3-dimensional flutter kinematic structural stability

Having recalled the kinematic structural stability (ki.s.s) issue and its solution for divergence-type instability, we address the same problem for flutter-type instability for the minimal required configuration of dimensions—meaning 3 degree of freedom systems. We first get a sufficient non optimal condition. In a second time, the com-plete issue is tackled by two different ways leading to same results. A first way using calculations on Grassmann and Stiefel manifolds that may be generalized for any dimensional configuration. A second way using the specific dimensional configura-tion is brought back to calculations on the sphere. Differences with divergence ki.s.s are highlighted and examples illustrate the results.


Introduction
This paper deals with the so-called kinematic structural stability (ki.s.s.) for the flutter of non conservative elastic discrete systems. In a previous recent paper (see [1]), the ki.s.s. problematic was formulated in its generality and the solution for the divergence criterion for conservative as well as for non conservative elastic discrete systems has also been given by use of two independent ways. The first one has been proposed for some years by using the formula of Schur's complements, using Lagrange multipliers for introducing the kinematic constraints (see for example [2,3]). The second approach [1] is based upon a variational formulation of the divergence criterion and the explicit elimination of Lagrange's multipliers associated to the additional kinematic constraints. Both approaches lead (fortunately!) to the same results: for conservative elastic systems, the ki.s.s. is universal (as it was for long time known) and can be proved by the use of Rayleigh's quotient and Courant's Minimax results: in fact, adding a kinematic constraint on a conservative 1. KI.S.S.

Divergence ki.s.s
Let Σ = Σ free be a mechanical system, q = (q 1 , . . . , q n ) a (local) coordinate system of its configuration space, p a loading parameter and q e an equilibrium configuration of Σ subjected to a load system characterized to simplify by a single dimensionless parameter p. In this paper, only linear stability is investigated. Suppose then that q e is linearly stable in the Lyapunov sense. That means that the dynamical system is Lyapunov stable where M is the symmetric positive definite mass matrix and K(p) the stiffness matrix is without any property and especially generically non symmetric. By using the square root S of M we convert (1) intoŸ +K(p)Y = 0 (2) whereK(p) = S −1 K(p)S −1 contains in a unique matrix the whole dynamics. The load parameter p is suppose monotone increasing and often we suppose that for p = 0 the system is conservative stable implying that K(0) is symmetric positive definite. If the system becomes unstable for p = p * , the domain of stability is then [0, p * [ with eventually p * = +∞. For divergence stability the equation becomes K(p)X = 0 or equivalentlyK(p)Y = 0.
A family of ℓ linear kinematic constraints is a family C = (C 1 , . . . , C ℓ ) of linear forms identified by the scalar product with column vectors so that these constraints are equivalent to a matrix C = mat(C 1 . . . C ℓ ) ∈ M nℓ (R). The constrained system will be denoted by Σ C and as mentioned in the introduction, 0 is supposed to be still an equilibrium position of Σ C . Analogously, [0, p * C [ is the stability interval of the configuration 0 for Σ C . The kinematic structural stability (ki.s.s.) refers to the preserving of the stability of any system Σ C produced from Σ by adding any family of kinematic constraints C to Σ . That means that for the same value p of the loading, q e is still an equilibrium configuration of the constrained system Σ C and that q e is still Lyapunov stable as equilibrium configuration of Σ C .
For elastic conservative systems, Courant Minimax results about Rayleigh's quotient may then be easily translated as universal ki.s.s. (for divergence and for flutter as well because the instability may only occur by divergence!): p * ≤ p * C ∀C. On the contrary, for non conservative elastic systems, the ki.s.s issue is more complicated because, at least for linear stability, two modes of instability may occur: divergence instability and flutter instability. We then separate the ki.s.s into two types of ki.s.s.: one for divergence and one other for flutter.
The recent results mentioned in the introduction show that the divergence ki.s.s. is conditional, the condition being that the symmetric part of the stiffness matrix remains positive definite: it is nothing else that the second order work criterion that then appears as the optimal criterion for divergence ki.s.s. (see [2,3,1]). Denoting by p * div the critical divergence load of Σ , by p * div,Σ C the critical divergence load of Σ C and by p * sw the critical value of p for the second order work criterion, we then get even though C = ∅ and the value p * sw is optimal. Finally, p * sw ≤ p * div (meaning C = ∅) implying that, as long as p < p * sw neither the free system Σ nor any constrained system Σ C may be divergence unstable.
We are here now concerned by the same issue for flutter (in)stability. We then have to compare the critical flutter load p * f l of Σ with the critical flutter load p * f l,C of any constrained system Σ C . If flutter ki.s.s. is conditional, we have also to find out an eventual value p * k,f l such that (p * f l is the critical flutter load of Σ and p * f l,C is the critical flutter load of Σ C ). Only the case of a free 3 d.o.f system Σ = Σ free is investigated and, in order to preserve the possibility for the constrained system Σ C to be flutter unstable, only one constraint (ℓ = 1) is possible.

Flutter ki.s.s. with n = 3
Let u = u(p) ∈ L(E) be the morphism of the euclidean n-dimensional vector space E = R n with matrix K(p) in the canonical basis of E. Generally, the flutter instability of the system Σ means that the operator u fails to be R-diagonalizable.
According to [1], the ki.s.s. is relative to the so-called compressions u F of u to the subspaces F of E defined by u F = p F • u |F ∈ L(F ) where u |F is the restriction of u to F and p F is the orthogonal projection on F . According to the investigated criterion, the ki.s.s. is equivalent to study if the considered property of u is preserved for all its compressions u F .
For divergence stability, the ki.s.s. issue and its solution can be so reformulated: the compressions of an invertible linear map u = u(p) ∈ L(E) to any (non nil) subspace F remain invertible if and only if the symmetric part u s (p) of u(p) is definite. Because for p = 0, u(0) is supposed symmetric definite positive, by continuity we are led to the second order work criterion. Recall that the compression of an operator u naturally arises for computing its numerical range W (u). The relationship between the numerical range W (K) of the stiffness matrix K and the second order work is direct: K satisfies the second order work criterion means that W (K) ⊂ R * + . For compressions and numerical range of matrices and operators see for example [9]. Supposed now u(p) R-diagonalizable with, to simplify, only simple eigenvalues. The flutter ki.s.s. issue then consists to know if there is a vector subspace F of E such that u F = p F • u |F ∈ L(F ) is no more R-diagonalizable and eventually to find such a candidate F k,f l . This issue in its great generality is very complicated first because there no general convenient criterion of R diagonalizability and secondly because, even with a practicable algebraic criterion for low dimensions, the issue remains, as we will see below, a real challenge.
Suppose that dim(E) = 3 (3 d.o.f. system Σ = Σ free ) and that the system is constrained by only one kinematic constraint. This constraint is described by a vector e 3 that can be chosen on the sphere S(E). In fact, e 3 or −e 3 represents the same constraint showing that in this case the good geometric structure is the projective space. However, in the general case, the constraint is multidimensional and represented by the vector space F ⊥ . That means that the well adapted structure to investigate the general issue is, as already mentioned in [1] and in the introduction, the one of Grassmann manifold Gr m,n (R) = Gr m (R n ) = Gr m (E) of all m-dimensional subspaces of E which is a m(n − m) dimensional compact manifold. To conclude this paragraph, let us remind that the R-diagonalizability used criterion for u F when dim(F ) = m = 2 is where χ u F is the characteristic polynomial of u F and that leads to the following flutter ki.s.s criterion: Because of the compactness of Gr 2 (E) and the continuity of the minimum exists and is reached for an element F k,f l . The corresponding constraint is then given by any vector e 3 ∈ F ⊥ k,f l ∩ S(E).

Geometric considerations and preliminary calculations
The aim of this paragraph is to transform (8) in order to solve (7). Indeed, the natural way to calculate F k,f l consists on differentiating φ for getting the critical points. The derivative of Φ is however difficult to be evaluated because the "points" are vector spaces and we then start by transforming Φ. To do it, we better view Gr 2 (E) as the quotient space of the Stiefel manifold St 2 (E) of all 2-uplets (e 1 , e 2 ) of orthonormal vectors of E by the orthogonal group 0 2 (R). We then will describe any point F of Gr 2 (E) and any function of F like Φ by its expression as function of any element (e 1 , e 2 ) of St 2 (E) such that the vector space spanned by (e 1 , e 2 ) is F without forgetting that it may be independent of this choice because of the quotient by 0 2 (R). Remark that, by this way, St 2 (E) appears as a principal fiber bundle with 0 2 (R) as group and with Gr 2 (E) as base space: Here, dimSt 2 (E) = 3, dim0 2 (R) = 1 and π : St 2 (E) → Gr 2 (E) is the projection map of the total space St 2 (E) of the principal fiber bundle on its base space Gr 2 (E) = St 2 (E)/0 2 (R) that at each family (e 1 , e 2 ) of two orthonormal vectors of E associates the vector space π(e 1 , e 2 ) spanned by these vectors. Reciprocally, if F ∈ Gr 2 (E) = St 2 (E)/0 2 (R), the set π −1 (F ) is the f1ber over F built by all the orthonormal bases (e 1 , e 2 ) of F . Let (e 1 , e 2 ) be an orthonormal basis of F = π((e 1 , e 2 )). We use the letter φ instead of the same capital letter Φ to refer to the function of the variables in Two expressions of Φ that will be used are given by: Proof. The following transformations hold: because p is self-adjoint and for the same reasons. Straightforward calculations show then that: which is exactly (9). Here (e 1 , e 2 ) is viewed as an element of π −1 (F ) ⊂ St 2 (E). Direct calculations may show that this expression of φ((e 1 , e 2 )) does not depend on the choice of (e 1 , e 2 ) as orthonormal basis of F meaning in the fiber over F and then justifying the notation Φ(F ). That may be directly checked at each step of the calculations but we will not do it.
To better understand the flutter as a competition between the symmetric part u s of u and its skew symmetric part u a and because the second order work criterion involves u s , we now transform the last expression by using the symmetry of u s and the skew symmetry of u a . Calculations give: leading to: which is exactly (10).
It may be checked again that this expression of φ((e 1 , e 2 )) does not depend on the choice of (e 1 , e 2 ) but only on its equivalence class under the group action by 0 2 (R) which justify the expression Φ(F ).
But there is a very significant choice. Let be e 3 any of the both unit vectors of F ⊥ . Suppose that e 3 ̸ ∈ ker u a (this last case will be handled separately and when n = 3, it is a one dimensional vector space). Then, 4 intrinsic quantities are involved in the issue: the three real eigenvalues of u s : and −β 2 (β > 0) the unique not nil eigenvalue of the symmetric linear map u 2 a . Indeed, because n = 3, rank u a = 2, ker u a = ker u 2 a and (ker u a ) ⊥ = Imu a = Imu 2 a = E −β 2 = ⟨w 1 , w 2 ⟩ with w 1 and w 2 two orthonormal eigenvectors of u 2 a so that u 2 a (w i ) = −β 2 w i for i = 1, 2. We then deduce, after having chosen w 3 ∈ ker u a , that u a (w 1 ) = βw 2 , u a (w 2 ) = −βw 1 , u a (w 3 ) = 0 ((w i ) i is still an orthonormal basis of E).
The quantities ∆ k (e 3 ) will play a significant rule and they can be explicitly evaluated: Proof. These results are obvious if x ∈ ker u a and because ∆ k is a 3 homogeneous function of x, we may suppose that x ∈ S(E) \ ker u a . Moreover, the minimal polynomial of u 2 a is π u 2 a = X(X + β 2 ) meaning that u 4 a = −β 2 u 2 a that leads to the second assertion 2. Moreover, Finally, the last assertion comes from the following calculation of ∆ 2 (x). By using the orthonormal basis Calculating now the square of the norm of the right-hand side and the left-hand side of the last equation, we get: and because ∆ 2 (e 3 ) ≥ 0: which proves the last assertion.

Sufficient conditions
First and foremost, remark that in the degenerated case where the second order work criterion (SOWC) fails with a 2 dimensional isotropic cone C, choosing F ⊂ C and the constraint in F ⊥ destabilizes the system and flutter ki.s.s. fails. Indeed, we then get tr(u s,F ) = 0 and, according to (10), we get Φ(F ) = −4(u a (e 1 ) | e 2 ) 2 ≤ 0.
On the contrary, suppose now to simplify the reasoning, that the SOWC holds meaning here that α 1 > 0. Then, without calculating the minimum of Φ (or φ, h), a sufficient flutter ki.s.s. condition may be got.

Proposition 2. As long as
Proof. Because of well-known results about Rayleigh's quotient for u s , α 1 ≤ (u s (x) | x) ≤ α 3 for all unit vector x and the extrema are respectively reached for the eigenvectors v 1 (minimum) and v 3 (maximum) associated to α 1 and α 3 . Then with a minimum when e 3 = v 3 and a maximum when e 3 = v 1 . Moreover, with a minimum when e 3 ∈ ker u ⊥ a and a maximum when e 3 = w 3 . Then: and we deduce that a flutter ki.s.s. sufficient condition reads as (17) meaning that, as long as the arithmetic mean of the both lowest eigenvalues of u s is greater than the square root of the not nil eigenvalue of u 2 a , no additional kinematic constraint may destabilize the system Σ and the flutter k.i.s.s. is ensured. Moreover, suppose as usually that, for p = 0, the system is elastic conservative stable. Then, α 1 (0) > 0, α 2 (0) > 0 and β(0) = 0. Thus, by continuity the minimal positive value p skf root of α 1 (p) + α 2 (p) − 2β(p) = 0 is >0. On [0, p skf ], the flutter ki.s.s. is ensured.
Suppose now that α 1 + α 2 ≤ 2β. Because both terms in competition are reached for e 3 = v 3 and for e 3 = w 3 and because v 3 ̸ = w 3 , there is no chance in order that this equality should be realized by a convenient constraint and the flutter k.i.s.s. can be still ensured. The sufficient condition (17) is then not necessary nor optimal and we now tackle the issue of necessary and sufficient flutter ki.s.s. conditions.

Calculation of the extrema of Φ and h
The aim of this section is then to calculate the minimum of Φ, φ, h. It is a significant challenge because of the nature of the variables, the non convexity of the function and the deep non linearity of the issue. It will be done separately for φ and h, the one for Φ resulting from those last both. It allows first to validate the results and secondly to highlight the power of the geometric tools like Grassmann or Stiefel manifolds which are the good tools for generalizing up to any dimension n ≥ 4 the problem supposed here tridimensional. The two conditions defining the critical points are themselves nonlinear because the functions Φ, φ, h are not quadratic. More specifically, they are roughly speaking 4-homogeneous. So, there is no hope of giving the solution by an algorithm involving only linear algebra instructions like for the divergence ki.s.s and the second order work criterion. The conditions have however a very nice expression and a significant geometrical meaning. Let us start.
With the same notations, critical points of Φ are characterized by the following. Proof. Usual derivative calculations give: But as u a is skew symmetric and u s symmetric, from (24) and (25)  By evaluating Φ at a critical point, we get the following flutter ki.s.s condition:

Proposition 4. Flutter ki.s.s holds as long as
or when for all units e 3 such that F = π((e 1 , e 2 )) = ⟨e 3 ⟩ ⊥ is a critical point of Φ.

Extremum of h
To validate the previous results and to present the calculations with more usual tools of differential geometry, we use the parametrization of the problem by the sphere S(E) of unit vectors of E meaning by the function h defined by (13) that is now recalled: The aim of this paragraph is then to calculate the critical points of h on the sphere S(E).

The mechanical system: Ziegler's column
We now apply the above results to the usual three d.o.f. Ziegler column Σ as the one used in [3] or [1] for investigating the divergence ki.s.s. Σ then consists in the three degree of freedom Ziegler system Σ as in Fig. 1 made up of three bars OA, AB, BC with OA = AB = AC = ℓ linked by three elastic springs of the same stiffness k. The nonconservative external action (the circulatory force) is the follower force ⃗ P . The elastic energy of the springs is and the virtual power of ⃗ P in any configuration θ = (θ 1 , θ 2 , θ 3 ) reads (P > 0 in compression): Put p = P ℓ k as dimensionless loading parameter and noting that (0, 0, 0) is the unique equilibrium configuration, the stiffness matrix then reads: Two cases of mass matrix will be investigated but we do not systematically give the corresponding mass distribution. Only the second one is associated to a uniform mass distribution. In each case, we give the numerical approximation of the solution e 3 (called here X with X T = (x 1 x 2 x 3 )) of Eqs. (43) and only for the first case, we give the expanded expressions of the quantities involved in Eqs. (43). We also validate the results by a direct numerical solution of the initial minimization problem. Because of the non convexity of the criterium function, a specific algorithm is devised. It relies on the conjunction of two classical numerical algorithms. First, a Nelder-Mead downhill simplex technique [10] is used so as to find the minimum of a function f (X, p) with X on the unit sphere for p fixed. Then, this first optimization algorithm is piloted by a dichotomy procedure on p which will converge to the value p * such that min X f (X, p * ) = 0. The sphere is parametrized by the usual spherical coordinates (x 1 = cos(ψ 1 ) · cos(ψ 2 ), x 2 = sin(ψ 1 ) · cos(ψ 2 ) and x 3 = sin(ψ 2 ) with −π ≤ ψ 1 ≤ π and −π/2 ≤ ψ 2 ≤ π/2) in order to lead to a minimization problem without constraint.
The critical values p * f l , p * s,k,f l and p * k,f l are then calculated in order to illustrate the above mechanical discussion. For the corresponding value of X T k,f l = (a b c) to p * k,f l , the destabilizing kinematic constraint then reads aθ 1 + bθ 2 + cθ 3 = 0. The above analytic results give only the first order equations for the critical points and then allows to find a ki.s.s. critical value of the load parameter. In order to have the exact set of equations and inequalities for the minimums points, a second order set of inequalities should be added to select among the critical points the minimum points. But, because of the non convexity of the problem, we have to evaluate h on the critical points or, in the best, on the set of the minimums to select the absolute one. The used simplex technique validates the analytic approach by using a complete different way which avoids any gradient method.

M = I 3
In this first case,K(p) = K(p) and

Conclusion
After having recalled the ki.s.s. issue and its solution for divergence of conservative and of non conservative systems as well, the 3 dimensional flutter ki.s.s. issue is investigated. First, using the usual algebraic criterion of flutter instability involving the discriminant of the characteristic polynomial, a sufficient condition involving the eigenvalues of the symmetric and the skew-symmetric parts of the operator is proposed that ensures a non optimal conditional flutter ki.s.s. characterized by the value p * s,k,f l of the load parameter. For getting necessary and sufficient conditions more advanced calculations are led. These conditions are brought back to an original minimization problem on the Grassmann manifold of the 2-planes Gr 2 (E). Two ways are used to tackle this issue. The first one uses differential calculation on this Grassmann manifold viewed as the base space of the principal fiber bundle of the Stiefel manifold St 2 (E) built by the families of two orthonormal vectors of E. It uses a more abstract point of view but leads to more compact calculations that may be generalized in higher dimensions. The second one uses the opportunity, in this three dimensional case, to parametrize the issue by the sphere because it is here a double cover of the involved Grassmann manifold Gr 2 (E). Calculations on the sphere are more usual and lead, after more tedious calculations, to the same results of a family of three explicit analytic nonlinear equations. The solutions of these equations lead to the optimal flutter ki.s.s. critical load p * k,f l and to the corresponding kinematic constraint. The mechanical consequences are then highlighted showing only a partial conditional flutter ki.s.s. understood in the sense of the safety of the structure. So, contrary to the divergence ki.s.s. -for the conservative and for the non conservative systems -the optimal flutter ki.s.s. condition does not ensure the flutter stability of the initial free system Σ = Σ free and the both cases p * k,f l > p * f l or p * k,f l < p * f l are possible. Finally, an illustration of these mechanical insights is done through a three degree of freedom Ziegler column with two distinct mass distributions.