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On the exact equation of inextensional, kinematically indeterminate assemblies
Computers and Structures 75 (2000) 145±155
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On the exact equation of inextensional, kinematically indeterminate assemblies T. Tarnai*, J. Szabo Technical University of Budapest, Department of Structural Mechanics, MuÈegyetem rkp. 3., H-1521, Budapest, Hungary Received 30 April 1998; accepted 3 March 1999
Abstract An exact equation describing the ®nite displacements of assemblies consisting of rigid bars and pin joints is presented. A numerical procedure is proposed for carrying out the calculations, and some examples are shown in order to illustrate the procedure. To depict the ®nite change of state, a system of non-linear algebraic equations must be solved. With linear approximation, ®nite node coordinate increments can be used. This is one of the advantages of the proposed procedure in contrast with the compatibility dierential equation description, where only in®nitesimal increments can be used. The procedure can be applied even for the analysis of pin-jointed space frames in the post-critical regime, also in the calculation of certain mechanisms of mechanical engineering. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Mechanism; Kinematically indeterminate structure; Bar-and-joint assembly; Iterative solution; Set of dierence equations
1. Introduction Assemblies of pin-jointed bars form mechanical models of a wide range of engineering structures. Some of them are kinematically indeterminate [1]. The analysis of such assemblies or mechanisms, which undergo large inextensional deformation, that is, the analysis of ®nite mechanisms (which has been excluded from structural engineering before) has become important. The reason why structural engineers have become interested in ®nite mechanisms is that there is an increasing number of areas where ®nite mechanisms are relevant. Such areas include, for instance, the erec-
* Corresponding author. Tel.: +36-1-463-1431; fax: +36-1463-1099. E-mail address: [email protected] (T. Tarnai).
tion technology by folding and lifting, such as the pantadome system [2]; structural form-®nding under dierent load patterns [3]; and deployable structures [4,5]. The description of the kinematic geometry of barand-joint assemblies, especially in the plane, is a classical problem in kinematics. However, the basic concepts of kinematics are dierent from those in structural analysis. Kinematics and structural analysis use dierent languages; and even for the same problem, they provide dierent descriptions and dierent methods of solution. In kinematics, for a long time, analysis based on geometric constructions was used exclusively [6], and only recently have vector±matrix methods been introduced [7], introducing generalized coordinates (signed distances and angles). In structural analysis, on the other hand, matrix methods have been used for a relatively long time; and
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 9 0 - 5
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
146
Nomenclature b B d dhi dr drg,n ei f, fi hi , ki i, g j j0 j1 `i r
number of bars compatibility matrix in the case of in®nitesimal change m j-dimensional vector of ®nite coordinate increments vector of ®nite displacement of node hi m j-dimensional vector of in®nitesimal coordinate increments in®nitesimal increment of the nth coordinate of node g unit vector of bar i, pointing from hi to ki : ei rhi ,ki =`i constraint function and its ith component starting point and end point of bar i, i 1,2, . . . ,b; hi 2 f1,2, . . . ,j g; ki 2 f1,2, . . . ,j0 g; hi 6 ki integers number of internal joints total number of joints: j0 j j1 number of external joints (attached to a rigid boundary) length of bar i vector of nodes composed of vectors rg :r rT1 rT2 . . . rTj0 T
for the investigation of the kinematic geometry of barand-joint assemblies, a system of compatibility dierential equations has been set up, applying Cartesian coordinates of the nodes [8]. The incremental technique with Newton±Raphson iteration, is an eective tool for solving the dierential equations with given initial conditions. However, if we are looking for not only a particular position of the assembly, but also for all possible positions (we want to describe the motion), then it is more eective to use the exact compatibility equations Ð known as constraint equations in kinematics [7], instead of the compatibility dierential equations. The aim of this paper is to present the exact equations describing the ®nite motions of mechanisms composed of rigid bars and pin joints, and a numerical procedure that is based upon them.
2. Compatibility equation of a kinematically indeterminate structure consisting of rigid bars and pin joints Let us consider a pin-jointed assembly which con-
rg
position vector of node g; g 1,2, . . . ,j0 ; the numbering of nodes starts with the internal nodes g 1,2, . . . ,j and ends with the external nodes g j 1,j 2, . . . ,j0 rg,n the nth component of the position vector of node g, n 1, . . . ,m r0g,n the nth component of the position vector of node g in the initial state rhi ,ki ith bar vector: rhi ,ki rki ÿ rhi t vector of real or ®ctitious bar elongations ti elongation of bar i (ith component of t); i 1,2, . . . ,b dg,n ®nite increment of the nth coordinate of node g Df, Dfi ®nite increment of f and its ith component e error tolerance m dimension of physical space (plane: m 2; space: m 3) n integer r B rank of matrix B n nth approximation in the procedure T transpose scalar product
sists of j internal joints, j1 external joints embedded in a rigid boundary, and b rigid bars. The initial state (in other words, the geometry) of the assembly is determined by the coordinates of the nodes, and the length of each bar is uniquely determined by them. The length of bar i is s m 2 X `i r0ki ,n ÿ r0hi ,n :
1
n1
Since the bars of the assembly are rigid, the bar lengths `i are constant. Consequently, the value of the expression fi `i ÿ
m X 1
` n1 i
2
rki ,n ÿ rhi ,n
2
can be equal to zero, if and only if the distance between the end points of the bar, calculated with the coordinates of the end points, is identical to the length of the bar calculated with Eq. (1). Equality fi 0 expresses the fact that the length of bar i is compatible with the coordinates of the end points of bar i. Let a
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
b-dimensional vector f be composed of elements fi : If f0
3
then the assembly is said to be in a compatible state. Eq. (3) (in scalar form, a system of nonlinear equations) is called the compatibility equation (system of equations) of the kinematically indeterminate structure (mechanism) consisting of rigid bars and pin joints. There may now be prescribed as many properly selected ®nite coordinate increments dg,n , in the domain of motion of the mechanism as the degrees of freedom; and the remainder of the compatible coordinate increments can be calculated. In this way the coordinates of the nodes in a new compatible state, satisfying Eq. (3) can be determined. In other words, a vector f composed of elements fi ` i ÿ
m X 1ÿ
` n1 i
rki ,n dki ,n ÿ rhi ,n ÿ dhi ,n
2
4
also satis®es Eq. (3). This will be illustrated in Appendix A with a trivial example. There are well-known algorithms and computer procedures for solving the system of equations (3). Nevertheless, it is worthwhile to seek a treatment of the continuous change of state of bar-and-joint mechanisms, based on the speci®c features of such assemblies. In the following sections, such an approach and a proposal for a numerical procedure will be presented.
3. The exact equation of the ®nite change of state of the assembly If the value of the constraint function in the state corresponding to Eq. (2) is denoted by f and in the state corresponding to Eq. (4) by f Df, and if both states are compatible states, then it is quite obvious that the dierence between Eqs. (4) and (2) results in f Df ÿ f Df 0
5
and this equation can be considered as the exact equation (system of quadratic equations) of the ®nite change of state of the assembly. The ith equation of the system of Eq. (5) is Dfi
m h X 1 2 rki ,n ÿ rhi ,n ` n1 i 2 i ÿ ÿ rki ,n dki ,n ÿ rhi ,n ÿ dhi ,n 0
whence, by reducing by 2, equation
6
m X 1
147
rki ,n ÿ rhi ,n
` n1 i ÿ
dhi ,n ÿ dki ,n
2 1ÿ ÿ dki ,n ÿ dhi ,n 0 2
7
is obtained, whose form more advantageous for computation is m X 1 dk ,n dh ,n ÿ dhi ,n ÿ dki ,n 0: 8 rki ,n i ÿ rhi ,n ÿ i 2 2 ` n1 i
4. Linear approximation of the exact equation of the ®nite change of state and the linear compatibility dierential equation By removing the term quadratic in d from the exact Eq. (7), we obtain an equation that can be considered as a linear approximation of Eq. (7): m X 1
` n1 i
ÿ rki ,n ÿ rhi ,n dhi ,n ÿ dki ,n 0
9
or in a more concise form: eTi dhi ÿ eTi dki 0;
ei 6 0:
10
This enables us to ascertain that the coecient matrix in the linear approximation of the exact equations is an extension of the coecient matrix of the system of compatibility dierential equations (valid for in®nitesimal changes) for ®nite displacements. However, while this matrix is exact for in®nitesimal change, it is only approximate for ®nite change. Namely, Eq. (10) makes it possible to ®ll an `empty' b jm matrix B 0 with m-dimensional unit rowvectors (subvectors) eTi and ÿeTi so that multiplying the obtained matrix B 6 0 by vector d, Eq. (10) is ful®lled for every i. Each row of the `empty' matrix is partitioned into m-dimensional subvectors. The formula of ®lling is the following: . the hi th zero subvector of row i is replaced with unit vector eTi , . the ki th zero subvector of row i is replaced with unit vector ÿeTi if ki Rj, and the zero subvector is preserved if ki > j: Considering the j-dimensional vector d containing the ®nite node displacement values d does not contain the zero displacements of the external joints, it is easy to see Ð if the ith rowvector of B is denoted by Bi Ð that Eq. (10) can be written in the form Bi d 0: For all Bi 's together:
11
148
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
Bd0
12
and taking the vector t of given real or ®ctitious bar elongations into account: B d t 0:
13
5. Inhomogeneous equation and successive approximation In this section we restrict ourselves to one-degree-offreedom assemblies, emphasizing, however, that the actual numerical studies have been extended also to particular multi-degree-of-freedom assemblies. Substituting (partly) arbitrarily selected coordinate increments dki ,n , dhi ,n into the system of equation (8), Dfi in general will not be equal to zero: m X 1 dk ,n dh ,n Dfi rki ,n i ÿ rhi ,n ÿ i ` 2 2 n1 i ÿ dhi ,n ÿ dki ,n ti : 14 The proposed procedure, which makes the absolute value of each component ti of the error vector t smaller than the speci®ed error tolerance e > 0, may be outlined as follows. In the 0th step of the procedure, the approximate values of the increments dg,n are determined by Eq. (12). Since we have a kinematically indeterminate assembly, we can suppose that B is not a full-columnrank matrix; and among its minor matrices of the largest rank, B11 has the largest-absolute-value determinant. With a suitable rearrangement of rows and/or columns of B and d, Eqs. (12) and (13) take the form B11 B12 d B11 d1 B12 d2 0: 15 If a ®ctitious change of bar length t is present, then we have B11 B12 d t B11 d1 B12 d2 t 0: 16 Here, the dependent coordinate increments are in vector d1, and the only independent coordinate increment is in d2 (so now it is one-dimensional). Prescribing the value of the independent coordinate increment d2 and that of t, the dependent coordinate increments in d1 can be determined. In the 0th approximation t 0: Therefore, according to Eq. (15): d 10 ÿBÿ1 11 B12 d2 : d 0 1
17
obtained in this way will be the 0th approximation of d1, and the 0th approximation of d will be " # 0 d 0 d1 : 18 d2
In the next steps of the procedure, d2 remains unchanged. In the 1st step of the procedure, by substituting the components of d 0 into the system of Eq. (14), the components of t 1 are obtained, and the dependent coordinate increments should be changed by the ®ctitious change of length t t 1 : Thus, according to Eq. (16) we have 1 ÿ1 d 11 ÿBÿ1 11 B12 d2 ÿ B11 t :
19
Since the ®rst term of the right side of Eq. (19) is d 0 1 , Eq. (19) can be written as 1 d 11 d 10 ÿ Bÿ1 11 t :
That suggests the heuristic formula of successive approximation nÿ1 n d n ÿ Bÿ1 11 t 1 d1
by which the nth approximation of d is obtained: " # n d n 1 d : d2
20
21
If the absolute values of all components of vector t n are not greater than the speci®ed error tolerance e > 0 jt n i jRe;
i 1,2, . . . ,b
22
then the procedure may be considered as ®nished in the nth step. If the convergence of the approximation is slow then, to achieve greater eciency, it is worth reconstructing B with the actual nodal coordinates and determining B11 and its inverse by pivoting. If the structure approaches kinematic bifurcation (the absolute value of the vector product of the unit vectors of two adjacent bars is close to zero), then keeping matrix B unchanged and using successive approximation we can reach the bifurcation up to any degree of accuracy. If the procedure is divergent, then by the selected independent coordinate increment(s) in the neighbourhood of the initial state, it is not possible to determine a state which could be considered as a new initial state, but with change of sign of the independent coordinate increment it may be possible.
6. Examples In this section we shall consider applications of the numerical procedure developed in Section 5. Example 1. The ®rst example is one of the simplest mechanisms in the plane: a four-bar linkage. For par-
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
ticular ratios of bar lengths, it can occur that bifurcation appears in the motion of these linkages [7]. We will show that even bifurcation does not cause any dif®culty in our numerical procedure. Fig. 1 shows a four-bar linkage with all bars of equal length `. This assembly has a 3 4 compatibility matrix B whose rank is r B 3; that indicates that the assembly is indeed a ®nite mechanism. During the continuous motion the assembly gets to the position shown by thin lines in Fig. 1(a). In this position, denoting x and y by 1 and 2, we write the exact coordinates of the nodes 1,2,3,4 in order of succession 11, 12, 21, 22, 31, 32, 41, 42 in the vector: rT a c 1 a a
c 0 0 1
0 `,
149
cent unit vectors (e.g., e1 and e2 je1 e2 j c > 0 but this value, in course of the further motion, would tend to zero. If, for instance, ` 1 and c 0:1 then in course of the further motion (even if we want to reach axis x in steps smaller than c ` we describe the change with the compatibility matrix corresponding to the values c 0:1; a 0:994987437:
11 12 21 22 3 1 1 0 ÿ1 0 B 4 ÿa ÿc 0 0 5 2 0 0 ÿa ÿc 3 2
p 1 ÿ c2 :
The absolute value of the vector product of two adja-
If d2 d12 is selected as free ®nite motion, then after rearrangement we have:
Fig. 1. A four-bar linkage in (a) a general compatible position, (b) a folded state (at a bifurcation of the motion). Scheme of calculation of motion at the folded state; motion due to the ®rst degree of freedom: the assembly in (c) an incompatible con®guration determining a ®ctitious assembly and (d) a new compatible position.
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150
11 22
3 ÿ1 1 0 40 ÿa 0 5; ÿa 0 ÿc 2 3 d21 6 d11 7 d1 7 d 6 4 d22 5; d2 d12
B11
2
21
2
12 3 0 4 ÿc 5; 0 2
B12
t
3 0 4 0:000038133 5 `; 0:000038133 2
2 d 12 d 11 ÿ Bÿ1 11 t
3 0:005012754 4 0:005012754 5 `; ÿ0:1
3 0 4 ÿ0:000000191 5 `; ÿ0:000000191 2
t
3
ÿ1 ÿ1=a 0 40 5; Bÿ1 ÿ1=a 0 11 a=c 1=c ÿ1=c
d12 ÿ0:1 ` d2 ;
` 1; 2
d 10
2
2
3 c=a 4 5 d2 ÿBÿ1 11 B12 d2 ` c=a ÿ1 2 3 0:010050 21 4 ` 0:010050 5 11 22 ÿ0:1
Let the error tolerance be speci®ed e 10ÿ6 `, then the error vectors in the successive steps of the iteration are: t 0 0; 2
3 0 t 1 4 ÿ0:005050505 5 `; ÿ0:005050505 2 3 0:004974430 1 0 1 ÿ1 d1 d1 ÿ B11 t 4 0:004974430 5 `; ÿ0:1
Fig. 2. Bifurcation of the compatibility path of the four-bar linkage in Fig. 1(a).
3
jt i 3 j < e 10ÿ6 `:
By adding the values d contained in d 2 to the previous components of vector rT , we obtain with accuracy of e the vector rT corresponding to the new position: rT 1 0 2 0 0 0 1 0 ` and the compatibility matrix is: 11 12 21 22 3 1 0 ÿ1 0 : B 4 ÿ1 0 0 05 0 0 ÿ1 0 2
This position of the assembly (Fig. 1(b)) can be called folded state. Further motion of the assembly, or unfolding of the folded assembly can be directed on the same principles. Here r B 2, that is the rank has decreased by 1. The change of rank indicates that in this position the motion has a bifurcation (Fig. 2). Locally, the assembly behaves like a two-degree-offreedom in®nitesimal mechanism; that is, it has two dierent independent in®nitesimal motions. The two independent in®nitesimal motions, however, together are ®ctitious, and only one of them can be real. That means the possibility of two independent excitations. So, we can prescribe arbitrary small, but ®nite coordinates r12 or r22 separately. Firstly, let us prescribe r12 ÿ0:1 ` and keep r22 0: In Fig. 1(c), there is a ®ctitious one-degree-offreedom assembly whose compatibility matrix is suitable to calculate the dependent coordinate increments corresponding to the independent coordinate increments of the real assembly with any degree of accuracy. The independent coordinate increment can be identical to r12 in Fig. 1(c), but can be a fraction of r12 (especially if points of the trajectories should be densely determined). The sub-procedure for unfolding the folded assembly is ended if the real assembly (and its compatibility matrix) ful®ls the requirements detailed above. The compatibility matrix composed of the unit vectors of the bars of the ®ctitious assembly takes the
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
form
B
are 2
11 12
21 22
3
ÿa ÿc 1 0 0 5 2 3 ÿ1 0
a c 4 ÿa c 0 0
a 0:995037190, c 0:099503719:
Let the independent coordinate increment be d2 d12 then partitioning B according to this we have 2
B11
B12
151
11 21
22
2
ÿa ÿc
1
2
3 0
3
2
d 10
3 ÿ0:01 11 4 5 21 ÿBÿ1 11 ` B12 d2 ` 0 ÿ0:2 22
Now the folded real assembly is considered the initial con®guration whose node coordinates are compiled in vector rT 1 0
3
11 7 ÿ1 5 21 a=c 22
0 ÿ1=a Bÿ1 11 6 0 40 ÿ1=c ÿ1=c
2
Let d2 d12 ÿ0:1 ` and e 5 10ÿ6 `, then
3
1 6 7 ; 0 5 2 4 ÿa 0 3 0 ÿ1 0 12 2 3 c 1 6 7 4 c 5 2 ; 0 3 a
1
3 0 1=a 0 5; 40 0 1 1=c 1=c ÿa=c 2 3 c=a 4 0 5; c=a 0:1: ÿBÿ1 11 B12 2 ÿBÿ1 11
2
2 0 0 0
1 0 `:
In the successive steps of the iteration we have the following error vectors which are arranged in a tabular form t 0 0):
t 1 =`
t 2 =`
t 3 =`
t 4 =`
t 5 =`
t 6 =`
ÿ0.015050 +0.004950 ÿ0.020000
+0.014861 +0.000013 +0.014648
+0.000338 0 +0.000486
ÿ0.000147 0 ÿ0.000142
ÿ0.000004 0 ÿ0.000006
ÿ0.000001 0 ÿ0.000001
(1 (2 (3
and similarly the dependent coordinate increment vectors: d 0 1 =`
d 1 1 =`
d 2 1 =`
d 3 1 =`
d 4 1 =`
d 5 1 =`
ÿ0.01 0 ÿ0.2
ÿ0.005025 ÿ0.020000 ÿ0.101504
ÿ0.005013 ÿ0.005352 ÿ0.098506
ÿ0.005013 ÿ0.004866 ÿ0.099966
ÿ0.005013 ÿ0.005008 ÿ0.100015
ÿ0.005013 ÿ0.005014 ÿ0.010000
The exact value is: d11 d21 ÿ0:005012563`
The matrices in the approximation formula nÿ1 n d n ÿ Bÿ1 11 t ; 1 d1
d 10 ÿBÿ1 11 B12 d2
t 0 0;
(11 (21 (22
It is seen that the absolute values of the components ÿ6 of t 6 are smaller than e: jt 6 `, i i j < e 5 10 1,2,3: Therefore, the iteration can be stopped. By adding the components of d obtained in the last step to the components of vector r of the nodes in the initial
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
152
compatible state, the coordinates of the nodes in the new compatible state are obtained: T
h
r `
d 115
d12
2`
d 215
d 225
0 0 `
i 0 :
The new compatible position of the assembly is shown in Fig. 1(d). Secondly, we analyse the motion due to the second degree of freedom. The initial state is the folded state of the assembly (Fig. 3(a)), where the initial node coordinates are rT ` 0
2` 0 0
0 `
0 :
Let us prescribe r22 ÿ0:1 ` and keep r12 0: In Fig. 3(b) there is a ®ctitious one-degree-of-freedom assembly whose compatibility matrix composed of the unit vectors of its bars takes the form 2
11 12
a B 4 ÿ1 0
ÿc 0 0
21 22 3 ÿa c 1 0 0 5 2 ÿa c 3
a 0:995037190, c 0:099503719:
Let the independent coordinate increment be d2 d22 then partitioning B according to this we have
2
B11
11 12
a 4 ÿ1 0 2
Bÿ1 11
21 3 ÿa 1 0 5 2 ; ÿa 3
ÿc 0 0
1
2
0 4 ÿ1=c 0
B12
22 2 3 c 40 5; c
3
3 11 ÿ1 0 ÿa=c 1=c 5 12 0 ÿ1=a 21
Let d2 d22 ÿ0:1 ` and e 10ÿ6 `, then the 0th approximation of the dependent coordinate increments is 2
d 10
ÿBÿ1 11
3 11 0 5 12 B12 d2 4 0 ÿ0:01 ` 21
In the successive steps of the aproximation we have the following error vectors: t 1 =`
t 2 =`
t 3 =`
0.004950000 0 0.004950000
0.000012685 0 0.000012685
0.000000001 0 0.000000001
(1 (2 (3
Fig. 3. Scheme of calculation of motion of the four-bar linkage at the folded state (a); motion due to the second degree of freedom: the assembly in (b) an incompatible con®guration determining a ®ctitious assembly and (c) a new compatible position.
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
153
Fig. 4. The Connelly±Servatius mechanism: (a) initial state (at the bifurcation of motion); (b) and (c) two possibilities of incompatible con®gurations de®ning a ®ctitious assembly for the determination of a new compatible position.
and the dependent coordinate increment vectors: d1 1 =`
d 2 1 =`
0 0 0.004974688
0 0 0.005012564
(11 (12 (21
Vector d 2 1 can be considered exact, because the absolute values of the components of t 3 are smaller than e 10ÿ6 `: Therefore, we have a compatible position of the assembly shown in Fig. 3(c). Fig. 5. A cusp type bifurcation of the compatibility path of the Connelly±Servatius mechanism.
Example 2. The second example, shown in Fig. 4(a),
154
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Fig. 6. Trajectories of the end points of the connecting bar (nodes 1 and 2 in Fig. 4).
is more complicated. This assembly has been analysed by Connelly and Servatius [9] and by GaÂspaÂr and Tarnai [10]. It consists of two equal Watt type mechanisms, connected by a horizontal bar. It is a one-degree-of-freedom ®nite mechanism, and in the position shown in Fig. 4(a) its motion has a bifurcation. The peculiar thing is that, in the kinematic parameter space, the point of bifurcation is such that the two branches of the compatibility path after their intersection have no continuation at the point of bifurcation: that is, the compatibility path has a cusp there (Fig. 5). In the symmetrical con®guration shown in Fig. 4(a), that we consider as the initial state, the rank of the compatibility matrix B is less than the number of columns of B by 2: 2j ÿ r B 2: That means that the assembly locally behaves like a two-degree-of-freedom in®nitesimal mechanism. Both degrees of freedom can be used separately for exciting the mechanism. To excite the folded structure, the procedure suggested in the previous example is applied. We move node 1 (or 2) downwards arbitrarily up to such a value r12 (while r22 is unchanged) for which the compatibility matrix B of the ®ctitious structure obtained in this way satis®es the conditions m j ÿ r B 1 and jdet Bjr0:1: However, these displacements (increments) can point only downwards because in upward direction there is no compatible state of the assembly in a small neighbourhood of the initial state. Let, for instance, r12 ÿ0:6 and r22 ÿ0:5 (or r12 ÿ0:5 and r22 ÿ0:6), then the compatibility matrix B of the ®ctitious structure shown in Fig. 4(b) is used in the successive approximation to describe the motions of the real structure as long as the real structure gets to a position where its actual compatibility matrix does not satisfy the above-mentioned conditions m j ÿ r B 1 and jdet Bjr0:1: (The variant in Fig. 4(c) can also be considered as a ®ctitious structure, that is obtained by taking node 3 of the left-hand Watt mechanism to the position of coordinate r32 :) After applying the numerical procedure developed in Section 5 we will ®nd that
in both newly obtained states, the rank r B is increased by 1, indicating that, indeed, the motion has a bifurcation in the starting position of the assembly. Since in the new states, B is a full-row-rank matrix, the assembly is a ®nite mechanism. By changing the coordinate increments we can proceed with the motion. Fig. 6 shows the trajectories of the end points of the connecting bar 1 (nodes 1 and 2), along which the points travel twice during a period of motion. The eciency of the computer program that we have developed, is shown also by the fact that even if we start the motion of nodes 1 or 2 upwards, the program ®nds a compatible state far from the initial state that would be reached by continuous motion.
7. Conclusions The numerical procedure presented here is suitable for determining compatible states of one or more degree-of-freedom mechanisms consisting of rigid bars and pin joints, and for describing their motion. The numerical procedure supplemented by equilibrium equations is also suitable for ®nding equilibrium forms of kinematically indeterminate assemblies under given load patterns; and if the bars are elastic, then it may be used for describing the behaviour of the assembly in the post-critical region where, after progressive buckling of bars, a bar-and-joint structure starts to behave like a ®nite mechanism. The procedure can also be applied in the analysis of kinematic geometry of certain mechanisms used in mechanical engineering.
Acknowledgements We thank Professors C.R. Calladine and Zs. GaÂspaÂr for helpful comments. The research reported here was done within the framework of the Hungarian±British Intergovernmental Science and Technology Cooperation Programme with the partial support of
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
OMFB and the British Council. Partial support by OTKA Grants No. T015860 and No. T0224037, and FKFP Grant No. 0391/1997 is also gratefully acknowledged. T.T. thanks the Head of the Engineering Department of the University of Cambridge for hospitality. Appendix A The model presented in Fig. 1 is suitable to show that, under evident restrictions, the trivial solutions always satisfy the system of exact compatibility equations. With the notations in the ®gure, the system of Eq. (8) takes the form: r21 d21 =2 ÿ r11 ÿ d11 =2 d11 ÿ d21 r22 d22 =2 ÿ r12 ÿ d12 =2 d12 ÿ d22 =` 0,
` 0 ÿ ` ÿ a=2 a ÿ 0 0 0 ÿ ` ÿ b=2 b ÿ 0 ÿa2 =2 ÿ `b ÿ b2 =2 0: 2. Let the initial state be the bifurcation state in Fig. 1(b). Now let us introduce a parameter b such that ÿ`RbR` arbitrary, and a parameter a such that a p 2 `2 ÿ b2 ÿ `: Moreover, let d21 a and d22 b, and the rest dhi ,n ,dki ,n 0: The vector of the initial node coordinates is rT ` 0 2` 0 0 0 ` 0 : Thus, Eqs. (A1)±(A3) take the form: 2` a=2 ÿ ` ÿ 0 0 ÿ a 0 b=2 ÿ 0 ÿ 0 0 ÿ b ÿ`a ÿ a2 =2 ÿ b2 =2 q q ÿ` `2 ÿ b2 ` `2 ÿ b2 b2 =2 ÿ b2 =2 0,
A1 0 0 ÿ ` ÿ 0 0 ÿ 0 0 0 ÿ 0 ÿ 0 0 ÿ 0 0,
r31 d31 =2 ÿ r11 ÿ d11 =2 d11 ÿ d31 r32 ` 0 ÿ 2` ÿ a=2 a ÿ 0 0 0 ÿ 0 ÿ b=2 b ÿ 0
d32 =2 ÿ r12 ÿ d12 =2 d12 ÿ d32 =` 0,
155
A2
ÿ`a ÿ a2 =2 ÿ b2 =2 0:
r41 d41 =2 ÿ r21 ÿ d21 =2 d21 ÿ d41 r42 d42 =2 ÿ r22 ÿ d22 =2 d22 ÿ d42 =` 0:
References A3
We show here two trivial cases. 1. Let the initial state be that in Fig. 1(a). Let us introduce a parameter a such that ÿ`RaR` arbitrary, p and a parameter b such that b 2 `2 ÿ a2 ÿ `: Moreover, let d11 d21 a and d12 d22 b, and the rest dhi ,n ,dki ,n 0: The vector of the initial node coordinates is rT 0 ` ` ` 0 0 ` 0 : Thus, Eqs. (A1)±(A3) take the form: ` a=2 ÿ 0 ÿ a=2 a ÿ a ` b=2 ÿ ` ÿ b=2 b ÿ b 0, 0 0 ÿ 0 ÿ a=2 a ÿ 0 0 ÿ 0 ÿ ` ÿ b=2 b ÿ 0 ÿa2 =2 ÿ `b ÿ b2 =2 p p ÿa2 =2 ÿ ` `2 ÿ a2 a2 =2 ` `2 ÿ a2 0,
[1] Tarnai T, GaÂspaÂr Zs. Improved packing of equal circles on a sphere and rigidity of its graph. Math Proc Cambridge Philos Soc 1983;93:191±218. [2] Kawaguchi M. Design problems of long span spatial structures. Eng Struct 1991;13:144±63. [3] Hangai Y, Kawaguchi K. Shape-®nding of unstable structures. Forma 1990;5:29±41. [4] You Z, Pellegrino S. Foldable bar structures. Int J Solids and Structures 1997;34:1825±47. [5] Escrig F, Valcarcel JP. Geometry of expandable space structures. Int J Space Structures 1993;8:71±84. [6] Hunt KH. Kinematic geometry of mechanisms. Oxford: Clarendon, 1978. [7] Haug EJ. Computer aided kinematics and dynamics of mechanical systems vol. 1: Basic methods. Boston: Allyn and Bacon, 1989. [8] Szabo J, KollaÂr L. Structural design of cable-suspended roofs. Budapest/Chichester: AkadeÂmiai KiadoÂ/Ellis Horwood, 1984. [9] Connelly R, Servatius H. Higher order rigidity Ð what is the proper de®nition? Discrete and Comp Geometry 1994;11:193±200. [10] GaÂspaÂr Zs, Tarnai T. Finite mechanisms have no higherorder rigidity. Acta Technica Acad Sci Hung 1994;106:119±25.
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On the exact equation of inextensional, kinematically indeterminate assemblies T. Tarnai*, J. Szabo Technical University of Budapest, Department of Structural Mechanics, MuÈegyetem rkp. 3., H-1521, Budapest, Hungary Received 30 April 1998; accepted 3 March 1999
Abstract An exact equation describing the ®nite displacements of assemblies consisting of rigid bars and pin joints is presented. A numerical procedure is proposed for carrying out the calculations, and some examples are shown in order to illustrate the procedure. To depict the ®nite change of state, a system of non-linear algebraic equations must be solved. With linear approximation, ®nite node coordinate increments can be used. This is one of the advantages of the proposed procedure in contrast with the compatibility dierential equation description, where only in®nitesimal increments can be used. The procedure can be applied even for the analysis of pin-jointed space frames in the post-critical regime, also in the calculation of certain mechanisms of mechanical engineering. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Mechanism; Kinematically indeterminate structure; Bar-and-joint assembly; Iterative solution; Set of dierence equations
1. Introduction Assemblies of pin-jointed bars form mechanical models of a wide range of engineering structures. Some of them are kinematically indeterminate [1]. The analysis of such assemblies or mechanisms, which undergo large inextensional deformation, that is, the analysis of ®nite mechanisms (which has been excluded from structural engineering before) has become important. The reason why structural engineers have become interested in ®nite mechanisms is that there is an increasing number of areas where ®nite mechanisms are relevant. Such areas include, for instance, the erec-
* Corresponding author. Tel.: +36-1-463-1431; fax: +36-1463-1099. E-mail address: [email protected] (T. Tarnai).
tion technology by folding and lifting, such as the pantadome system [2]; structural form-®nding under dierent load patterns [3]; and deployable structures [4,5]. The description of the kinematic geometry of barand-joint assemblies, especially in the plane, is a classical problem in kinematics. However, the basic concepts of kinematics are dierent from those in structural analysis. Kinematics and structural analysis use dierent languages; and even for the same problem, they provide dierent descriptions and dierent methods of solution. In kinematics, for a long time, analysis based on geometric constructions was used exclusively [6], and only recently have vector±matrix methods been introduced [7], introducing generalized coordinates (signed distances and angles). In structural analysis, on the other hand, matrix methods have been used for a relatively long time; and
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 9 0 - 5
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
146
Nomenclature b B d dhi dr drg,n ei f, fi hi , ki i, g j j0 j1 `i r
number of bars compatibility matrix in the case of in®nitesimal change m j-dimensional vector of ®nite coordinate increments vector of ®nite displacement of node hi m j-dimensional vector of in®nitesimal coordinate increments in®nitesimal increment of the nth coordinate of node g unit vector of bar i, pointing from hi to ki : ei rhi ,ki =`i constraint function and its ith component starting point and end point of bar i, i 1,2, . . . ,b; hi 2 f1,2, . . . ,j g; ki 2 f1,2, . . . ,j0 g; hi 6 ki integers number of internal joints total number of joints: j0 j j1 number of external joints (attached to a rigid boundary) length of bar i vector of nodes composed of vectors rg :r rT1 rT2 . . . rTj0 T
for the investigation of the kinematic geometry of barand-joint assemblies, a system of compatibility dierential equations has been set up, applying Cartesian coordinates of the nodes [8]. The incremental technique with Newton±Raphson iteration, is an eective tool for solving the dierential equations with given initial conditions. However, if we are looking for not only a particular position of the assembly, but also for all possible positions (we want to describe the motion), then it is more eective to use the exact compatibility equations Ð known as constraint equations in kinematics [7], instead of the compatibility dierential equations. The aim of this paper is to present the exact equations describing the ®nite motions of mechanisms composed of rigid bars and pin joints, and a numerical procedure that is based upon them.
2. Compatibility equation of a kinematically indeterminate structure consisting of rigid bars and pin joints Let us consider a pin-jointed assembly which con-
rg
position vector of node g; g 1,2, . . . ,j0 ; the numbering of nodes starts with the internal nodes g 1,2, . . . ,j and ends with the external nodes g j 1,j 2, . . . ,j0 rg,n the nth component of the position vector of node g, n 1, . . . ,m r0g,n the nth component of the position vector of node g in the initial state rhi ,ki ith bar vector: rhi ,ki rki ÿ rhi t vector of real or ®ctitious bar elongations ti elongation of bar i (ith component of t); i 1,2, . . . ,b dg,n ®nite increment of the nth coordinate of node g Df, Dfi ®nite increment of f and its ith component e error tolerance m dimension of physical space (plane: m 2; space: m 3) n integer r B rank of matrix B n nth approximation in the procedure T transpose scalar product
sists of j internal joints, j1 external joints embedded in a rigid boundary, and b rigid bars. The initial state (in other words, the geometry) of the assembly is determined by the coordinates of the nodes, and the length of each bar is uniquely determined by them. The length of bar i is s m 2 X `i r0ki ,n ÿ r0hi ,n :
1
n1
Since the bars of the assembly are rigid, the bar lengths `i are constant. Consequently, the value of the expression fi `i ÿ
m X 1
` n1 i
2
rki ,n ÿ rhi ,n
2
can be equal to zero, if and only if the distance between the end points of the bar, calculated with the coordinates of the end points, is identical to the length of the bar calculated with Eq. (1). Equality fi 0 expresses the fact that the length of bar i is compatible with the coordinates of the end points of bar i. Let a
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
b-dimensional vector f be composed of elements fi : If f0
3
then the assembly is said to be in a compatible state. Eq. (3) (in scalar form, a system of nonlinear equations) is called the compatibility equation (system of equations) of the kinematically indeterminate structure (mechanism) consisting of rigid bars and pin joints. There may now be prescribed as many properly selected ®nite coordinate increments dg,n , in the domain of motion of the mechanism as the degrees of freedom; and the remainder of the compatible coordinate increments can be calculated. In this way the coordinates of the nodes in a new compatible state, satisfying Eq. (3) can be determined. In other words, a vector f composed of elements fi ` i ÿ
m X 1ÿ
` n1 i
rki ,n dki ,n ÿ rhi ,n ÿ dhi ,n
2
4
also satis®es Eq. (3). This will be illustrated in Appendix A with a trivial example. There are well-known algorithms and computer procedures for solving the system of equations (3). Nevertheless, it is worthwhile to seek a treatment of the continuous change of state of bar-and-joint mechanisms, based on the speci®c features of such assemblies. In the following sections, such an approach and a proposal for a numerical procedure will be presented.
3. The exact equation of the ®nite change of state of the assembly If the value of the constraint function in the state corresponding to Eq. (2) is denoted by f and in the state corresponding to Eq. (4) by f Df, and if both states are compatible states, then it is quite obvious that the dierence between Eqs. (4) and (2) results in f Df ÿ f Df 0
5
and this equation can be considered as the exact equation (system of quadratic equations) of the ®nite change of state of the assembly. The ith equation of the system of Eq. (5) is Dfi
m h X 1 2 rki ,n ÿ rhi ,n ` n1 i 2 i ÿ ÿ rki ,n dki ,n ÿ rhi ,n ÿ dhi ,n 0
whence, by reducing by 2, equation
6
m X 1
147
rki ,n ÿ rhi ,n
` n1 i ÿ
dhi ,n ÿ dki ,n
2 1ÿ ÿ dki ,n ÿ dhi ,n 0 2
7
is obtained, whose form more advantageous for computation is m X 1 dk ,n dh ,n ÿ dhi ,n ÿ dki ,n 0: 8 rki ,n i ÿ rhi ,n ÿ i 2 2 ` n1 i
4. Linear approximation of the exact equation of the ®nite change of state and the linear compatibility dierential equation By removing the term quadratic in d from the exact Eq. (7), we obtain an equation that can be considered as a linear approximation of Eq. (7): m X 1
` n1 i
ÿ rki ,n ÿ rhi ,n dhi ,n ÿ dki ,n 0
9
or in a more concise form: eTi dhi ÿ eTi dki 0;
ei 6 0:
10
This enables us to ascertain that the coecient matrix in the linear approximation of the exact equations is an extension of the coecient matrix of the system of compatibility dierential equations (valid for in®nitesimal changes) for ®nite displacements. However, while this matrix is exact for in®nitesimal change, it is only approximate for ®nite change. Namely, Eq. (10) makes it possible to ®ll an `empty' b jm matrix B 0 with m-dimensional unit rowvectors (subvectors) eTi and ÿeTi so that multiplying the obtained matrix B 6 0 by vector d, Eq. (10) is ful®lled for every i. Each row of the `empty' matrix is partitioned into m-dimensional subvectors. The formula of ®lling is the following: . the hi th zero subvector of row i is replaced with unit vector eTi , . the ki th zero subvector of row i is replaced with unit vector ÿeTi if ki Rj, and the zero subvector is preserved if ki > j: Considering the j-dimensional vector d containing the ®nite node displacement values d does not contain the zero displacements of the external joints, it is easy to see Ð if the ith rowvector of B is denoted by Bi Ð that Eq. (10) can be written in the form Bi d 0: For all Bi 's together:
11
148
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
Bd0
12
and taking the vector t of given real or ®ctitious bar elongations into account: B d t 0:
13
5. Inhomogeneous equation and successive approximation In this section we restrict ourselves to one-degree-offreedom assemblies, emphasizing, however, that the actual numerical studies have been extended also to particular multi-degree-of-freedom assemblies. Substituting (partly) arbitrarily selected coordinate increments dki ,n , dhi ,n into the system of equation (8), Dfi in general will not be equal to zero: m X 1 dk ,n dh ,n Dfi rki ,n i ÿ rhi ,n ÿ i ` 2 2 n1 i ÿ dhi ,n ÿ dki ,n ti : 14 The proposed procedure, which makes the absolute value of each component ti of the error vector t smaller than the speci®ed error tolerance e > 0, may be outlined as follows. In the 0th step of the procedure, the approximate values of the increments dg,n are determined by Eq. (12). Since we have a kinematically indeterminate assembly, we can suppose that B is not a full-columnrank matrix; and among its minor matrices of the largest rank, B11 has the largest-absolute-value determinant. With a suitable rearrangement of rows and/or columns of B and d, Eqs. (12) and (13) take the form B11 B12 d B11 d1 B12 d2 0: 15 If a ®ctitious change of bar length t is present, then we have B11 B12 d t B11 d1 B12 d2 t 0: 16 Here, the dependent coordinate increments are in vector d1, and the only independent coordinate increment is in d2 (so now it is one-dimensional). Prescribing the value of the independent coordinate increment d2 and that of t, the dependent coordinate increments in d1 can be determined. In the 0th approximation t 0: Therefore, according to Eq. (15): d 10 ÿBÿ1 11 B12 d2 : d 0 1
17
obtained in this way will be the 0th approximation of d1, and the 0th approximation of d will be " # 0 d 0 d1 : 18 d2
In the next steps of the procedure, d2 remains unchanged. In the 1st step of the procedure, by substituting the components of d 0 into the system of Eq. (14), the components of t 1 are obtained, and the dependent coordinate increments should be changed by the ®ctitious change of length t t 1 : Thus, according to Eq. (16) we have 1 ÿ1 d 11 ÿBÿ1 11 B12 d2 ÿ B11 t :
19
Since the ®rst term of the right side of Eq. (19) is d 0 1 , Eq. (19) can be written as 1 d 11 d 10 ÿ Bÿ1 11 t :
That suggests the heuristic formula of successive approximation nÿ1 n d n ÿ Bÿ1 11 t 1 d1
by which the nth approximation of d is obtained: " # n d n 1 d : d2
20
21
If the absolute values of all components of vector t n are not greater than the speci®ed error tolerance e > 0 jt n i jRe;
i 1,2, . . . ,b
22
then the procedure may be considered as ®nished in the nth step. If the convergence of the approximation is slow then, to achieve greater eciency, it is worth reconstructing B with the actual nodal coordinates and determining B11 and its inverse by pivoting. If the structure approaches kinematic bifurcation (the absolute value of the vector product of the unit vectors of two adjacent bars is close to zero), then keeping matrix B unchanged and using successive approximation we can reach the bifurcation up to any degree of accuracy. If the procedure is divergent, then by the selected independent coordinate increment(s) in the neighbourhood of the initial state, it is not possible to determine a state which could be considered as a new initial state, but with change of sign of the independent coordinate increment it may be possible.
6. Examples In this section we shall consider applications of the numerical procedure developed in Section 5. Example 1. The ®rst example is one of the simplest mechanisms in the plane: a four-bar linkage. For par-
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
ticular ratios of bar lengths, it can occur that bifurcation appears in the motion of these linkages [7]. We will show that even bifurcation does not cause any dif®culty in our numerical procedure. Fig. 1 shows a four-bar linkage with all bars of equal length `. This assembly has a 3 4 compatibility matrix B whose rank is r B 3; that indicates that the assembly is indeed a ®nite mechanism. During the continuous motion the assembly gets to the position shown by thin lines in Fig. 1(a). In this position, denoting x and y by 1 and 2, we write the exact coordinates of the nodes 1,2,3,4 in order of succession 11, 12, 21, 22, 31, 32, 41, 42 in the vector: rT a c 1 a a
c 0 0 1
0 `,
149
cent unit vectors (e.g., e1 and e2 je1 e2 j c > 0 but this value, in course of the further motion, would tend to zero. If, for instance, ` 1 and c 0:1 then in course of the further motion (even if we want to reach axis x in steps smaller than c ` we describe the change with the compatibility matrix corresponding to the values c 0:1; a 0:994987437:
11 12 21 22 3 1 1 0 ÿ1 0 B 4 ÿa ÿc 0 0 5 2 0 0 ÿa ÿc 3 2
p 1 ÿ c2 :
The absolute value of the vector product of two adja-
If d2 d12 is selected as free ®nite motion, then after rearrangement we have:
Fig. 1. A four-bar linkage in (a) a general compatible position, (b) a folded state (at a bifurcation of the motion). Scheme of calculation of motion at the folded state; motion due to the ®rst degree of freedom: the assembly in (c) an incompatible con®guration determining a ®ctitious assembly and (d) a new compatible position.
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
150
11 22
3 ÿ1 1 0 40 ÿa 0 5; ÿa 0 ÿc 2 3 d21 6 d11 7 d1 7 d 6 4 d22 5; d2 d12
B11
2
21
2
12 3 0 4 ÿc 5; 0 2
B12
t
3 0 4 0:000038133 5 `; 0:000038133 2
2 d 12 d 11 ÿ Bÿ1 11 t
3 0:005012754 4 0:005012754 5 `; ÿ0:1
3 0 4 ÿ0:000000191 5 `; ÿ0:000000191 2
t
3
ÿ1 ÿ1=a 0 40 5; Bÿ1 ÿ1=a 0 11 a=c 1=c ÿ1=c
d12 ÿ0:1 ` d2 ;
` 1; 2
d 10
2
2
3 c=a 4 5 d2 ÿBÿ1 11 B12 d2 ` c=a ÿ1 2 3 0:010050 21 4 ` 0:010050 5 11 22 ÿ0:1
Let the error tolerance be speci®ed e 10ÿ6 `, then the error vectors in the successive steps of the iteration are: t 0 0; 2
3 0 t 1 4 ÿ0:005050505 5 `; ÿ0:005050505 2 3 0:004974430 1 0 1 ÿ1 d1 d1 ÿ B11 t 4 0:004974430 5 `; ÿ0:1
Fig. 2. Bifurcation of the compatibility path of the four-bar linkage in Fig. 1(a).
3
jt i 3 j < e 10ÿ6 `:
By adding the values d contained in d 2 to the previous components of vector rT , we obtain with accuracy of e the vector rT corresponding to the new position: rT 1 0 2 0 0 0 1 0 ` and the compatibility matrix is: 11 12 21 22 3 1 0 ÿ1 0 : B 4 ÿ1 0 0 05 0 0 ÿ1 0 2
This position of the assembly (Fig. 1(b)) can be called folded state. Further motion of the assembly, or unfolding of the folded assembly can be directed on the same principles. Here r B 2, that is the rank has decreased by 1. The change of rank indicates that in this position the motion has a bifurcation (Fig. 2). Locally, the assembly behaves like a two-degree-offreedom in®nitesimal mechanism; that is, it has two dierent independent in®nitesimal motions. The two independent in®nitesimal motions, however, together are ®ctitious, and only one of them can be real. That means the possibility of two independent excitations. So, we can prescribe arbitrary small, but ®nite coordinates r12 or r22 separately. Firstly, let us prescribe r12 ÿ0:1 ` and keep r22 0: In Fig. 1(c), there is a ®ctitious one-degree-offreedom assembly whose compatibility matrix is suitable to calculate the dependent coordinate increments corresponding to the independent coordinate increments of the real assembly with any degree of accuracy. The independent coordinate increment can be identical to r12 in Fig. 1(c), but can be a fraction of r12 (especially if points of the trajectories should be densely determined). The sub-procedure for unfolding the folded assembly is ended if the real assembly (and its compatibility matrix) ful®ls the requirements detailed above. The compatibility matrix composed of the unit vectors of the bars of the ®ctitious assembly takes the
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
form
B
are 2
11 12
21 22
3
ÿa ÿc 1 0 0 5 2 3 ÿ1 0
a c 4 ÿa c 0 0
a 0:995037190, c 0:099503719:
Let the independent coordinate increment be d2 d12 then partitioning B according to this we have 2
B11
B12
151
11 21
22
2
ÿa ÿc
1
2
3 0
3
2
d 10
3 ÿ0:01 11 4 5 21 ÿBÿ1 11 ` B12 d2 ` 0 ÿ0:2 22
Now the folded real assembly is considered the initial con®guration whose node coordinates are compiled in vector rT 1 0
3
11 7 ÿ1 5 21 a=c 22
0 ÿ1=a Bÿ1 11 6 0 40 ÿ1=c ÿ1=c
2
Let d2 d12 ÿ0:1 ` and e 5 10ÿ6 `, then
3
1 6 7 ; 0 5 2 4 ÿa 0 3 0 ÿ1 0 12 2 3 c 1 6 7 4 c 5 2 ; 0 3 a
1
3 0 1=a 0 5; 40 0 1 1=c 1=c ÿa=c 2 3 c=a 4 0 5; c=a 0:1: ÿBÿ1 11 B12 2 ÿBÿ1 11
2
2 0 0 0
1 0 `:
In the successive steps of the iteration we have the following error vectors which are arranged in a tabular form t 0 0):
t 1 =`
t 2 =`
t 3 =`
t 4 =`
t 5 =`
t 6 =`
ÿ0.015050 +0.004950 ÿ0.020000
+0.014861 +0.000013 +0.014648
+0.000338 0 +0.000486
ÿ0.000147 0 ÿ0.000142
ÿ0.000004 0 ÿ0.000006
ÿ0.000001 0 ÿ0.000001
(1 (2 (3
and similarly the dependent coordinate increment vectors: d 0 1 =`
d 1 1 =`
d 2 1 =`
d 3 1 =`
d 4 1 =`
d 5 1 =`
ÿ0.01 0 ÿ0.2
ÿ0.005025 ÿ0.020000 ÿ0.101504
ÿ0.005013 ÿ0.005352 ÿ0.098506
ÿ0.005013 ÿ0.004866 ÿ0.099966
ÿ0.005013 ÿ0.005008 ÿ0.100015
ÿ0.005013 ÿ0.005014 ÿ0.010000
The exact value is: d11 d21 ÿ0:005012563`
The matrices in the approximation formula nÿ1 n d n ÿ Bÿ1 11 t ; 1 d1
d 10 ÿBÿ1 11 B12 d2
t 0 0;
(11 (21 (22
It is seen that the absolute values of the components ÿ6 of t 6 are smaller than e: jt 6 `, i i j < e 5 10 1,2,3: Therefore, the iteration can be stopped. By adding the components of d obtained in the last step to the components of vector r of the nodes in the initial
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
152
compatible state, the coordinates of the nodes in the new compatible state are obtained: T
h
r `
d 115
d12
2`
d 215
d 225
0 0 `
i 0 :
The new compatible position of the assembly is shown in Fig. 1(d). Secondly, we analyse the motion due to the second degree of freedom. The initial state is the folded state of the assembly (Fig. 3(a)), where the initial node coordinates are rT ` 0
2` 0 0
0 `
0 :
Let us prescribe r22 ÿ0:1 ` and keep r12 0: In Fig. 3(b) there is a ®ctitious one-degree-of-freedom assembly whose compatibility matrix composed of the unit vectors of its bars takes the form 2
11 12
a B 4 ÿ1 0
ÿc 0 0
21 22 3 ÿa c 1 0 0 5 2 ÿa c 3
a 0:995037190, c 0:099503719:
Let the independent coordinate increment be d2 d22 then partitioning B according to this we have
2
B11
11 12
a 4 ÿ1 0 2
Bÿ1 11
21 3 ÿa 1 0 5 2 ; ÿa 3
ÿc 0 0
1
2
0 4 ÿ1=c 0
B12
22 2 3 c 40 5; c
3
3 11 ÿ1 0 ÿa=c 1=c 5 12 0 ÿ1=a 21
Let d2 d22 ÿ0:1 ` and e 10ÿ6 `, then the 0th approximation of the dependent coordinate increments is 2
d 10
ÿBÿ1 11
3 11 0 5 12 B12 d2 4 0 ÿ0:01 ` 21
In the successive steps of the aproximation we have the following error vectors: t 1 =`
t 2 =`
t 3 =`
0.004950000 0 0.004950000
0.000012685 0 0.000012685
0.000000001 0 0.000000001
(1 (2 (3
Fig. 3. Scheme of calculation of motion of the four-bar linkage at the folded state (a); motion due to the second degree of freedom: the assembly in (b) an incompatible con®guration determining a ®ctitious assembly and (c) a new compatible position.
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
153
Fig. 4. The Connelly±Servatius mechanism: (a) initial state (at the bifurcation of motion); (b) and (c) two possibilities of incompatible con®gurations de®ning a ®ctitious assembly for the determination of a new compatible position.
and the dependent coordinate increment vectors: d1 1 =`
d 2 1 =`
0 0 0.004974688
0 0 0.005012564
(11 (12 (21
Vector d 2 1 can be considered exact, because the absolute values of the components of t 3 are smaller than e 10ÿ6 `: Therefore, we have a compatible position of the assembly shown in Fig. 3(c). Fig. 5. A cusp type bifurcation of the compatibility path of the Connelly±Servatius mechanism.
Example 2. The second example, shown in Fig. 4(a),
154
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
Fig. 6. Trajectories of the end points of the connecting bar (nodes 1 and 2 in Fig. 4).
is more complicated. This assembly has been analysed by Connelly and Servatius [9] and by GaÂspaÂr and Tarnai [10]. It consists of two equal Watt type mechanisms, connected by a horizontal bar. It is a one-degree-of-freedom ®nite mechanism, and in the position shown in Fig. 4(a) its motion has a bifurcation. The peculiar thing is that, in the kinematic parameter space, the point of bifurcation is such that the two branches of the compatibility path after their intersection have no continuation at the point of bifurcation: that is, the compatibility path has a cusp there (Fig. 5). In the symmetrical con®guration shown in Fig. 4(a), that we consider as the initial state, the rank of the compatibility matrix B is less than the number of columns of B by 2: 2j ÿ r B 2: That means that the assembly locally behaves like a two-degree-of-freedom in®nitesimal mechanism. Both degrees of freedom can be used separately for exciting the mechanism. To excite the folded structure, the procedure suggested in the previous example is applied. We move node 1 (or 2) downwards arbitrarily up to such a value r12 (while r22 is unchanged) for which the compatibility matrix B of the ®ctitious structure obtained in this way satis®es the conditions m j ÿ r B 1 and jdet Bjr0:1: However, these displacements (increments) can point only downwards because in upward direction there is no compatible state of the assembly in a small neighbourhood of the initial state. Let, for instance, r12 ÿ0:6 and r22 ÿ0:5 (or r12 ÿ0:5 and r22 ÿ0:6), then the compatibility matrix B of the ®ctitious structure shown in Fig. 4(b) is used in the successive approximation to describe the motions of the real structure as long as the real structure gets to a position where its actual compatibility matrix does not satisfy the above-mentioned conditions m j ÿ r B 1 and jdet Bjr0:1: (The variant in Fig. 4(c) can also be considered as a ®ctitious structure, that is obtained by taking node 3 of the left-hand Watt mechanism to the position of coordinate r32 :) After applying the numerical procedure developed in Section 5 we will ®nd that
in both newly obtained states, the rank r B is increased by 1, indicating that, indeed, the motion has a bifurcation in the starting position of the assembly. Since in the new states, B is a full-row-rank matrix, the assembly is a ®nite mechanism. By changing the coordinate increments we can proceed with the motion. Fig. 6 shows the trajectories of the end points of the connecting bar 1 (nodes 1 and 2), along which the points travel twice during a period of motion. The eciency of the computer program that we have developed, is shown also by the fact that even if we start the motion of nodes 1 or 2 upwards, the program ®nds a compatible state far from the initial state that would be reached by continuous motion.
7. Conclusions The numerical procedure presented here is suitable for determining compatible states of one or more degree-of-freedom mechanisms consisting of rigid bars and pin joints, and for describing their motion. The numerical procedure supplemented by equilibrium equations is also suitable for ®nding equilibrium forms of kinematically indeterminate assemblies under given load patterns; and if the bars are elastic, then it may be used for describing the behaviour of the assembly in the post-critical region where, after progressive buckling of bars, a bar-and-joint structure starts to behave like a ®nite mechanism. The procedure can also be applied in the analysis of kinematic geometry of certain mechanisms used in mechanical engineering.
Acknowledgements We thank Professors C.R. Calladine and Zs. GaÂspaÂr for helpful comments. The research reported here was done within the framework of the Hungarian±British Intergovernmental Science and Technology Cooperation Programme with the partial support of
T. Tarnai, J. Szabo / Computers and Structures 75 (2000) 145±155
OMFB and the British Council. Partial support by OTKA Grants No. T015860 and No. T0224037, and FKFP Grant No. 0391/1997 is also gratefully acknowledged. T.T. thanks the Head of the Engineering Department of the University of Cambridge for hospitality. Appendix A The model presented in Fig. 1 is suitable to show that, under evident restrictions, the trivial solutions always satisfy the system of exact compatibility equations. With the notations in the ®gure, the system of Eq. (8) takes the form: r21 d21 =2 ÿ r11 ÿ d11 =2 d11 ÿ d21 r22 d22 =2 ÿ r12 ÿ d12 =2 d12 ÿ d22 =` 0,
` 0 ÿ ` ÿ a=2 a ÿ 0 0 0 ÿ ` ÿ b=2 b ÿ 0 ÿa2 =2 ÿ `b ÿ b2 =2 0: 2. Let the initial state be the bifurcation state in Fig. 1(b). Now let us introduce a parameter b such that ÿ`RbR` arbitrary, and a parameter a such that a p 2 `2 ÿ b2 ÿ `: Moreover, let d21 a and d22 b, and the rest dhi ,n ,dki ,n 0: The vector of the initial node coordinates is rT ` 0 2` 0 0 0 ` 0 : Thus, Eqs. (A1)±(A3) take the form: 2` a=2 ÿ ` ÿ 0 0 ÿ a 0 b=2 ÿ 0 ÿ 0 0 ÿ b ÿ`a ÿ a2 =2 ÿ b2 =2 q q ÿ` `2 ÿ b2 ` `2 ÿ b2 b2 =2 ÿ b2 =2 0,
A1 0 0 ÿ ` ÿ 0 0 ÿ 0 0 0 ÿ 0 ÿ 0 0 ÿ 0 0,
r31 d31 =2 ÿ r11 ÿ d11 =2 d11 ÿ d31 r32 ` 0 ÿ 2` ÿ a=2 a ÿ 0 0 0 ÿ 0 ÿ b=2 b ÿ 0
d32 =2 ÿ r12 ÿ d12 =2 d12 ÿ d32 =` 0,
155
A2
ÿ`a ÿ a2 =2 ÿ b2 =2 0:
r41 d41 =2 ÿ r21 ÿ d21 =2 d21 ÿ d41 r42 d42 =2 ÿ r22 ÿ d22 =2 d22 ÿ d42 =` 0:
References A3
We show here two trivial cases. 1. Let the initial state be that in Fig. 1(a). Let us introduce a parameter a such that ÿ`RaR` arbitrary, p and a parameter b such that b 2 `2 ÿ a2 ÿ `: Moreover, let d11 d21 a and d12 d22 b, and the rest dhi ,n ,dki ,n 0: The vector of the initial node coordinates is rT 0 ` ` ` 0 0 ` 0 : Thus, Eqs. (A1)±(A3) take the form: ` a=2 ÿ 0 ÿ a=2 a ÿ a ` b=2 ÿ ` ÿ b=2 b ÿ b 0, 0 0 ÿ 0 ÿ a=2 a ÿ 0 0 ÿ 0 ÿ ` ÿ b=2 b ÿ 0 ÿa2 =2 ÿ `b ÿ b2 =2 p p ÿa2 =2 ÿ ` `2 ÿ a2 a2 =2 ` `2 ÿ a2 0,
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