Constrained Buckling of Spatial Elastica: Application of Optimal Control Method

Abstract

A post-buckling analysis of a constant or variable length spatial elastica constrained by a cylindrical wall is performed for a first time by adopting an optimal control methodology. Its application in a constrained buckling analysis is shown to be superior when compared to other numerical techniques, as the inclusion of the unilateral constraints is feasible without the need of any special treatment or approximation. Furthermore, the formulation is simple and the optimal configurations of the spatial elastica can be also obtained by considering the minimization condition of the Hamiltonian. We first present the optimal control formulation for the constrained buckling problem of a constant length spatial elastica, including its associated necessary optimality conditions that constitute the Pontryagin's minimum principle. This fundamental constrained buckling problem is used to validate the proposed methodology. The general buckling problem of a variable length spatial elastica is then analyzed that consists of two parts; (1) the solution of the optimal control problem that involves the inserted elastica inside the conduit and (2) the derivation of the buckling load by taking into account the generation of the configurational or Eshelby-like force at the insertion point of the sliding sleeve. A variety of examples are accordingly presented, where the effects of factors, such as the presence of uniform pressure, the clearance of the wall, and the torsional rigidity, on the buckling response of the spatial elastica, are investigated.

A Frictional Contact-Pattern-Based Model for Inserting a Flexible Shaft Into Curved Channels

“…The system is optimized for the unknown inputs with respect to the zero-value constraints and non-negative objective function at a given insertion length. We choose the non-linear least-square solver (LSQNONLIN) from MATLAB for easy implementation and a good balance between efficiency and accuracy [27]. Moreover, the shooting method [18] and the collocation method [21] could be applied to solve the system equations regarding the constraints.…”

“…In contrast, the FEM model evenly meshes the shaft and uses the penalty method to calculate the contact forces at the nodes only. The shaft is modeled from one end to another using the chain algorithm [26] for both FEM [9], [14], and CPB methods [21], [27]. FEM methods simulate the insertion process with high fidelity and heavy computation load.…”

Configurational forces and nonlinear structural dynamics

“…Configurational forces, introduced in solid mechanics by Eshelby [15,16,17,18] to model interactions between dislocations or forces driving crack propagation, have been recently shown to be possible in structural mechanics too [6]. The action of configurational forces on structures have been exploited to provide unexpected quasi-static response [5,7,8] and propulsion [4,12], have been explained through a material force balance [25,35,36,41] and have been used to investigate constrained buckling problems [29,30,31]. It is also worth noting that the recent research on configurational forces in structural mechanics has eventually inspired a new interpretation of their action in solids [3].…”

Nested Bloch waves in elastic structures with configurational forces

“…In this case only configurational forces at the end of the sliding sleeve exerted on the substructure B are generated. Considering that the substructure B oscillates transversally under mode n B , the angular frequency ω can be obtained from eqn (27) as…”

“…In particular, the sliding of an elastic rod through a frictionless sleeve generates at both constraint ends an 'Eshelby-like' force of amount proportional to the square of the bending moment and direction parallel to that of sliding. Configurational forces may also be derived through an asymptotic approach [5,6] or a material force balance [7][8][9][10] and have been so far exploited in a series of novel applications [11][12][13][14][15][16][17][18][19][20]. The action of Eshelby-like forces was recently disclosed in a dynamic framework and the case of a falling mass attached to an end of an elastic rod investigated to reveal a complex and often counterintuitive motion [21].…”