تجزیه و تحلیل حساسیت و اصلاح خصوصیات دینامیکی سازه ها با استفاده از تقریب مرتبه دوم

This paper presents a formulation in the form of an inverse eigen value problem for modification of vibration behavior of structures. The proposed method which is based on the second order approximation in Taylor expansion is expressed in terms of variables relating to stiffness or mass matrix parameters in a finite element formulation. An initial sensitivity analysis identifies the regions within the structure where the modifications would yield the required changes in the structures dynamic characteristics. An algorithm is developed which allows efficient modification of structural dynamics characteristics without iterations. These modifications are conducted locally so that only elemental stiffness and matrices are affected. The algorithm is applied to four case studies and it is found that large modification of natural frequencies of up to 10% can be realized with an induced error of less than 5% for truss structures, and less than 3% for plane problems.

The common industrial practice for optimising the vibration behavior of structures is to conduct a series of modifications on the FEA simulations of the structure in order to achieve the required eigenfrequencies. This approach, known as the forward variation approach or design load analysis cycle is extremely time consuming, expensive and rarely yields to an optimum solution. The vibration optimisation problem can be defined as an inverse engineering problem. The inverse engineering refers to problems where the desired response of the system is known or decided but the physical system is unknown. These problems are difficult because a unique solution is rarely possible. The current state-of-the-art in the inverse approach to the vibration problem is only limited to structures modelled using simple linear springs, dampers and point masses. Very little attention has been given to formulating the inverse eigenvalue problem for two and three dimensional and higher order finite elements which are most commonly used in simulation of real structures. Optimisation of vibration characteristics is defined as an inverse eigenvalue problem or problem of designing systems in order to produce the desired response. To eliminate the need to re-analyse the whole structure modelled by finite elements, an inverse approach is required in order to find the exact modified parameters in various finite element formulations which yield the required natural frequencies. Early work in tackling the inverse eigenvalue problem by other researchers [1] and [2] utilised the 1st order terms of Taylor’s series expansion and is based on Rayleigh’s work. Others such as Chen and Garba [3] used the iterative method to modify structural systems. Recently Baldwin and Hutton [4] presented a detailed review of structural modification techniques. These were classified into categories of the techniques based on small modification, techniques based on localised modification and those based on modal approximation. Further research on structural modification was carried out by Tsuei et al. [5], [6] and [7] who presented a method of shifting the desired eigenfrequencies using the forced response of the system. The method is based on modification of either the mass or stiffness matrix by treating the modification of the system matrices as an external forced response. This external forced response is formulated in terms of the modification parameters, thus creating a modified eigenvalue problem. More recently Zhang and Kim [8] investigated the use of mass matrix modification to achieve desired natural frequencies. McMillan and Keane [9] investigated a method of shifting eigenfrequencies of a rectangular plate by adding concentrated mass elements. Sivan and Ram [10] and [11] extended further the research on structural modification by studying the construction of a mass spring system with prescribed natural frequencies. They obtained stiffness and mass matrices using the orthogonality principles. They [12] developed a new algorithm based on Joseph’s work [13] which involves the solution of the inverse eigen value problem. In the last few years the work on the inverse problem by Gladwell [14] started to be taken seriously by engineers and researchers interested in this field of engineering. Mottershead [15] also considered the problem of resonance in the forced vibration of machines and structures by the design of physical modifications to achieve targeted natural frequencies. His technique of achieving the required system include structural modifications by adding a point mass, a grounded spring or by a spring connecting two co-ordinates. Li et al. [16] considered optimising dynamic behavior of a multi body system by conducting modifications on its mass and stiffness matrices. The above techniques have predominantly been applicable to discrete systems made up of simple linear spring and mass elements. Even with these simple elements the problem of mapping of the ‘physically viable’ stiffness and mass matrix to a real structure has not been fully resolved and the challenging problem of applying the inverse vibration problem to continuous finite elements has not yet been addressed. The method proposed in this paper is based on a matrix treatment procedure for modifying stiffness and mass matrices of the finite elements most commonly used in modelling structures as continuous systems. Our earlier work [17] and [18] was focused on bar and beam elements. This was then extended [20] and [21] to two dimensional elements. The new formulation significantly improves the previous work by using the second order Taylor approximation in the inverse formulation. Moreover, the proposed technique conducts the modification on the mass and stiffness matrices at a local level thence reducing the computational effort considerably.

An inverse eigenvalue formulation based on the second order approximation has been developed in order to determine the required geometrical and material modifications for pre-defined natural frequency values of structures. The method has been validated by applying it to four case studies. The results are compared against the previous results from a first order formulation and exact solutions and it is shown that large modifications of frequencies can be conducted very efficiently with an acceptable level of accuracy. In the case of the truss structure the natural frequencies can be shifted by up to 10% with a maximum error of 6%. In the case of the two dimensional elastic problem a natural frequency shift of 10% was obtained within an error bound of 3 to 4% in plane stress and plane strain applications. The results obtained in these case studies indicate that using the second order approach the model can be modified up to about 10% with much improved accuracy in comparison with the linearised first order approach. Moreover, it is found that better accuracy can be obtained in the cases involving planner problems compared to models made up of truss elements. The great advantage of the proposed formulation is that the modification is conducted only on small parts of the global stiffness and mass matrices, requiring minimal computational processing time and memory. The modifications can therefore be conducted on the stiffness and mass matrices in their assembled form. This is a very important feature of the proposed technique which allows it to be used as an add-on tool to most commercial FEA codes which do not necessarily make the assembled stiffness and mass matrices accessible. When compared with the classical approach to optimization techniques used in finite elements, it is found the proposed model always yields the exact required eigen frequencies with minimal numerical effort. This is a distinct advantage over the classical techniques where the design space is iteratively searched until a near optimum solution is realized. Moreover, since the proposed algorithm always works with the components of stiffness and mass matrix of individual elements it is much more computationally economical. The other important feature of the proposed technique is that the optimisation is carried in one step with no iterations.