Lamb waves received great attention in structural health monitoring (SHM), but it is complicated by the multi-mode character and the dispersion effect, which impacts the damage positioning and high-resolution imaging. Warped frequency transform(WFT) can warp the frequency axis by setting up a reasonable warped map. Warped frequency transform based on the group velocity dispersion curves of Lamb waves can be employed to process Lamb waves. Based on this direct dispersion compensation idea, WFT is adopted here to process the measured signals to suppress dispersion. A new method of high-resolution damage imaging is proposed based on WFT, and the propagation of Lamb waves in damaged aluminum pate is simulated by finite element software ABAQUS. Results show that WFT can effectively compress dispersed wave-packets, and high-resolution damage imaging can be obtained by the proposed method.
A method was presented based on 1.5-dimensional energy spectrum. This method combines 1.5-dimensional spectrum and teager energy operator. To solve the fault diagnosis problem of rolling bearings and improve the signal to noise ratio, a new diagnosis method based on EEMD de-noising and 1.5-dimensional energy spectrum was proposed. Firstly, the fault signal was decomposed by ensemble empirical mode decomposition method and a set of intrinsic mode function components were obtained. Secondly, filtered the components by correlation coefficient - kurtosis criteria and the selected IMF components were used to reconstruct signal. Finally, in order to get bearing fault characteristic frequency information, the 1.5-dimensional energy spectrum of reconstructed signal was calculated. Simulation inner fault signal and measured inner fault signal of rolling bearings were analyzed by the proposed method, and the diagnosis results were satisfactory.
The simultaneous field measurement on wind speed, wind direction and wind-induced acceleration response were conducted on the super tall building-Litong plaza in Guangzhou when typhoon Vicente affected this building. The wind filed characteristics atop of this building were obtained through statistical analysis. Meanwhile the frequency-spectrum characteristics of wind-induced response, the dynamics characteristic were identified from the measured wind-induced acceleration data. The nonlinear damping ratio was also obtained by the random decrement technique. On the other hand the displacements atop of the building were previously measured by GPS system under ambient condition. The multi-path effect of the measured displacement was investigated and its removing method was studied in this paper. The comparisons of the obtained natural frequencies from the different methods were conducted. The effectiveness of the identified natural frequencies was therefore verified by the compared results.
Considering the nonlinearity and uncertainty of the high-speed flexible parallel mechanism, a robust model predictive control strategy is presented to suppress vibration response of the system. The actuator is piezoceramic patch, the sensor is strain gauge. The finite element method and modal truncation technique are applied to obtain the dynamic model of the mechanism. The nonlinear, coupling factors and high-order modal effects existed in the dynamic model of the system are considered as disturbance. The modal force is treated as uncertain disturbance, and the measurement noise is considered, the predictive model of the system dynamic response is obtained, and the future output is obtained from the model. The Kalman filter estimator is designed to estimate the state variables of the system. A standard quadratic programming optimization problem is formed where the performance index function minimizes a quadratic performance function. The constraints are the control input voltage and its change rate. The optimization problem is solved to obtain the optimal control output voltages. The rolling optimization control system is constructed to suppress the vibration response of the system. The optimum placements of the actuators and sensors are determined by using the controllability index featured as actuating energy and the observability index featured as signal energy. With a novel 2-DoF parallel mechanism as an example, the experimental modal method is applied to obtain the first two natural frequency and damping ratio. And comparing with the results obtained by the finite element method, it shows that the model is not accurate. Using dSPACE real-time simulation system and MATLAB/Simulink, the control system is built based on the model and the vibration control experimental study is carried out. The experimental results showed the proposed controller can effectively suppress the vibration response, and effectiveness and robustness of the controller is verified.
The period and its property of a Velocity-Dependent-Frequency nonlinear oscillator with cubic restoring force has not been discussed in literatures, what’s more, the traditional standard methods, i.e. perturbation methods or harmonic balance methods for determining approximations to the periodic solutions for this kind of oscillators may breakdown in their corresponding first-order calculations. In particular, the frequencies become singular for finite values of the amplitude. This paper firstly gives the exact period expression of this oscillator; based on the integrability condition of period integral expression the exact solution of special class of initial conditions for the oscillator is obtained by using harmonic balance method. The property of the period of the oscillator is studied, and the approximate analytical expression of the period is given by the complete elliptic integral of the first kind, and then the phenomenon and reason of somewhat like the square waves of the oscillation are analyzed. It shows that the period of the oscillator decays to zero finally when the amplitude increases to infinity, and the reason of square waves is the parameter of the system , the square waves will be squarer when the amplitude increases. At last, a Hermite interpolation method is presented for the periodic solution of the oscillator, the time variable is transformed into a new harmonically oscillating time of which the frequency is one-half the one of the oscillator, with the corresponding governing differential equation transforms into a form suitable for Hermite interpolation analysis. The solutions are compared with the numerical solution and results show good agreement.
In order to diagnose fault effectively by using vibration signal, a bearing fault diagnosis model based on neighborhood adaptive locality preserving projections was proposed. Firstly, the bearing vibration signal was decomposed into several smooth intrinsic mode functions (IMFs) by EMD and the auto-regressive (AR) model of IMF was established to construct original characteristic subset. Then, the algorithm of neighborhood adaptive locality preserving projections was used to reduce the dimension of original characteristic subset to gain low-dimension eigenvectors and projection matrix. The best reduced dimension and the best corresponding projection matrix could be determined by studying the relationship between fault recognition rate and dimension of the low-dimension eigenspace with the low-dimension eigenvectors as inputs and least square support vector machine(LS-SVM)as classifier. At last Low-dimension eigenvectors converted from original characteristic subset based on the best reduced dimension were put into LS-SVM for recognizing the conditions and fault states of bearing. Results of the test indicated that the proposed model diagnosed bearing fault with high accuracy.
A pedestrian bridge suspended to a vehicular bridge to form a combined bridge, can provide passages not only for vehicles but pedestrians. However, the vibration comfort of the suspended pedestrian bridge under vehicle excitation from the vehicular bridge still needs to be discovered. In this paper, fundamental frequency estimation formulas applied to the suspended pedestrian bridge are proposed and verified through finite element method. Results show that fundamental frequencies of the suspended pedestrian bridge depend largely on dynamic characteristics of the vehicular bridge. The vehicle induced vibration of a long-span suspended pedestrian bridge is analyzed through separated iteration methods. Results show that the vibration responses become larger with worse road condition or higher density traffic flow. The vibration responses have no evident relationship with vehicle speed. The vertical acceleration peak of the pedestrian bridge in mid-span under vehicle excitation exceeds half of that excited by pedestrian. Therefore, the vehicle induced vibration effect should be considered in the comfort evaluation of pedestrian bridges suspended to vehicular bridges, which will decrease the pedestrian comfort.
Aiming at the aero-engine fault, an abrupt fault diagnosis method based on affinity propagation clustering was proposed. Firstly abrupt fault historical monitoring data are used to establish faults database, then through affinity propagation clustering to find all exemplars of abrupt faults in the database, when need to diagnose new test data, through affinity propagation clustering to find the exemplar of the new collected data, and we can obtain the fault type by center matching with the centers obtained from the faults database. After applied this fault diagnosis method on the aero-engine abrupt fault diagnosis, the simulation and experiment results show that this method can diagnose abrupt fault, and compared with other methods, it took shorter time and produced lower error.
In this paper, rubber test data were fitted with hyperelastic models in ABAQUS finite-strain package. On the basis of the fitting results, a proper model was chosen. Furthermore, dynamic response of chiral honeycomb rubber claddings subjected to UNDEX has been investigated, and the influence of main design parameters including height, porosity and panel material on shock resistance was discussed. It is found that as cladding’s height, porosity and panel wave impedance increase, its shock resistance and buffering effect will be improved. The results show not only play an instructive role in understanding the protective mechanism and performance of chiral honeycomb claddings subjected to underwater explosion, but also provide a guideline to further optimal design.
In order to accurately describe the dynamic model of hydraulic excavator’s mechanical arm , the modal functions are adopted to describe elastic deformation of the mechanical arm ,Lagrange theorem and the principle of virtual work is used to establish rigid-flexible coupling nonlinear dynamic equations of arm frame system. The dynamic equations are numerically solved by MATLAB, the rigid-flexible coupling model of hydraulic excavator’s mechanical arm is established and simulated by the simulation software ADAMS and NASTRAN , it is showed that the modeling method of dynamic equations adopted in this paper is correct by contrasting the both results. The modal and dynamic response is calculated by applying the numerical solution method ,the sensitivity of the first natural frequency for the Geometric parameters related is solved, the main modal parameters are analyzed which influence dynamic characteristics of mechanical arm, the basis is provided for the further optimization of excavator structure and motion error control.
According to the various components characteristics of the bridge deflection, the non-linear relationship of temperature and the temperature deflection effect was established. In order to improve the regressive ability to fit the temperature deflection effect, a multiple least square support vector machine (M-LS-SVM) regressive model was presented. The subtractive clustering was adopted to divide the input space into several sub-spaces, and sub-models were built by LS-SVM in every sub-space. In order to minimize the severe correlation among sub-models and to improve the accuracy and robustness of the model, the sub-models were combined by principal components regression (PCR). Experimental and analytical results show that the method can separate the temperature effect from deflection monitoring signals and provide the basis data for damage detection from the long-term monitoring signals.
Bearing fault characteristic frequencies contain weak energy when the early fault occurs, and they are often overwhelmed in noise. According to the research results on the fault mechanism of rolling bearings, the demodulation of bearing resonance frequency is helpful in improving SNR and extracting the characteristic frequencies. Based on time-frequency analysis and information entropy, Frequency Band Entropy (FBE) is introduced to find the bearing resonance frequency. At first, the Short Time Fourier Transform (STFT) is utilized to calculate the time-frequency distribution of signal. Then the amplitude spectrum entropy of every frequency is estimated, which discover the complexity of every frequencies changing with time. This property provides a way of blindly designing of optimal filter parameters. The bearing characteristic frequencies can be extracted effectively from the signal after filtering. The simulation and experiment verify the validity of the proposed method.
Combined with comprehensive test of the pilot section of the Beijing-Shanghai high-speed railway and focused on the structure key parts of the both ends of the end thorns structure which adjacent to the road and bridge structure connected position,the vibration characteristics test on CRTSⅡ slab ballastless track bridge-subgrade transition were developed. The vibration response of spatial change characteristics along the road longitudinal and vertical direction and the relationship between the speed of train and the vibration response are analyzed.Test study show as follows:①Along the road longitudinal direction the maximum value of vibration response is located at the junction of the end thorns structure transition board side and the subgrade bearing layer,showing that the support stiffness of the former is less than the latter and reflecting that the end thorn structure transition board are not preferably achieve gradually transition from high to low of the stiffness;②On the vertical multi-storey structure layers the vibration response is relatively large when the horizontal direction structure is discontinuity and show a decreasing trend along the depth,but the vibration characteristics is significantly improved when the track slabs are longitudinal connected;③With the train speed increased the vibration displacement show a linear increase while the vibration velocity and vibration acceleration show a nonlinear accelerated increase relationship.
missile-launcher coupling and system dynamic response for bare missile vertical thermal launch were mainly researched. Simplifying and equivalent modeling were discussed for track-based missile system. The theory analysis of miss
ile taking off was discussed. The rigid-flexible coupling multi-body dynamics model was created based on dynamics and finite software. The dynamics simulation and analysis were carried on for missile launch. The results were compared with result of the two kind models. The numerical simulation results indicate that the coupling is obvious between the missile and launcher during missile taking off. The vibration of system has influence on the taking off attitude. Vibration damping characteristics of launchers are suitable. The rigid-flexible coupling model could reflect the dynamic performance of missile launch well compared with the rigid model.
Key words: vertical thermal launch; coupling; taking off attitude; dynamics simulation
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Orifice-type viscous dampers exhibit a hysteresis behavior which is explained by additional stiffness. When mixed with air, its efficiency and additional stiffness would change. To study the dynamic characteristics of the damper mixed with air, calculated the nonlinear damping force and stiffness in a series model. A linear model in parallel which was convenient in practical application was established based on equivalent damping and stiffness, then researched on the efficiency and additional stiffness of dampers mixed with air in different frequencies and amplitudes. Experimental results of viscous dampers imply this method and model is correct.
The complex modal approximate analysis is developed to the vibration of foundation beams, studying transverse vibrations of three-parametric viscoelastic Pasternak foundation beams. Complex frequency equations and modal function expressions are obtained for clamped-clamped,clamped-pinned and free-free ends. The numerical methods are employed to approximately solve the transcendental equations, and the differential quadrature scheme is applied to verify the approximate result. The effect of the boundary conditions, the stiffness and the viscoelastic coefficients on natural frequencies and modal functions are analyzed in numerical examples. The numerical results obtained with the differential quadrature and the complex mode methods are in good agreement.
Because natural frequency of the rope-load oscillation system affects the stability and performance of the control system, this paper proposes a simple robust controller to achieve this problem. Firstly, a linear dynamic model of a rotary crane is derived by using a disturbance observer. The model has robustness with respect to varying parameters such as joint friction, load mass and velocity of horizontal boom motion. Next, a state feedback controller with integrator is designed based on the model, and controller gains are determined by using linear matrix inequality (LMI) optimization for achieving robustness with respect to rope length variance. Finally, comparative simulations and experimental results demonstrate the effectiveness of the proposed method. Therefore, the crane can be easily operated without sensor systems for measuring rope length, consequently, the structure of the crane can be simplified and implementation cost can be reduced.
The metal corrugated core sandwich panels with superior mechanical performance can be used as crashworthiness and protective structures. Take the V-type corrugated core sandwich panel for example; Analysis of the deformation mode and curves of crushing force versus displacement are made by quasi-static crushing experiment. At the same time, the lateral crushing mechanical behavior and performance of energy absorption is taken with the finite element software Abaqus. The parameters of crushing force, deformation mode, energy absorption and mean crushing strength are compared. The results show that the crushing performance obtained from the experiment is consistent with that from numerical simulation. The deformation mode of the sandwich core structures has great effect on the crushing performance. The specific energy and mean crushing strength of mode I is better than that of mode II.
The Free Hanging Catenary shape and Lazy-Wave shape are designed for an deepwater umbilical based on the theoretical equations of the two kinds of shapes. The Static and dynamic responses of this umbilical under both shapes are calculated, as well as the distribution characteristics of effect tension and curvature. The results show that the maximum effect tensions all happen at the top hang points of the Free Hanging Catenary shape and Lazy-Wave shape, but the maximum curvatures happen at the TDP of the Free Hanging Catenary shape and the lift section’s endpoint of the Lazy-Wave shape. In addition,the Lazy-Wave shape can reduce the effect tension and the dynamic interference to the umbilical caused by the motion of the top float and the wave.
Based on the differential quadrature method, the transverse vibration and stability concerning of spinning circular plate is investigated. According to the plane stress problem and Kirchhoff’s thin plate theory, the middle internal forces of axisymmetric spinning circular plate with homogeneity and constant thickness are obtained under the action of linearly distributed normal inertial force along the radial direction of the spinning circular plate, and the differential equation of motion with variable coefficients for the axisymmetric spinning circular plate is established in the form of polar coordinates. The differential quadrature method is employed to discretize the differential equation with variable coefficients and corresponding to some bound-ary conditions. Through numerical calculation for spinning circular plate with simply supported edge, clamped edge (non-radial displacement at the external boundary) and completely free edge, the real parts and imaginary parts of the first two order complex frequency versus the angular speed are analyzed, and their sta-bility types and corresponding to critical angular speed are obtained respectively.
According to the “code for design of concrete structures” (GB 50010-2010), the user subroutine VUMAT that can be implemented in the advanced finite element program ABAQUS is coded, in which the strain rate effect of material can be considered. The simulated results of a reinforced concrete column under static and dynamic loading modes are compared with experimental results, which proves the reliablity of the coded subroutine. Incremental dynamic analysis of a four-storey reinforced concrete frame structure is conducted to study the effects of strain rate on the maximum top displacement, maximum base shear, maximum storey drift, capacity curve, collapse-resistant capacity and collapse mode. The results show that the maximum top displacement and maximum storey drift decrease after considering strain rate effect; the maximum storey drift is more sensitive to strain rate than the maximum top displacement; As the intensity of earthquake increases, the influence of strain rate to the structural response is more obvious; the collapse process is basically unchanged after considering strain rate effect, but the collapse-resistant capacity increases obviously.
Aiming at defects of different denoising methods, a minimum description length denoising method based on the improved normalized maximum likelihood (INML) was proposed. This method encodes the model class and adding to the original code length and adaptively defines the threshold in signal denoising. Numerical simulation signals and experimental signals of rolling element bearings were used to test and compare the performances of the proposed method with VisuShrink, BayesShrink and RNML etc. The results showed that the INML-based noise cancellation method had more effective denoising performance and higher SNR, not only in eliminating random noise, but in preserving interested components of the signals. INML method enriched the theory of MDL denoising and enhanced its performance.
A very large floating oil storage vessel which is the main structure of ocean oil reserve is the Very Large Floating Structure (VLFS), thus hydroelastic response of the oil storage vessel is an important part of research work. By using of sine mode shape function, first the radiation potential and hydroelastic response was expressed in terms of the modal superposition method. Then the velocity potential and fluid stress was solved on body surface of a quarter model according to the structure symmetry by means of the Green function method based on 3D potential theory. Finally, hydroelastic response of the oil storage vessel, deflection of the vessel bottom and bending moment were calculated by use of the hydroelastic equation of motion. Hydrodynamic parameters also analyzed with the water depth and wavelength, which provide a method to future design and work.
Continuous contact force model is built in order to investigate joint clearance effects on the dynamics of two-axis hydraulic vibration test system. Test system with clearance joint is imported into the software ADAMS for dynamic simulation, and an experimental set-up is designed and built to achieve some experimental validations under the simple harmonic excitation inputs with different phase, results indicate that coupling vibration effect between two exciters leads to the apparent fluctuation of steady response of accelerations in the case of certain clearance size and excitations out of phase, meanwhile, rapid increase of peak acceleration happens when the excitation frequency and excitation amplitude increase. Thus, selection of clearance size in revolute joint for two-axis hydraulic vibration test system has a critical role in eliminating the non-linear influence。
Residual displacement is one of the most important parameters to assess the repairability and usability of bridge structures after an earthquake, while the P△; effect has significant influences on the residual displacement of bridge structures. Probabilstic characteristics and empirical prediction equations of residual displacements for bridge structures were quantificationally investigated based on large numbers of nonlinear time history analyses by taking the P-△; effect and randomness of strong ground motion into account. The governing differential equation of motion for bridge structures including the P△; effect was developed firstly based on the dynamic balance theory. The influences of the P-△; effect, the post-yield stiffness ratio, the normalized yield strength and the natural period of vibration on the statistical characteristics of residual displacement were quantificationally investigated based on the inelastic dynamic responses of bridge structures under 69 selected strong ground motions. The correlation between the peak and residual displacements was discussed and the prediction equation for the residual-to-peak displacement ratio was also proposed. The results show that the P-△; effect, the post-yield stiffness ratio and the normalized yield strength have significant impact on the mean of residual displacement; the residual displacement can be described as the Lognormal or the Weibull distribution variable; the correation between the residual and peak displacements was remarkable and the mean (dispersion) of the residual-to-peak displacement ratio increases (decreases) with the increase of the natural period of vibration and the stability factor.
Based on multi-fractal theory, a method of on-line condition monitoring of tool wear was presented. The generalized fractal dimensions of acoustic emission (AE) signals in cutting process were calculated using box-counting method. The generalized dimension spectrums of AE signals to different tool wear condition were gained, and the relation between tool wear condition and generalized dimensions was analyzed. The feature distances and correlation coefficients of generalized dimensions of AE signals were calculated. The classification for tool wear condition was made through comparing the values of correlation coefficients of generalized dimensions. The experimental results show that the method can be used effectively for on-line condition monitoring of tool wear.
When a gear has a local fault, the vibration signals of gear may contain amplitude and phase modulation. Their spectrums contain meshing frequencies, harmonics, and coupling frequencies generated by modulation. According to the characteristics of vibration signals from the gears with faults, a feature extraction method based on the envelope and time-frequency image of S transformation was proposed. Firstly, vibration signals from gears with differ worn degrees were collected by fault simulation in an experimental setup. Then, envelopes were obtained by Hilbert transform from vibration signals and envelopes of time-frequency contour maps were achieved by S transformation. Finally the features were extracted by calculating statistic parameters based on the grey-scale matrixes (GLCMs) of the maps. The result shows that the proposed method can effectively extract gear fault feature.
Impact sound synthesis method is given based on a ball-plate collision model, to synthesize impact sounds coming from arbitrary material plates. The impact sound synthesis method for any virtual material is given in the two-dimensional perception space, through interpolation functions among aluminum, glass and wood. Two subjective evaluation experiments are designed and carried out in this study. The effect of the using of synthetic sound and recording for material identification is studied in Experiment 1. The sound continua are synthesized with varied material properties, and corresponding material control strategies are evaluated by Experiment 2. The results show that listeners can identify materials of plates better using synthetic sounds in comparison to recordings, and the material identification results has the same variation trend with damping, thus the validity of the impact sound synthesis method is confirmed.
On the basic of dynamic model of vehicle-bridge coupled vibration, the influencial parameters on the impact factor of each member in butterfly-shaped arch bridge were studied by the self-compiling computing program, such as velocity of vehicle, distance of eccentric load, structural damping, road roughness, traffic direction, wheelbase, vehicle-to-vehicle distance, vehicle number and structural design parameters. The results show that: (1) as a whole, the impact factors of main cable forces will increase with vehicle velocity; (2) the impact factor of transverse displacement at vault of major arch ribs on eccentric side will increase with the increment of the distance of eccentric load, however, the impact factor of vertical displacement will decrease rapidly; (3) the increment of the structure damping will make impact effect weaken; (4) the impact factor varies along with the grade of road roughness; (5) there is great difference of the impact factors on the symmetric members when the driving direction is different; (6) there is an optimum rise/span ratio that minimizes the impact factor of mid-span deflection; (7) the impact factor will be reduced obviously when the nielsen system is used.
It has been calculated the hydrodynamic pressures of gravity dams subjected to vertical earthquake component in the frame of the Scaled Boundary Finite Element Method (SBFEM) considered the water compressibility and the absorption of the reservoir bottom, and it has given the solution of gravity dam-reservoir interaction not only in the frequency domain but also in the time domain. Numerical examples show the proposed method is capable of the high degree of accuracy. It is concluded that the hydrodynamic pressures caused by vertical ground motion are of special important analysis of gravity dams under the prescribed assumptions. Besides, it also has been studied the hydrodynamic effects of water compressibility and the absorption of the bottom.
Since permutation entropy (PE) algorithm can better magnify tiny change of a time series data, is computed simply and shows good quality in real-time application, which has given us a good application prospect in sudden change detection of a signal, but whose parameters that are embedding dimension and delay time are still determined by experience or trial, this problem has been a bottle-neck of the PE algorithm for engineering application. For theory of the PE algorithm, method based on reconstructing optimal phase space of time series is put forward to determine model parameters. Considering to two views on phase space reconstruction, basic theories of independent and joint determination methods are introduced to determine delay time and embedding dimension. Then validation and comparison of methods are carried out by simulated signal and whole life data of rolling bearing, it is concluded that the independent determination of model parameters was better than joint determination for abnormality detection.
The system modal parameters identification under ambient excitation becomes the focus of the structural health monitoring. The improved ITD method was put forward based on stochastic subspace method considering ITD method has worse identification accuracy. Through the agency of four-story steel frame model experiment,it was proved that the improved ITD method was more accurate for the model of damping ratio and frequency identification than the ITD method. The improved ITD method could be applied to structural modal parameter identification under ambient excitation. Compared with the SSI method, the precision has not reduced, simultaneously reduced the computing time, It could provide the possibility for the application of this method to real-time monitoring of the structures.
The technology of operation response analysis and laser scanning vibrometer with type of PSV-400-B are used to get the operation response parameters of train automatic door, including frequency and vibration mode.The experimental program and the system diagram, which are for the test to obtain the operation response parameters under GB/T 21563-2008 simulating long-life random vibration environment, are offered. The accuracy of the laser scanning vibrometer is proved. Three kinds of scanning measuring point arrangement for the type of train automatic door, which is ①door including glass window, ②door not including glass window, ③glass window only, are selected. The 1st operation response parameters, including operation response frequency and its vibration mode, are obtained from the three kinds of scanning objects.
Some improvements and parallel computing studies were carried out with the Jacobi-Davidson(J-D) method. Some strategies, such as the spectral transformation technique, restart and deflation techniques, were added and integrated with the J-D method to make it suitable to the large-scale modal analysis. A parallel modal solving system based on PANDA framework was created using the improved J-D algorithm and various numerical software packages. Utilizing the solving system and parallel computers, the parallel scalability of the J-D algorithm was studied via numbers of tests on an engineering structure. The maximum computing scale is over 10 million degrees of freedom, and the maximum numbers of parallel CPU processors attain 128. The influences of inner iteration steps and numbers of restarted vectors on the convergence velocity of outer iterations were studied, and the speedup curves for different scales were obtained. The results show that the improved J-D method is competent for the large-scale modal analysis, the memory costs increase linearly with the computing scales and only 39.4 GB of memory is needed for the modal analysis of 10.25 million scales. Also, the improved J-D method takes on an excellent parallel scalability that the speedup curves are almost linear within 128 testing processors and the curve is gradually close to the ideal speedup one as the accretion of computing scales. The parallel efficiency of 10.25 million scales at 128 processors attain 88.1 %.
The deducible process was given to solve the probability distribution density associated with an individual asperity contact spot of area a. The probability distribution density is applicable to the mechanical processing rough surface in spite of grinding, milling or turning. The approach of computing the fractal dimension and fractal roughness was embodied from the structure function of whole flexible joint surface. The theoretical computation and analysis from reference [1] were quantitatively experimentally validated. The experimental evidence displays that under a certain condition of normal loading heavy object, the absolute warp between the theoretical static friction coefficients and experimental ones distributes from -0.03302 to 0.01926, and the relative windage between them fluctuates from -8.323% to 5.512%.