MS1.3 - Advances in Computational Structural Dynamics

The influence of uncertainties arising from material properties and manufacture in engineering structures leads to increasing variation in the response characteristics with frequency. This is in combination with the likelihood that within a complex system, localised nonlinearities can also present themselves, rendering commonplace linear high frequency techniques such as the Hybrid Finite Element (FE) - Statistical Energy Analysis (SEA) method inapplicable. The alternative being to run vastly expensive huge degree of freedom Monte Carlo simulations in the time domain in order to fully capture the nonlinear dynamics, becomes prohibitive for large built-up systems. The development of a linearisation procedure applicable to an ensemble of random systems featuring deterministic nonlinearity is firstly considered in this work, which is centred around the technique of Equivalent Linearisation. This presents a relation between the order of both the nonlinearity in the system and response moment required to determine linearised parameters. With the proposed linearisation being developed around deterministic nonlinearity it is exploited by the Hybrid FE-SEA method by describing the nonlinearity as part of the master system, but for this an extension to hybrid variance theory is necessary. The linearised Hybrid FE-SEA method is then assessed against a series of Monte Carlo benchmark simulations adopting a Lagrange Rayleigh-Ritz model that employ different linearisation procedures including that developed for the hybrid method but also the full nonlinear solution.

Rank deficiency of the dynamic stiffness matrix at a given frequency is an indicator of resonance. In order to achieve resonance at several eigenfrequencies, the sum of ranks of the dynamic stiffness matrix at these frequencies can be used as a heuristic optimization objective. This objective avoids the explicit computation of natural frequencies and eigenmodes. The rank of the matrices is not minimized directly. Instead, a surrogate function for matrix rank is used, e.g. log-det function. This surrogate function is known to work well in the case of affine dependency of the matrix on parameters, like the dynamic stiffness matrix of truss structures depending on cross-sectional areas. Reducing the rank of the dynamic stiffness matrix for higher frequencies implies that the matrix is not semi-positive definite. For this case, the log-det function is valid with a combination of interior-point methods and Fazel's semi-definite embedding via linear matrix inequalities. Further constraints on the fundamental frequency and compliance can be easily added within the framework as additional linear matrix inequalities. This describes the heuristic design procedure. However, an obtained design is not guaranteed to show resonance at each frequency. Thus, any design should be checked after the optimization procedure. Several successful numerical examples illustrate the performance of the approach.

Scalar constant coefficients are generally found in conventional integration methods and it has been shown that there is no conventional integration method that can have explicit formulation and unconditional stability simultaneously. Alternatively, a novel type of solution methods is proposed for solving equation of motion. A concept of mode can give a fundamental basis for the successful development of this type of methods. The profile of the methodology contains three major stages: (1) To decouple a coupled system of n equations of motion (i.e., the problem under analysis) into a set of n uncoupled equations of motion, (2) To develop an eigen-dependent method for solving each uncoupled equation of motion, and (3) To convert all the n eigen-dependent methods into a problem-dependent method by using a reverse procedure of an eigen-decomposition technique. Although the novel type of integration methods is derived from a concept of mode, it does not involve solving any eigenvalue problem during the step-by-step integration procedure. This type of integration methods is characterized by problem dependen-cy since the coefficients can be functions of the initial physical properties (i.e., the mass matrix, damping coefficient matrix and stiffness matrix) for defining the problem under analysis. A fully explicit method is developed and it has an explicit displacement difference equation and an ex-plicit velocity difference equation. Notice that the well-known Newmark family method has an implicit displacement difference equation and an implicit velocity difference equation. The de-velopment details are presented and each major stage is theoretically justified by a concept of mode. Numerical properties of stability and accuracy for the proposed fully explicit method are ana-lytically explored. As a result, it is concluded that it can have unconditional stability and second order accuracy. An unconditional stability allows it select a step size without stability considera-tion and a second order accuracy allows it choose an appropriate step size for conducting the step-by-step integration procedure. Since the proposed fully explicit method has explicit dis-placement and velocity difference equations, both the displacement and velocity of the next step can be directly calculated without involving any nonlinear iterations. This method is simple in logic and structure and thus coding is much easier than implicit methods. Consequently, it can easily handle complex nonlinearities. Numerical tests for solving a coupled system of 250, 500 and 1000 equations of motion have been performed and the CPU demand for the proposed fully explicit method is only about 3.3%, 2.2% and 1.4% of that consumed by the Newmark family method ( beta = 0.25 and gamma = 0.5), respectively. As a consequence, it is affirmed that the proposed fully explicit method is of high computational efficiency in the solution of general structural dynamic problems in contrast to conventional integration methods, such as the Newmark family method.