Determination of the natural frequencies of axially moving beams by the method of multiple scales

The natural frequencies of an axially moving beam were determined by using the method of multiple scales. The method of second-order multiple scales could be directly applied to the governing equation if the axial motion of the beam is assumed to be small. It can be concluded that the natural frequencies affected by the axial motion are proportional to the square of the velocity of the axially moving beam. The results obtained by the perturbation method were compared with those given with a numerical method and the comparison shows the correctness of the multiple-scale method if the velocity is rather small.

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION ANDTHE ASYMPTOTICS OF SOLUTIONS (Ⅰ)

In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.

THE METHOD OF MULTIPLE SCALES APPLIED TO THE NONLINEAR STABILITY PROBLEM OF A TRUNCATED SHALLOW SPHERICAL SHELL OF VARIABLE THICKNESS WITH THE LARGE GEOMETRICAL PARAMETER

Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger, the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.

Outcome assessment of stroke convalescence using multiple scales based on different dimensions

BACKGROUND: Convalescence is an important stage of stroke treatment. A lot of patients have somatic and mental disorders at various degrees. The primary standard can only reflect partial conditions of somatic disorder; in addition, multiple dimensions of patients at the phase of stroke convalescence are further observed by using a lot of standards, such as signs and symptoms of traditional Chinese medicine, daily activity and psychological status. OBJECTIVE: To analyze the outcome assessments of the cases of stroke convalescence measured with different criteria consisting of various dimensions by a cross-sectional investigation of the condition of stroke convalescent patients. DESIGN: Scale evaluation. SETTING: Departments of Clinical Epidemiology Exploratory Development and Neurology, the Second Affiliated Hospital of Guangzhou University of Traditional Chinese Medicine; National Center for Training of Design, Measurement and Evaluation in Clinical Research,Guangzhou University of Traditional Chinese Medicine. PARTICIPANTS: A total of 194 stroke convalescent patients treated in the Second Affiliated Hospital of Guangzhou University of Traditional Chinese Medicine from July 26, 2000 to February 28, 2001 were taken as subjects of the study. There were 126 males and 68 females aged from 40 to 89 years, and the illness course ranged from 14 to 181 days. All patients met diagnosis-treatment criteria of stroke (the second version)[DTCS(V2.0)] and various kinds of diagnostic points of cerebrovascular diseases; moreover, all patients provided confirmed consents. METHODS: They were assessed by assessment methods including the following assessment instruments: DTCS(V2.0), self-designed scale of traditional Chinese medicine (TCM) symptoms (28 symptoms and physical signs were scored as 0, 1, 2 marks from none to severity), modified Edinburgh-Scandinavia stroke scale (a total of 45 marks, 0 to 15 marks as mild defect, 16 to 30 as moderate defect, 31 to 45 as severe defect), modified Barthel activities of daily life (ADL) index (a total of 100 marks, less than 60 marks as unable self-care), vitality and mental health (subscales derived from Health Survey Questionnaire, SF-36). The collected data from scales and inter-scale correlation were processed by the statistic methods mainly including descriptive analysis, Spearmen correlation analysis, factor analysis, etc. MAIN OUTCOME MEASURES: ① Average scores of scales and criteria; ② correlation between modified Edinburgh-Scandinavia stroke scale and other scales. RESULTS: All of the patients completed the assessment, and analyzed in the final analysis. ① The average scores of the scales and criteria: The average scores of DTCS(V2.0), self-designed scale of TCM symptoms, modified Edinburgh-Scandinavia stroke scale, modified Barthel ADL index, vitality and mental health scales were 6.51±6.29, 13.73±6.97, 7.56±7.35, 63.58±23.68, 52.79±23.32 and 62.83±22.75 respectively. ② Correlation between modified Edinburgh-Scandinavia stroke scale and other scales: The Spearman correlation coefficients (R ’) of modified Edinburgh-Scandinavia stroke scale with diagnosis-treatment criteria of stroke, scales of TCM symptoms, modified Barthel ADL index, vitality scale and mental health scale were 20.885, 0.302, -0.824, -0.294 and -0.258 respectively. CONCLUSION: The modified Edinburgh-Scandinavia stroke scale and DTCS(V2.0) shared the same assessment dimension, so they can be mutually alternated in some clinical practices. Discrepancy in measurements of health status was gained due to the diverse dimensions applied in outcome assessments. It is necessary to build up a multi-dimensional assessment criteria system, such as signs and symptoms, daily activities and psychological status, for assessing the stroke convalescent cases in a more comprehensive scope and reflecting the efficacy of TCM treatment scientifically.

Multiple scales method for analyzing a forced damped rotational pendulum oscillatorwithgallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces.The multiple scales method(MSM)is applied to solve the proposed problem.Several types of rotational pendulum oscillators are studied and talked about in detail.These include the forced damped rotating pendulum oscillator with gallows,the damped standard simple pendulum oscillator,and the damped rotating pendulum oscillator without gallows.The MSM first-order approximations for all the cases mentioned are derived in detail.The obtained results are illustrated with concrete numerical examples.The first-order MSM approximations are compared to the fourth-order Runge-Kutta(RK4)numerical approximations.Additionally,the maximum error is estimated for the first-order approximations obtained through the MSM,compared to the numerical approximations obtained by the RK4 method.Furthermore,we conducted a comparative analysis of the outcomes obtained by the used method(MSM)and He-MSM to ascertain their respective levels of precision.The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

COMPUTER COMPUTATION OF THE METHOD OF MULTIPLE SCALES-DIRICHLET PROBLEM FOR A CLASS OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .

Analytical approximations to a generalized forced damped complex Duffing oscillator:multiple scales method and KBM approach

In this investigation,some different approaches are implemented for analyzing a generalized forced damped complex Duffing oscillator,including the hybrid homotopy perturbation method(H-HPM),which is sometimes called the Krylov-Bogoliubov-Mitropolsky(KBM)method and the multiple scales method(MSM).All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling the original problem.All obtained approximations are discussed graphically using different numerical values to the relevant parameters.Moreover,all obtained approximate solutions are compared with the 4thorder Runge-Kutta(RK4)numerical approximation.The maximum residual distance error(MRDE)is also estimated,in order to verify the high accuracy of the obtained analytic approximations.

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.

MATHEMATICAL MODELING OF QUINTIC DUFFING EQUATION AND NUMERICAL SIMULATION USING MULTIPLE SCALES MODIFIED LINDSTEDT–POINCARE METHOD

A perturbation algorithm Multiple Scales Modified Lindstedt–Poincare(MSMLP),combination of method of Multiple Scales and modified Lindstedt–Poincare is proposed for the solution of Quintic Duffing equation which combines the advantages of both the methods.Solution obtained by the MSMLP method is compared with the Multiple Scales method and accurate closed form approximate solution of the Quintic Duffing equation.The proposed method produces better results for a wide range of amplitude values of oscillations and strong nonlinearities.Numerical simulation has been performed in MATHEMATICA 7.0.

An optimal strategy for coordinating and dispatching “source-load” in power system based on multiple time scales

Due to the phenomenon of abandoning wind power and photo voltage(PV)power in the“Three Northern Areas”in China,this paper presents an optimal strategy for coordinating and dispatching“source-load”in power system based on multiple time scales.On the basis of the analysis of the uncertainty of wind power and PV power as well as the characteristics of load side resource dispatching,the optimal model of coordinating and dispatching“source-load”in power system based on multiple time scales is established.It can simultaneously and effectively dispatch conventional generators,wind plant,PV power station,pumped-storage power station and load side resources by optimally using three time scales:day-ahead,intra-day and real-time.According to the latest predicted information of wind power,PV power and load,the original generation schedule can be rolled and amended by using the corresponding time scale.The effectiveness of the model can be verified by a real system.The simulation results show that the proposed model can make full use of“source-load”resources to improve the ability to consume wind power and PV power of the grid-connected system.

Free vibration of vibrating device coupling two pendulums using multiple time scales method

A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established using Lagrange's equation. In order to obtain approximate solution, multiple time scales method, one of perturbation technique, was applied. Cases of non-resonant and 1:1:2:2 internal resonant were discussed. In the non-resonant case, the validity of multiple time scales method is confirmed, comparing numerical results derived from fourth order Runge-Kutta method with analytical results derived from first order approximate expression. In the 1:1:2:2 internal resonant case, modal amplitudes of Aa1 and Ab2 increase, respectively, from 0.38 to 0.63 and from 0.19 to 0.32, while the corresponding frequencies have an increase of almost 1.6 times with changes of initial conditions, indicating the existence of typical nonlinear phenomenon. In addition, the chaotic motion is found under this condition.

Multiple time scales of fluvial processes—theory and applications

Fluvial processes comprise water flow,sediment transport and bed evolution,which normally feature distinct time scales.The time scales of sediment transport and bed deformation relative to the flow essentially measure how fast sediment transport adapts to capacity region in line with local flow scenario and the bed deforms in comparison with the flow,which literally dictates if a capacity based and/or decoupled model is justified.This paper synthesizes the recently developed multiscale theory for sediment-laden flows over erodible bed,with bed load and suspended load transport,respectively.It is unravelled that bed load transport can adapt to capacity sufficiently rapidly even under highly unsteady flows and thus a capacity model is mostly applicable,whereas a non-capacity model is critical for suspended sediment because of the lower rate of adaptation to capacity.Physically coupled modelling is critical for fluvial processes characterized by rapid bed variation.Applications are outlined on very active bed load sediment transported by flash floods and landslide dam break floods.

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method,particle methods,and other spatial methods on a single body sub-dividedintomultiple subdomains.This is in conjunctionwithimplementing thewell known Generalized Single Step Single Solve(GS4)family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework.In the current state of technology,the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as theNewmarkmethods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier.However,the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses,overcomes such barriers and allows a wide variety of arbitrary implicit-implicit,implicit-explicit,and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement,velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains.By selecting an appropriate spatialmethod and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body,the proposed work has the potential to better capture the physics of a given simulation.The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility.The accuracy and efficacy of the present work have not yet been seen in the current field,and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses:discrete and continuum models

The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses.The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix.The inclusions are idealized by lumped masses,and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses.Additionally,the model is capable of depicting the wave propagation in bi-material bars,wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses.The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered,and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method(LPM).Next,a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique.The subsequent use of the second-order method of multiple scales(MMS)facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form.The novelties of the present study consist of(i)considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains;(ii)developing the second-order LPM for the wave propagation in the discrete chains;and(iii)deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses.Finally,a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models.These parameters include the ratio of the spring mass to the lumped mass,the nonlinear stiffness coefficient of the spring,and the wave amplitude.

Multiple Resonance and Stability of a Motor-elastic Linkage Mechanism System

The dynamics of a three-phase AC motor-elastic linkage mechanism system is considered. Taking the drive motor and the linkage mechanism as an integrated system, the coupling dynamic equations of the system are established by the finite element method. The multiple resonance and its stability of the system are studied using the method of multiple scales. The first order approximate solutions of the multiple resonance of the system are obtained. An algorithm for determining the stability of resonance is derived. The studies show that the multiple resonance and its stability of the system are not only related to the structure parameters of the linkage mechanism, but also to the electromagnetism parameters of the motor. At last, an experiment is given to verify the results of the theoretical analysis.

Multi-Time Scale Operation and Simulation Strategy of the Park Based on Model Predictive Control

Due to the impact of source-load prediction power errors and uncertainties,the actual operation of the park will have a wide range of fluctuations compared with the expected state,resulting in its inability to achieve the expected economy.This paper constructs an operating simulation model of the park power grid operation considering demand response and proposes a multi-time scale operating simulation method that combines day-ahead optimization and model predictive control(MPC).In the day-ahead stage,an operating simulation plan that comprehensively considers the user’s side comfort and operating costs is proposed with a long-term time scale of 15 min.In order to cope with power fluctuations of photovoltaic,wind turbine and conventional load,MPC is used to track and roll correct the day-ahead operating simulation plan in the intra-day stage to meet the actual operating operation status of the park.Finally,the validity and economy of the operating simulation strategy are verified through the analysis of arithmetic examples.

Multiple Time Scale Analysis of River Runoff Using Wavelet Transform for Dagujia River Basin, Yantai, China

Based on monthly river runoff and meteorological data, a method of Morlet wavelet transform was used to analyze the multiple time scale characteristics of river runoff in the Dagujia River Basin, Yantai City, Shandong Province. The results showed that the total annual river runoffin the Dagujia River Basin decreased significantly from 1966 to 2004, and the rate of decrease was 48× 10^6ma/10yr, which was higher than the mean value of most rivers in China. Multiple time scale characteristics existed, which accounted for different aspects of the changes in annual river runoff, and the major periods of the runofftime series were identified as about 28 years, 14 years and 4 years with decreasing levels of fluctuation. The river runoff evolution process was controlled by changes in precipitation to a certain extent, but it was also greatly influenced by human activities. Also, for different time periods and scales, the impacts of climate changes and human activities on annual river runoff evolution occurred at the same time. Changes in the annual river runoffwere mainly associated with climate change before the 1980s and with human activities after 1981.

Assessing the quality of chlorophyll-a concentration products under multiple spatial and temporal scales

Abstract The chlorophyll-a concentration data obtained through remote sensing are important for a wide range of scientific concerns.However,cloud cover and limitations of inversion algorithms of chlorophyll-a concentration lead to data loss,which critically limits studying the mechanism of spatial-temporal patterns of chlorophyll-a concentration in response to marine environment changes.If the commonly used operational chlorophyll-a concentration products can offer the best data coverage frequency,highest accuracy,best applicability,and greatest robustness at different scales remains debatable to date.Therefore,in the present study,four commonly used operational multi-sensor multi-algorithm fusion products were compared and subjected to validation based on statistical analysis using the available data measured at multiple spatial and temporal scales.The experimental results revealed that in terms of spatial distribution,the chlorophyll-a concentration products generated by averaging method(Chl1-AV/AVW)and GSM model(Chl1-GSM)presented a relatively high data coverage frequency in Case Ⅰ water regions and extremely low or no data coverage frequency in the estuarine coastal zone regions and inland water regions,the chlorophyll-a concentration products generated by the Neural Network algorithm(Chl2)presented high data coverage frequency in the estuarine coastal zone Case 2 water regions.The chlorophyll-a concentration products generated by the OC5 algorithm(ChlOC5)presented high data coverage frequency in Case I water regions and the turbid Case Ⅱ water regions.In terms of absolute precision,the Chl1-AV/AVW and Chl1-GSM chlorophyll-a concentration products performed better in Class I water regions,and the Chl2 product performed well only in Case Ⅱ estuarine coastal zones,while presenting large errors in absolute precision in the Case Ⅰ water regions.The ChlOC5 product presented a higher precision in Case Ⅰ and Case Ⅱ water regions,with a better and more stable performance in both regions compared to the other products.

The Biorthogonality of Multiple Vector-valued Bivariate Wavelet Packets

The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.

MULTIPLE TIME SCALE ANALYSIS OF CLIMATE VARIATION IN Macao DURING THE LAST 100 YEARS

The multiple time scale climate changes are studied and calculated with statistical analysis and wavelet transformation on the basis of daily series of observed data over the period 1901-2007 in Macao.The result shows that statistically significant oscillations with 2 to 5 years of period generally exist in the series of climate variables(e.g.annual mean surface air temperature and precipitation as well as evaporation etc.),but with obvious locality in time domain.The variation of annual mean surface air temperature has a quasi 60-year period.The phases of the 60-year variation approximately and consistently match that of Atlantic Multidecadal Oscillation(AMO).The oscillations of seasonal mean surface air temperature in summer and winter have the periods of quasi 30-year and quasi 60-year,respectively.These two periods of oscillations have statistically significant correlation with Pacific decadal oscillation(PDO) and AMO,individually.The multidecadal variations of the precipitation of the annually first flood period and annual evaporation are dominated by periods of quasi 30-year and quasi 50-year,respectively.