equal to zero (

===

123

0QQQ

), from (22.2.21) it

results

===

123

***

0QQQ ; starting from the constraint generalized forces

123

tan , , 0RRRλθ λ==−=, the formulae (22.2.21') lead to ==

0RR ,

Dynamics of Non-holonomic Mechanical Systems

447

=−

/cosR λθ

. Observing that

/ , 1,2, 3, / 0

ii j

TMi Tππ π∂∂= = ∂∂= ,

= 1, 2, 3j , one obtains the Boltzmann–Hamel equations

()

−= + =− =

123 213 3

0, , 0

cos

πππ πππ π

       ,

(22.2.43)

which lead to the same solution (22.2.39) – (22.2.39'''), with the initial conditions

(22.2.38'). Indeed, we get

===

123

,0,vt tπππω;

(22.2.43')

as well, we notice that

0π is a true co-ordinate.

22.2.3.2 Motion of a Heavy Circular Disc on a Plane

Let be a heavy homogeneous circular disc of mass M and of radius R , which rolls

on the plane

, in which are situated the axes

, Ox Ox

′′ ′′

of the fixed frame of

reference

′′′ ′

123

Oxx x , considered in Sect. 22.1.1.1; the position of the disc is specified by

five parameters, i.e.:

,,,xxψθ

′′

and ϕ (see Fig. 22.1), linked by the non-holonomic

and scleronomic constraint relations (22.1.3) or (22.1.3') (relations between four of the

five parameters), in case of the slidingless rolling of the disc on the rough plane

(case in which the relative velocity of the contact point

vanishes). Hence, the

considered system has only three degrees of kinematic freedom.

If we denote by

O the centre of the disc (the centre of the movable frame

123

Ox x x ,

non-rigidly linked to the disc, of axes

Ox along OI ,

Ox in the plane of the disc,

normal to

OI and parallel to the plane Π and

Ox normal at O to the disc, so that the

frame of reference be right-handed), we can write

′′ ′′ ′

=− =+ =

12 3

cos sin , cos cos , sin

OO O

xxR xxR xRθψ θψ θ ,

introducing thus five generalized co-ordinates

,,,,

xxψθϕ

′′

; differentiating with

respect to time, we get

′′

=+ −

′′

=− −

′

sin sin cos cos ,

sin cos cos sin ,

cos ,

xxR R

θθψ ψθψ

θθ





so that the constraint relations (22.1.3') read

()

′

+−+=

′

+−+=

cos cos sin sin cos 0,

cos sin sin cos sin 0.

θψψ θψθ ψϕ

θ ψψ θ ψθ ψϕ



(22.2.44)

MECHANICAL SYSTEMS, CLASSICAL MODELS

448

The kinetic energy of the mass centre

O , where we assume that the whole mass M

is concentrated, with respect to the pole

′

O , is given by

′′′

222

123

(1/2) ( )

OOO

Mx x x

and the kinetic energy with respect to the pole

O is given by

=++

12 3

0222

(1/2)[ ( ) ]TJ Iωω ω

, where ==

IIJ and

I are central principal

moments of inertia

(/4, /2)JMR I MR==; thus, taking into account the

relations (5.2.35) between the vector

ω and Euler’s angles, one obtains

()()

()

′′′

=++ +

⎡

⎣

++ + +

⎤

⎦

22 2 22

22 2

1cos

14cos 2 4cos ,

TMxx MR θψ

θθ ϕ θψϕ





(22.2.45)

the potential energy being given by

=− = sinVUMgRθ .

(22.2.45')

If we take the generalized co-ordinates in the order

′′

qxqx,

===

,,qqqψθϕ

, by comparing the constraint relations (22.2.44) with (22.2.14),

we get

== ==

=− =

====

10 11 12 13

14 15

20 21 22 23

24 25

0, 1, 0, cos cos ,

sin sin , cos ,

0, 0, 1, sin cos ,

cos sin , sin ;

aa aaR

aR aR

aaaaR

aR aR

ψθ

ψθ ψ

ψθ

ψθ ψ

starting from the given force

′

==−

33 3

,FF Mg, and from the position vector

′′′′′′′

=++

riii

123

12 3

OO O O

xxx, we can calculate the given generalized forces

==== =−

123 4

0, cosQQQQ Q MgRθ .

Lagrange’s equations with multipliers (22.2.13) are written in the form (for

= 1,2, 3,5j )

Mx Mxλλ

′′

==  ,

(22.2.46)

()

sin 2 cos cos cos sin cos ,

cos cos sin ;

ψθ ψθϕθ λψλψθ

ψθϕ λ θλ ψ

++=+

⎡⎤

⎣⎦

+= +





(22.2.46')

instead of the equation corresponding to the index

= 4j , we use the first integral of

the mechanical energy

sin , constTMgR hhθ

′

=− + =

(22.2.46'')

Multiplying the equations (22.2.46) by

cos ψ and sin ψ , respectively, summing and

taking into account the constraint relations (22.2.44), we get the relation

Dynamics of Non-holonomic Mechanical Systems

449

()

⎡

⎤

+= − +

⎢

⎥

⎣

⎦

cos sin sin cos

λψλψ θϕθ ψθϕ



;

replacing the left member of this relation in the equations (22.2.46'), we obtain

()

=+=

sin 2 , 3 cos 2 sin

ψθ θϕ ψθϕ ψθθ

    



wherefrom

()

sin 2 , 3 cos 2 sin

θϕψθϕψθ

θθ

=+=





(22.2.47)

Eliminating

ϕ between these last two equations, we obtain the non-linear differential

equation

+−

dd10

3cot

ψψ





(22.2.48)

the unknown function being

= ()ψψθ



; by the change of independent variable

cosu θ , we get the differential equation of second order

−+−−=

d1 d5

(1 ) (1 5 ) 0

2d6

uu u

ψψ





(22.2.48')

This equation is of the type of Gauss’s hypergeometric differential equation

[

]

′′ ′

− +−++ − =(1 ) ( 1) 0xxy xy yγαβ αβ

(22.2.49)

with the general integral of the form

−

= + +− +− −

() (,,;) ( 1 , 1 ,2 ;)yx AF x Bx F x

αβγ α γβ γ γ ,

(22.2.49')

where

A and B are constant and (,,;)Fxαβγ is Gauss’s hypergeometric series

( 1)...( 1) ( 1)...( 1)

(,,;) 1 ,| | 1,

! ( 1)...( 1)

Fx xx

αα α ββ β

αβγ

γγ γ

∞

++−++−

=+ <

++−

∑

(22.2.49'')

In case of the equation (22.2.48') we have

3/2, 5/6αβ αβ+= = and = 1/2γ ,

so that the general integral of this equation is given by

()()

=+++

1113

, , ;cos , , ;cos cos

2222

AF BFψαβθ αβ θθ



(22.2.48'')

MECHANICAL SYSTEMS, CLASSICAL MODELS

450

A and B being arbitrary constants.

To study the equation (22.2.48), one can use also the change of independent variable

−=

(1 2 ) cosv θ , which leads to the equation ( = 2γ )

−+−−=

dd10

(1 ) 2(1 2 ) 0

vv v

ψψ





(22.2.50)

the general integral of which is given by

()()

′′

=+−−

222

,,2;sin 1, 1,0;sin cos

222

AF BF

θθθ

ψαβ αβ



(22.2.50')

or by

()()

′′ ′′

=+−−

222

, ,2;cos 1, 1, 0;cos cosec

222

AF BF

θθθ

ψαβ αβ



(22.2.50'')

where

3, 10/ 3αβ αβ+= = , while

′ ′ ′′ ′′

,,,ABA B

are arbitrary constants.

Analogously, by means of the substitution

sinu θ one obtains an equation of

Gauss with

= 2γ , the general integral of which is

()( )

′′′ ′′′

=+−−

222

,,2;sin 1, 1,0;sin cosecAF BFψαβθ αβ θθ



(22.2.51)

with

3/2, 5/ 6, ,ABαβ αβ

′′′ ′′′

+= = being arbitrary constants.

The first equation (22.2.47) allows now to express

= ()ϕϕθ by hypergeometric

series too.

Taking into account (22.2.44), we can express the kinetic energy (22.2.45) in the

form

()

′

=+ +++

⎡⎤

⎣⎦

22222

15cos 5 6 12 cos

TMR θψ θ ϕ ψϕ θ

 



;

(22.2.45'')

the conservation theorem (22.2.46'') leads to the equation

()

22 2 2

15cos 5 6 12 cos ( sin), ,

RMgR

θψ θ ϕ ψϕ θ θ++++=−=

 



(22.2.52)

which, after replacing



and ϕ , allows to obtain, by a quadrature, the time t as a function

of the angle

θ . One can thus calculate = ()tθθ and then determine the generalized

co-ordinates

11 22

( ), ( ), ( )

OO OO

xx xxθθψψθ

′′ ′′

=== and = ()ϕϕθ, by quadratures

too; one obtains thus

11 22

(), (), ()

OO OO

xxtxxt tψψ

′′ ′′

=== and = ()tϕϕ .

The kinetic energy and the potential energy being given by (22.2.45'') and (22.2.45'),

respectively, one can write Lagrange’s kinetic potential

′

=+TU

L in the form

Dynamics of Non-holonomic Mechanical Systems

451

()

⎡⎤

=+++−

⎢⎥

⎣⎦

2222

6cos 5 sin sin

ψθϕ θψ θ θ





L ,

(22.2.53)

where the generalized co-ordinates are

( ), ( ), ( )tt tψψ θθ ϕϕ===. We obtain

thus Lagrange’s equations

()

++ =

⎡⎤

⎣⎦

++− +=

12 cos cos 2 sin 0,

10 12 cos sin 2 sin cos cos 0,

12 cos 0,

ψθϕ θψ θ

θψθϕψθψθθ θ

ψθϕ



   



which lead to the first integrals

()

++==

+= =

6cos cos sin , const,

cos , const;

ψθϕ θψ θ

ψθϕ





(22.2.54)

the first of these relations can be written also in the form

6cos sinbaθψ θ



(22.2.54')

while the second equation of Lagrange becomes

+− +=

56sin sincos cos 0

θψθψθθ θ

  

(22.2.54'')

Eliminating



between the relations (22.2.54') and (22.2.54''), we get

++ −+ + =

32223

5 sin 6 (1 cos ) ( 36 )cos sin cos 0

ab a b

θθ θ θ θθ



;

multiplying by

2θ



and integrating, it result

() ()

222

cos 36 cos 8

5 6 ln tan 6 ln tan sin

sin sin sin

ab g

ab ab c

θθ θθ

θθ

θθθ

−−+−++=



()

++ − + ==

22 2

181

5 36 12 cos , const

sin

abab cc

θθ



(22.2.55)

hence a first integral of the previous equation. This equation with separate variables can

be integrated by a quadrature, obtaining the time

t as a function of θ , and then

= ()tθθ . Returning to the relation (22.2.54') and then to the second relation (22.2.54),