.

,

5. 6. 7.

.

.

Yu. M. Dal and Z. N. Litvinenkova,"Supercritical deformation of a plate with a crack," Prikl. Mekh., i_~i, No. 3, 64-71 (1975). M. Sh. Dyshel' and O. B. Milovanova, "Method of experimental investigation of buckling of plates with a slit," Prikl. Mekh., 13, No. 5, 99-104 (1977). Z. N. Litvinenkova, "Buckling of a tensioned plate with an internal crack," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 148-151 (1973). B. L. Clarkson, "The propagation of fatigue cracks in a tensioned plate subjected to acoustic loads," in: Acoustical Fatigue in Aerospace Structures, Syracuse University Press (1965), pp. 361-388. J. R. Dixon and J. S. Stranningan, "Stress distribution and buckling in thin sheets with central slits," in: Proceedings of the Second International Conference on Fracture (Brighton, 1969), Chapman and Hall, London (1969), pp. 105-118. G. F. Zielsdorff and R. L. Carlson, "On the buckling of thin tensioned sheets with cracks and slots," Eng. Fract. Mech., 4, 939-950 (1972).

MOTIONS OF A NONHOLONOMIC SYSTEM UDC 531.01

N. V. Nikitina

The present article is a continuation of [4], where it was shown that theWagnar integration method [I] can be applied to the problems in which certain active forces are taken into consideration. Here we investigate different cases of motion of simplified models of a wheel carriage, in which the transmission of breaking forces to one or the other pair of wheels is anticipated with their blocking at the axis of rotation. Here it is undesirable to have turns of the carriage frame that would increase with time. The object of this work is to compare the variations in the angles of rotation of the carriage frame for different cases of realization of some nonholonomic couplings, using the Wagner method with the supplementation indicated in [4] for the solution of equations of motion of nonholonomic systems. w We consider the motion of the simplest model of a carriage on identical rigid wheels of radius r, which are represented by equivalent rear and front wheels [5]. Let the front wheel slip withoutrotating; the rear wheel rolls without slipping. The position of the system is determined by the parameters x, y, ~, 0, ~z; the kinetic energy is equal to

The coupling equations have the form

:o;

=o.

(1.1)

We introduce new coordinates: xl=

x~ = l r M g ;

I'Mx;

x3|'O~A~;

x, : F~OO; xb = I T % .

(1.2)

In terms of (1.2) coupling equations (i.i) become --xx sin c x , + xz co~ cx, - - bxa = O; xz cos c x , + xe sin cx a - - a x 5 = O,

where

a=]/

bl;

,;

We introduce two unit vectors Sin CX~

-

- CX4 COS

- - 7~

b

FI~

-

'

-

cos

cx4

ea = i I ] / - 1 ~

which are orthogonal to the nonholonomic manifold in space. thogonal unit vectors lying in the nonholonomic manifold:

s i n cx4 + ~' l/]"-'t- a'

i6

a l / ~

'

We construct three mutually or-

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 14, No. i, pp. 131-134, January, 1978. Original article submitted November 19, 1976. ]04

0038-5298/78/1401-0104507.50

9 1978 Plenum Publishing Corporation

a cos cxa

a sin cx 4

V-~

V ~

1

= + 7 ~ V l + b~

We then define the angle of rotation of the unit vectors: abcx4

V-~.~Vi+~

v=-

We shall assume that we have incomplete dissipation given by the Rayleigh function

2R = k,q~,2 + k~O~,

(l.

3)

where kx is the coefficient of viscous friction of the medium appearing during the change in angle ~ ; k2 is the coefficient of viscous friction of the medium appearing during the change of angle O. We obtain the first integrals after separating the variables in the equations of motion: 9

~, =

ck 2 -

-~

Vi + ,,~ r

abcx4

V-i-~-~ ~ ] =+ b--~+ Cl;

ck 2 . / :-

.

n2 : - - ~

abcx 4

t' i d- a z sin V - i ~ - a ~ 1/1,F-+ b~ + Cv

We put the constants of integration C:, Cz equal to Cl=,4sina;

C~:Acosa.

Finally we get

ckz VI,---+ a~ ~

Xi

cos cxi

J

Aab

cos (? + ~z)

dx 4 +

cos cx4 tg (V + ~x)ax~ + 0__ccos cx4 + Cs;

-r ~ xz=

ck~F~l~ Aab

sincx4 .dx4 + , COS (? + e.)

S

(1.4)

_1 a V-1--"+-'~ f sin cx, tg (V + a) dx 4 + b sin cx 4 5- C,;

V ~

c

hi

X3 =

C5

t

O2A

+ C~e

Aabct

x~=

|"'l--+--h-~l/-l~ abc

2arctg(CTe(l+b')l~)---~---~

xa=-- ke (I -t"Aa2b b2) 1''2 1 'l- am In tg [ 21_~_(~ +

a) +

-~-] - - ~-~

;

In cos (? + a) + C 8.

w We discard the energy of the rotational motion of the wheel; energy of the system will be 27 = M (~2 + ~2) +

Let one condition, the system

i.e.,

then the kinetic

O2Aq~~_ 062.

(2.1)

the absence of slip of the front axis, be imposed on the motion of -- x sin 9 + Y cos9 + 6ai cos(9 -- O) = 0

and

~ = 9o,

~ = 0 for t = 0.

Then in variables

(i. 2) coupling equation

- - xl sin 90 + xa cos 90 + x,ao (q% - - cx4) = O,

(2.2) (2.2) becomes (2.3)

where a0 =

The coupling equation (2.3) is integrated. sipation (1.3) into consideration:

~1. We shall use the Wagner method and take dis-

105

v" O/J.o OJo~ 0,078

Z

8035

o,f

#,2

0,3

"S #,~ t, ~

Fig. 1

t--~ S T ] / l

+ a2 cosZ(~o--cx4)dx~+ C~;

xt=---~-sinqDoSin(~o--cx~)-{-ClCOS~o ao

x2 = c

cos q~o sin (q:o - -

cx4) + CI sin

q%

(2.4)

V1 +a2ocos~(~o--cx4)dxa+Cs;

~ 1 7

] / 1 + a~ cos~ (q% - -

cx4) dx~ + C~,

where X4

J

=

-

.. V 1 + a~co~'(~o - c u )

0

w In the analysis of motion of a braked carriage it is important to compare different regimes, in which blocking of one or the other pair of wheels is anticipated. It is undesirable to have turns of the carriage frame that would increase with time [2]. The change of angle of rotation 9 in different cases is shown in Fig. i. Curve I corresponds to (2.4) and shows the change in 0 (t) when a single nonholonomic c o u p l i n g 0 c c u r s ; i.e., there is no lateral slippage of the front axis with the wheels. Curve 2 takes the energy dissipation (1.3) into consideration. Curves 3 and 5 correspond to the change of ~(t) in the absence of rear axis lateral slippage; the energy dissipation (1.3) has been taken into consideration in constructing 3. Curve 4 corresponds to solution (1.4), where the energy of the rotational motion of the wheels has been taken into consideration. Numerical data from [3] were used in constructing

curves i, 2, 3, 4.

The following data were used for the first case of the motion: x = 0, y = 0, @ = 0, ~ = 0.175 rad; x = ii m/sec, ~ = 0, y = 0.

k2 = 35 kgf/m at t = O.

For the second case we used: k2 = i00 kgf/m at t = 0, x = 0, y = O, 0, ~ = ii m/sec, 0 = 0. i sec -I, ~ = 0. Curve 2 was constructed by the approximate

Cauchy--Euler method

~ = 0.175 rad,

~=

[6, 7].

The braking regime with anticipation of the blocking of front wheels is the least hazardous, since in this case the growth of 8 is limited. LITERATURE i.

2. 3. 4. 5. 6. 7.

106

CITED

V . V . Wagner, "Geometrical interpretation of motion of nonholonomic dynamic systems," in: Proceedings of a Seminar on Vector and Tensor Analysis (Scientific-Research Institure of Mathematics) [in Russian], No. 5 (1941), pp. 301-327. B . B . Genbom, V. A. Dem'yanyuk, and T. G. Myskiv, "On the stability of motion of a braked automobile," Avtomob. Prom-st, No. 3, 22-25 (1974). L . G . Lobas, "Some aspects of general formulation of the problem of motion of an automobile," Mekh. Tverd. Tela (Resp. Mezhved. Sb.), No. 8, 98-106 (1976). N . V . Nikitina, "Interpretation of equations of motion of nonholonomic systems with partial energy dissipation," Prikl. Mekh., 12, No. 8, 119-122 (1976). N . V . Nikitina, "System control stability after elimination of nonholonomic couplings," D0Pov. Akad. Nauk UkrSSR, Ser. A, No. 9, 806-810 (1976). V . I . Smirnov, A Course in Higher Mathematics [in Russian], Vol. i, Nauka, Moscow (1974). E. Jahnke, F. Emde, and F. LDsch, Tables of Higher Functions, 6th ed., McGraw-Hill (1960).