A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.     

Prediction of in-process frequency response function and chatter stability considering pose and feedrate in robotic milling

Chatter occurs easily during robotic milling owing to the low structural stiffness of industrial robots and can degrade the machining quality or even cause robot failure. The accurate frequency response function (FRF) of the robot is essential for predicting chatter stability and selecting the appropriate process parameters. However, the FRF of a robot is affected by multiple factors, such as pose, operating state, and external excitation. In this study, an in-process FRF prediction method considering robot pose and feedrate was developed and used to predict chatter stability. Firstly, the static FRFs were obtained from the experimental modal analysis for different robot poses and used to train a Gaussian process regression (GPR) model. Subsequently, the static FRF predicted using GPR and the modal parameters identified by operational modal analysis (OMA) were used to calculate the in-process FRFs of the robot in the operation state. After removing the harmonic components of the vibration signals using a matrix notch filter, OMA was conducted using the least-squares complex frequency. Furthermore, the FRF of the robot was transformed from the robot flange coordinate system into the engagement coordinate system using the kinematics model and the tool path. The dynamic milling model, considering tool and robot modes was used for predicting stability. Finally, the proposed method was demonstrated by time-domain simulation of the robot-tool system and milling tests, and the effects of the running state and feed direction on chatter stability considering robot mode were analyzed.     

含间隙弹簧振动系统的非线性模态特性

针对含间隙的两自由度弹簧-质量分段振动系统的非线性模态开展了研究.首先,解析确定了分段保守自治系统发生同相和反相模态运动的初始位移,并采用加权平均方法确定了分段振动系统的模态频率,及其在位形空间模态曲线.然后,采用数值方法求解了系统的非线性模态曲线和模态频率,与本文获得的解析模态频率比较,说明本文的结果较等效模态频率有更好的精度.研究结果表明:在一定的参数条件下,系统的非线性模态个数会高于系统的自由度数目,系统可能发生内共振,而产生多余模态.多余模态运动是两振子同向振动中含有异向振动,说明多余模态是在同相模态运动和反相模态运动之间转换的模态.     

An approach to predict lower-order dynamic behaviors of a 5-DOF hybrid robot using a minimum set of generalized coordinates

This paper presents an effective semi-analytical approach for predicting lower-order dynamics of a five degrees-of-freedom (DOF) hybrid robot named TriMule, which is composed of a 3-DOF parallel mechanism plus a 2-DOF A/C wrist. In this method, the governing equations of motion of limbs within the parallel mechanism are first formulated by finite element analysis (FEA) and then reduced to super-element models. This is followed by exploiting a general stiffness model of multiple DOF joints connecting the super-elements. These two threads lead to the reduced dynamic model of the parallel mechanism while keeping the full set of lower-order modes retained. Finally, the dynamic model of entire system is established by merging the models of parallel mechanism and wrist. The computational results show that the lower-order natural frequencies, mode shapes of the entire system, and the frequency response functions (FRFs) of the robot tool center point (TCP) estimated by the proposed approach have very good agreement with those obtained by a full order FE model and experimental modal tests. The merits of this approach lie in that the established model allows the full set of lower-order dynamics of the entire system to be predicted effectively and accurately by only using fourteen generalized coordinates.     

Applications of hybrid optimization techniques for model updating of rotor shafts

This paper presents an approach by combining the genetic algorithm (GA) with simulated annealing (SA) algorithm for enhancing finite element (FE) model updating. The proposed algorithm has been applied to two typical rotor shafts to test the superiority of the technique. It also gives a detailed comparison of the natural frequencies and frequency response functions (FRFs) obtained from experimental modal testing, the initial FE model and FE models updated by GA, SA, and combination of GA and SA (GA–SA). The results concluded that the GA, SA, and GA–SA are powerful optimization techniques which can be successfully applied to FE model updating, but the appropriate choice of the updating parameters and objective function is of great importance in the iterative process. Generally, the natural frequencies and FRFs obtained from FE model updated by GA–SA show the best agreement with experiments than those obtained from the initial FE model and FE models updated by GA and SA independently.     

System identification of torsionally coupled buildings

In this study, an extended random decrement method, which considers the correlation among measurements, was employed to reduce the measured dynamic responses of general torsionally coupled multi-story building under random excitations. The Ibrahim time domain technique was then applied to calculate the modal frequencies and damping ratios based on only a few floor response measurements. To obtain the complete mode shapes, an interpolation method was developed to estimate the mode shape values for the locations without measurements. The seismic responses at floors with and without measurements were also calculated. Numerical results through a seven-story torsionally coupled building under ambient random excitations demonstrated that the proposed method is able to identify structural dominant modal parameters accurately even with highly coupled modes and noise contamination. A small number of response measurements, no requirement for input excitation measurements and simple on-line calculations make the proposed method favorable for implementation.     

结构中频振动分析的混合方法

针对结构振动的中频问题,提出了一种新的混合分析方法.具有低模态密度的子结构利用有限元建模,高模态密度子结构利用波动方法建模,并利用边界处的位移连续和力平衡条件进行求解.以耦合梁结构为例,给出了具体的计算过程,通过解析方法进行了仿真验证.结果表明了此混合方法的有效性.进一步地计算了高频子结构的能量密度响应,并且通过对比说明,此方法在计算边界位置的能量密度响应时可以得到精确度更高的结果.     

Effect of non-linear modal interaction on the dynamic instability of axially excited cylindrical shells

The non-linear vibrations and instabilities of cylindrical shells under pulsating axial loads are investigated using Donnell’s shallow shell equations. Based on physical and mathematical reasoning, a meaningful low dimensional model is derived and used together with Galerkin method to obtain a set of coupled non-linear equations of motion. Particular attention is paid to the investigation of modal interaction between non-linear vibration modes with equal or nearly equal natural frequencies and to its influence on parametric instability and snap-through buckling. A parametric analysis using continuation techniques identifies the relevant bifurcations connected with the modal interaction and highlights their relevance in design.     

The role of actual modal-contributions versus frequency to optimize the structure

In modal frequency response analysis, the dynamic analyst is often faced with the structure’s dynamic behavior, the modal contributions included, over a frequency window rather than at a single frequency. Therefore a new method in modal frequency response analysis has been developed for computing both complex modal-contributions and real, actual modal-contributions over a frequency range. Contributions from normal modes to displacement, velocity, or acceleration of a set of selected evaluation points (grid-component combinations) are considered. The focus lies on identifying the major actual-contributions from normal modes and the frequency range they are active in. The method is valid for all branches of mechanical engineering. With the thorough knowledge of the dominant modal-contributions to the physical motion response of relevant structure locations and the modal contributions’ frequency history, the traditional design process can substantially be enhanced. It is worthwhile to notice that by the use of the presented procedure the dynamic analyst may find innovative redesigns which the automatic structural optimizers are not able to find. Examples are given to demonstrate the application, the strength of the coupling between modes, the influence of base and force excitation on the modal contributions and, finally, some recommendations on how to reduce undesired structural responses.     

Robustness of active modal damping of large flexible structures

The control of large flexible systems is an area of growing concern for control engineers. In particular, the robustness of any control design for a flexible structure is of great interest because of the inexact knowledge of the structure, the necessity of model truncation in the control design or the variability of the structure itself. Numerous techniques have been offered as solutions for one or more of the above problems, but few applications have been produced. This present paper reviews the method of AMD and presents as a design example the pinhole/occulter facility (P/OF). This system is a large space system comprised of a flexible beam (30 m long by 0·3 m wide), a gimbal-pointing system and an optical alignment system mounted in the shuttle cargo bay and excited by typical shuttle disturbances. The flexible modes are assumed known, but only to within about 10%. A controller was designed for the nominal case and the effects of variations of the modal frequencies and modal damping factors on the system performance studied. A series compensator was designed that also resulted in a stable system. The performance effects for the same modal variations are presented for this system as a comparison. The ADM system out-performs the series-compensated system for all cases except the design case.     

Dynamic response analysis of linear structural systems subject to component changes

A method of analysis is developed for determining transient responses of large multiply-connected structural systems subjected to changes of structural components. A dynamic system is divided into two subsystems: the support which remains unaltered and the branch which is liable to change. The response characteristics of an original system are used as a basis for evaluating the new response of the altered system. The responses of the support interface coordinates due to external excitations on it are called base motion. The equations of motion of the system are formulated using subsystem modal properties such that the base motions are the generalized forcing functions. This enables incorporation of the alternative properties in the new analysis without having recourse to the entire system modal properties. The method is applied to a 16-storey building rigid frame model. The methods gives response results comparable with the conventional integrated system analysis. Approximations due to modal truncation are the same as component mode substitution method.     

A method for topology optimization of structures under harmonic excitations

This paper deals with topology optimization of large-scale structures with proportional damping subjected to harmonic excitations. A combined method (CM) of modal superposition with model order reduction (MOR) for harmonic response analysis is introduced. In the method, only the modes corresponding to a frequency range which is a little bigger than that of interest are used for modal superposition, the contribution of unknown higher modes is complemented by a MOR technique. Objective functions are the integral of dynamic compliance of a structure, and that of displacement amplitude of a certain user-defined degree of freedom in the structure, over a range of interested frequencies. The adjoint variable method is applied to analyze sensitivities of objective functions and the accuracy of the sensitivity analyses can also be ensured by CM. Topology optimization procedure is illustrated by three examples. It is shown that the topology optimization based on CM not only remarkably reduce CPU time, but also ensure accuracy of results.     

Eigenvalue analysis for coupled large linear damped structures

A method is described for the eigenvalue analysis of large linear damped systems that can be broken down into a number of coupled subsystems. The method uses a normal mode representation of the subsystem receptances at the coupling stations. The method allows the neglect of unimportant higher mode contributions to the subsystem receptances and results in a reduced order approximate characteristic equation. An iteration method gives eigenvalue corrections to any desired accuracy.     

Rotation angle analyses of plate ultrasonic motor under dual-mode coupling drive

Speed improvements of plate ultrasonic motors could be achieved by choosing suitable excitation frequencies; however, the combination of these two frequencies is hard to be determined. Based on the linear superposition of vibrations at two independent mode frequencies, the rotation angles and amplitudes of modal vibrations are proposed to select the possible dual-frequency excitation combinations in this paper. Experimental prototype is a plate ultrasonic motor using single-phase asymmetric excitation, which can work under a single vibration or multiple vibration modes. The speed characteristics of prototype driven by dual-mode coupling are used to verify the coupling effects of dual-mode drive, in which each rotation angle of two modal vibrations is not close to vertical phase, and the two modal vibrations with larger amplitudes and closer rotation angles could improve the rotation speed of motor.     

Prediction of the Posture-Dependent Tool Tip Dynamics in Robotic Milling Based on Multi-Task Gaussian Process Regressions

Chatter vibration is one of the main factors that limit the productivity and quality of the robotic milling process. To predict the robotic milling stability, it is essential to obtain the tool tip frequency response function (FRF). The tool tip dynamics of a robot heavily depend on its postures and used tools. A state-of-art methodology of combining the regression model with the Receptance Coupling Substructure Analysis (RCSA) method is proved to be effective in predicting tool tip FRFs of machine tools for different positions and tools. However, for the milling robot, the cross coupling FRFs have an obvious influence on the dynamic property of the milling robot, thereby greatly affecting the milling stability boundary. It is of great challenge to directly integrate the effect of the cross coupling FRFs into the state-of-art approach to predict the tool tip dynamics. To tackle this challenge, in this paper, we propose an approach to predict the posture-dependent tool tip dynamics for different tools in robotic milling considering the cross coupling FRFs. First, a more comprehensive RCSA procedure is adopted to include the cross coupling FRFs. Then, the impact test is designed to measure the required FRF matrix. By fitting the measured FRF matrix with the multiple-degree-of-freedom (MDOF) model, the number of modal parameters is significantly reduced. Next, the Multi-Task Gaussian Process (MTGP) regression model is employed to mine the physical correlations between different modal parameters. Compared to the ordinary Gaussian Process regression model, the number of required regression models in MTGP is reduced and the prediction performance is improved in terms of accuracy and robustness. Furthermore, the effectiveness of the proposed approach is validated by the impact test and milling experiment on an industrial robot.     

A novel order reduction method for nonlinear dynamical system under external periodic excitations

The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper.Using the modal transformation,a large nonlinear dynamical system is split into a 'master' subsystem,a 'slave' subsystem,and a 'negligible' subsystem.Accordingly,a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the 'negligible' subsystem and slaving the 'slave' sub...     

柔怀臂的主极点设置控制

柔性臂控制是一个活跃的研究课题,已经有不少结果,最多用的方法是通过模态分析以控制柔性臂低价振型（一般为首一或二个振型）本文提出了一种“主极点”（ＤｏｍｉｎａｎｔＰｏｌｅｓ）控制器,它可以控制任何数目的低价振型而不失稳定性,该控制器可以方便地转变自适应控制器以应会中模态频率与阻尼系数与实际柔性臂之间的误差,本文用李雅普诺夫理论证明本控制系统的全局稳定性（Ｇｌｏｂａｌｓｔａｂｉｌｉｔｙ）。     

A mode-based meta-model for the frequency response functions of uncertain structural systems

Frequency response functions (FRFs) play an important role in the assessment of the structural response of linear systems subjected to dynamic forces. In this work a functional relation (meta-model) between the uncertain structural parameters and the model properties is presented. The meta-model provides a computationally fast solution to approximate the eigenfrequencies and mode shapes needed thereafter for evaluating the FRFs. The meta-model circumvents the repeated solution of the eigenvalue problem for each set of uncertain structural input parameters. The provided relations can be helpful in the design stage to control the dynamic response within certain frequency bands. Moreover, the meta-model can be used for optimization and reliability assessments based on Monte Carlo sampling procedures. Numerical examples show the application of the method focusing mainly on the variability of the FRFs. The efficiency and accuracy of the meta-model to compute approximate FRFs is assessed by a comparison with the solution by Finite Element (FE) analyses.     

Hybrid statistical modelling of the frequency response function of industrial robots

Models that predict the Frequency Response Function (FRF) of six degree-of-freedom (6-dof) industrial robots used for machining operations such as milling are usually built using Experimental Modal Analysis (EMA) of vibration data obtained from modal impact hammer tests performed at a finite number of points in the robot's workspace corresponding to specific arm configurations. While modal impact hammer tests are not constrained by the operating conditions of the robot, such as specific arm configurations allowed by part fixturing, they are limited by the number of workspace points that can be practically sampled and the associated robot downtime. Alternatively, the process of determining robot FRFs from on-line machining process data (e.g., forces and vibration) through Operational Modal Analysis (OMA) enables a denser sampling of the robot's workspace without requiring robot downtime. However, OMA may require several long tool paths and one or more complex part setups to enable sampling of a sufficiently large number of locations/arm configurations. This paper presents an efficient hybrid statistical modelling methodology that combines the two approaches, thus enabling possible optimization of sampling density and robot downtime, to efficiently determine the robot FRFs as a function arm configuration. The approach consists of first calibrating a Gaussian Process Regression (GPR) model with FRF data derived from EMA conducted at a small number of discrete locations in the robot's workspace. Then, FRFs calculated from OMA of milling forces and tool tip vibration data derived from robotic milling tests are used to update the initial GPR model using Bayesian inference and efficient hyperparameter updating. The proposed hybrid robot FRF modelling method is experimentally validated and shown to yield accurate predictions of the robot FRF while being computationally efficient.