Modelling the spindle–holder taper joint in machine tools: A tapered zero-thickness finite element method

This study presents a tapered zero-thickness finite element model together with its parameter identification method for modelling the spindle–holder taper joint in machine tools. In the presented model, the spindle and the holder are modelled as solid elements and the taper joint is modelled as a tapered zero-thickness finite element with stiffness and damping but without mass or thickness. The proposed model considers not only the coupling of adjacent degrees of freedom but also the radial, tangential and axial effects of the spindle–holder taper joint. Based on the inverse relationship between the dynamic matrix and frequency response function matrix of a multi-degree-of-freedom system, this study proposes a combined analytical–experimental method to identify the stiffness matrix and damping coefficient of the proposed tapered zero-thickness finite element. The method extracts those parameters from FRFs of an entire specimen that contains only the spindle–holder taper joint. The simulated FRF obtained from the proposed model matches the experimental FRF quite well, which indicates that the presented method provides high accuracy and is easy to implement in modelling the spindle–holder taper joint.     

A new experimental modal analysis method based on double-exponential windowing and application to lightly damped bladed wheels

In this paper, a new method is presented for experimental modal analysis for lightly damped structures such as bladed wheels. Like most of existing methods, the one proposed here is also based on frequency response functions (FRFs). An FRF can be calculated to be the ratio of the Fourier transforms of a response and a force. For a pair of force–response time-histories generated by a force hammer impacting on a lightly damped structure, an exponential window is required to apply on both the force- and response-time histories when performing the Fourier transforms. This results in an approximate of the true FRF. Using a different decaying rate in the exponential window, a second approximated FRF is produced. These two approximated FRFs, instead of one as in existing experimental modal analysis methods, are then combined to abstract modal information and recover the true FRF.     

Accurate identification of the frequency response functions for the rotor-bearing-foundation system using the modified pseudo mode shape method

In this paper, an identification technique in the dynamic analyses of rotor-bearing-foundation systems called the pseudo mode shape method (PMSM) was improved in order to enhance the accuracy of the identified dynamic characteristic matrices of its foundation models. Two procedures, namely, phase modification and numerical optimisation, were proposed in the algorithm of PMSM to effectively improve its accuracy. Generally, it is always necessary to build the whole foundation model in studying the dynamics of a rotor system through the finite element analysis method. This is either unfeasible or impractical when the foundation is too complicated. Instead, the PMSM uses the frequency response function (FRF) data of joint positions between the rotor and the foundation to establish the equivalent mass, damping, and stiffness matrices of the foundation without having to build the physical model. However, the accuracy of the obtained system's FRF is still unsatisfactory, especially at those higher modes. In order to demonstrate the effectiveness of the presented methods, a solid foundation was solved for its FRF by using both the original and modified PMSM, as well as the finite element (ANSYS) model for comparisons. The results showed that the accuracy of the obtained FRF was improved remarkably with the modified PMSM based on the results of the ANSYS. In addition, an induction motor resembling a rotor-bearing-foundation system, with its housing treated as the foundation, was taken as an example to verify the algorithm experimentally. The FRF curves at the bearing supports of the rotor (armature) were obtained through modal testing to estimate the above-mentioned equivalent matrices of the housing. The FRF of the housing, which was calculated from the equivalent matrices with the modified PMSM, showed satisfactory consistency with that from the modal testing.     

Non-linear parameter estimation with Volterra series using the method of recursive iteration through harmonic probing

Volterra series provides a platform for non-linear response representation and definition of higher order frequency response functions (FRFs). It has been extensively used in non-parametric system identification through measurement of first and higher order FRFs. A parametric system identification approach has been adopted in the present study. The series response structure is explored for parameter estimation of polynomial form non-linearity. First and higher order frequency response functions are extracted from the measured response harmonic amplitudes through recursive iteration. Relationships between higher order FRFs and first order FRF are then employed to estimate the non-linear parameters. Excitation levels are selected for minimum series approximation error and the number of terms in the series is controlled according to convergence requirement. The problem of low signal strength of higher harmonics is investigated and a measurability criterion is proposed for selection of excitation level and range of excitation frequency. The procedure is illustrated through numerical simulation for a Duffing oscillator. Robustness of the estimation procedure in the presence of measurement noise is also investigated.     

Prediction of uncertain frequency response function bounds using polynomial chaos expansion

A probabilistic method is developed to predict the uncertainty bounds on Frequency Response Functions (FRFs) developed from Finite Element models. A non-intrusive Polynomial Chaos Expansion (PCE) method is used to predict uncertainty regression models of the various parameters that make up a curvefit of the FRF: natural frequencies, damping ratios, complex amplitudes, mass and stiffness residuals, by making use of an efficient Latin Hypercube technique. These uncertainty models are then combined to efficiently determine PDFs of the parameters and also the uncertainty bounds of the FRFs. The approach is demonstrated using two examples; a simple beam containing uncertainty in Young's Modulus, and a full-scale aircraft composite wing model containing uncertainties in both Young's modulus and the shear modulus. The results were compared with Monte Carlo Simulation (MCS) and it was found that the parameter PDFs and FRF error bounds obtained using a 2nd-order PCE model agreed very well whilst requiring significantly less computation.     

Vibration testing based on impulse response excited by laser ablation

This paper proposes an innovative vibration testing method based on impulse response excited by laser ablation. In conventional vibration testing using an impulse hammer, high-frequency elements of over tens of kilohertz are barely present in the excitation force. A pulsed high-power YAG laser is used in this study for producing an ideal impulse force on a structural surface. Illuminating a point on a metal with the well-focused YAG laser, laser ablation is caused by generation of plasma on the metal. As a result, an ideal impulse excitation force generated by laser ablation is applied to the point on the structure. Therefore, it is possible to measure high-frequency FRFs due to the laser excitation. A water droplet overlay on the metal is used to adjust the force magnitude of laser excitation. An aluminum block that has nine natural frequencies below 40 kHz is employed as a test piece. The validity of the proposed method is verified by comparing the FRFs of the block obtained by the laser excitation, impulse hammer, and finite element analysis. Furthermore, the relationship between accuracy of FRF measurements and sensitivity of sensors is investigated.     

Multiple flux difference effect in the lattice hydrodynamic model

Considering the effect of multiple flux difference, an extended lattice model is proposed to improve the stability of traffic flow. The stability condition of the new model is obtained by using linear stability theory. The theoretical analysis result shows that considering the flux difference effect ahead can stabilize traffic flow. The nonlinear analysis is also conducted by using a reductive perturbation method. The modified KdV (mKdV) equation near the critical point is derived and the kink-antikink solution is obtained from the mKdV equation. Numerical simulation results show that the multiple flux difference effect can suppress the traffic jam considerably, which is in line with the analytical result.     

Nonlinear finite element model updating of an infilled frame based on identified time-varying modal parameters during an earthquake

A model updating methodology is proposed for calibration of nonlinear finite element (FE) models simulating the behavior of real-world complex civil structures subjected to seismic excitations. In the proposed methodology, parameters of hysteretic material models assigned to elements (or substructures) of a nonlinear FE model are updated by minimizing an objective function. The objective function used in this study is the misfit between the experimentally identified time-varying modal parameters of the structure and those of the FE model at selected time instances along the response time history. The time-varying modal parameters are estimated using the deterministic–stochastic subspace identification method which is an input–output system identification approach. The performance of the proposed updating method is evaluated through numerical and experimental applications on a large-scale three-story reinforced concrete frame with masonry infills. The test structure was subjected to seismic base excitations of increasing amplitude at a large outdoor shake-table. A nonlinear FE model of the test structure has been calibrated to match the time-varying modal parameters of the test structure identified from measured data during a seismic base excitation. The accuracy of the proposed nonlinear FE model updating procedure is quantified in numerical and experimental applications using different error metrics. The calibrated models predict the exact simulated response very accurately in the numerical application, while the updated models match the measured response reasonably well in the experimental application.     

Direct mode-shape expansion of a spatially incomplete measured mode by a hybrid-vector modification

A direct estimation method for expanding incomplete experimental mode shapes is presented. The approach adopts a hybrid vector which includes measured data at master degrees of freedom (dofs) and constant values at slave dofs. The constant values are refined by a set of mode-correction factors. Modelling errors between the analytical model and tested structure are also considered by introducing a series of model-correction factors. Initial-guess values of the mode-correction factors are used to decouple the coupled constructed equations, and an iterative technique for solving these equations is proposed. The results from a five-degree-of-freedom mass–spring system indicate that the proposed approach provided a better performance than the commonly used existing expansion methods and can reliably estimate unmeasured components of mode shapes, even in cases with limited modal measurements and severe measurement noise. The performance of the proposed method was also investigated using real measurements from a steel cantilever-beam experiment. Experimental data were measured by 20 accelerometers mounted at the cantilever beam: among these accelerometers, three of these were assumed to be measured, and the others were used to check the estimation accuracy of the proposed method. The results show that the unmeasured components in the mode shapes were properly estimated by implementing the proposed method, even for high-frequency modes.     

Active vibration control on a quarter-car for cancellation of road noise disturbance

In this paper, a methodology is presented for the cancellation of road noise, from the analysis of vibration transmission paths for an automotive suspension to the design of an active control system using inertial actuators on a suspension to reduce the vibrations transmitted to the chassis. First, experiments were conducted on a Chevrolet Epica LS automobile on a concrete test track to measure accelerations induced on the suspension by the road. These measurements were combined with experimental Frequency Response Functions (FRFs) measured on a quarter-car test bench to reconstruct an equivalent three dimensional force applied on the wheel hub. Second, FRFs measured on the test bench between the three-dimensional driving force and forces at each suspension/chassis linkage were used to characterize the different transmission paths of vibration energy to the chassis. Third, an experimental model of the suspension was constructed to simulate the configuration of the active control system, using the primary (disturbance) FRFs and secondary (control) FRFs also measured on the test bench. This model was used to optimize the configuration of the control actuators and to evaluate the required forces. Finally, a prototype of an active suspension was implemented and measurements were performed in order to assess the performance of the control approach. A 4.6 dB attenuation on transmitted forces was obtained in the 50–250 Hz range.     

Identification of the dynamic properties of joints using frequency–response functions

The dynamic properties of joints are extremely difficult to model accurately using a purely analytical approach. However, these properties can be extracted from experimental data. In this paper we present a method for establishing a theoretical model of a joint from the substructures and assembly frequency–response function (FRF) data. The identification process considers not only translational, but also rotational degrees of freedom (RDOFs). The validity of the proposed method is demonstrated numerically and experimentally. A combined numerical–experimental approach was used to identify the mass, stiffness and damping effects of a real bolted joint. Using the least-squares method, data from the wide frequency range were used. A substructure synthesis method with the joint effects included was used to check the extracted values.     

Non-equilibrium thermal entanglement for a three spin chain

The dynamics of a chain of three spins coupled at both ends to separate bosonic baths at different temperatures is studied. An exact analytical solution of the master equation in the Born-Markov approximation for the reduced density matrix of the chain is constructed. It is shown that for long times the reduced density matrix converges to the non-equilibrium steady state. Dynamical and steady state properties of the concurrence between the first and the last spin are studied.     

Mobility measurement in six DOFs applied to the hub of a car

Complex models are usually needed to predict functional performance of mechanical devices such as noise and vibration levels. A typical model is divided into substructures where each substructure is described either as a simulation or as an experimental model. When coupling substructures, information is needed that describes the boundary conditions at the attachment points of the various substructures. This can be hard to achieve, especially when full structural behavior including rotational degrees of freedom (DOFs) are needed. The objective of this study was to obtain the mechanical mobility matrix in six DOFs for a wheel hub of car, as an example of a complex mechanical structure, through an experimental approach based on the multiple-input multiple-output technique. Further, evaluation of the measurement quality was conducted without requirements for numerical simulation comparisons. A specially designed brake disc was fabricated for direct attachment of shakers and transducers. The quality of the 6-DOF mechanical mobility matrix was evaluated using reciprocities, coherence functions, and random error in the gain-factor estimates. The results showed good quality in the reciprocities, and the multiple coherences in each degree of freedom were close to one in the frequency range of 25–500 Hz. The random error can be reduced by increasing the number of averages.     

FINITE ELEMENT MODEL UPDATING USING ANTIRESONANT FREQUENCIES

This paper uses antiresonant frequencies in the finite element model updating of an experimental 6-m aluminum truss and analyzes the physical correctness of the updated model by using it to detect damage. Rigid elements are used to simplify the modelling of welded joints, and their dimensions are used as parameters in an iterative update based on eigenvalue and antiresonance sensitivities. An update using both natural frequencies and antiresonant frequencies is shown to produce a 48% better correlation to experimental frequency response functions (FRFs) than an update that uses only natural frequencies. The antiresonant updated model is used to predict FRFs for the truss in 112 damaged configurations. Pattern classification and curve-fit algorithms for damage detection are tested. The curve-fit method correctly identified damage 92·6% of the time compared to 76·1% for the pattern classifier. The high quality of the model is attributed to the use of rigid elements that are updated using antiresonant frequencies.     

Chaos synchronization and parameter estimation of single-degree-of-freedom oscillators via adaptive control

This paper addresses the problem of synchronizing a class of single-degree-of-freedom oscillators with uncertain parameters. A modified adaptive control scheme is proposed to achieve globally asymptotic stable synchronization between the master and slave oscillators with arbitrary different initial conditions based on the Barbalat's Lemma. One of the advantages of the method is that it requires only a scalar driving signal which may be easily designed in practical applications. On the other hand, the method can ensure that the unknown parameters in the slave oscillator would be completely estimated with an adaptive updating law. Numerical simulations are performed on two examples to verify the analytical results.     

Debonding detection of honeycomb sandwich structures using frequency response functions

A vibration-based non-destructive evaluation (NDE) method is proposed to determine the location and size of debonding in honeycomb sandwich beams. Although most of the existing vibration-based NDE methods need many measurement points, the method proposed here only utilizes the frequency response function (FRF) measured at one point. A parameterized damaged Timoshenko beam model is developed with the method of reverberation-ray matrix (MRRM) for the first time, and combined with the genetic algorithm (GA) to inverse the damage parameters from the measured FRF. The detection of a honeycomb sandwich beam can be divided into two steps: (1) identifying the equivalent elastic moduli and other parameters of the intact sandwich beam. (2) Identifying the debonding location and size of the damaged sandwich beam with the predetermined parameters. It is demonstrated experimentally that the method can inverse damage parameters with acceptable precision.     

Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Structural model updating using incomplete transfer function of strain data

In this paper a model updating algorithm is presented to estimate structural parameters at the element level utilizing frequency domain representation of the strain data. Sensitivity equations for mass and stiffness parameters estimation are derived using decomposed form of the strain-based transfer functions. The rate of changes of eigenvectors and a subset of measured natural frequencies are used to assemble the sensitivity equation of the strain-based transfer function. Solution of the derived sensitivity equations through the least square method resulted in a robust parameters estimation method. Numerical examples using simulated noise polluted data of 2D truss and frame models confirm that the proposed method is able to successfully update structural models even in the presence of mass modeling errors.     

Regularised finite element model updating using measured incomplete modal data

This paper presents an effective approach for directly updating finite element model from measured incomplete vibration modal data with regularised algorithms. The proposed method is based on the relationship between the perturbation of structural parameters such as stiffness change and the modal data measurements of the tested structure such as measured mode shape readings. In order to adjust structural parameters at detailed locations, structural updating parameters will be selected at critical point level to reflect the modelling errors at the connections of structural elements. These updating parameters are then evaluated by an iterative or a direct solution procedure, which gives optimised solutions in the least squares sense without requiring an optimisation technique. In order to reduce the influence of modal measurement uncertainty, the Tikhonov regularisation method incorporating the L-curve criterion is employed to produce reliable solutions for the chosen updating parameters. Numerical simulation investigations and experimental studies for the laboratory tested space steel frame structure are undertaken to verify the accuracy and effectiveness of the proposed methods for adjusting the stiffness at the joints of structural members. The results demonstrate that the proposed methods provide reliable estimates of finite element model updating using the measured incomplete modal data.     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势, 具有良好的工程应用前景.