Eigensensitivity Analysis Research Articles

Eigen-sensitivity analysis of a two-stage planetary transmission

A lateral-torsional coupling dynamics model of a two-stage planetary transmission is established based on lumped parameter method. The inherent frequency as well as vibration mode of studied planetary transmission are analysed, and the vibration mode can be classified into four categories, they are rigid body vibration mode, central component translational vibration mode, central component torsional vibration mode as well as planet gear vibration mode. In order to find out the main parameters affecting the inherent characteristics, the perturbation method is used to calculate the eigen-sensitivity of mass parameter as well as stiffness parameter on inherent frequency. The simulation results showed that the main mass parameter affecting the inherent frequency is the mass of planet gear. While as for the stiffness, the main parameters are different according to the different orders.

Prediction of mistuning effect of bladed disks using eigensensitivity analysis

Vibration modes with repeated eigenvalues often occur in engineering design and analysis practices due to geometrical symmetries of structural systems such as the cyclic symmetry of bladed disk assemblies of turbomachinery. However, the degeneration of eigenvector space and the resulting discontinuities have prevented useful applications of eigensensitivities to vibration analysis of a wide class of problems with repeated modes. To overcome such difficulties, a new concept of a global design variable is developed in which all intended multivariate design modifications are grouped into a single global design variable. Eigensensitivities have then been formulated for repeated eigenvalues from which accurate predictions of vibration characteristics can be made. Further, second order eigenvalue derivatives are also employed to further improve the accuracy of predicted vibration properties. Such newly formulated eigensensitivity analysis has been effectively applied for the first time to the prediction of mistuning effect of bladed disk assemblies. Numerical results from a realistic discrete parameter model of a bladed disk have demonstrated that not only natural frequencies and mode shapes can be predicted very accurately, but also blade vibration responses under engine order excitations. In addition, the proposed eigensensitivty analysis can predict the statistical variations of blade vibrations under random mistuning. Finite element modeling and vibration testing of a practical bladed disk structure have been carried out to demonstrate the practical potential of the proposed method to be possibly integrated into finite element analysis for structural modification predictions, especially for structures with repeated eigenvalues such as bladed disks.

An iterative method for exact eigenvalues and eigenvectors of general nonviscously damped structural systems

A novel and computationally efficient iterative method is proposed for the exact eigenvalues and eigenvectors of nonviscously damped vibration systems. General nonviscous damping model is assumed in which damping forces depend on the past motion history via convolution integrals over exponentially decaying kernel functions. The presence of nonviscous damping leads to complex frequency dependent eigenvalue problem whose solution requires either extensively augmented state-space formulation or iterative full complex eigensolutions on mode by mode basis, both of which are computationally expensive when practical systems with large dimensions are considered. By simply solving the eigenvalue problem of the underlying undamped vibration system, the real eigenvalues and eigenvectors can then be combined with the nonviscous damping matrix to develop an iterative procedure from which required complex eigenvalues and eigenvectors of the damped system can be computed. First order perturbation is employed to further improve the starting estimates of the desired eigenvalues and eigenvectors and the convergence is generally very fast. The proposed method is mathematically developed based on progressive eigensensitivity analysis and convergence is ensured when the norm of the damping matrix is of second order when compared with that of the stiffness matrix. Representative numerical examples of discrete mass spring damping model as well as practical finite element model with nonviscous damping are given to demonstrate and validate the accuracy and efficiency of the proposed iterative method.

Modification of natural frequencies of an automotive belt drive system based on eigen-sensitivity analysis of its configuration parameters

Abstract A methodology is presented to simultaneously modify natural frequencies of an automotive belt drive system via configuration modification to match natural frequencies of the system with its working conditions. In this methodology, the configuration modification is regarded as an inverse eigenvalue problem in which necessary changes of a set of configuration parameters are determined for achieving a desired natural frequency variation relative to the original design value. Natural frequencies and mode shapes of the system are obtained by solving an eigenvalue problem that is directly derived from established governing equations of motion of a typical automotive belt drive system. Explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters are derived by differentiating the eigenvalue equation. Several configuration parameters are selected as design variables and their corresponding modification regions and directions are given based on frequency sensitivity analysis. In the inverse eigenvalue problem, variations of natural frequencies caused by changes of multiple design variables are approximated using the first and second order Taylor series expansions, which are based on explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters. By solving the inverse eigenvalue problem, necessary changes of design variables are determined and the desired natural frequencies are achieved so that resonance conditions are successfully avoided.

Eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method

In this paper, free vibrations of a rotating clamped Euler–Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange׳s method. The kinetic and strain energy expression are derived from Rayleigh–Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem.

Reduced thin-layer elements for modeling the nonlinear transfer behavior of bolted joints of automotive engine structures

Damping properties of assembled structures are largely influenced by frictional damping between joint interfaces. Therefore, these effects must be considered during the modeling process. Applying thin-layer elements (TLEs) with a linear, orthotropic material model on mechanical interfaces to incorporate joint damping has shown good agreement with experimental modal analysis in previous work. In the TLE model, constant hysteretic damping is assumed. The damping and stiffness parameters for the TLEs are experimentally identified on an isolated lap joint. Imprecisions caused by model simplifications and parameter uncertainty are addressed by model updating or uncertainty analysis. This requires multiple evaluations of models that are equivalent in all respects but their TLE parameterization. In this work, a model reduction technique for the TLE modeling approach is presented which significantly reduces computational cost for the re-calculation of eigenvalues after joint parameters are changed. The reduction is based on an eigensensitivity analysis and results in a single, linear equation for each eigenvalue. The presented approach is applied to a model updating example. Here, the model reduction allows for a much larger number of design variables which means experimental data can be reproduced more accurately with a physically more meaningful model.

Response Sensitivity and the Assessment of Nonlinear Vibration Using a Nonlinear Lateral–Torsional Coupling Model of Vehicle Transmission System

The eigensensitivity analysis does not meet the increasing industrial requirements of the dynamic performance of a vehicle transmission system. To reduce vibration, it is necessary to include response sensitivity in the guideline in the design stage. In this study, we developed a nonlinear lateral–torsional coupling spur gear system model considering the effect of time-varying mesh stiffness, clearance, mass eccentricity, and transmission error. Then the dynamic response sensitivity to system parameters was systematically analyzed by taking the shaft torsional stiffness, for example. The equation of response sensitivity was deduced by a direct method (DM) based on the fitting of the clearance function curve using a polynomial function. In allusion to the characteristic of the aperiodicity of response sensitivity curves of the nonlinear system in the time domain, a novel assessment method—differential sensitivity based on the root mean square (RMS) of response is proposed. This method provides statistical results in a certain range, thus avoiding the inaccuracy of the partial amplitude. The vibrational energy of modified system (MS) can also be estimated. All the abovementioned characteristics make it possible to provide the theoretical support for dynamic modification, model updating, and optimal design.

Efficient and accurate calculation of sensitivity of damped eigensystems

The exact methods (the algebraic method and Nelson’s method) to calculate eigensensitivities need a matrix decomposition for each eigensensitivity, and therefore are time-consuming. The modal method may be the most efficient method for eigensensitivity analysis, but suffers from modal truncation error. A method is presented by exactly expressing the modal truncation error as a sum of available modes and system matrices. The proposed method maintains original space without having to use the state-space form. It is shown that the proposed method yields good trade-off between accuracy and computational complexity in terms of the theoretical analysis and numerical studies.

Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2, …, λr of the matrix satisfy |λ1| ⩽ … ⩽ |λr| and |λs| < |λs+1| (s ⩽ r−1), then associated with any eigenvalue λi (i ⩽ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2, …, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.

Eigensensitivity analysis of damped systems with distinct and repeated eigenvalues

This paper considers the computation of eigensolution sensitivity of viscously damped eigensystems with distinct and repeated eigenvalues. To simplify the computation, a combined normalization, which combines two traditional normalizations, is presented. Based on the combined normalization, a method for sensitivity analysis of eigenvalues and eigenvectors is studied. In the case of distinct eigenvalues, the proposed method can determine the eigenvector derivatives directly and is robust since the components of coefficient matrices are all of the same order of magnitude. The computational cost of the second-order sensitivities of eigenvectors can be reduced remarkably since the matrix decomposition of the coefficient matrix is available from the computation process of the first-order eigensensitivities. In the case of repeated eigenvalues, an algorithm is presented for computing the eigensolution sensitivities. The algorithm maintains N-space without using state-space equations such that the computational cost is reduced. The method is accurate, compact, numerically stable and easy to be implemented. Finally, three numerical examples have demonstrated the validity of the proposed method. The capacity of predicting the changes of eigensolutions with respect to the changes of design parameters in terms of the first- and second-order eigensensitivities is studied with application to the analysis of a two-stage floating raft isolation system.

About eigen sensitivity analysis of mechanical structures

Several methods for a calculation of derivatives of eigenvectors with respect to design parameters are described here. These are the finite-difference method, the modal method, a modified modal method, Nelson's method, an improved first-order approximation of eigenvalues and eigenvectors and an iterative method. By combining the other structural reanalysis techniques and one of these sensitivity methods, it is possible to enhance the efficiency and the accuracy of structural optimization techniques for determining the optimum condition of mechanical structure specified by an analyst. The sensitivity approach is based on the prior selection of updating parameters (design variables) in the initial FE model.

Approximate method for eigensensitivity analysis of a defective matrix

Based on the exact modal expansion method, an arbitrary high-order approximate method is developed for calculating the second-order eigenvalue derivatives and the first-order eigenvector derivatives of a defective matrix. The numerical example shows the validity of the method. If the different eigenvalues μ ( 1 ) , … , μ ( q ) of the matrix are arranged so that | μ ( 1 ) | ≤ ⋯ ≤ | μ ( q ) | and satisfy the condition that | μ ( q 1 ) | < | μ ( q 1 + 1 ) | for some q 1 < q , and if the approximate method only uses the left and right principal eigenvectors associated with μ ( 1 ) , … , μ ( q 1 ) , then associated with μ ( h ) ( h ≤ q 1 ) the errors of the eigenvalue and eigenvector derivatives by the p th-order approximate method are nearly proportional to | μ ( h ) / μ ( q 1 + 1 ) | p + 1 .

Characterizing the vibration of an elastically point supported rectangular plate using eigensensitivity analysis

Normalized frequencies are computed for a rectangular, isotropic plate resting on elastic supports. The normalized frequencies are determined using eigensensitivity analysis, which approximates the eigenparameters in a Mauclarin series, yielding an approximate closed-form expression. One benefit of the approximate closed-form expression is its computational efficiency and yet another is its application of re-analysis. Accuracy of the approximate expression is assessed by comparing results with the widely used Rayleigh–Ritz method using orthogonal polynomials and beam shape functions in both approaches. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. Results indicate that the accuracy of higher frequencies computed by the sensitivity approach is highly dependent upon choice of basis function.

Modeling of Configuration-Dependent Flexible Joints for a Parallel Robot

This paper provides a method to determine the variable flexible joint parameters which are dependent on configurations for a PRS Parallel Robot. Based on the continuous force approach, virtual springs were used between the joint components to simulate the joint flexibility. The stiffness matrix of the joint virtual springs was derived. The method uses system dynamic characteristics in different configurations to set the virtual spring stiffness for all the joints in the system. Modal testing was conducted on a set of selected robot configurations to determine the system natural frequencies and mode shapes along with their variation. To obtain the virtual spring stiffness, the system was condensed at the joint nodal coordinates. Then eigen-sensitivity analysis was conducted on the condensed system matrices with respect to the stiffness parameters of the joint virtual springs. Thus, the virtual spring parameters in the model can be set to match the variation of the system dynamic responses with the robot configuration changes. The virtual spring parameters between the selected robot configurations were obtained by interpolation. The research indicates that the method is effective and relatively easy to conduct, compared to other methods. The variable flexible joint model is applicable to flexible multibody systems with variable configurations.

Identification of nonlinear boundary effects using nonlinear normal modes

Local nonlinear effects due to micro-slip/slap introduced in boundaries of structures have dominant influence on their lower modal model. This paper studies these effects by experimentally observing the behavior of a clamped–free beam structure with local nonlinearities due to micro-slip at the clamped end. The structure is excited near one of its resonance frequencies and recorded responses are employed to identify the nonlinear effects at the boundary. The nonlinear response of structure is defined using an amplitude-dependent nonlinear normal mode identified from measured responses. A new method for reconstructing nonlinear normal mode is represented in this paper by relating the nonlinear normal mode to the clamped end displacement-dependent stiffness parameters using an eigensensitivity analysis. Solution of obtained equations results equivalent stiffness models at different vibration amplitudes and the corresponding nonlinear normal mode is identified. The approach results nonlinear modes with efficient capabilities in predicting dynamical behavior of the structure at different loading conditions. To evaluate the efficiency of the identified model, the structure is excited at higher excitation load levels than those employed in identification procedures and the observed responses are compared with the predictions of the model at the corresponding input force levels. The predictions are in good agreement with the observed behavior indicating success of identification procedure in capturing the physical merits involve in the boundary local nonlinearities.

Model Calibration of Anisotropic Rotordynamic Systems With Speed-Dependent Parameters

In this paper, a method for calibrating rotordynamic models of speed-dependent systems with anisotropic support is presented. The method is based on the comparison between the calculated eigenvalues and those extracted from the measured synchronous frequency response functions. An eigensensitivity analysis is conducted to calculate the sensitivity of the computed eigenvalues to the selected elements to be updated. This method is suitable for field application since it requires simple coastdown tests. The method is illustrated on a test rig with fluid film bearings and is shown to be effective in the calibration of rotordynamic models at the speeds of the modes excited within the operating speed range.

Forced vibration analysis for dynamic coupling of finite-element spring-reduced suspension system

The current paper studies the dynamic behaviour of a coil spring in a vehicle suspension system. A portion of the coil spring is contacted with the top and bottom spring pads. The contact interface is modelled as a set of linear spring elements on the contact nodes of the coil spring. The eigensensitivity analysis is conducted to investigate the frequency variation of the stiffness-coupled system with respect to the contact stiffness variation. In the forced vibration analysis, the transfer function through the coil spring is obtained. The contribution of the coil spring modes to the load transfer is calculated by the modal contribution factor. It is shown that some of the longitudinal and bending vibration modes participate on the load transfer.

Reduced‐Order Model Development for Airfoil Forced Response

Two new reduced‐order models are developed to accurately and rapidly predict geometry deviation effects on airfoil forced response. Both models have significant application to improved mistuning analysis. The first developed model integrates a principal component analysis approach to reduce the number of defining geometric parameters, semianalytic eigensensitivity analysis, and first‐order Taylor series approximation to allow rapid as‐measured airfoil response analysis. A second developed model extends this approach and quantifies both random and bias errors between the reduced and full models. Adjusting for the bias significantly improves reduced‐order model accuracy. The error model is developed from a regression analysis of the relationship between airfoil geometry parameters and reduced‐order model error, leading to physics‐based error quantification. Both models are demonstrated on an advanced fan airfoil′s frequency, modal force, and forced response.

Two-stage damage diagnosis approach for steel braced space frame structures

In order to diagnose the location and extent of damage in steel braced space frame structures, a two-stage damage diagnosis approach is proposed. This approach is comprised of the damage locating vectors method and eigensensitivity analysis. By deriving formulas used to calculate characterizing stresses in space frame members, and by defining characterizing stresses in connections, the damage locating vectors method is extended to locate damage in space frame members and connections. In addition, the simplified calculation of modal mass-normalization constants for damaged structures is improved. The first- and second-order sensitivities of the modal parameter discrepancies with respect to the structural model parameters and the stiffness matrix of beam elements with one damaged end are utilized. To verify the effectiveness of the proposed approach, numerical simulation analysis and experimental testing of a steel braced space frame model are performed. Ten and seven damage patterns are simulated in the numerical example and experimental testing, respectively. Modal parameters of the undamaged and damaged structures are extracted from the acceleration data using the natural excitation technique (NExT) and the eigensystem realization algorithm (ERA). The extended damage locating vectors method is utilized to determine potentially damaged elements. Based on the identified modal information, the extent of damage of the potentially damaged elements is estimated using the second-order eigensensitivity analysis. It is demonstrated that the two-stage damage diagnosis approach is effective when the damage of the members or connections in steel braced space frame structures reaches a certain level.

Buckling of simply supported rectangular plates under combined bending and compression using eigensensitivity analysis

In this paper, an approximate quadratic closed-form expression is presented for the critical buckling analysis of a plate subjected to combined bending and compression. The formula is developed by expanding the eigenvalue, the critical buckling load, for a plate under combined bending and compression in a Mauclaurin's series about a plate subjected only to compression. The general expression can be used for all combinations of simply supported and clamped rectangular plates boundary conditions. An explicit formula in terms of the plate aspect ratio R and plate load parameter α is evaluated for simply supported plates. Compared with the Rayleigh–Ritz method, this approximate expression provides an excellent comparison when the load parameter α⩽1.52 for plate aspect ratio between 0.2⩽ R⩽2.8.