Support position optimization with minimum stiffness for plate structures including support mass
Introduction
A flexural plate with intermediate simple or point supports is one of the most commonly used structural elements in civil, aerospace, marine, electronic and mechanical engineering applications. Usually, these supports are used to hold the plate structure statically. Often, they are also employed to improve the structural characteristics and performance by the optimal design of the supports’ stiffnesses and attachment points, especially when other structural design modifications cannot be effectively performed in practical problems [1], [2], [3]. Thus far, a great number of publications are available in the literature investigating the dynamic properties of plates with various boundary conditions resting on fixed or movable point supports. Usually, an exact solution of the transverse vibration is not available even for a thin (Kirchhoff model) plates with general (elastic or rigid) point supports. Therefore, various numerical approaches, for example based on the finite element method (FEM) or the Rayleigh-Ritz method, have been developed in order to determine the dynamic behaviors, typically the natural frequencies, mode shapes of the plate system and its vibration response to a general excitation [1], [2], [3], [4], [5], [6], [7].
It is well known that both the restraint stiffness and the attachment location of an elastic support are very important in engineering applications. Small changes to either the stiffness or position of an intermediate support can dramatically affect the dynamic properties of a beam or plate structure [5,[8], [9], [10], [11]]. Thus, these parameters are often utilized on purpose to modify the vibration characteristics or the critical buckling load of the structure [1,4,8,9]. Moreover, there exists an exact optimum position for a point support, at which a certain or critical value of the stiffness can essentially raise a natural frequency of interest to a preset target value or to its upper limit [1,12]. Olhoff and Akesson [8] highlighted that attaining the minimum stiffness of a structure gives a much more efficient design of the support in practice, because both the economic and material costs of a flexible support are directly related to its restraint stiffness. Therefore, estimating the minimum stiffness of the flexible support enables designers and engineers to obtain the minimum weight design of a structural system in practical engineering. In addition, previous studies [1,12] have shown that the optimal support position to maximize a specified natural frequency may be non-unique once the restraint stiffness of the additional support is beyond a critical or threshold value. Besides, above this minimum stiffness, the target natural frequency cannot be raised further by increasing the support restraint stiffness, but the associated mode shapes of the beam or plate structure are modified, primarily due to mode switching between two consecutive modes [9].
A survey of the early literature reveals that an elastic transverse support is typically modeled as a massless translational spring simply connected at a point with a finite or infinite stiffness [2,4,[7], [8], [9]]. Thus, the mass or inertia properties of the spring support are neglected or excluded in the dynamic analyses of the beam or plate structure. The massless support assumption also means that the support stiffness is not fully correlated with its material or economic expenditure, which is not realistic in engineering practice [8]. However, it is well recognized in general that the restraint stiffness of a spring support is closely associated with its material cost or mass. Moreover, from the structural vibration theory [13], it is commonly known that part of the elastic support mass does virtually participate in the transverse vibration of the structure, and therefore affects its dynamic properties, including its natural frequencies. In other words, the additional mass of a point support should be incorporated into the support position optimization to achieve the minimum stiffness required, or to maximize a natural frequency of interest. Such a problem has practical importance in structural designs, but to the authors’ knowledge, has not been addressed as yet in the available literature.
The problem under investigation in this paper is to optimize the positions of elastic point supports in order to maximize a natural frequency, particularly the fundamental frequency, of a flexural plate structure. This is because in many cases of engineering applications, the structural dynamic behavior is highly dependent on the first few natural frequencies and the relevant mode shapes. Raising a natural frequency of a structure as far away as possible from the driving frequency of an external load can significantly reduce its vibration response. Damping is not considered in the present analysis, even though the response amplitude of a structure near resonance is mainly determined by the modal damping. However, the concept here is to ensure that the natural frequencies and the excitation frequencies are well separated, and in this case the damping has little influence on the response. To obtain more realistic results, both the stiffness and mass of a simple support are considered simultaneously in the plate vibration analysis to obtain the corresponding minimum stiffness mainly due to its practical significance. To achieve this, the frequency sensitivity analysis with respect to an elastic support location is first conducted using the finite element (FE) approach [12]. Since the dynamic analysis most commonly uses FEM, such a derivation of the design sensitivity is fully consistent with the numerical modal analysis of a structure. Second, the minimum stiffness of the interior support required for a certain target natural frequency is estimated at a point of attachment to the plate. For the general bending vibration of a plate structure, the determination of the minimum stiffness of the flexible support can be simply formulated as a generalized eigenvalue problem [5,14], and therefore the optimum stiffness may be obtained numerically as the lowest positive eigenvalue.
Afterwards, a heuristic optimization procedure, called the evolutionary shift method [12], is implemented to determine the optimal support location as well as the corresponding minimum restraint stiffness for maximization of the structural natural frequency of interest. Initially, the optimization of the support location assumes that the attachment occurs only at the nodes of the FE model, with the contribution of the support mass included. On the basis of the design sensitivity analysis, the support location will be shifted in the specified direction with a step size given by the element size, to gradually reach to the approximate optimal position for the design task. However, the optimal support position is unlikely to occur exactly at an FE node, and will usually occur within an element. To gain a more accurate estimate of the optimal position, without discretizing the local region near the solution with a very fine FE mesh, the equivalent stiffness and mass matrices of an elastic point support located within an element is used to efficiently obtain the optimal position and make the design solution insensitive to the FE mesh [5,15]. Finally, the feasibility and effectiveness of the proposed optimization algorithm is demonstrated by three benchmark examples of rectangular plates. The optimal results are compared to the traditional solutions that neglect the mass of the spring support [5,12,14] to demonstrate the effect of the support mass inclusion on the optimal design of the intermediate spring supports.
Section snippets
Derivative of natural frequency with respect to support position
In structural dynamic analysis, the characteristic equation of an undamped system in the discrete form is [16]where [K] and [M] are the global stiffness and mass matrices, respectively. ωi denotes the ith natural frequency in radians and {ϕ}i is the associated vibration mode of the structure, which has been mass normalized. Notice that ωi is an implicit function of the support parameters.
As is well known, design sensitivity analysis determines the effect of a design variable
Evolutionary procedure for support position optimization
Often, a structural design optimization is performed based on size, shape and topology. However, when the structural design parameters cannot be altered due to some design limitations, changes in the restraint conditions of a structure can also be utilized to effectively improve the structural static or dynamic behaviors [1]. At present, the positions of the spring point supports in the plate structure are optimally designed to raise a natural frequency of interest to a target value or to its
Illustrative examples
The validity of the formulation for the frequency derivative calculation and the effectiveness of the proposed optimization approach to obtain both the minimum stiffness and the optimal position of internal point supports will be demonstrated with several examples. In this section, different boundary conditions of the rectangular plate structures are explored and the optimal results are compared with those obtained in the literature that ignore the support mass [5,14] to illustrate the effects
Conclusions
In this work, simple translational supports are optimally designed to raise a natural frequency of the rectangular flexural plate structure to a target value or to its upper limit. For engineering applications, the effective mass of the spring support should be included in the vibration analysis of the plate structure to achieve a more practical design of the additional support. First, the frequency derivative formulation with respect to the support movement is developed using FE analysis,
CRediT authorship contribution statement
D. Wang: Methodology, Software, Writing – original draft. M.I. Friswell: Conceptualization, Data curation, Visualization, Validation, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (grant number 51975470) and the Natural Science Foundation of Shaanxi Province, China (2020JM-114).
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