Multi-step transversal and tangential linearization methods applied to a class of nonlinear beam equations

Abstract

A novel and continuously parameterized form of multi-step transversal linearization (MTrL) method is developed and numerically explored for solving nonlinear ordinary differential equations governing a class of boundary value problems (BVPs) of relevance in structural mechanics. A similar family of multi-step tangential linearization (MTnL) methods is also developed and applied to such BVP-s. Within the framework of MTrL and MTnL, a BVP is treated as a constrained dynamical system, i.e. a constrained initial value problem (IVP). While the MTrL requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable, the essential difference of the present MTrL method from its previous version [Roy, D., Kumar, R., 2005. A multistep transversal linearization (MTL) method in nonlinear structural dynamics. J. Sound Vib. 17, 829-852.] is that it has the flexibility of treating nonlinear damping and stiffness terms as time-variant damping and stiffness terms in the linearized system. The resulting time-variant linearized system is then solved using Magnus' characterization [Magnus, W., 1954. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7, 649-673.]. Towards numerical illustrations, response of a tip loaded cantilever beam (Elastica) is first obtained. Next, the response of a simply supported nonlinear Timoshenko beam is obtained using a variationally correct (VC) model for the beam [Marur, S., Prathap, G., 2005. Nonlinear beam vibration problems and simplification in finite element model. Comput. Mech. 35(5), 352-360.]. The new model does not involve any simplifications commonly employed in the finite element formulations in order to ease the computation of nonlinear stiffness terms from nonlinear strain energy terms. A comparison of results through MTrL and MTnL techniques consistently indicate a superior quality of approximations via the transversal linearization technique. While the usage of tangential system matrices is common in nonlinear finite element practices, it is demonstrated that the transversal version of linearization offers an easier and more general implementation, requires no computations of directional derivatives and leads to a consistently higher level of numerical accuracy. It is also observed that higher order versions of MTrL/MTnL with Lagrangian interpolations may not work satisfactorily and hence spline interpolations are suggested to overcome this problem.

Comparison of two linearization schemes for the nonlinear bending problem of a beam pinned at both ends

“…The results included in this paper may be regarded as an extension of that obtained in Ramachandra and Roy (2001a) and Viswanath and Roy (2007) for the simply supported beam problem, with special emphasis in its coherence with the analytical solution. In the current approach, solving procedure has been modified by using a centered finite difference method with multi-point linear constraints, introduced by Keller (1969).…”

“…Multi-step transversal linearization method (Ramachandra and Roy, 2001a;Ramachandra and Roy, 2001b;Ramachandra and Roy, 2002;Viswanath and Roy, 2007) consists of replacing the nonlinear vector field by a set of conditionally linear ODE systems in such manner that transversal intersections of the solution manifolds of the nonlinear problem and the linearized one are provided on nodal points.…”

“…The general expression for the Lagrangian of the problem, adopting a variational approach (for more details, see Viswanath and Roy (2007) and Monleón (1999)) will be:…”

“…The equality of derivatives increases the closeness between nonlinear and LTL-based solution paths. This new scheme of linearization is called LTL-first level by Ramachandra and Roy (2001a) and tangential linearization (MTnL) by Viswanath and Roy (2007). In the present study, a modified MTnL is applied without performing the derivation of the ODE-s system.…”

“…Later, Viswanath and Roy (2007) apply the MTrL and MTnL methods to a tip loaded cantilever beam with constant cross-section and compare the output results with the analytical ones (Mattiason, 1981). Likewise, they compare the two linearization techniques on a constant cross-section beam pinned at both ends with a uniformly distributed load.…”