Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium

This paper deals with topological design optimization of vibrating bi-material elastic structures placed in an acoustic medium. The structural vibrations are excited by a time-harmonic external mechanical surface loading with prescribed excitation frequency, amplitude and spatial distribution. The design objective is minimization of the sound pressure generated by the vibrating structures on a prescribed reference plane or surface in the acoustic medium. The design variables are the volumetric densities of material in the admissible design domain for the structure. A high frequency boundary integral equation is employed to calculate the sound pressure in the acoustic field. This way the acoustic analysis and the corresponding sensitivity analysis can be carried out in a very efficient manner. The structural damping is considered as Rayleigh damping. Penalization models with respect to the acoustic transformation matrix and/or the damping matrix are proposed in order to eliminate intermediate material volume densities, which have been found to appear obstinately in some of the high frequency designs. The influences of the excitation frequency and the structural damping on optimum topologies are investigated by numerical examples. Also, the problem of maximizing (rather than minimizing) sound pressures in points on a reference plane in the acoustic medium is treated. Many interesting features of the examples are revealed and discussed.     

Damping optimization of viscoelastic laminated sandwich composite structures

Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.     

Frequency-dependent noise analysis and damping in MEMS

The paper presents a combined noise and damping analysis for MEMS structures, based on frequency-dependent behavioral models extracted from finite element simulations. The design of high sensitivity MEMS-based microsystems needs to consider the frequency noise shaping induced by damping phenomena on micro scale motion, for its contribution can be significant in the system level noise analysis. Frequency-dependent behavioral models for squeeze-film damping are generated from finite element analysis simulations. Unlike existing noise analysis reported in literature, based on frequency-independent damping assumption, the paper integrates the damping and noise aspects using the same frequency-dependent models. The results can be used for a noise-based optimization procedure applied to the design of resonating microsensors, as illustrated for the case of a MEMS-based gyroscope.     

On topology optimization of damping layer in shell structures under harmonic excitations

This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.     

Finite element-based analysis of shunted piezoelectric structures for vibration damping

Piezoelectric patches shunted with passive electrical networks can be attached to a host structure for reduction of structural vibrations. This approach is frequently called “shunted piezo damping” and has the advantage of guaranteed stability and low complexity in implementation. For numerical treatment of such structures, a finite element modelling methodology is presented that incorporates both the piezoelectric coupling effects of the patches and the electrical dynamics of the connected passive electrical circuits. It allows direct computation of the achieved modal damping ratios as a major design criterion of interest. The damping ratios are determined from the eigenvalue problem corresponding to the coupled model containing piezoelectric structure and passive electrical circuit. The model includes local stiffening and mass effects as a result of the attached patches and, therefore, enables accurate prediction of the natural frequencies and corresponding modal damping ratios. This becomes crucial for choosing the patch thickness to achieve optimal modal damping for a given host structure. Additionally, structures with complex geometry or spatially varying material properties can easily be handled. Furthermore, the use of a finite element formulation for the coupled model of piezoelectric patches and a host structure facilitates design modifications and systematic investigations of parameter dependencies. In this paper, the impact of parameters of the passive electrical network on modal damping ratios as well as the variation of the patch thickness are studied. An application of this modelling method is realized by commercial software packages by importing fully coupled ANSYS^© – models in MATLAB^©. Afterwards, modal truncation is applied, the dynamic equations of the passive electrical network are integrated into the piezoelectric model and eigenvalue problems are solved to extract the increase in modal damping ratios. The numerical results are verified by experiments.     

大型挤压油膜阻尼器非线性动力学分析与优化研究进展

挤压油膜阻尼器（Squeeze Film Dampers,SFDs）是旋转机械中常用的一类支承阻尼结构装置,能够改善转子系统的动力特性.当前工程实际中已经大量使用的两类不同结构形式的挤压油膜阻尼器,仍然存在着减振效果不稳定甚至会导致转子失稳,以及阻尼器动力学机制不清楚、建模和分析精度差、设计方法欠缺等理论技术难题.本文首先介绍两类典型SFD的结构形式和主要失效模式,然后详细叙述SFD动力学分析与优化设计方法在发展过程中的代表性研究成果,涉及SFD动力学特性、转子 SFD系统动力学特性研究、SFD动力学设计与优化等几个方面的研究,并对SFD的试验测试技术方面的成果进行评述.在此基础上,探讨目前先进航空发动机用大型挤压油膜阻尼器亟须开展的基础研究任务,特别强调了数据驱动与动力学解析模型融合的SFD动力学建模、分析与设计优化的发展方向.     

Engineering MEMS Resonators With Low Thermoelastic Damping

This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices     

Experimental validation of compact damping models of perforated MEMS devices

Measured damping coefficients of six different perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the surface shape is rectangular, and the perforation is uniform validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24 to 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130–220 kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this underestimation are discussed by studying the various flow components in the models.     

Fully stressed design of passive controllers in framed structures for seismic loadings

This paper addresses the optimal design problem of added damping in framed structures. Interstory performance indices for linear and nonlinear structures are chosen and restricted to allowable values under the excitation of an ensemble of realistic ground motion records. Optimality criteria are formulated based on fully stressed characteristics of the optimal solution, and a simple analysis/redesign procedure is proposed for attaining optimal designs. Results of four examples presented compare well to those obtained using formal gradient-based optimization.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Hysteretic damping of structures vibrating at resonance: An iterative complex eigensolution method based on damping-stress relation

In this paper the damping is examined as an engineering property used in analysis and design of structures and machines. The design engineer needs to know not only the stresses of his structure or machine, under steady state conditions but also the stresses under resonance conditions. Then the material damping, as a function of the stress of the structure, has an important role to play and ignoring the damping the calculated stresses are far from reality. The nonlinearity here is due to the dependence of the hysteretic damping on the stress of the structure. Specifically here two problems are investigated in the following way:Firstly the direct problem is solved. The direct problem is to find the maximum bending stress at the resonance when the relation of the dissipating energy (or of the hysteretic damping) vs. the bending stress is known in advance. To perform this calculation, a useful tool for the design engineer, the structure is modelled using the continuum mechanics analytical approach or the finite elements (FE) method. Then the eigenvalues are calculated and using an iterative procedure the real stress. The procedure presented here is called iterative complex eigensolution method (ICEM). Secondly the inverse problem is solved. The inverse problem is to find the relation between the hysteretic damping and the bending stress. For this purpose the logarithmic decrement is experimentally measured, the eigenvalues and the maximum bending stress of the structure, excited at the eigenvalue, when the damping is the same as the measured one, are computed using the finite elements method. Once the bending stresses are found in each discrete element of the structure, then the mathematical expression of the relation of the dissipating energy and the stresses can be specified by minimizing a suitably formed objective function.     

Integrating structure and control design to achieve mixed H2/H performance

Present technology in structure design (smart structures, civil structures and aerospace structures) includes the use of feedback control. While retrofitting such active elements can be useful in existing structures, future designs will require something more than retrofitting technology. Future technology will certainly require a more systematic integration of the design of a structure and its active elements. This paper provides a step in that direction. We seek to integrate the design of the structure with its active elements to achieve mixed H2/H performance for the controlled structural system (closed loop system). More specificaly, the approach presented here solves a mixed passive control (structure design) and active control (feedback control law design) problems with performance characterized by system norms such that H performance bounds are guaranteed with les active energy. This approach al ows us to answer the question 'what is an optimal distribution of mass, stiffness, damping and control energy throughout a structure?' The main conclusion drew in this paper is that control design tools can be useful for structure design.     

嵌入多层黏弹性胶膜复合材料阻尼工字梁的多目标设计优化

针对嵌入多层黏弹性胶膜的复合材料阻尼工字梁,传统的混合单元法在进行结构动态分析与设计优化时存在很大困难的问题,采用基于离散层理论的多层梁单元建模分析复合材料阻尼工字梁.通过对正交各向异性铺层和腹板进行等效处理的方法,对工字梁凸缘嵌入单层阻尼层模型进行参数化分析;分别对嵌入多层或单层阻尼层的模型建立多目标优化模型,优化目标为模态损耗因子和固有频率最大化,设计变量为阻尼层层数（厚度）及其嵌入位置;应用多目标遗传算法进行优化求解.结果表明：基于离散层理论的阻尼梁单元计算精度好且易于优化,对于嵌入单层较厚阻尼和嵌入多层较薄阻尼的复合材料工字梁,获得的阻尼效果与动刚度损失基本相当,但对于高阻尼的方案,前者比后者的动刚度损失更大.     

Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.     

Influence of Boundary Conditions on the Dynamic Characteristics of Squeeze Films in MEMS Devices

Micromechanical structures that have squeeze-film damping as the dominant energy dissipation mechanism are of interest in this paper. For such structures with narrow air gap, the Reynolds equation is used for calculating squeeze-film damping, which is generally solved with trivial pressure boundary conditions on the side walls. This procedure, however, fails to give satisfactory results for structures under two important conditions: 1) for an air gap thickness comparable to the lateral dimensions of the microstructure and 2) for nontrivial pressure boundary conditions such as fully open boundaries on an extended substrate or partially blocked boundaries that provide side clearance to the fluid flow. Several formulas exist to account for simple boundary conditions. In practice, however, there are many micromechanical structures such as torsional microelectromechanical system (MEMS) structures that have nontrivial boundary conditions arising from partially blocked boundaries. Such boundaries usually have clearance parameters that can vary due to fabrication. These parameters, however, can also be used as design parameters if we understand their role on the dynamics of the structure. We take a MEMS torsion mirror as an example device that has large air gap and partially blocked boundaries due to static frames. We actuate the device and experimentally determine the quality factor Q from the response measurements. Next, we model the same structure in ANSYS and carry out computational fluid dynamics analysis to evaluate the stiffness constant K, the damping constant D, and the quality factor Q due to the squeeze film. We compare the computational results with experimental results and show that without taking care of the partially blocked boundaries properly in the computational model, we get unacceptably large errors.     

Estimation of squeeze film damping in artificial hair-sensor towards the detection-limit of crickets’ hairs

The filliform hairs of crickets are among the most sensitive flow sensing elements in nature. The high sensitivity of these hairs enables crickets in perceiving tiny air-movements which are only just distinguishable from noise. This forms our source of inspiration to design highly-sensitive array system made of artificial hair sensors for flow pattern observation i.e. flow camera. The realization of such high-sensitive hair sensor requires designs with low thermo-mechanical noise to match the detection-limit of crickets’ hairs. Here we investigate the damping factor in our artificial hair-sensor using different methods, as it is the source of the thermo-mechanical noise in MEMS structures. The theoretical analysis was verified with measurements in different conditions to estimate the damping factor. The results show that the damping factor of the artificial hair sensor as estimated in air is in the range of 10^?12 N m/rad s^?1, which translates into a 93 μm/s threshold airflow velocity.     

Modal analysis of nonlinear systems with classical and non-classical damping

In the dynamic analysis and design of structures and equipment, the behavior of various components beyond the linear range is often of interest. A nonlinear vibration analysis is time consuming, particularly if many configurations or loading conditions have to be considered in order to arrive at representative values for design. The selection of an efficient and yet accurate analysis procedure is therefore extremely important.

Two mode-superposition procedures are presented for the dynamic analysis of nonlinear structures with classical (proportional) and non-classical (non-proportional) damping. The nonlinearity at each time step is treated as a pseudo force. Undamped eigensolution and complex modes are used to uncouple the equations of motion for classical and non-classical damping cases. A recursive procedure based on the exact solution of the differential equation is used to obtain the modal responses. The advantages of the proposed method of computing the response over the existing integration techniques are: (a) the simplicity of the procedure, (b) a substantial reduction in computational time, (c) the possibility of using fewer modes to achieve the desired accuracy, and (d) the adaptability of the procedure to parallel processing machines which will further reduce the computational time.     

Hybrid variable damping control: design,simulation, and optimization

This paper presents the design, formulation, and performance optimization of a new hybrid electromagnetic damper in response to the demand for a tunable, regenerative and fail-safe damping device for various applications. Damping in a multitude of engineering applications has a variable threshold requirement based on system excitation. Since system excitation is also variable; dampers are such that an adequate amount of damping is provided, opposed to an optimal amount as a function of excitation. In this research it was shown that, by implementing a hybrid damper design based on a bias component provided through a hydraulic medium and a variable component provided by electromagnetics, an optimal damping quantity can be obtained for a given excitation. The produced damping force and electrical power were formulated based on the structure’s geometry and input displacement. The presented design was optimized for a scooter scaled application and it was shown that the damping and regenerative characteristics can be adjusted for different requirements. Furthermore, it was illustrated that this design has the potential to be scaled for other applications as well.     

常压封装下隧道磁阻微陀螺的高灵敏结构设计

针对现有微陀螺难以实现常压封装下高灵敏检测的难题,设计了两种应用隧道磁阻效应检测的微陀螺结构,分别采用面内检测与离面检测,本文从阻尼与检测磁场两方面对二者性能进行分析。首先通过对二者阻尼的计算,得到常压下面内检测结构的灵敏度为35.18 nm/(°/s),离面检测结构的灵敏度为3.19 nm/(°/s),面内结构比离面结构的灵敏度高约10倍,从阻尼方面考虑,采用面内检测的结构更优;其次设计了应用于两种结构中的检测磁场,得到了面内检测磁场相比于离面检测磁场具有更高的磁场变化率,更好的磁场一致性,从磁场方面考虑,同样得到面内检测的微陀螺结构更优。因此,应用面内检测结构可以实现隧道磁阻微陀螺在常压下的高灵敏检测。     

Evolutionary topology optimization of elastoplastic structures

We have recently proposed in (Fritzen et al., Int J Numer Methods Eng 106(6):430–453, 2016) an evolutionary topology optimization model for the design of multiscale elastoplastic structures, which is in general independent of the applied material law. Facing the variability of the final design for minor parameter changes when dealing with plastic structural designs, we further improve the robustness and the effectiveness of the BESO optimization procedure in this work by introducing a damping scheme on sensitivity numbers and by progressively reducing the sensitivity filtering radius. The damping scheme constraining the variance of the sensitivity numbers stabilizes the topological evolution process in particular for dissipative structural designs. By setting initially a large filter radius value and reducing it gradually, the emergence of the redundant structural branches, which are to be eliminated afterwards and are the main reasons deteriorating the design process, could be avoided. The robustness and the effectiveness of the improved model has been validated by means of benchmark numerical examples of conventional homogeneous structures.