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The instantaneous distribution of acceleration of a spherically moving system
Y. Meeha~ms Vol. 1, pp. 23-42. Pergamon Press 1966. Printed in Great Britain
THE INSTANTANEOUS DISTRIBUTION OF ACCELERATION OF A SPHERICALLY MOVING SYSTEM Prof. Dr. -Ing. W. MEYER ZUR CAPEff.LRN* and Dr. -Ing. G. D ~ C H Institut f'dr Getriebelehre, Technische Hochschule, Aachen (Original German version received 21 June 1965)
AlcOtt--The case of the accelerated movement of a rigid body in space is considered, for which one point of tim body is fixed. Tho~ points of the body are sought for which one or another component of their total acceleration vanishes; these define two cones, which are analogous to the Bresse circles of plane kinematics. 1. INTRODUCTION IF a body is rotating around a fixed point O of itself, the velocity and acceleration vectors o f all points on a random line through 0 will be parallel and their magnitude proportional to the distance between the point and the fixed point 0. Therefore it will be sufficient to consider only those points of the body, which are situated on a sphere of the radius R r = 1 unit of length. The velocity and acceleration of points having an arbitrary distance R r ~ 1 from the fixed point therefore need to be multiplied by R K. Trihedrals o f vectors
The spherical system is taken to be rotating instantaneously round the instantaneous rotation axis O P with an angular velocity to, for which the magnitude co#0, Fig. l(a).
--t,+ .....
!,' j
/
.,.=
Fxo. l(a). Nomenclature. * Director, Imfitut fi}r Getriebelehre, Technische Hochschule, Aachen. 23
24
W. MEYER ZUR CAPELLENand G. DITTRICH
The point P, one of the two points of penetration of the instantaneous rotation axis through the unit sphere, shall be given an orthogonal trihedral of vectors, consisting of the position vector p--OP and the unit vectors t and n according to the vector equation
nfpxt
(1)
The vectors p and t are situated in the plane tangent to the slant side of the pole cone, and of these t gives the direction in which the instantaneous rotation axis changes. An axis OC of the spherically moving body is chosen now at random; let it lie in a plane through the pole axis, which is inclined to the tangential plane of the pole cone by the angle tp. Let the angle between the axes OP and OC, that is to say, the spherical distance between the points P and C on the unit sphere be r. As the body moves, the point C will describe a spherical curve kc, Fig. l(b), with the spherical radius of curvature p=CoC
FIo. 1Co). The osculating cone. at the considered location. OCo becomes the center-line of a circular cone (osculating cone), which will touch the path cone on at least three infinitesimally adjacent axes. Let the spherical distance between the center of curvature C o and the instantaneous center of rotation P be ro. Therefore the spherical radius of curvature is p=ro-r.
(2)
To the moving point C we now ascribe an accompanying trihedral of vectors, consisting of the position vector O~C= C, the vector along its path tangent T, and the vector of its path normal N. The vectors T and N will form the plane tangent to the sphere through the point C. Relative directions are governed by N-- C x T.
(3)
Coordinate systems For description of the geometry of motion, Cartesian frames O X Y Z are introduced, such that the Z-axis is coincident with OP and the XZ-plane agrees with the tangential plane of pole cone formed by the vectors p and t, see Fig. l(a). The XZ-plane and the YZ-
The instantaneous distribution of acceleration of a spherically moving systan
25
plane meet the unit sphere in two orthogonal great circles, which may be considered as the coordinate lines of a spherical xy-system with the origin M at P. The conjunction between the Cartesian coordinates XYZ and the spherical coordinates x y can be derived easily from Fig. 2 as follows: X tanxfy, Y
tan y = ~
(4a) (4b)
with X 2 + 7 2 + Z ~ --Rx 2 -- 1. The equations between the spherical coordinates xy and the spherical polar coordinates (r, cp) follow from Fig. 3 by means of the Napier rules for spherical geometry
I
Ly
I Fie. 3. Spherical eoordlnate system.
26
W. MEYER ZUR CAPELLEN and G. DITrRICH tan x f t a n
r
cos ~p,
(5a)
tan y = t a n r sin ~o,
(Sb)
and from these tan2r =tan2x + tan2y.
(5c)
Furthermore, from Figs. 3 and 2 the following relations can be found X=sin rcos ¢p,
(6a)
Y= sin r sin ~,
(6b)
Z--cos r.
(6c)
2. THE ACCELERATION The angular acceleration $ measures the instantaneous change of the angular velocity vector, or dm e=-=/o (7) dt Herein dm consists of the two components- do~. p and - ¢ods. t, Fig. 4. The first of these components, characterizing the change of magnitude of the angular velocity vector, may be expressed by d~o ~, = - (s) dt
0 ~
FIe. 4. Mvafion of the angular a c c ~ t i o n .
~~
The instantaneous distribution of acceleration of a spherically moving system
27
and the second component, characterizing its change of direction, may be expressed as ds u--~.
(9)
The latter measures the velocity of change of position of the pole. In these terms the magnitude of angular acceleration is given by
8 = x/(cb2 + O32u2)
(I0)
but if the abbreviations 8:
j=~-~,
(11a)
~b ~b=~-~
(lib)
and u
0=-
(11c)
j=~/(~2 +02).
(12)
O3
are used, equation (10) willappear as
The angular acceleration ~ has the direction of dee, which may be specified by OP, (Fig. 4), and therefore it must be situated within the tangential plane of pole cone. The angle g between the direction OP, of the angular acceleration and the instantaneous axis of rotation can be calculated in the following way
), o3u sin 0-------
8
0
4(
2+02) '
(13a)
(13b)
and thus
tang=
0
.
(13c)
28
W. M E Y E R Z U R CAPELLEN and G. D r I T R I C H
The velocity v of the point C may then be written v--v'T
(14)
in which v--o~" sinr gives the magnitude and T the direction of the vector v. The acceleration vector can be found by taking the time derivative of equation (14): dv a--~---O "T+v "T.
(15)
Should the path element ds* be introduced instead of time, then 'i" dT =
dT ds*
d'-'t'--'ds"-;dt
=v •
T'
(16)
is obtained. The primes here signify derivatives with respect to the path element ds*. In terms of the trihedral of any point moving on a unit sphere, the following equations (refer to [1-3]) for the derivatives of its motion are always available
ETIi0 N' C'
=
ot 0jEll
-cotp
0
1
(17)
0
and this T'=
1
-N-C tanp
(18)
can be deduced. By combining equations (18), (16) and (15), the acceleration can now be written v2 a -- 6T + - - N -
tanp
vzC
(19)
or
affi~+aN+ac
(20)
in which the first term is the tangential acceleration, aTf6T.
(21a)
The next term designates the normal acceleration p2
amffitan/~N and the last term is the radial acceleration directed toward the fixed point 0
(21b)
The instantaneous distribution of acceleration of a sphericallymoving system ac-- - vzC.
29 (21c)
The normal acceleration and tlle radial acceleration form together the total normal acceleration of the spatially moving point. Before investigating the geometrical loci where the components of acceleration will vanish singularly or all together, the acceleration of an arbitrary point C should be looked at from another point of view [4]. The velocity of the point C may be written in the form v-c0x C
(22)
a--6xC+mxC
(23a)
a = ~ x C + o~x v.
(23b)
and therefore the acceleration is
or using @ •8 and C--dC/dtffiv
It is evident, that the acceleration can be expressed by the sum of its components
affia,+a .
(24)
In this, a~ =E x C
(25a)
is an acceleration normal at the point C to the plane formed by s and C and is only dependent on s, whilst a~ • m x v = m x (mx C) =(mC)m- m2C
(25b)
ties within the plane PCO through the pole axis and is solely dependent on o~. 3. THE CONSTANT SPEED CONE The surface of the "constant speed cone" k o is defined as containing those axes of the body, whose points Co momentarily possess no tangential acceleration. That means, that the acceleration vector a, must lie in the plane CgPO of the path normal, i.e. the plane Col',O must be perpendicular to the plane through the pole axis, as in Fig. 5. In the rectangular spherical triangle PCoP. with hypotenuse PP, it follows from Napier's rules of spherical geometry that the equation for the constant speed cone is tan r--tan g cos cp,
(26)
in which g is defined by equation (13c). In view of equations (5a, b) the equation (26) can be written in spherical xy-coordinates: tan2x-6 tan2y - tan g tan x -- 0.
(27)
30
W. MEYER ZUR CAPELLEN and G. DITTRICH
Y Isss'''-
f,', i...--_-J-X !
II
~
-
//
-
I
/
ss
•
Fro. 5. The "constant speed cone", defined as the locus of all points without tangential acceleration. Introducing X,. Y, Z as homogeneous coordinates we get
X•2
/ y \ 2 _(X)tano__ 0
(280
or
X 2 + y2 _ XZtan O = 0
(28b)
The constant speed cone is therefore a cone of second order. In order to investigate whether this cone penetrates the equatorial plane which is perpendicular to the instantaneous rotation axis OP, the coordinate Z is made zero. Then equation (28b) reduces to X 2+ y2 _ 0
(29a)
(X + i iO(X - i tO=O
(29b)
or
with i = x / ( - 1). Thus the constant speed cone and the equatorial plane have in common the mutual imaginary axes
X+iY=O and
X - i Y=O
(30a/b)
with the directions Y tantpx, 2 = 2 = + f .
(31)
These axes correspond to the two imaginary circle-points which occur in planar kinematics.
The instantaneous distribution of acceleration of a spherically moving system
31
By the above deduction it is evident, that the constant speed cone is symmetric both to the tangential plane PP,O of the pole cone and to a plane through the midpoint axis OM o of k o, the plane being inclined at an angle g/2 with the normal plane of the pole cone. The constant speed cone must touch the normal plane of the pole cone in the instantaneous rotation axis OP according to equation (26), see Fig. 5. The instantaneous center of rotation P cannot have an acceleration caused by e), because v is always zero (see equation 25b). Therefore the acceleration of the instantaneous center of rotation may be derived from equation (25a) by putting C = p = t x n , which yields ap =a, = s x (t x n) = ( s n ) t - (st)n.
(32)
The first term of this sum will be zero because s is normal to n. It may then be recalled that, according to Fig. 4, the angle between 8 and t is equal to (~/2 + g); as a result, the acceleration of the instantaneous center of rotation can be written ap •e sin 0" n
(33)
Its direction is coincident with that of the normal to the pole path. Equations (33) and (13b) give its magnitude ap =o~u=Oo~2.
(34)
In order to find out more of the geometrical features of the constant speed cone, the curve of its intersection with the tangential plane Z = Rk = 1 touching the sphere in the point P will be determined. Equation (28b) leads to X2+ y 2 - Xtang-~O or
1
• \2
The curve is therefore a circle with diameter tan g, of which the center is displaced by ½ tan g in the X-direction. Any plane which is perpendicular to the central axis OMg and which does not pass through the fixed point O, cuts the cone in an ellipse. The constant speed cone therefore must be an elliptic cone, which intersects the sphere in a spherical ellipse. Without demonstration (refer to [5, 7] let it be said here, that when the half of the smaller angle of aperture is taken to be b •g/2, the bigger one can be obtained from the relation: tan a = ~ mIn ~ . tga n g .
The focal axes lie in the tangential plane of the pole cone and are inclined by the angle e to the central axis OM o. Then this angle e may be found from the relation: sin e---+ tan 2 -g 2"
32
W. MEYER ZUR CAPELLEN and G. DITTRICH 4. THE INFLECTION CONE
The condition that the normal acceleration aN should vanish (see equation 21b), is that the spherical radius of curvature becomes p = 7r/2. At the same time this is the condition for those points of the system, which instantaneously will have an inflection point or flat point in their path. The path therefore has at least three mutual infinitesimally adjacent points of contact with a great circle. Now the Euler-Savary formula of spherical kinematics [1, 5] is 1 tanr
1 1 t a n r o Osin~o
(35)
and if we introduce into this the radius of curvature according to equation (2) together with the abbreviation tan w -- 0 sin tp,
(36)
tan2r tan p = tan w tan2r-- tan r + tan w"
(37)
there is obtained
When furthermore the condition p = ~/2 is satisfied, the equation
(38)
tan w t a n 2 r - tan r + tan w = 0
is found and therewith the equation defining the "inflection cone" kw is known. This is the geometrical locus of all points, whose path is without normal acceleration. Its equation can be written (tan2x + tan2y)(0 tan y - 1) + 0 tan y = 0
(39)
in spherical xy-coordinates, or alternatively
,] +
o
(4Oa)
in Cartesian coordinates. The latter can be written ( X 2 + I"2)(0 Y - Z ) + 0 Y Z 2 =0.
(40b)
The inflection cone is thus a cone of third order which intersects the equatorial plane (Z--0) in the X-axis (Y=0) and on the two imaginary equatorial axes given by equations (30a, b). Solving equation (38) now for tan r, we get tan r -- 2ta~w [1 -I-~/(1 - 4 tan2w)].
(41)
The instantaneousdistributionof accelerationof a sphericallymovingsystem
33
and it is then evident, that real values of tan r will exist only for
12tanwl_~l or that is to say, for 1 sm~0_~.
(42)
For example, a limiting angle cp~30 ° is obtained in the case 0 = 1 . But where 0_~0.5 the values of tan r are always real. The equation (38) for the inflection cone may also be simplified to the form sin (2r) -- 20 sin rp.
(43)
This equation shows, that the inflection cone is symmetrical to the normal plane of the pole cone: (9=~/2) and that it touches the tangential plane of the pole cone (~p--0) at the instantaneous rotation axis OP. T h e normal plane of the pole cone cuts the infle~don cone along the inflection pole axes O P w. The inclination angles 6 of these axes relative to the tangential plane of the pole cone are found to be sin26 =20
(44)
because we may set rffi6 and ~ =~/2 in equation (43). Three cases therefore arise: if 0>0.5 there exist no real inflection pole axes; for 0ffi0.5 there is only one axis with the inclination angle 6---45°; and for 0<0.5 there are two real inflation pole axes because the inflection cone disintegrates into two different cones. Y
s ~ .
Fro. 6. The "inflectioncone", being the locus of all points withoutnormal n_~ocderation. Parameter values0----1, ½and ½. Figure 6 shows the intersection curves of three characteristic inflection cones kw with the unit sphere. Furthermore, the intersection curves of the inflection cones kw of the kinematic inversion are drawn, but these will not be discussed here in further detail.
34
W. MEYER ZUR CAPELLEN and G. DITTRICH 5. THE AC~I.EitATION AXES
Since with to # 0 and e # 0 there exists no axis in a spherically moving body along which the total acceleration vanishes, we shall take those lines of the body, whose points have neither tangential nor normal acceleration to be named the instantaneous acceleration axes OJ [6]. They occur as the lines of intersection of the constant speed cone and the inflection cone. Two cones of second and third order, with mutual cusps, in general have six slant lines in common, but in this case two of them are always coincident with the two imaginary equatorial axes and one with the instantaneous rotation axis OP: the latter has already been identified as a singular axis. Therefore there remain three lines of intersection, the position of which and reality conditions for which still need to be discussed. Taking into consideration equation (13), equation (26) is substituted in equation (38), and thus an equation for the directions ~p of the acceleration axes is derived: tan aq~-~1 tan2tp + ~ (~b2+ 02) tan q~_~ = 0 .
(45)
This equation of third order will in general have three solutions which either are all real or two of them are imaginary. Equation (45) can be reduced by substituting tan,=2+
1 ~-~.
(46)
Thereupon it assumes the form 23 +p2 + q =0,
(47)
where 1 / 2
"2
1~
p=-7"~l O~p_\+¢P ---3)
(48a)
and
1 1'02 2 2 2\
(48b)
The roots of such a cubic equation are real if and only if the discriminant /'q'~2
A=~)
1'p~3
+~)
~0
(49)
that is, when
2~2+@3~o.
(~)
It follows from these equations (48a, b) that the three acceleration axes are real, if and only if F-- 4(~b2 + 02)3 _ (~b2+ 02)2 _ 18~b2(~b2 + 02) + 27~b4+ 4~b2 _~0.
(51)
The instantaneous distribution of acceleration of a sphericallymoving system
35
Using the abbreviations
{/)2-~-02=~ and
~b2---t/,
(52a/b)
equation (51) assumes the form F_= 4~3 _ ~2 _ 18~/+ 27~/2+ 4q ~ 0
(53)
The reality of the roots depends on the parameters ~ and ~/. We shall now consider where in the ~/plane these points are located, when the equation (53) is taken to have an equal sign. Equation (53) describes a cubic curve kl,. For any chosen value of ~ =~b2+ 02, however, a quadratic equation in t/remains: F = 27~/2+ 2 ( 2 - 90q + ~2(4~- 1).
(54)
Its discriminant is A* = 4[(2- 902 - 27(4~ - 1)~2] or more simply A* = 16(I - 3 0 s.
(55)
From this it follows that the roots for t/=~b 2 are real only if (56a) For ~ffi~
(56b) If ~ and r/written
~__/~+1
(57a)
~/=v+~-7,
(57b)
and
F*--4#3 + 3(/x-v) 2
(58)
follows. It is easy to deduce from this, that the curve k~,* described by this function contains a cusp at the point (#--0, v - 0 ) with two coincident tangents, the equation of which is /z-3v=O
(59)
¢-3n- =o.
(6o)
but for the curve ke
36
W. MEYER ZUR CAPELLEN and G. DITTRICH
The curve kF touches the f-axis in the origin (0, 0) and intersects it at the point 0 / 4 , 0). Within the zone of interest the curve has the shape shown in Fig. 7. It can be seen that the points (~, ~/) which describe an instantaneous movement with three discrete real acceleration
•.
0~4
11127 *:w-~003 .L2 ~Z
/
__002
0-01
(>1
0
o
0~2 ~/4
0"3
I
-0.01
I I ! !
-002
F/o. 7. Diagram for reckoninR the number of real acceleration axes. In the first q~djant there exist three discrete acceleration axes for pairs of values within the curved triangle, on the two b m n c ~ of the curve two coincident and one discrete axes while at the cusp three coincident axes can be found; in the remaining area only one real acceleration axis occurs.
axes appear only in the first quadrant within the triangular region limited by the curve k F. Here is A < 0 and F < 0 . The directions of the three acceleration axes in this case may be found from equation (46) by setting gj = 2 x / ( - P/3) • coslg + ~ (j - 1)~1 ,
] = 1, 2, 3
(61a)
a n d t¢ f r o m
cos 3K =
q 2x/(_p/3)3.
(61b)
An example of an instantaneous movement with three discrete real acceleration axes is shown in Fig. 8(a). The points on k r ( F = 0 ) belong to instantaneous movements which have
The instantaneous distribution of acceleration of a spherically moving system
37
Y
!
I~
,
eI
I
i
X
I
Fxo. 8(a). Movement of a spherical system with [~b2 ~ 0.023, 02=0.27] which causes three discrete real acceleration axes.
one discrete and two coincident acceleration axes, the positions of which may be got from equation (46) by substituting in turn
21 =2~/(-q/2)
(62a)
22 =23 = - ~/(-q/2).
(62b)
and
One example is shown in Fig. 8(b). Then for the special c a s e r / = ~ b 2 = 1 / 2 7 and ~ = ~ 2 + 02-- 1/3, i.e. 02 =8/27 all three acceleration axes are coincident, Fig. 8(c), and in this case
FIG. 8(b). Movement of a spherical system with [~b2 ~ 0.028, 02 =0.27] which causes one discrete and two coincident real acceleration axes.
38
W. MEYER ZUR CAPELLEN and G. DITTRICH
I F1o. 8(c). Movement of a spherical system with [~b2= 1/27, 02 = 8/27] and thus three coincident r~fi acceleration axes inclined at an angle ~=60 ° with the spherical x-coordinate. equation (45) simplifies to (tan
¢p -
~/3) a = 0
with the solution tan ~p1.2, 3 = n/3 that is, ~Pl. 2, 3 = 6 0 ° .
The three coincident acceleration axes are inclined at an angle ~p= 60 ° relative to the spherical x-coordinate. In all the other cases for which ( A > 0 , F > 0 ) there exists only one real acceleration axis, as in Fig. 8(d). The plane t h r o u g h the instantaneous rotation axis
IY
F[o. 8(d). Movement of a spherical system having [~b2 =0"27, 02=0.27] with one real acceleration axis.
The imtantanoous distribution of acceleration of a spherically moving system
39
which contains also this real acceleration'axis is again defined by equation (46), if
'#1'1=~1 i "l"/~2*
(63a)
is used in it, where
z~,'*2 = ~ / [ - (q/2) + ~A]
(63b)
I n the Figs. (9a-c) and (10a-c) the acceleration axes OJ are drawn as chain lines between the inflection cone kw and the constant speed cone kg for further characteristic pairs o f the parameters 0 and ~b.
Y
/
I,/
L, ,'.----/
\ xt.t
-
,~ #I ]
l! i
#
#
I
FIo. 9(a). Moveznent of a spherical system having
[~2 ~0"016, 02=0-25] with three discrete real acceleration axes.
Fxo. 9(b). Movement of a spherical system having
[~2 = 0 ~ ,
a2=o.25]
with one discrete and two coincident real acceleration axes.
40
W. MEYER Z U R CAPELLEN and (3. DITFRICH
Y
f//I /I ~11 I
/
,
/
X
FIo. 9(¢). Movement of a spherical system having
[4)2--o.25, 02=0.2.~] with one real ac~leration axis.
Y °
• ' I " ;I
l
I ! // /
j
N o . 10(a). Movement of a spherical system having
[ ~ o ~ o 2 , e2--1/9] with three discrete real acceleration axes.
The instantaneous distribution of acceleration of a spherically moving system
41
,Y
FIG. 10(b). Movement of a spherical system having [~b2 ~,0.004, O2ffi1/9] with one discrete and two coincident real acceleration axes.
~Y
X
....
L-2 ..... I/
FIO. 10(c). Movement of a spherical system having
[~2 =1/9, 02=I/9] with one real ac~eration axis.
6. SUMMARY In order to survey the distribution of accelerations o f an accelerated spherically moving body the acceleration of a random point has been sprit up into a radial component directed toward the fixed point 0 and into tangential and normal components with respect to the spherical path. As an analogy to plane kinematics, the points whose tangential acceleration vanishes have been found t o be situated on the "constant speed cone" while the points without normal acceleration are placed on the "inflection cone". There m a y occur up to three discrete real acceleration axes as lines of intersection of the constant speed and the
42
W. MEYER ZUR CAPELLEN and G. DITTRICH
inflection cone; for all points on these lines both the normal acceleration and the tangential acceleration vanish. Other than the fixed point 0 there are no points of the accelerated spherically moving body for which the total acceleration is zero.
REFERENCES
[1] H. R. MOLLEI~Sph~rische Kinematik. VEB Dcutscher Verlag dcr W i ~ h a f t e n (1962). [2] E. I ¢ , a t ~ o , Di~'erentialgeometrle. Akad. V~'lagsg~,,llschaft Gcest und Portig K.-G. (1957). [3] M. I.~GJ~J~Y and W. F'n~NZ, Vorlesungen fiber Vektorrechnung. Akad. Verlagsgescllschaft Gcest und e o r ~ K.-G. (1959). [4] R. Brn~, Technische Raumkinematik. Springer-Vcxlag (1963). [5] G. D, lr~CH, Ueber die momentane Bewegungsgeometrie eines sph[uisch bewegten starren Systems. Dr. Diss. TH Aachen (1964). [6] O. BOTTEMA,Acceleration Axes in Spherical Kinematics. ASME Paper No. 64-Mech-9. [7] W. MEY~ zuR C A P ~ , G. DITI~CH and B. J A ~ S ~ , Systematik und Kinematik ebener und sphiirischer Viergelenkgetriebe. Forschungsbericht des Landes Nordrhein-Westfalen.
THE INSTANTANEOUS DISTRIBUTION OF ACCELERATION OF A SPHERICALLY MOVING SYSTEM Prof. Dr. -Ing. W. MEYER ZUR CAPEff.LRN* and Dr. -Ing. G. D ~ C H Institut f'dr Getriebelehre, Technische Hochschule, Aachen (Original German version received 21 June 1965)
AlcOtt--The case of the accelerated movement of a rigid body in space is considered, for which one point of tim body is fixed. Tho~ points of the body are sought for which one or another component of their total acceleration vanishes; these define two cones, which are analogous to the Bresse circles of plane kinematics. 1. INTRODUCTION IF a body is rotating around a fixed point O of itself, the velocity and acceleration vectors o f all points on a random line through 0 will be parallel and their magnitude proportional to the distance between the point and the fixed point 0. Therefore it will be sufficient to consider only those points of the body, which are situated on a sphere of the radius R r = 1 unit of length. The velocity and acceleration of points having an arbitrary distance R r ~ 1 from the fixed point therefore need to be multiplied by R K. Trihedrals o f vectors
The spherical system is taken to be rotating instantaneously round the instantaneous rotation axis O P with an angular velocity to, for which the magnitude co#0, Fig. l(a).
--t,+ .....
!,' j
/
.,.=
Fxo. l(a). Nomenclature. * Director, Imfitut fi}r Getriebelehre, Technische Hochschule, Aachen. 23
24
W. MEYER ZUR CAPELLENand G. DITTRICH
The point P, one of the two points of penetration of the instantaneous rotation axis through the unit sphere, shall be given an orthogonal trihedral of vectors, consisting of the position vector p--OP and the unit vectors t and n according to the vector equation
nfpxt
(1)
The vectors p and t are situated in the plane tangent to the slant side of the pole cone, and of these t gives the direction in which the instantaneous rotation axis changes. An axis OC of the spherically moving body is chosen now at random; let it lie in a plane through the pole axis, which is inclined to the tangential plane of the pole cone by the angle tp. Let the angle between the axes OP and OC, that is to say, the spherical distance between the points P and C on the unit sphere be r. As the body moves, the point C will describe a spherical curve kc, Fig. l(b), with the spherical radius of curvature p=CoC
FIo. 1Co). The osculating cone. at the considered location. OCo becomes the center-line of a circular cone (osculating cone), which will touch the path cone on at least three infinitesimally adjacent axes. Let the spherical distance between the center of curvature C o and the instantaneous center of rotation P be ro. Therefore the spherical radius of curvature is p=ro-r.
(2)
To the moving point C we now ascribe an accompanying trihedral of vectors, consisting of the position vector O~C= C, the vector along its path tangent T, and the vector of its path normal N. The vectors T and N will form the plane tangent to the sphere through the point C. Relative directions are governed by N-- C x T.
(3)
Coordinate systems For description of the geometry of motion, Cartesian frames O X Y Z are introduced, such that the Z-axis is coincident with OP and the XZ-plane agrees with the tangential plane of pole cone formed by the vectors p and t, see Fig. l(a). The XZ-plane and the YZ-
The instantaneous distribution of acceleration of a spherically moving systan
25
plane meet the unit sphere in two orthogonal great circles, which may be considered as the coordinate lines of a spherical xy-system with the origin M at P. The conjunction between the Cartesian coordinates XYZ and the spherical coordinates x y can be derived easily from Fig. 2 as follows: X tanxfy, Y
tan y = ~
(4a) (4b)
with X 2 + 7 2 + Z ~ --Rx 2 -- 1. The equations between the spherical coordinates xy and the spherical polar coordinates (r, cp) follow from Fig. 3 by means of the Napier rules for spherical geometry
I
Ly
I Fie. 3. Spherical eoordlnate system.
26
W. MEYER ZUR CAPELLEN and G. DITrRICH tan x f t a n
r
cos ~p,
(5a)
tan y = t a n r sin ~o,
(Sb)
and from these tan2r =tan2x + tan2y.
(5c)
Furthermore, from Figs. 3 and 2 the following relations can be found X=sin rcos ¢p,
(6a)
Y= sin r sin ~,
(6b)
Z--cos r.
(6c)
2. THE ACCELERATION The angular acceleration $ measures the instantaneous change of the angular velocity vector, or dm e=-=/o (7) dt Herein dm consists of the two components- do~. p and - ¢ods. t, Fig. 4. The first of these components, characterizing the change of magnitude of the angular velocity vector, may be expressed by d~o ~, = - (s) dt
0 ~
FIe. 4. Mvafion of the angular a c c ~ t i o n .
~~
The instantaneous distribution of acceleration of a spherically moving system
27
and the second component, characterizing its change of direction, may be expressed as ds u--~.
(9)
The latter measures the velocity of change of position of the pole. In these terms the magnitude of angular acceleration is given by
8 = x/(cb2 + O32u2)
(I0)
but if the abbreviations 8:
j=~-~,
(11a)
~b ~b=~-~
(lib)
and u
0=-
(11c)
j=~/(~2 +02).
(12)
O3
are used, equation (10) willappear as
The angular acceleration ~ has the direction of dee, which may be specified by OP, (Fig. 4), and therefore it must be situated within the tangential plane of pole cone. The angle g between the direction OP, of the angular acceleration and the instantaneous axis of rotation can be calculated in the following way
), o3u sin 0-------
8
0
4(
2+02) '
(13a)
(13b)
and thus
tang=
0
.
(13c)
28
W. M E Y E R Z U R CAPELLEN and G. D r I T R I C H
The velocity v of the point C may then be written v--v'T
(14)
in which v--o~" sinr gives the magnitude and T the direction of the vector v. The acceleration vector can be found by taking the time derivative of equation (14): dv a--~---O "T+v "T.
(15)
Should the path element ds* be introduced instead of time, then 'i" dT =
dT ds*
d'-'t'--'ds"-;dt
=v •
T'
(16)
is obtained. The primes here signify derivatives with respect to the path element ds*. In terms of the trihedral of any point moving on a unit sphere, the following equations (refer to [1-3]) for the derivatives of its motion are always available
ETIi0 N' C'
=
ot 0jEll
-cotp
0
1
(17)
0
and this T'=
1
-N-C tanp
(18)
can be deduced. By combining equations (18), (16) and (15), the acceleration can now be written v2 a -- 6T + - - N -
tanp
vzC
(19)
or
affi~+aN+ac
(20)
in which the first term is the tangential acceleration, aTf6T.
(21a)
The next term designates the normal acceleration p2
amffitan/~N and the last term is the radial acceleration directed toward the fixed point 0
(21b)
The instantaneous distribution of acceleration of a sphericallymoving system ac-- - vzC.
29 (21c)
The normal acceleration and tlle radial acceleration form together the total normal acceleration of the spatially moving point. Before investigating the geometrical loci where the components of acceleration will vanish singularly or all together, the acceleration of an arbitrary point C should be looked at from another point of view [4]. The velocity of the point C may be written in the form v-c0x C
(22)
a--6xC+mxC
(23a)
a = ~ x C + o~x v.
(23b)
and therefore the acceleration is
or using @ •8 and C--dC/dtffiv
It is evident, that the acceleration can be expressed by the sum of its components
affia,+a .
(24)
In this, a~ =E x C
(25a)
is an acceleration normal at the point C to the plane formed by s and C and is only dependent on s, whilst a~ • m x v = m x (mx C) =(mC)m- m2C
(25b)
ties within the plane PCO through the pole axis and is solely dependent on o~. 3. THE CONSTANT SPEED CONE The surface of the "constant speed cone" k o is defined as containing those axes of the body, whose points Co momentarily possess no tangential acceleration. That means, that the acceleration vector a, must lie in the plane CgPO of the path normal, i.e. the plane Col',O must be perpendicular to the plane through the pole axis, as in Fig. 5. In the rectangular spherical triangle PCoP. with hypotenuse PP, it follows from Napier's rules of spherical geometry that the equation for the constant speed cone is tan r--tan g cos cp,
(26)
in which g is defined by equation (13c). In view of equations (5a, b) the equation (26) can be written in spherical xy-coordinates: tan2x-6 tan2y - tan g tan x -- 0.
(27)
30
W. MEYER ZUR CAPELLEN and G. DITTRICH
Y Isss'''-
f,', i...--_-J-X !
II
~
-
//
-
I
/
ss
•
Fro. 5. The "constant speed cone", defined as the locus of all points without tangential acceleration. Introducing X,. Y, Z as homogeneous coordinates we get
X•2
/ y \ 2 _(X)tano__ 0
(280
or
X 2 + y2 _ XZtan O = 0
(28b)
The constant speed cone is therefore a cone of second order. In order to investigate whether this cone penetrates the equatorial plane which is perpendicular to the instantaneous rotation axis OP, the coordinate Z is made zero. Then equation (28b) reduces to X 2+ y2 _ 0
(29a)
(X + i iO(X - i tO=O
(29b)
or
with i = x / ( - 1). Thus the constant speed cone and the equatorial plane have in common the mutual imaginary axes
X+iY=O and
X - i Y=O
(30a/b)
with the directions Y tantpx, 2 = 2 = + f .
(31)
These axes correspond to the two imaginary circle-points which occur in planar kinematics.
The instantaneous distribution of acceleration of a spherically moving system
31
By the above deduction it is evident, that the constant speed cone is symmetric both to the tangential plane PP,O of the pole cone and to a plane through the midpoint axis OM o of k o, the plane being inclined at an angle g/2 with the normal plane of the pole cone. The constant speed cone must touch the normal plane of the pole cone in the instantaneous rotation axis OP according to equation (26), see Fig. 5. The instantaneous center of rotation P cannot have an acceleration caused by e), because v is always zero (see equation 25b). Therefore the acceleration of the instantaneous center of rotation may be derived from equation (25a) by putting C = p = t x n , which yields ap =a, = s x (t x n) = ( s n ) t - (st)n.
(32)
The first term of this sum will be zero because s is normal to n. It may then be recalled that, according to Fig. 4, the angle between 8 and t is equal to (~/2 + g); as a result, the acceleration of the instantaneous center of rotation can be written ap •e sin 0" n
(33)
Its direction is coincident with that of the normal to the pole path. Equations (33) and (13b) give its magnitude ap =o~u=Oo~2.
(34)
In order to find out more of the geometrical features of the constant speed cone, the curve of its intersection with the tangential plane Z = Rk = 1 touching the sphere in the point P will be determined. Equation (28b) leads to X2+ y 2 - Xtang-~O or
1
• \2
The curve is therefore a circle with diameter tan g, of which the center is displaced by ½ tan g in the X-direction. Any plane which is perpendicular to the central axis OMg and which does not pass through the fixed point O, cuts the cone in an ellipse. The constant speed cone therefore must be an elliptic cone, which intersects the sphere in a spherical ellipse. Without demonstration (refer to [5, 7] let it be said here, that when the half of the smaller angle of aperture is taken to be b •g/2, the bigger one can be obtained from the relation: tan a = ~ mIn ~ . tga n g .
The focal axes lie in the tangential plane of the pole cone and are inclined by the angle e to the central axis OM o. Then this angle e may be found from the relation: sin e---+ tan 2 -g 2"
32
W. MEYER ZUR CAPELLEN and G. DITTRICH 4. THE INFLECTION CONE
The condition that the normal acceleration aN should vanish (see equation 21b), is that the spherical radius of curvature becomes p = 7r/2. At the same time this is the condition for those points of the system, which instantaneously will have an inflection point or flat point in their path. The path therefore has at least three mutual infinitesimally adjacent points of contact with a great circle. Now the Euler-Savary formula of spherical kinematics [1, 5] is 1 tanr
1 1 t a n r o Osin~o
(35)
and if we introduce into this the radius of curvature according to equation (2) together with the abbreviation tan w -- 0 sin tp,
(36)
tan2r tan p = tan w tan2r-- tan r + tan w"
(37)
there is obtained
When furthermore the condition p = ~/2 is satisfied, the equation
(38)
tan w t a n 2 r - tan r + tan w = 0
is found and therewith the equation defining the "inflection cone" kw is known. This is the geometrical locus of all points, whose path is without normal acceleration. Its equation can be written (tan2x + tan2y)(0 tan y - 1) + 0 tan y = 0
(39)
in spherical xy-coordinates, or alternatively
,] +
o
(4Oa)
in Cartesian coordinates. The latter can be written ( X 2 + I"2)(0 Y - Z ) + 0 Y Z 2 =0.
(40b)
The inflection cone is thus a cone of third order which intersects the equatorial plane (Z--0) in the X-axis (Y=0) and on the two imaginary equatorial axes given by equations (30a, b). Solving equation (38) now for tan r, we get tan r -- 2ta~w [1 -I-~/(1 - 4 tan2w)].
(41)
The instantaneousdistributionof accelerationof a sphericallymovingsystem
33
and it is then evident, that real values of tan r will exist only for
12tanwl_~l or that is to say, for 1 sm~0_~.
(42)
For example, a limiting angle cp~30 ° is obtained in the case 0 = 1 . But where 0_~0.5 the values of tan r are always real. The equation (38) for the inflection cone may also be simplified to the form sin (2r) -- 20 sin rp.
(43)
This equation shows, that the inflection cone is symmetrical to the normal plane of the pole cone: (9=~/2) and that it touches the tangential plane of the pole cone (~p--0) at the instantaneous rotation axis OP. T h e normal plane of the pole cone cuts the infle~don cone along the inflection pole axes O P w. The inclination angles 6 of these axes relative to the tangential plane of the pole cone are found to be sin26 =20
(44)
because we may set rffi6 and ~ =~/2 in equation (43). Three cases therefore arise: if 0>0.5 there exist no real inflection pole axes; for 0ffi0.5 there is only one axis with the inclination angle 6---45°; and for 0<0.5 there are two real inflation pole axes because the inflection cone disintegrates into two different cones. Y
s ~ .
Fro. 6. The "inflectioncone", being the locus of all points withoutnormal n_~ocderation. Parameter values0----1, ½and ½. Figure 6 shows the intersection curves of three characteristic inflection cones kw with the unit sphere. Furthermore, the intersection curves of the inflection cones kw of the kinematic inversion are drawn, but these will not be discussed here in further detail.
34
W. MEYER ZUR CAPELLEN and G. DITTRICH 5. THE AC~I.EitATION AXES
Since with to # 0 and e # 0 there exists no axis in a spherically moving body along which the total acceleration vanishes, we shall take those lines of the body, whose points have neither tangential nor normal acceleration to be named the instantaneous acceleration axes OJ [6]. They occur as the lines of intersection of the constant speed cone and the inflection cone. Two cones of second and third order, with mutual cusps, in general have six slant lines in common, but in this case two of them are always coincident with the two imaginary equatorial axes and one with the instantaneous rotation axis OP: the latter has already been identified as a singular axis. Therefore there remain three lines of intersection, the position of which and reality conditions for which still need to be discussed. Taking into consideration equation (13), equation (26) is substituted in equation (38), and thus an equation for the directions ~p of the acceleration axes is derived: tan aq~-~1 tan2tp + ~ (~b2+ 02) tan q~_~ = 0 .
(45)
This equation of third order will in general have three solutions which either are all real or two of them are imaginary. Equation (45) can be reduced by substituting tan,=2+
1 ~-~.
(46)
Thereupon it assumes the form 23 +p2 + q =0,
(47)
where 1 / 2
"2
1~
p=-7"~l O~p_\+¢P ---3)
(48a)
and
1 1'02 2 2 2\
(48b)
The roots of such a cubic equation are real if and only if the discriminant /'q'~2
A=~)
1'p~3
+~)
~0
(49)
that is, when
2~2+@3~o.
(~)
It follows from these equations (48a, b) that the three acceleration axes are real, if and only if F-- 4(~b2 + 02)3 _ (~b2+ 02)2 _ 18~b2(~b2 + 02) + 27~b4+ 4~b2 _~0.
(51)
The instantaneous distribution of acceleration of a sphericallymoving system
35
Using the abbreviations
{/)2-~-02=~ and
~b2---t/,
(52a/b)
equation (51) assumes the form F_= 4~3 _ ~2 _ 18~/+ 27~/2+ 4q ~ 0
(53)
The reality of the roots depends on the parameters ~ and ~/. We shall now consider where in the ~/plane these points are located, when the equation (53) is taken to have an equal sign. Equation (53) describes a cubic curve kl,. For any chosen value of ~ =~b2+ 02, however, a quadratic equation in t/remains: F = 27~/2+ 2 ( 2 - 90q + ~2(4~- 1).
(54)
Its discriminant is A* = 4[(2- 902 - 27(4~ - 1)~2] or more simply A* = 16(I - 3 0 s.
(55)
From this it follows that the roots for t/=~b 2 are real only if (56a) For ~ffi~
(56b) If ~ and r/written
~__/~+1
(57a)
~/=v+~-7,
(57b)
and
F*--4#3 + 3(/x-v) 2
(58)
follows. It is easy to deduce from this, that the curve k~,* described by this function contains a cusp at the point (#--0, v - 0 ) with two coincident tangents, the equation of which is /z-3v=O
(59)
¢-3n- =o.
(6o)
but for the curve ke
36
W. MEYER ZUR CAPELLEN and G. DITTRICH
The curve kF touches the f-axis in the origin (0, 0) and intersects it at the point 0 / 4 , 0). Within the zone of interest the curve has the shape shown in Fig. 7. It can be seen that the points (~, ~/) which describe an instantaneous movement with three discrete real acceleration
•.
0~4
11127 *:w-~003 .L2 ~Z
/
__002
0-01
(>1
0
o
0~2 ~/4
0"3
I
-0.01
I I ! !
-002
F/o. 7. Diagram for reckoninR the number of real acceleration axes. In the first q~djant there exist three discrete acceleration axes for pairs of values within the curved triangle, on the two b m n c ~ of the curve two coincident and one discrete axes while at the cusp three coincident axes can be found; in the remaining area only one real acceleration axis occurs.
axes appear only in the first quadrant within the triangular region limited by the curve k F. Here is A < 0 and F < 0 . The directions of the three acceleration axes in this case may be found from equation (46) by setting gj = 2 x / ( - P/3) • coslg + ~ (j - 1)~1 ,
] = 1, 2, 3
(61a)
a n d t¢ f r o m
cos 3K =
q 2x/(_p/3)3.
(61b)
An example of an instantaneous movement with three discrete real acceleration axes is shown in Fig. 8(a). The points on k r ( F = 0 ) belong to instantaneous movements which have
The instantaneous distribution of acceleration of a spherically moving system
37
Y
!
I~
,
eI
I
i
X
I
Fxo. 8(a). Movement of a spherical system with [~b2 ~ 0.023, 02=0.27] which causes three discrete real acceleration axes.
one discrete and two coincident acceleration axes, the positions of which may be got from equation (46) by substituting in turn
21 =2~/(-q/2)
(62a)
22 =23 = - ~/(-q/2).
(62b)
and
One example is shown in Fig. 8(b). Then for the special c a s e r / = ~ b 2 = 1 / 2 7 and ~ = ~ 2 + 02-- 1/3, i.e. 02 =8/27 all three acceleration axes are coincident, Fig. 8(c), and in this case
FIG. 8(b). Movement of a spherical system with [~b2 ~ 0.028, 02 =0.27] which causes one discrete and two coincident real acceleration axes.
38
W. MEYER ZUR CAPELLEN and G. DITTRICH
I F1o. 8(c). Movement of a spherical system with [~b2= 1/27, 02 = 8/27] and thus three coincident r~fi acceleration axes inclined at an angle ~=60 ° with the spherical x-coordinate. equation (45) simplifies to (tan
¢p -
~/3) a = 0
with the solution tan ~p1.2, 3 = n/3 that is, ~Pl. 2, 3 = 6 0 ° .
The three coincident acceleration axes are inclined at an angle ~p= 60 ° relative to the spherical x-coordinate. In all the other cases for which ( A > 0 , F > 0 ) there exists only one real acceleration axis, as in Fig. 8(d). The plane t h r o u g h the instantaneous rotation axis
IY
F[o. 8(d). Movement of a spherical system having [~b2 =0"27, 02=0.27] with one real acceleration axis.
The imtantanoous distribution of acceleration of a spherically moving system
39
which contains also this real acceleration'axis is again defined by equation (46), if
'#1'1=~1 i "l"/~2*
(63a)
is used in it, where
z~,'*2 = ~ / [ - (q/2) + ~A]
(63b)
I n the Figs. (9a-c) and (10a-c) the acceleration axes OJ are drawn as chain lines between the inflection cone kw and the constant speed cone kg for further characteristic pairs o f the parameters 0 and ~b.
Y
/
I,/
L, ,'.----/
\ xt.t
-
,~ #I ]
l! i
#
#
I
FIo. 9(a). Moveznent of a spherical system having
[~2 ~0"016, 02=0-25] with three discrete real acceleration axes.
Fxo. 9(b). Movement of a spherical system having
[~2 = 0 ~ ,
a2=o.25]
with one discrete and two coincident real acceleration axes.
40
W. MEYER Z U R CAPELLEN and (3. DITFRICH
Y
f//I /I ~11 I
/
,
/
X
FIo. 9(¢). Movement of a spherical system having
[4)2--o.25, 02=0.2.~] with one real ac~leration axis.
Y °
• ' I " ;I
l
I ! // /
j
N o . 10(a). Movement of a spherical system having
[ ~ o ~ o 2 , e2--1/9] with three discrete real acceleration axes.
The instantaneous distribution of acceleration of a spherically moving system
41
,Y
FIG. 10(b). Movement of a spherical system having [~b2 ~,0.004, O2ffi1/9] with one discrete and two coincident real acceleration axes.
~Y
X
....
L-2 ..... I/
FIO. 10(c). Movement of a spherical system having
[~2 =1/9, 02=I/9] with one real ac~eration axis.
6. SUMMARY In order to survey the distribution of accelerations o f an accelerated spherically moving body the acceleration of a random point has been sprit up into a radial component directed toward the fixed point 0 and into tangential and normal components with respect to the spherical path. As an analogy to plane kinematics, the points whose tangential acceleration vanishes have been found t o be situated on the "constant speed cone" while the points without normal acceleration are placed on the "inflection cone". There m a y occur up to three discrete real acceleration axes as lines of intersection of the constant speed and the
42
W. MEYER ZUR CAPELLEN and G. DITTRICH
inflection cone; for all points on these lines both the normal acceleration and the tangential acceleration vanish. Other than the fixed point 0 there are no points of the accelerated spherically moving body for which the total acceleration is zero.
REFERENCES
[1] H. R. MOLLEI~Sph~rische Kinematik. VEB Dcutscher Verlag dcr W i ~ h a f t e n (1962). [2] E. I ¢ , a t ~ o , Di~'erentialgeometrle. Akad. V~'lagsg~,,llschaft Gcest und Portig K.-G. (1957). [3] M. I.~GJ~J~Y and W. F'n~NZ, Vorlesungen fiber Vektorrechnung. Akad. Verlagsgescllschaft Gcest und e o r ~ K.-G. (1959). [4] R. Brn~, Technische Raumkinematik. Springer-Vcxlag (1963). [5] G. D, lr~CH, Ueber die momentane Bewegungsgeometrie eines sph[uisch bewegten starren Systems. Dr. Diss. TH Aachen (1964). [6] O. BOTTEMA,Acceleration Axes in Spherical Kinematics. ASME Paper No. 64-Mech-9. [7] W. MEY~ zuR C A P ~ , G. DITI~CH and B. J A ~ S ~ , Systematik und Kinematik ebener und sphiirischer Viergelenkgetriebe. Forschungsbericht des Landes Nordrhein-Westfalen.