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Bifurcations of Equilibria in Potential Systems at Bimodal Critical Points
Alexei A. Mailybaev
e-mail: mailybaev@imec.msu.ru

Alexander P. Seyranian
e-mail: seyran@imec.msu.ru Institute of Mechanics, Moscow State Lomonosov University, Michurinsky prospect 1, 119192 Moscow, Russia

Bifurcations of equilibria at bimodal branching points in potential systems are investigated. General formulas describing postbuckling paths and conditions for their stability are derived in terms of the original potential energy. Formulas describing unfolding of bimodal branching points due to a change of system parameters are given. A full list of possible cases for postbuckling paths, their stability, and unfolding depending on three system coefficients is presented. In order to calculate these coefficients, one needs the derivatives of the potential energy and eigenvectors of the linearized problem taken at the bifurcation point. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of an articulated elastic column having four degrees of freedom and depending on three problem parameters (stiffness coefficients at the hinges). For some of the bimodal critical points, numerical results are obtained illustrating influence of parameters on postbuckling paths, their stability, and unfolding. A surprising phenomenon that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode is recognized. An example with a constrained sum of the stiffnesses of the articulated column shows that the maximum critical load (optimal design) is attained at the bimodal point. DOI: 10.1115/1.2793136

1 Introduction
This paper is devoted to analysis of bimodal branching points of stable trivial equilibrium in multiple degrees-of-freedom potential systems with the symmetry. These points were studied in a number of books and papers 1­5 . In the books 1,2 , a rather general method how to analyze postbuckling paths and their stability is presented. This method involves diagonalization procedure of the potential energy and elimination of passive coordinates, i.e., some transformations of the original potential energy are needed. References 3,4 deal with the unfolding of bimodal branching points of general two degrees-of-freedom systems with symmetry. The bimodal critical points and their unfolding for two degrees-of-freedom systems with double symmetry were studied in Chap. X of the well-known book 5 on bifurcation theory. However, in these works, the full list of possible bifurcations was not given. Some early examples on bimodal critical points were presented in Refs. 6­8 . It turns out that bimodal branching points are closely related to structural optimization problems 1 . Bimodal optimal columns in continuous formulation were recognized in Ref. 9 . Since that time, bi- and multimodality multiplicity of eigenmodes at the same critical load became a popular topic in structural optimization under stability constraints 10­14 . In this paper, we intend to give a complete theory of bimodal bifurcations in potential systems with symmetries. We present the full classification of possible cases for postbuckling paths and their stability depending on three coefficients. It is important that all the formulas derived in this paper are given in terms of the original potential energy of the system with multiple degrees of freedom. Then, we study unfolding of bimodal branching points due to change of problem parameters. Our approach is straightforward, explicit, and practical allowing to analyze bifurcations and stability of postbuckling paths, as well as their unfolding, based on calculation of the derivatives of the potential energy and eigenContributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 25, 2006; final manuscript received August 20, 2007; published online February 26, 2008. Review conducted by Edmundo Corona.

vectors of the linearized problem, taken at the bifurcation point. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of an articulated bimodal elastic column having four degrees of freedom and depending on three parameters.

2

Potential Systems
with a state vector q a system are determined by function V q at which first respect to the state vector is 1

Consider a potential system = q1 , q2 , ... , qn . Equilibria of such critical points of the potential energy variation of the potential energy with zero:

V q =0

An equilibrium is stable if it is a minimum of the potential. The sufficient stability condition is that the second variation of the potential is positive for all small variations q:
2

Vq
2

0

2

with the unstrict inequality V q 0 giving the necessary condition. The equilibrium condition 1 can be written in the form V =0 = q ,
1

q

, ... ,
2

q

3
n

The stability condition 2 requires positive definiteness of the Hessian matrix
2 2

V/ q ]

2 1 2

2

V/ q1 q
2

2

¯ ¯

2 2

V/ q1 q V/ q2 q ]
2

n n

Cq =
2

V/ q1 q V/ q1 q

V/ q ]

2 2

04

n

2

V/ q2 q

n

¯

V/ q

2 n

with the second derivatives taken at the equilibrium point q. For 1 the second variation of the potential, one has 2V = 2 C q · q, n where a dot denotes the inner product in R . The symmetric matrix C is called the stiffness matrix for elastic systems. We consider systems with the potential V q having the property MARCH 2008, Vol. 75 / 021016-1

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V q =V -q

5

This means that the system is symmetric under the inversion of the state vector q -q pendulum systems, straight beams, plates, etc. . Clearly, q = 0 is an equilibrium for such systems. Moreover, the Taylor expansion of the potential V q in the neighborhood of q = 0 contains only even order terms.

3 Unimodal (Pitchfork) Bifurcation
Consider a system with the potential smoothly dependent on a parameter such that the trivial equilibrium q = 0 is stable for 0 and unstable for 0. For example, is a deviation of the loading parameter from a critical value. At = 0, the stability condition 4 is violated, and the stiffness matrix C0 = C 0 becomes singular and positive semidefinite C0 0 . In the case of unimodal pitchfork bifurcation, there is only one eigenvector u = u1 , u2 , ... , un satisfying the equation C0u =0 6

Fig. 1 Pitchfork bifurcation: ,,a... supercritical ,,v1111 > 0... and ,,b... subcritical ,,v1111 < 0.... Stable equilibria are shown by solid lines.

v

11

u·

2

V

n

=
i, j =1

V qi q j

3

u iu

j

12

n here, u · = i=1ui / qi is the derivative along the direction u in state space . Nonzero solutions of Eq. 11 are

This eigenvector u corresponds to the zero eigenvalue of the matrix C0 and is defined up to an arbitrary nonzero scalar factor. Properties of the unimodal bifurcation are well known. However, in this section, we provide the derivation that facilitates the further analysis of the bimodal case. For small q and , the potential is given by the Taylor expansion 1 V= 2 +
n

=

-

v11 6 v1111

13

i, j=1

1 V q iq j + 4! qi q j V qi q j
3

2

n

i, j,k,l=1

V q iq j q kq l + ¯ qi q j qk ql 7

4

1 2

n

q iq j + ¯

i, j=1

where all the derivatives are taken at q = 0 and = 0. Here, we used condition 5 implying that all terms of odd order in q vanish; an arbitrary constant term of the potential is taken to be zero. The main second order term in expansion 7 can be represented as 1 V = 2 C0q · q + ¯. By using this expression in the equation for equilibria 3 , we find V = C0q + ¯ =0 Hence, according to Eq. 6 , nontrivial equilibria for small given asymptotically by q u 8 are

Nontrivial equilibria exist only if the expression under the square root is positive. Thus, if v11 / v1111 0, then two nontrivial solu0. If v11 / v1111 0, then two nontrivial solutions exist for 0. These two cases are called supercritical and tions exist for subcritical bifurcations, respectively 5 . Let us study stability of the equilibria q = u for small . The equilibrium is stable if the stiffness matrix C is positive definite 1 or, equivalently, the second variation 2V = 2 C q · q is positive for all q. Here, the stiffness matrix C q is evaluated at q = u. Up to zero order terms, C q = C0 + ¯ with the positive semidefinite matrix C0 such that C0 q · q = 0 only for q u. Hence, the stability condition must be checked only along the degenerate direction q = u:
2

1 V= 2

n

i, j =1 n

V qi q

2

qi q
j q= u, 4

j

1 4 +

i, j ,k,l=1

V qi q j qk q V qi q j
3

u iu
l q=0, =0

j

u

k

u

l

9

1 2

n

u iu
q=0, =0

j

14

i, j =1

where is an unknown function of . In order to find , consider the equation u · V = 0 following directly from Eq. 3 . By using expansion 7 , we obtain 1 u· V= 3! =0
n

where we used the expansion similar to Eq. 7 and neglected higher order terms. By using notation 12 , we write the stability condition 2V 0 in the form
v
1111 2

i, j,k,l=1

V q iq j q ku l + qi q j qk ql

4

n

i, j=1

V qi q j

3

q iu j + ¯ 10

2

+v

11

0

15

In Eq. 10 , the two lowest order terms are presented, and the term u · C0q = C0u · q vanishes due to condition 6 . Substituting Eq. 9 into Eq. 10 and neglecting higher order terms, we obtain the equation for as
v
1111 3

For the trivial equilibrium =0 , the stability condition yields 0. Since we assumed that the trivial equilibrium is stable v11 0, one obtains for
v
11

0 13 , we substitute =-v

16
1111

For the nontrivial solutions 6v11 into Eq. 15 and get
v
1111

/

6 where the coefficients v
v
1111
4

+v

11

=0 are V u iu j u ku qi q j qk ql
4

11

2

3 and v
n

0

17

11

1111

u·

V=
i, j,k,l=1

l

This gives the well-known property of the pitchfork bifurcation 1,5 : The nontrivial equilibrium is stable in supercritical bifurcations v1111 0 and unstable in subcritical bifurcations v1111 0 , see Fig. 1. Transactions of the ASME

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4 Bifurcation at a Bimodal Critical Point
The goal of this paper is to study bifurcation at a so-called bimodal critical point, when there are two linearly independent eigenvectors u1 and u2 unstable modes satisfying Eq. 6 . For small q and , Eq. 8 gives asymptotic form of the bifurcating equilibria. Hence, q is a null-space vector of C0 given by an arbitrary linear combination of u1 and u2: q u1 + u
2

The second equality in Eq. 22 yields the quartic equation for as c
4 4

+c

3

3

+c

2

2

+c

1

+ c0 =0

23

with the coefficients c0 = v
1222v22

18

-v

2222v12

c1 =3v
1112v22

1122v22

-2v

1222v12

-v

2222v11

where and are unknown functions of . Consider the equations u1 · V = 0 and u2 · V = 0 following directly from Eq. 3 . Similar to the unimodal case, by using expressions 10 and 18 and neglecting higher order terms, we obtain the equations for and as
v
11

c2 =3v c3 = v
1111v22

-3v

1222v11

24 +2v
1112v12

-3v

1122v11

+v

12

+

v

1111

3

6
v
1112 3

+

v

1112

2

2
v
1122 2

+

v

1122

2

2
v
1222 2

+

v

1222

3

6
v
2222 3

=0 19

c4 = v

1111v12

-v

1112v11

v

12

+v

22

+

6

+

2

+

2

+

6

=0

Here, we introduced the notation
v
abcd

= ua ·

ub ·

uc ·

ud ·

V

v

ab

= ua ·

ub ·

V 20

with the derivatives evaluated at q = 0 and =0 for comparison, see Eq. 12 . Equations 19 can be solved as follows. Expressing from either of Eq. 19 , we find =c where c =- =-
v v
1111 3 2

21

+3v1112 6 v11 +3v1122 6 v12

2

+3v1122 + v + v12 +3v1222 + v + v22

1222

1112

3

2

2222

22

and = / . It is also possible to express through from Eqs. 19 with a coefficient depending on the inverse ratio 1 / = / .

Equation 23 has two or four real roots, see Sec. 5 for the proof that the situation when all four roots are complex is impossible, i.e., isola point does not exist. The vanishing leading coefficient c4 =0 corresponds to 1 / = / = 0, which yields =0. The obtained results can be summarized as follows. THEOREM 1. Nontrivial equilibria near a bimodal critical point =0 have the asymptotic form q u1 + u2, with = and = / c. There exist two or four branches of nontrivial equilibria given by two or four real solutions of quartic equation (23), and c given by expression (22). Each branch determines two symmetric equilibria, which differ by the sign; the branch is sub0) and supercritical if critical if c 0 (equilibria appear for 0). c 0 (equilibria appear for We note that the maximum number of postbuckling paths was counted 5 but formulas for the coefficients 21 ­ 24 are new. Let us study stability of the equilibria q = u1 + u2 for small . The equilibrium is stable if the stiffness matrix C is positive defi1 nite or, equivalently, the second variation 2V = 2 C q · q is positive for all q with the stiffness matrix C q evaluated at q = u1 + u2. As in the unimodal case see Sec. 3 , the stability condition must be checked only along the degenerate directions. In the bimodal case, degenerate directions are q = au1 + bu2 with arbitrary constants a and b. Up to lowest order terms, we have

2

1 V= 2

n

i, j=1 n

V qi q

2

qi q
j 3 u1+ u2,

j

1 4

n

i, j,k,l=1

V qi q j qk q

4

au1i + bu
l q=0, =0

2i

au1 j + bu

2j

u 1k + u

2k

u 1l + u

2l

1 + 2 =

i, j=1

V qi q j + +
v
1111 2

au1i + bu
q=0, =0

2i

au1 j + bu
2

2j

1 v 2

11

2
v
1112 2

+v

1112

+ +

v

1122

2
v
1222 2

a

2

+v

12

2

+v

1122

2

ab +

1 v 2

22

+

v

1122

2

2

+v

1222

+

v

2222

2

2

b

2

25

where we used expansion 7 and notation 20 ; u1i and u2i are the components of the vectors u1 and u2. Then, the stability condition 2 V 0 for arbitrary nonzero a and b takes the form of positive definiteness of the 2 2 matrix

v v

11 12

+v +v

1111

/2 /2

2 2

1112

+v +v

1112 1122

+v +v

1122

/2 /2

2 2

v v

12 22

1222

+v +v

1112

/2 /2

2 2

1122

+v +v

1122 1222

+v +v

1222

/2 /2

2 2

0

26

2222

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It was assumed that the trivial equilibrium = =0 is stable 0. In this case, the stability condition 26 yields the infor equalities

v

11

0

v

22

0

v

11

v
2

22

-v

2 12

0

27

For nontrivial equilibria lent to

=

, =c

, condition 26 is equiva-

v v

11 12

c+ v c+ v

1111

/2 /2

2 2

1112

+v +v

1112 1122

+v +v

1122

/2 /2

v v

12 22

1222

c+ v c+ v

1112

/2 /2

2 2

1122

+v +v

1122 1222

+v +v

1222

/2 /2

0

28

2222

Thus, an equilibrium with the branch corresponding to a given is stable if the matrix 28 is positive definite. If this matrix has a negative eigenvalue, the equilibrium is unstable.

According to symmetry condition 30 , the coefficients 20 do not change if we substitute u1 and u2 by S u1 = u1 and S u2 =-u2, respectively. The following coefficients vanish:
v
12

5 No Isola Point Exists
We show that it is impossible to have four the quartic equation 23 . First, we choose the vectors u1 and u2 such 19 and 20 v12 = 0. With this choice, the 2 elements vij is reduced to the diagonal form. Eq. 24 , we have c2 v =3 c4 v
1222 1112

=v

1112

=v

1222

=0

31

complex roots

of

that in expressions 2 matrix with the Then, according to
v v
22 11

since they change their sign under the substitution u2 -u2. Note that Eq. 19 with conditions 31 coincides with the corresponding equation for a two degrees-of-freedom system with double symmetry studied earlier 3­5 . For the sake of convenience, we introduce the normalization conditions for the vectors u1 and u2 such that
v
11

=-1

v

v -3 v

22 11

c0 v =- c4 v

22

=-1

32

1222 1112

29

Now, let us assume that all four roots of the polynomial 23 are complex and equal to x1 iy 1, x2 iy 2. Then, c0 / c4 = x2 + y 2 x2 1 1 2 + y 2 0. Hence, from conditions 27 and expressions 29 , we 2 obtain v1222 / v1112 0 and c2 / c4 0. Under these inequalities, it is easy to show that c2 / c4 2 36c0 / c4. On the other hand, c2 / c4 = x2 + y 2 + x2 + y 2 +4x1x2. Since c2 / c4 0, we have c2 / c4 2 1 1 2 2 4x1x2 2 =16x2x2 16c0 / c4. But, this contradicts to the inequal12 ity c2 / c4 2 36c0 / c4 derived above. Therefore, Eq. 23 always has real roots. This means that there is no isola point, i.e., there exist nontrivial paths bifurcating from the trivial state at the bimodal critical point.

which is possible since according to Eq. 27 v11 0 and v22 0. Solving system 19 with Eqs. 31 and 32 gives the unknown and corresponding to three types of nontrivial equilibria:
2

=

6 v1111
2

=0 6 v2222 =
v v
1111

33

=0
v v
2222

=

34 -3v1122 6 2 2222 -9v1122

2

=

1111

v

-3v1122 6 2 2222 -9v1122

2

1111

v

35

6 Symmetric Systems
In many practical problems, a system possesses an additional symmetry represented by the following invariance condition for the potential: V q =V S q 30 with a linear map S q satisfying the relation S S q = q of course, S is assumed to be different from q -q . This condition may reflect axial or spatial symmetry of the system. For example, consider a beam of variable cross section with the material distribution symmetric with respect to the middle and identical boundary conditions taken at x = a, where x is the axial coordinate with the origin at the beam center. Then, S w x = w -x , where q w x is a deflection function of the beam. Keeping in mind this example, we say that q is a symmetric or antisymmetric form if S q = q or S q =-q, respectively. If S q q, we say that the form is of mixed type. In a unimodal pitchfork bifurcation, the unstable mode u must be either symmetric or antisymmetric: u = S u . It cannot be of mixed type since that would automatically provide two linearly independent unstable modes u and S u . Let us consider a bimodal bifurcation. We can always choose the vectors u1 and u2 to be symmetric or antisymmetric: S u1,2 = u1,2. We assume that u1 is symmetric, while u2 is antisymmetric. 021016-4 / Vol. 75, MARCH 2008

Solutions 33 , 34 , and 35 with different signs of and define two symmetric, two antisymmetric, and four mixed-type equilibria 18 , respectively. Symmetric equilibria are subcritical or supercritical for negative and positive values of v1111, respectively. The type of antisymmetric equilibria is determined similarly by the sign of v2222. Mixed-type equilibria 35 exist if the quantities v2222 -3v1122 and v1111 -3v1122 have the same sign. Under this condition, mixed-type equilibria are subcritical or supercritical for negative and positive signs of the fractional factor in Eq. 35 , respectively. The stability condition 26 takes the form - +v
1111 2

/2+ v

1122

2

/2 - +v

v
1122

v

1122

1122 2

/2+ v

2222

2

/2

0 36

For symmetric equilibria 33 , eigenvalues of the matrix 36 are
1

=

v

1111

2

3 0

2

=

3v

1122

-v 6

1111

2

37

By using inequalities 27 , we obtain the stability conditions as
v
1111

3v

1122

-v

1111

0

38

For antisymmetric equilibria 34 , the eigenvalues are
1

=

v

2222

2

3

2

=

3v

1122

-v 6

2222

2

39

and the stability conditions become Transactions of the ASME

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Table 1 Classification of bifurcations at a bimodal critical point No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
v
1111

v

2222

v

1111v2222

-9v

2 1122

v

1111

-3v

1122

v

2222

-3v

1122

bifurcation. The parameters 2 , ... , m can be treated as imperfections that keep the system symmetry. Nontrivial equilibria in this case are described by the asymptotic formula 18 . The unknown coefficients and are determined by the equations - +~ v - +~ v where
n 11

+v +v

1111

3

/6+ v

1122

2

/2=0 /6=0 43

22

1122

2

/2+ v

2222

3

~ v

11

=
k=2

u1 ·

2

V
k k

n

~ v

22

=
k=2

u2 ·

2

V
k k

44

with the derivatives taken at 1 = 2 = ¯ = m = 0 and q = 0. Equations 43 differ from Eqs. 19 and 31 only by small constants ~ 11 and ~ 22 dependent on the unfolding parameters 2 , ... , m. v v By solving system 43 , we find the coefficients and corresponding to nontrivial equilibria. Similar to Eqs. 33 ­ 35 , there can be symmetric, antisymmetric, and mixed-type solutions:
v
2 2222

0
2

3v

1122

-v
2

2222

0
22

40
2 22 1122

2

=

6

Finally, for mixed-type equilibria 35 , we find the eigenvalues
1,2

- ~ 11 v v1111
2

=0 -~ v
v

45

=

v

1111

+v

2222

v

1111

-v

2222

+36v

6 41
2

=0 -~ v

=

6

22

46
v

2222

and the stability conditions
v
1111

=6

11

v

2222

-3

-~ v

0 and v

2222

0

v

1111v2222

-9v

2 1122

v

0

42
2

1111v2222

-9v

22 2 1122

1122

47 =6 -~ v
22

The obtained results allow classifying all types of bimodal bifurcations by the signs of specific quantities depending on derivatives of the potential evaluated at = 0 and q = 0, see Table 1. We note that only three numbers, namely, v1111, v2222, and v1122, govern the postbuckling behavior. Bifurcation diagrams corresponding to 16 cases of Table 1 are shown in Figs. 2 and 3, see the diagrams corresponding to =0. Due to the symmetry with respect to the planes = 0 and =0, we 0 of the , , space. In show only the quarter domain , the figures, stable equilibria are shown by thick lines. Thin solid and dashed lines correspond to unstable equilibria with one and two negative eigenvalues of the matrix 36 , respectively. S, A, and M are abbreviations for symmetric, antisymmetric, and mixed-type equilibria, respectively. Pictures in Figs. 2 and 3 =0 are based on relations 33 ­ 36 , 38 , 40 , and 42 . One can see from Figs. 2 and 3 =0 that stable nontrivial equilibria exist in six cases the Cases 1­3, 6, 11, and 12 . An equilibrium of any type can be stable: symmetric, antisymmetric, or mixed type. In the remaining ten cases, all nontrivial equilibria are unstable. These cases describe limit points leading to dynamic snaps since beyond these critical points, there is no stable solution. Nontrivial supercritical equilibria can be unstable for bimodal bifurcations, while for unimodal bifurcations, they are always stable. However, stable nontrivial equilibria are always supercritical. If symmetric or antisymmetric equilibrium is stable, mixedtype equilibrium is unstable, and if mixed-type equilibrium is stable, the symmetric and antisymmetric equilibria are unstable. Note that Cases 1, 6, 7, 11, and 12 of Table 1 were recognized and qualitatively described 5 .

v

1111

-3

-~ v

11

v

1122

v

1111v2222

-9v

2 1122

If ~ 11 v libria means mixed points s:

~ 22 , the branches of symmetric and antisymmetric equiv 45 and 46 do not intersect in the space , , . This that the bimodality is destroyed. As for the equilibria of type 47 , they coincide with the symmetric ones 45 at the 6~ v
v
11

2 s

=-

1111

-~ v -3v

22 s 1122

=0

s

=

~ v

22

v v

1111 1111

-3~ v -3v

11

v

1122

1122

48 Similarly, mixed-type equilibria 47 coincide with the antisymmetric ones 46 at the points a:
a

=0

2 a

=

6~ v
v

11

2222

-~ v -3v

22 1122

,

a

=

~ v

11

v v

2222 2222

-3~ v -3v

22

v

1122

1122

49 At these points, the secondary postcritical bifurcations occur. Critical points 48 and 49 exist if the quantities 2 and 2 determined by the corresponding expressions are positive. With a change of parameters 2 , ... , m, the bimodal bifurcation splits into a series of unimodal bifurcations. For understanding the structure of the bifurcating equilibria, let us plot solutions 45 ­ 47 in the , 2 , 2 space. Each of these solutions is represented by a straight line, Fig. 4. The line corresponding to the symmetric equilibria lies in the , 2 plane, the line corresponding to the antisymmetric equilibria lies in the , 2 plane, and the line corresponding to the mixed-type equilibria intersects the two previous lines. Of course, only the 2 0, 2 0 part of the space has physical meaning. Therefore, we can distinguish four qualita2 tively different situations. If s 0 and 2 0, then the mixeda type equilibrium line does not intersect the physical domain equi2 libria of mixed type do not exist . If s 0 and 2 0, then the a mixed-type equilibrium half-line belongs to the physical domain equilibria of mixed type exist and appear in the bifurcation of 2 symmetric equilibria ; this is the case shown in Fig. 4. If s 0 MARCH 2008, Vol. 75 / 021016-5

7 Unfolding of Bifurcations at Bimodal Critical Points
Now, let us consider a symmetric system, as in the previous section, with the potential smoothly depending on m parameters 1 , 2 , ... , m. We assume that, for fixed 2 = ¯ = m = 0, the bimodal bifurcation takes place in one-parameter system = 1, as described above. For small nonzero but fixed values of the parameters 2 , ... , m, the system behavior depending on = 1 can change qualitatively, i.e., we can observe unfolding of the bimodal Journal of Applied Mechanics

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Fig. 2 Unfolding of the bimodal bifurcation: Cases 1­8

021016-6 / Vol. 75, MARCH 2008

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Fig. 3 Unfolding of the bimodal bifurcation: Cases 9­16

Journal of Applied Mechanics

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Fig. 5 Elastic articulated column loaded by an axial force

First, consider the trivial equilibrium eigenvalues of the matrix 36 become
1

= = 0. In this case, the
22

=- + ~ v

11

2

=- + ~ v

51

Fig. 4 Structure of nontrivial equilibria for nearly bimodal critical point

and 2 0, then the mixed-type equilibrium half-line belongs to a the physical domain equilibria of mixed type exist and appear in 2 the bifurcation of antisymmetric equilibria . Finally, if s 0 and 2 a 0, then the mixed-type equilibrium segment between the 2 2 points s , s , s and a , 2 , 2 belongs to the physical domain a a equilibria of mixed type exist and are bounded by the bifurcations of symmetric and antisymmetric equilibria . space, the equilibrium lines become curves, In the , , which are orthogonal to the planes = 0 and = 0. Eliminating from Eqs. 47 , we obtain
v
2222

-3v

1122

2

-v

1111

-3v

1122

2

=6

=~ v

11

-~ v

22

50 , plane, this is a hyperbola, ellipse, or empty set On the depending on the signs of the coefficients v1111 -3v1122, v2222 -3v1122, and . Unfolding of bifurcations for 16 cases of Table 1 is depicted in Figs. 2 and 3. These figures are based on relations 45 ­ 50 . Note that the bifurcation points of the trivial equilibrium correspond to = ~ 11 for the symmetric path and = ~ 22 for the antisymmetric v v path. For the sake of simplicity, in the figures, we took ~ 11 0 v and ~ 22 0, which does not change the pictures qualitatively. v As an example, let us consider unfolding of the first case in Table 1. The unperturbed situation ~ 11 = ~ 22 =0 , is shown in v v Fig. 2 Case 1, =0 . From Table 1, it follows that the denominators in formulas 48 and 49 are positive. Hence, if = ~ 11 v - ~ 22 0, then there are only intersections bifurcations between v symmetric and mixed-type equilibrium branches at the point 48 , 0 . If 0, then only antisymmetric and Fig. 2 Case 1, mixed-type equilibrium branches intersect at the point 49 , Fig. 2 0 . Therefore, in the unfolding picture, the mixedCase 1, type equilibria appear due to the secondary bifurcation of sym0 or antisymmetric 0 equilibria. metric It should be noted that the unfolding of a bimodal critical point is qualitatively different for systems without symmetry property 30 . In the latter case, typically, there are no secondary pitchfork bifurcations. In multiparameter case, stability criterion for the equilibria is the condition of positive definiteness of the matrix 36 , where one must substitute - by - + ~ 11 and - + ~ 22 in the first and v v second diagonal elements, respectively. 021016-8 / Vol. 75, MARCH 2008

The bimodal critical point is defined by the conditions 1 = 2 =0. With the use of Eq. 44 , these conditions define a plane of codimension 2 in parameter space 1 , ... , m . Hence, the codimension of a bimodal critical point equals 2 this critical point can be typically found by adjusting values of two parameters . Here, the symmetry 30 is very important: Due to this symmetry, the offdiagonal elements ~ 12 vanish. These elements are nonzero in sysv tems without symmetry 30 or if both unstable modes are symmetric or antisymmetric . In that case, the codimension of a bimodal critical point equals 3, which agrees with general results of the singularity theory 15 . Stability of nontrivial equilibria can be studied similarly by computing eigenvalues of the 2 2 second variation matrix. However, in the perturbed case, we can avoid these computations by using known properties of unimodal bifurcations Sec. 3 , and the properties of postcritical paths for large 1 2 , ... , m at these values of 1, the stability type of a postcritical path is the same as for 2 = ¯ = m =0 . The results of stability analysis are shown in Figs. 2 and 3. Recall that stable equilibria are shown by thick lines, while thin solid and dashed lines correspond to unstable equilibria with one and two negative eigenvalues of the matrix 0 , the first 36 , respectively. For example, in Fig. 2 Case 1, bifurcation is supercritical symmetric equilibria are stable , and antisymmetric equilibria appear when the unstable trivial equilibrium bifurcates antisymmetric equilibria are unstable . After the secondary bifurcation, antisymmetric equilibria become stable as in the bimodal picture =0 , and unstable mixed-type equilibria appear. We can see that for higher values of , the stability properties of all the equilibria are the same as for the bimodal bifurcation for =0. We remark that the unfoldings in Cases 5 and 10, as well as 15 and 16, are similar from the physical point of view since the unstable paths differ only by degrees of instability. Note that for symmetric two degrees-of-freedom systems, classification of four cases with respect to the parameters v1111 -3v1122 and v2222 -3v1122 was given in 3 , and Cases 1, 7, 11, 0 of Figs. 2 and 3 were drawn in 4 , while we have and 12 recognized 16 different cases, each of them corresponding to different pictures in 3D space.

8

Mechanical Example

As a mechanical example, we consider an elastic articulated column with elastically clamped ends loaded by an axial force P, Fig. 5. The column consists of five segments of length L connected by six elastic hinges with the bending stiffnesses b0 , b1 , ... , b5. Linear stability problem for the straight equilibrium of the column has been treated in 10 . We consider a symmetric structure with symmetric boundary conditions, so that b0 = b5, b1 = b4, and b2 = b3. Deflection of the column is determined by the vector of coordinates q = q1 , q2 , q3 , q4 , which are related to the angles between the segments and the horizontal axis as Transactions of the ASME

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q
5

i+1

- qi = L sin

i

i = 0, ... , 4

q0 = q5 =0

52

The potential function of the column is V=
i=0

bi 2

i

-

i-1

2

- PL 1 - cos

i

-1

=0

5

=0 53

For the sake of simplicity, we introduce nondimensional quantities ~i = q qi L ~ = PL P b* ~ = bi bi b* ~= V V b* 54

where b* is a reference stiffness. Substituting Eq. 52 into Eq. 53 with the use of Eq. 54 and omitting tildes, we obtain the nondimensional potential function as V= b0 arcsin q 2 +
1 2

+

b1 arcsin q2 - q1 - arcsin q 2
1 2

1

2

b2 arcsin q3 - q2 - arcsin q2 - q 2
2 2

+

b2 arcsin q4 - q 2
3 2

3

Fig. 6 Stiffness parameters for columns undergoing bimodal buckling

- arcsin q3 - q b0 arcsin q + 2

+

b1 arcsin q4 + arcsin q4 - q 2
2 1 1

4

2

- P 5- 1- q - 1- q2 - q - 1- q4 - q
3 2

2

2 2b2 - b0 + b2 -2b0b2 +4b2 + b 0 1 5 55 =

2 2

- 1- q3 - q

2

2

- 1- q

4

2

1 9b2 +40b0b1 -90b0b2 + 100b2 - 200b1b2 + 225b 0 1 5

2 2

For small values of the coordinates qi, the potential can be expanded in Taylor series V= b0 2 b1 q+ q2 -2q 21 2 + b1 -2q4 + q 2
3 2 3 2 1 2

60 This equation defines a surface in three-dimensional space of the column stiffnesses b0 , b1 , b2 shown in Fig. 6. Each point on this surface corresponds to a column with the bimodal critical buckling load. Note that for rigid clamping of the column as b0 tends to infinity , Eqs. 58 and 59 furnish the buckling loads Ps = b1 + b
2

+

b2 q3 -2q2 + q 2

1

2

+

b2 q4 -2q3 + q 2
2

2

2

+

b0 2 P 2 q - q + q2 - q 24 21

1

+ q3 - q

2

2

+ q4 - q

+ q2 + ¯ 4

56

Pa =

The second order terms given in Eq. 56 define the stiffness matrix C. Equation 6 for the linear buckling problem takes the form b0 +4b1 + b2 -2 P u1 + -2b1 -2b2 + P u2 + b2u3 =0 -2b1 -2b2 + P u1 + b1 +5b2 -2 P u2 + -4b2 + P u3 + b2u4 =0 57 b2u1 + -4b2 + P u2 + b1 +5b2 -2 P u3 + -2b1 -2b2 + P u4 =0 b2u2 + -2b1 -2b2 + P u3 + b0 +4b1 + b2 -2 P u4 =0 Due to symmetry of the column, Eq. 57 possesses symmetric and antisymmetric solutions. For the symmetric solution, we take u4 = u1, u3 = u2. Then, from the first two or the last two equations 57 , we get the quadratic equation for buckling loads
2 Ps - Ps b0 +2b1 + b2 + b0b1 + b0b2 + b1b2 =0

b1 +3b 3

2

61

Thus, for rigid clamping, the bimodality condition is b1 =3b2. This means that the bimodal surface tends to the plane b1 =3b2 for the stiffness b0 tending to infinity, see Fig. 6. Let us study postbuckling behavior of the symmetric column for the parameters b0 =1, b1 = 0.25, b2 = 1, satisfying the bimodality condition 60 . According to Eqs. 57 ­ 59 , we compute the bimodal critical buckling load P = 1 and the corresponding eigenmodes eigenvectors u1 = 1,2,2,1 and u2 = 1 , 0.4 , -0.4 , -1 . Expanding the potential function 55 up to fourth order terms and using Eq. 20 , we compute the coefficients v11 = -4.0 and v22 = -3.36. Then, we normalize the eigenvectors u1 and u2 dividing them by -v11 and -v22 , respectively, so that the condition 32 is satisfied. Using normalized eigenvectors in Eq. 20 , we calculate the coefficients
v
1111

58

= 0.25

Both roots of this equation are positive, and the smaller root gives the critical buckling load if buckling is symmetric. For the antisymmetric solution, we take u4 =-u1, u3 =-u2 and similarly obtain the quadratic equation P2 - P a
a

v

2222

= 0.38605

v

1122

= 0.25

62

3 b0 +2b1 +3b 5

2

1 9 + b0b1 + b0b2 +5b1b2 =0 59 5 5

The bifurcation belongs to Type 1 in Table 1. It means that both symmetric and antisymmetric solutions are supercritical and stable while the mixed-type solution is supercritical and unstable, see Fig. 2 Case 1, =0 . The nontrivial equilibria according to Eqs. 33 ­ 35 are given asymptotically as qs = qa = q
m1

0, 2.4494, 4.8989, 4.8989, 2.4494, 0 0, 2.1507, 0.8602, - 0.8602, - 2.1507, 0 63

The smaller root of this equation yields the critical buckling load if buckling is antisymmetric. The condition of bimodality is that the smaller Ps is equal to the smaller Pa. So, we have Journal of Applied Mechanics

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0, 2.4665, 2.7184, 1.6110, - 0.3018, 0 MARCH 2008, Vol. 75 / 021016-9

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Fig. 7 Stiffnesses and buckling modes of the elastic column ,,b0 =1, b1 =0.25 , b2 =1...

q

m2

=

0, - 0.3018, 1.6110, 2.7184, 2.4665, 0
Fig. 9 Critical load depending on stiffness parameters

The stiffnesses of the bimodal column and corresponding new equilibrium states, divided by , are presented in Fig. 7. Let us study unfolding of this bifurcation due to change of the stiffness b0. According to Eqs. 44 and 56 , we find =~ v
11

=~ v

11

-~ v

22

=2 b

1

u12 -2u

11

2

- u22 -2u

21

2

= 1.3358 b

1

-~ v

22

=2 b0 u - u

2 11

2 21

= - 0.0952 b

0

64

67 Thus, if we decrease the stiffness b1 0, then the antisymmetric form of instability becomes critical, and the corresponding unfold0 . If the stiffness is increased ing is shown in Fig. 2, Case 6 b1 0, then the symmetric form of instability becomes critical 0. with the unfolding is shown in Fig. 2, Case 6 8.1 Bimodal Optimal Column. Let us consider columns under the condition b0 + b1 + b2 = const 68 This equality resembles the fixed total volume constraint for a continuous column. Figure 9 shows dependence of the critical load on b1 and b2 with b0 given by Eq. 68 with const = 1; due to homogeneity of Eqs. 58 and 59 , the plot for any const can be obtained from Fig. 9 by scaling. Columns with bimodal critical loads correspond to edges, where the surfaces Pa and Ps intersect. The analysis similar to the one given above shows that the left bimodal arch contains two big parts corresponding to Bifurcations 1 and 6 according to the classification in Fig. 2; between these two parts, there is a tiny part corresponding to Bifurcation 11 not shown in the figure . The right arch corresponds to the bifurcation of Type 1. The maximal critical load Pmax = 0.4465 is attained at the bimodal point b0 = 0.4717, b1 = 0.1021, b2 = 0.4263 with the bifurcation of Type 1. We note that the postbuckling behavior of the articulated optimal column is similar to that of the continuous optimal column 12 . Clearly, a bimodal optimal solution is the generic phenomenon. In different optimization problems, the bimodal solutions were found 6,10­14 .

Hence, if we decrease the stiffness b0 0, then the antisymmetric form of instability becomes critical, and the corresponding 0 . If the stiffness is unfolding is shown in Fig. 2, Case 1 increased b0 0, then the symmetric form of instability becomes 0. critical with the unfolding shown in Fig. 2, Case 1 For the stiffnesses b0 =1, b1 = 0.15, b2 = 0.76465, we compute the bimodal critical buckling load P = 0.84167, the corresponding and u2 = 1 , 0.3888 , eigenmodes u1 = 1 , 3.0554 , 3.0554 , 1 -0.3888 , -1 , and the coefficients v11 = -10.449 and v22 = -3.3517. Then, we normalize the eigenvectors and calculate the coefficients
v
1111

= 0.29058

v

2222

= 0.35097

v

1122

= 0.05163

65

The bifurcation belongs to Type 6 in Table 1. This means that symmetric and antisymmetric solutions are supercritical and unstable while the mixed-type solution is supercritical and stable, see Fig. 2 Case 6, =0 . Thus, we have recognized a surprising effect that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode! According to Eqs. 33 ­ 35 , the bifurcating equilibria are given asymptotically as qs = qa = q q
m1 m2

0, 1.4056, 4.2949, 4.2949, 1.4056, 0 0, 2.2583, 0.8780, - 0.8780, - 2.2583, 0 66

= =

0, 2.9661, 4.3570, 2.9848, - 0.5632, 0 0, - 0.5632, 2.9848, 4.3570, 2.9661, 0

9

Conclusion

The stiffnesses of the bimodal column and corresponding nontrivial equilibrium states, divided by , are presented in Fig. 8. If we study unfolding of this bifurcation due to change of the stiffness b1, then according to Eqs. 44 and 56 , we get

Fig. 8 Stiffnesses and buckling modes of the elastic column ,,b0 =1, b1 =0.15 , b2 = 0.76465...

For general potential systems with symmetry having multiple degrees of freedom, we studied bifurcations at bimodal branching points. Formulas describing postbuckling paths and conditions for their stability are derived. We presented the full list of possible cases for postbuckling paths and their stability depending on three system coefficients v1111, v2222, and v1122. In order to calculate these coefficients, we need to know the derivatives of the potential energy and eigenvectors of the linearized problem taken at the bifurcation point. Then, we studied unfolding of bimodal branching points due to change of system parameters. Classification and analysis of all possible cases given in Table 1 with Figs. 2 and 3 constitute the central result of the paper. It is remarkable that all the formulas derived in this paper are given in terms of the original potential energy. The presented theory is illustrated by a mechanical example on stability and postbuckling behavior of a bimodal articulated elastic column having four degrees of freedom and depending on three stiffnesses at the hinges problem parameters . It is shown that Transactions of the ASME

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bimodal critical points are described by smooth surfaces in parameter space. Numerical results are presented illustrating influence of problem parameters on postbuckling paths, their stability and unfolding. Two different kinds of postbuckling behavior are demonstrated. One is associated with stable symmetric and antisymmetric modes, and unstable mixed-type modes, while the second one is associated with stable mixed-type modes and unstable symmetric and antisymmetric modes. Thus, a surprising phenomenon that a symmetric bimodal column loaded by an axial force can buckle with a stable asymmetric mode is recognized. A considered example with the constrained sum of the stiffnesses of the articulated column shows that the maximum critical load optimal design is attained at the bimodal point with the postbuckling behavior similar to that of the continuous optimal column. We remark that we have studied bimodal bifurcations of the stable stability path of the potential system with increasing load parameter. Certainly, the case when the trivial equilibrium is unstable on both sides of the bifurcation point could also be useful. It would be interesting to recognize more physical systems and phenomena related to bimodal bifurcations.

References
1 Thompson, J. M. T., and Hunt, G. W., 1973, A General Theory of Elastic Stability, Wiley, London. 2 Thompson, J. M. T., and Hunt, G. W., 1984, Elastic Instability Phenomena, Wiley, Chichester, UK. 3 Supple, W. J., 1967, "Coupled Branching Configurations in the Elastic Buckling of Symmetric Structural Systems," Int. J. Mech. Sci., 9, pp. 97­112. 4 Supple, W. J., 1973, "Coupled Buckling Modes of Structures," Structural Instability, W. J. Supple, ed., IPC Science and Technology Press, Guildford, UK, pp. 28 ­53. 5 Golubitsky, M., and Schaeffer, D., 1985, Singularities and Groups in Bifurcation Theory, Springer, New York. 6 Augusti, G., 1964, "Stabilita' di Strutture Elastiche Elementari in Presenza di Grandi Spostamenti," Atti Accad. Sci. Fis. Mat., Napoli, Serie 3a, 4 5 . 7 Koiter, W. T., 1969, "The Nonlinear Buckling Problem of a Complete Spherical Shell Under Uniform External Pressure," Proc. K. Ned. Akad. Wet., Ser. B: Phys. Sci., 72, pp. 40­123. 8 Bauer, L., Keller, H. B., and Reiss, E. L., 1975, "Multiple Eigenvalues Lead to Secondary Bifurcation," SIAM Rev., 17 1 , pp. 101­122. 9 Olhoff, N., and Rasmussen, S. H., 1977, "On Single and Bimodal Optimum Buckling Loads of Clamped Columns," Int. J. Solids Struct., 13, pp. 605­ 614. 10 Prager, S., and Prager, W., 1979, "A Note on Optimal Design of Columns," Int. J. Mech. Sci., 21, pp. 249­251. 11 Seyranian, A. P., Lund, E., and Olhoff, N., 1994, "Multiple Eigenvalues in Structural Optimization Problems," Struct. Optim., 8, pp. 207­227. 12 Seyranian, A. P., and Privalova, O. G., 2003, "The Lagrange Problem on an Optimal Column: Old and New Results," Struct. Multidiscip. Optim., 25, pp. 393­ 410. 13 Seyranian, A. P., and Mailybaev, A. A., 2003, Multiparameter Stability Theory With Mechanical Applications, World Scientific, River Edge, NJ. 14 Atanackovic, T. M., and Novakovic, B. N., 2006, "Optimal Shape of an Elastic Column on Elastic Foundation," Eur. J. Mech. A/Solids, 25, pp. 154 ­165. 15 Arnold, V. I., 1978, Mathematical Methods of Classical Mechanics, Springer, New York.

Acknowledgment
The authors are grateful to Wolfhard Kliem for his valuable assistance. This work was delivered at the Seventh World Congress on Structural and Multidisciplinary Optimization in May, 2007 in Seoul, Korea. It was supported by the grant of President of Russian Federation No. MK-2012.2006.1 and by INTAS Grant No. 06-1000013.9019.

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