Abstract

     Controlling the frequency response of an engineering component or structure is important in the aerospace and automotive sectors and is a key consideration when seeking a new and more efficient design for a given component. In this contribution, the standard truss layout optimization procedure is modified to incorporate semidefinite constraints to limit the minimum value of the first natural frequency. Since this increases the computational expense and reduces the scale of the problem that can be solved, a bespoke algorithm incorporating an adaptive ‘member adding’ procedure is proposed and applied to a number of benchmark example problems. It is demonstrated that this allows problems to be solved with relatively fine numerical discretization, allowing modified structures with an acceptable minimum first natural frequency response to be successfully identified.

Introduction

     In the design of modern engineering components, many considerations need to be taken into account including safety, cost, weight and manufacturability. The most prominent of these is safety, taking account of the regime of applied stresses to be sustained over the life of the component. Safety is influenced by the properties of the material employed, which may change as the design evolves. Additionally, when considering structures that include slender elements in compression, it is necessary to check for buckling instability to ensure safety is maintained. Another key parameter in the aerospace sector is the harmonic frequency of a structure. This should lie outside the frequency bands of surrounding components. Should a fundamental frequency of one component (e.g. a bracket) overlap with those of its attached neighbours, then resonance in the component may occur, also referred to as forced vibration. Forced vibration and resonance can then lead to High Cycle Fatigue (HCF) in the component, affecting its serviceable life and reducing its time to failure. It should be noted that a component may exhibit multiple resonant frequencies, each corresponding to a mode of vibration; repeated exposure to these frequencies may reduce the life of the component. However, this phenomenon is beyond the scope of the current contribution. Considering component manufacture, it is important to note that traditional manufacturing methods may limit the design freedom available; however, in the present contribution, it is assumed that Additive Layer Manufacturing (ALM) methods are available. The use of ALM means that complex truss forms can potentially be fabricated, beyond the scope of traditional subtractive manufacturing methods.

Conclusions

     Numerical layout optimization provides an efficient means of generating optimal truss structures for a given set of design requirements. However, traditional linear programming-based formulations are limited, and cannot for example accommodate frequency constraints. In this contribution, extended semi-definite programming-based formulations are considered that allow the minimum first natural frequency of a structure to be specified. The main conclusions are as follows.

     The use of a two phase approach, in which the traditional LP layout optimization formulation is used in the first phase and an SDP size optimization is used in the second phase, provides a computationally efficient means of generating solutions satisfying a specified frequency constraint. However, the solutions obtained are likely to be sub-optimal, with the resulting structures having higher than necessary volume.

     Alternatively, a constraint on frequency can be introduced in the optimization directly, furnishing layouts that satisfy both structural performance and first natural frequency requirements. However, when using a fully connected ground structure and a standard SDP solver, the computational cost and memory requirements have been found to be high, severely limiting the scale of problem that can be tackled.