Applied Mathematics and Mechanics Published by HUST Press, (English Edition, Vol.4 No.2, Apr. (1983) Wuhan, China

VECTOR ANALYSIS OF SPATIAL MECHANISWS

PART 1 CONFIGURATION ANALYSIS OF SPATIAL

~:ECHANIS~{S BY THE ~(ETHOD

OF VECTOR DECO~IPOSITION

Yu Xin(~ ~ ) (South China Institute of Technology)

(Received June 18, 1982)

ABSTRACT

In this serles of four papers we present pure vectorial me- thods in establishing the configuration, kinematics and dyna- mics of some of the four-link spatial mechanisms. In Part I, the method of vector decomposition is used to establish the configurations of the R-G-G-R and H-R-G-R mechanisms, respec-

tively.

I. Introduction

A certain class of four-link spatial mechanisms can be analysed by means of

vector methods as was first demonstrated by Chace in [ I] and [2]. The virtue of

this method lies in its direct geometric appeal and simplicity of concept so im-

portant in the design of spatial mechanisms; moreover, vector analysis is the na-

tural language of Newtonian mechanics and the mechanics of spatial mechanisms is

just a branch of it and its language must be consonant with the vector.Although

many subsequent authors (e.g. Bagci [ 3]and others) offer alternative methods,none

of them maintains the geometric spirit of the Gibbsian vector, nor its simplicity.

Nonetheless, the way Chace employed vectors often tarnishes this stated geometric

virtue in that his manipulative procedure depends heavily on tedious scalar alge-

bra and trigonometric identities while his solutions depend implicitly on the so-

lutions of polynomial equations (to a maximum) of the eighth -degree. In the pre-

sent series of four papers we offer alternative vectorial procedures: we obtain

explicit vector solutions by using purely vectorial methods. In Part i we treat

the configurations of the R-G-G-R and the H-R-G-R mechanisms by the methods of

vector decomposition. In Part 2 the configurations of the P-P-GIC and the R-G-C-

R mechanisms respectively are established by the method of vector equations. In

parts3 and 4 we present the kinematic and dynamic analysis of the above mechanisms

using purely vectorial methods, giving explicit vector solutions.

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

186 Yu Xin

II. Configuration of the R-G-G-R Mechanism

The configuration of the mechanism is represented by the series of line

vectors shown in Fig.l, where O and /~ are on the O

central lines of the shafts and p and Q are the P

centres of the respective ball-and-socket joints , k / OP is the crank and QR is the followers.

Now put aj

OP=ps P O = s a , OR=q~, (2.t) R O = r ~

where (%,~,~,~) are unit vectors.

\

FIG. I.

Evidently, the quantities

�9 a~=m, ~.5~=n (2 .21

are given constants; in the case where the crank and follower are perpendicular

to their respective shafts, m=0 and n=0. Henceforth, we shall call the equa-

tions of the type (2.2) the "structural equations".

From the given configuration we have

p~ +s~+q~ 4-r9 = 0 (2.3)

which is usually called the "loop equation". The problem is now reduced to the

determination of ~ and ~ subjected to the conditions (2.2) and (2.3).

Now let

'=~; ( 2 . 4 ) (~A~) A~

I= ~(~A~,) A~, t

The unit vectors (8,,), j)

Or since ~.-~=m, we have

~ - m ~ , = (~.0i+(~.Dl

On squaring both sides of (2.6) we have

1-m~= (~ .02+ (7~ . ~) 2

Hence, we may put

Z = m~ + ./-(i - m 2) ( c o s 6 / + s i , ~ ' )

where @ is the "input angle"-

Since @ is given, ~ is completely determined.

Now let

= -- (pZ +r~)

evidently form an orthogonal set and we may put

(2.5)

(2.6)

(2.7)

( 2 . 8 )

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

Vector Analysis of Spatial Mechanisms---(1)

and (2 .3 ) becomes

s a + q ~ =5 (2.9)

The problem is to solve ~ and ~ in (2.9).

Now put ~ (2.]0)

and,

where x, y, z are unknown scalars to be determined.

Eq.(2.11) is the formula for vector decomposition for the unknown vector ~ .

Now, [~A (~A~)" (2. l l ) ]

and [i~2A(a~A~)'(2.11)]

Moreover, (2.9)

~A (@2A#) -~ @2.~ -- (# .az) (~.~) X= ~A(~2A~).~ 2 -- (~2A~) ~ (2. t 2)

u = ~ q A ( ~ ) y : ~ . - = la2l#)~ -- (2. J3)

s~ =5--qN

which gives, on squaring both sides of the equation,

s 2 = c~ + q2_ 2cq~"

i.e. ,

(2 . J 4)

187

~,h~, ~ (2.17) z ~ f

and z is completely determined since we have the well-known vector identity(See

III below):

Once x, y, z are determined, ~ is given by the vector decomposition formula(2.11).

In III we shall present another way of decomposing the unknown vector.

To determine z , we have [(a2A~)'(2.11)],

. f i = c~Wq~- - s~ (2 .15) 2c2q

in view of (2.10).

Hence, substituting (2.15) into (2.12) and (2.13) respectively we get(since

~ , ~ = n ) ,

. - (c'+q~--s~) F" ~ x = 2c~q (~ 'A~) ~ (2 .16)

( cZ ..I- q2 - - s 2) - - nl 2 . ~.~

Y = 2cZq(a2A~.) ~

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

188 Yu Xin

A more traditional method would have been to put

qA~ z (~A@:) A a z it = I~Aa~I , Y,= l (u

and in the manner of (2.7), we have

= n a ~ + ~ / 7 1 - . q (cosCL + s i n e / , )

where O is the "output angle".

Then, on substituting (2.20) into (2.14) we obtain

where

( 2 . 1 9 )

( 2 . 2 0 )

.,4 (0) sine q -B (0) cosr = E (0)

.,4(0) = 2 ~ / ( l - - n 2) ( ~ - ] , ) 1

E (0) = c ~- q2_ s ~_ 2n (~. 8 2)

(2 .e l )

( 2 . 2 2 )

The quantities A(O).B(O) and E(@) are known, and hence the "input-output" rela-

tion (2.21) may be solved for r In this case, the traditional method appears

to be simpler.

III. The Configuration of

% _

FZG -I 2 ( a )

the H-R-G-R Pechanism

rzc.12 (b)

For the H-R~G-R mechanism shown, the input screw displacement at the heli-

cal pair (S)determines the line ~ completely and hence, given the lengths'RA,

the line segment z/B is also known. The problem is to determine the unit vec-

tors ~ and ~', the lengths a and b having been given.

Now z~AGB gives,

b-s =l;{--a~ (3. ~) where

l = I A B I

Moreover, we have the identity

dA (~/,#) +~A (~^a) +~ A (~A~,) = C" ( 3 . 2 )

[2] Hence, �9 (3.1)

I"- + c ~ - - b ~ ( 3 . 3 ) ~" ~ = - - "2a:

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

Vector Analysis of Spatial Mechanisms---( l j 189

and.

and (~ .~)

by (3 .4 ) i f ( 4 - ~ A ~ ) can be o b t a i n e d . But [~.(3.4)~1

(;[ ̂ ~) ~= (4 . t A~) '+ [ (~'.~) - - (# .~) ( X - ~ ) ] (4"X) +[ (,~-~) -- (~t , ~ ) ( I - ~ ) ] ( a - ~ )

which gives, on transposing terms,

(d.IA~)=+~/ [1--(d.X)'--(7,.g)'--(g.r~)~+2(a.X)(I-g)(g'd)]

i.e.,

(~.X^~) = _+~[ z - (a Z),- (I .~)']

in view of (3.5).

Once a is determined, ~ can be obtained from (3.1).

[ (X^~) ^ (3.2) ]

(~A~)24= (d.IA~)~A~+[ (~.4) -- (~-@) (~-~) 3~+[ (~-4)-- (~.~) (~-d) ]~ (3.4)

Now ~.~=0(by construction) (3.5)

(~.~), are known quantities and hence, ~ is completely determined

(3.6)

(3.7)

IV. Discussions

It is clear from the above example that it is not easy; nor is it neces-

sary, to first determine the ?input-output" relation of the type iI (2.2&) be-

fore establishing the configuration. We have here established the configura-

tion directly in vector form, using the method of vector decomposition, which

is the essence of eq.(3.4) and we have used vector algebra throughout, without

resorting to tedious scalar algebraic and trigonometric identities. Chace's

solutions depend on the solutions of polynomial equations of the fourth degree

or higher and it is known that no general solutions of such equations are ob-

tainable (See, e.g., Burnside and Panton [4]) and numerical procedures would

have to be resorted to; they are solvable for specific cases only. In the au-

thor's opinion, insistence on establishing the "input-output" relations first

hampers the progress of the solution. This course seems to have been followed

by subsequent authors, whatever other mathematical systems they may have employ-

ed. We here bypass the "input-output" relations which, in any case, can be

easily obtained once the configuration is established.

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

190 Vu Xin

References

1. Chace, M.A., Vector analysis of linkages, J. Eng. for Ind., Trans ASME, Series B, Voi.85 (]963), 289-297.

2. Chace, M.A., Development and application of vector mathematics for kinema- tic analysis of tl~ree-dimensional mechanisms, PH.D Thesis, The University. of Michigan, Ann Arbor, Mich., (1964).

3. Bagci, C., Oync~nic force and torque analysis of meches~isms using dual vectors and 3X3 screw matrix, J. Eng. for Ind., Trans ASME, Voi.94, May (1972) , 738-745.

4. Burnside, W.S., and A.W. Panton, Theory of Eq'~ations, Dublin University Press, (1881) .

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products