Based on the characteristics of membrane structures and the air influence factors,this paper presented a method to simulate the air aerodynamic force effects including the added air mass,the acoustic radiation damping...Based on the characteristics of membrane structures and the air influence factors,this paper presented a method to simulate the air aerodynamic force effects including the added air mass,the acoustic radiation damping and the pneumatic stiffness.The infinite air was modeled using the acoustic fluid element of commercial FE software and the finite element membrane roof models were coupled with fluid models.A comparison between the results obtained by FE computation and those obtained by the vibration experiment for a cable-membrane verified the validity of the method.Furthermore,applying the method to a flat membrane roof structure and using its wind tunnel test results,the analysis of nonlinear wind-induced dynamic responses for such geometrically nonlinear roofs,including the roof-air coupled model was performed.The result shows that the air has large influence on vibrating membrane roofs according to results of comparing the nodal time-history displacements,accelerations and stress of the two different cases.Meantime,numerical studies show that the method developed can successfully solve the nonlinear wind-induced dynamic response of the membrane roof with aerodynamic effects.展开更多
Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they a...Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.展开更多
The dynamic influence of joints in aero-engine rotor systems is investigated in this paper.Firstly,the tangential stiffness and loss factor are obtained from an isolated lap joint setup with dynamic excitation experim...The dynamic influence of joints in aero-engine rotor systems is investigated in this paper.Firstly,the tangential stiffness and loss factor are obtained from an isolated lap joint setup with dynamic excitation experiments.Also,the influence of the normal contact pressure and the excitation level are examined,which revel the uncertainty in joints.Then,the updated Thin Layer Elements(TLEs)method with fitted parameters based on the experiments is established to simulate the dynamic properties of joints on the interface.The response of the rotor subjected to unbalance excitation is calculated,and the results illustrate the effectiveness of the proposed method.Meanwhile,using the Chebyshev inclusion function and a direct iteration algorithm,a nonlinear interval analysis method is established to consider the uncertainty of parameters in joints.The accuracy is proved by comparison with results obtained using the Monte-Carlo method.Combined with the updated TLEs,the nonlinear Chebyshev method is successfully applied on a finite model of a rotor.The study shows that substantial attention should be paid to the dynamical design for the joint in rotor systems,the dynamic properties of joints under complex loading and the corresponding interval analysis method need to be intensively studied.展开更多
Errors will be caused in calculating the fatigue damages of details in liquid cargo tanks by using the traditional spectral analysis method which is based on linear system, for the nonlinear relationship between the d...Errors will be caused in calculating the fatigue damages of details in liquid cargo tanks by using the traditional spectral analysis method which is based on linear system, for the nonlinear relationship between the dynamic stress and the ship acceleration. An improved spectral analysis method for the assessment of the fatigue damage in detail of a liquid cargo tank is proposed in this paper. Based on assumptions that the wave process can be simulated by summing the sinusoidal waves in different frequencies and the stress process can be simulated by summing the stress processes induced by these sinusoidal waves, the stress power spectral density(PSD) is calculated by expanding the stress processes induced by the sinusoidal waves into Fourier series and adding the amplitudes of each harmonic component with the same frequency. This analysis method can take the nonlinear relationship into consideration and the fatigue damage is then calculated based on the PSD of stress. Take an independent tank in an LNG carrier for example, the accuracy of the improved spectral analysis method is proved much better than that of the traditional spectral analysis method by comparing the calculated damage results with the results calculated by the time domain method. The proposed spectral analysis method is more accurate in calculating the fatigue damages in detail of ship liquid cargo tanks.展开更多
In order to improve the office paper feeder design, and eliminate paper jam fault in running office equipment, the static deformation and dynamic response of paper were analyzed by use of the Finite Element Method (FE...In order to improve the office paper feeder design, and eliminate paper jam fault in running office equipment, the static deformation and dynamic response of paper were analyzed by use of the Finite Element Method (FEM). In the analysis, the three nodes mangle plate and shell element were employed, and finite element incremental formulations were derived on the basis of Updated Lagrangian (U.L) description. The newmark method was used to analyze the transient response of paper. All the results calculated in this article coincide with those by experiments.展开更多
This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.Thi...This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.展开更多
To effectively select random variable in nonlinear dynamic reliability analysis,the extremum selection method(ESM)is proposed.Firstly,the basic idea was introduced and the mathematical model was established for the ES...To effectively select random variable in nonlinear dynamic reliability analysis,the extremum selection method(ESM)is proposed.Firstly,the basic idea was introduced and the mathematical model was established for the ESM.The nonlinear dynamic reliability analysis of turbine blade radial deformation was taken as an example to verify the ESM.The results show that the analysis precision of the ESM is 99.972%,which is almost kept consistent with that of the Monte Carlo method;moreover,the computing time of the ESM is shorter than that of the traditional method.Hence,it is demonstrated that the ESM is able to save calculation time and improve the computational efficiency while keeping the calculation precision for nonlinear dynamic reliability analysis.The present study provides a method to enhance the nonlinear dynamic reliability analysis in selecting the random variables and offers a way to design structure and machine in future work.展开更多
This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This...This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing polynomial approximation interval methods. Interval arith- metic using the Chebyshev basis and interval arithmetic using the general form modified affine basis for polynomials are developed to obtain tighter bounds for interval computation. To further reduce the overestimation caused by the ＂wrap- ping effect＂ of interval arithmetic, the derivative information of dynamic responses is used to achieve exact solutions when the dynamic responses are monotonic with respect to all the uncertain variables. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effective- ness of the proposed hybrid interval method, in particular, its ability to effectively control the overestimation for specific timepoints.展开更多
The analysis of cable structures is one of the most challenging problems for civil and mechanical engineers.Because they have highly nonlinear behavior,it is difficult to find solutions to these problems.Thus far,diff...The analysis of cable structures is one of the most challenging problems for civil and mechanical engineers.Because they have highly nonlinear behavior,it is difficult to find solutions to these problems.Thus far,different assumptions and methods have been proposed to solve such structures.The dynamic relaxation method(DRM)is an explicit procedure for analyzing these types of structures.To utilize this scheme,investigators have suggested various stiffness matrices for a cable element.In this study,the efficiency and suitability of six well-known proposed matrices are assessed using the DRM.To achieve this goal,16 numerical examples and two criteria,namely,the number of iterations and the analysis time,are employed.Based on a comprehensive comparison,the methods are ranked according to the two criteria.The numerical findings clearly reveal the best techniques.Moreover,a variety of benchmark problems are suggested by the authors for future studies of cable structures.展开更多
Nonlinear partial differential equations(PDEs)are crucial for modeling complex fluid dynamics and are foundational to many computational fluid dynamics(CFD)applications.However,solving these nonlinear PDEs is challeng...Nonlinear partial differential equations(PDEs)are crucial for modeling complex fluid dynamics and are foundational to many computational fluid dynamics(CFD)applications.However,solving these nonlinear PDEs is challenging due to the vast computational resources they demand,highlighting the pressing need for more efficient computational methods.Quantum computing offers a promising but technically challenging approach to solving nonlinear PDEs.Recently,Liao[arXiv:2406.15821]proposed a framework that leverages quantum computing to accelerate the solution of nonlinear PDEs based on the homotopy analysis method(HAM),a semi-analytical technique that transforms nonlinear PDEs into a series of linear PDEs.However,the no-cloning theorem in quantum computing poses a major limitation,where directly applying quantum simulation to each HAM step results in exponential complexity growth with the HAM truncation order.This study introduces a“quantum-compatible linearization”approach that maps the whole HAM process into a system of linear PDEs,allowing for a one-time solution using established quantum PDE solvers.Our method preserves the exponential speedup of quantum linear PDE solvers while ensuring that computational complexity increases only polynomially with the HAM truncation order.We demonstrate the efficacy of our approach by applying it to the Burgers'equation and the Korteweg-de Vries(KdV)equation.Our approach provides a novel pathway for transforming nonlinear PDEs into linear PDEs,with potential applications to fluid dynamics.This work thus lays the foundation for developing quantum algorithms capable of solving the Navier-Stokes equations,ultimately offering a promising route to accelerate their solutions using quantum computing.展开更多
Quasi-static testing is the primary seismic research method employed.The method proposed in this study utilizes the neural network(NN)algorithm for restoring force identification to extend the hysteretic performance o...Quasi-static testing is the primary seismic research method employed.The method proposed in this study utilizes the neural network(NN)algorithm for restoring force identification to extend the hysteretic performance of nonlinear complex components obtained from quasi-static tests shared or performed at a lower cost to the time history analysis of the seismic response of the entire structure.This approach enables accurate analysis of the seismic performance of the structure under real earthquake ground motions at a relatively low experimental costs.At the level of restoring force model recognition,the eight-path hysteresis model recognition theory and the corresponding complete set of input and output variables in the NN algorithm are proposed.The NN restoring force model was established using input and output parameters that characterize hysteresis state features,with a two-hidden-layer NN architecture.The case study results indicate that the prediction results of the NN restoring force model align well with the target values when trained on samples obtained under both seismic and quasi-static loading conditions.At the level of the nonlinear dynamic analysis of structures,the hybrid analysis method of structural seismic response based on NN restoring force model is proposed.In this method,the potentially severe nonlinear and elastic parts of the structure are divided into several NN substructures and principal numerical substructure,respectively.The pseudo-static test data of nonlinear regions were used to train the proposed NN restoring force model to identify the restoring force of NN substructures in the same region under time-history dynamic analysis.The platform was built to complete the data interaction between several NN substructures and principal numerical substructures,and a precise integration method was used to program the dynamic equation solving module,gradually completing dynamic response analysis of the entire structure.A multi-degree-offreedom nonlinear frame case study indicate that the proposed method has good accuracy and can effectively analyze the structural nonlinear seismic response.展开更多
The endurance time (ET) method is a time history based dynamic analysis in which structures are subjected to gradually intensifying excitations and their performances are judged based on their responses at various exc...The endurance time (ET) method is a time history based dynamic analysis in which structures are subjected to gradually intensifying excitations and their performances are judged based on their responses at various excitation levels. Using this method, the computational effort required for estimating probable seismic demand parameters can be reduced by an order of magnitude. Calculation of the maximum displacement or target displacement is a basic requirement for estimating performance based on structural design. The purpose of this paper is to compare the results of the nonlinear ET method with the nonlinear static pushover (NSP) method of FEMA 356 by evaluating performances and target displacements of steel frames. This study will lead to a deeper insight into the capabilities and limitations of the ET method. The results are further compared with those of the standard nonlinear response history analysis. We conclude that results from the ET analysis are in proper agreement with those from standard procedures.展开更多
For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods...For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .展开更多
The performance of a classical damping matrix, constructed either from the use of initial structural properties or current structural properties, in the step-by-step solution of a nonlinear multiple degree of freedom ...The performance of a classical damping matrix, constructed either from the use of initial structural properties or current structural properties, in the step-by-step solution of a nonlinear multiple degree of freedom （MDOF） system is analytically evaluated. The analytical results are confirmed by numerical examples. Consequently, some conclusions are drawn from these analytical results that might be considered as rough guidelines for practical applications. It is found that a classical damping matrix constructed from initial structural properties is adequate for practical applications, since it has approximately the same damping effect as obtained by current structural properties and is more efficient in terms of computing.展开更多
文摘Based on the characteristics of membrane structures and the air influence factors,this paper presented a method to simulate the air aerodynamic force effects including the added air mass,the acoustic radiation damping and the pneumatic stiffness.The infinite air was modeled using the acoustic fluid element of commercial FE software and the finite element membrane roof models were coupled with fluid models.A comparison between the results obtained by FE computation and those obtained by the vibration experiment for a cable-membrane verified the validity of the method.Furthermore,applying the method to a flat membrane roof structure and using its wind tunnel test results,the analysis of nonlinear wind-induced dynamic responses for such geometrically nonlinear roofs,including the roof-air coupled model was performed.The result shows that the air has large influence on vibrating membrane roofs according to results of comparing the nodal time-history displacements,accelerations and stress of the two different cases.Meantime,numerical studies show that the method developed can successfully solve the nonlinear wind-induced dynamic response of the membrane roof with aerodynamic effects.
文摘Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.
基金supported by the National Natural Science Foundation of China(Nos.51575022,11772022 and 51475021).
文摘The dynamic influence of joints in aero-engine rotor systems is investigated in this paper.Firstly,the tangential stiffness and loss factor are obtained from an isolated lap joint setup with dynamic excitation experiments.Also,the influence of the normal contact pressure and the excitation level are examined,which revel the uncertainty in joints.Then,the updated Thin Layer Elements(TLEs)method with fitted parameters based on the experiments is established to simulate the dynamic properties of joints on the interface.The response of the rotor subjected to unbalance excitation is calculated,and the results illustrate the effectiveness of the proposed method.Meanwhile,using the Chebyshev inclusion function and a direct iteration algorithm,a nonlinear interval analysis method is established to consider the uncertainty of parameters in joints.The accuracy is proved by comparison with results obtained using the Monte-Carlo method.Combined with the updated TLEs,the nonlinear Chebyshev method is successfully applied on a finite model of a rotor.The study shows that substantial attention should be paid to the dynamical design for the joint in rotor systems,the dynamic properties of joints under complex loading and the corresponding interval analysis method need to be intensively studied.
基金financially supported by the National Natural Science Foundation of China(Grant No.51279102)the High-Technology Ship Research Project of the Ministry of Industry and Information Technology of China(Grant No.2012-534)
文摘Errors will be caused in calculating the fatigue damages of details in liquid cargo tanks by using the traditional spectral analysis method which is based on linear system, for the nonlinear relationship between the dynamic stress and the ship acceleration. An improved spectral analysis method for the assessment of the fatigue damage in detail of a liquid cargo tank is proposed in this paper. Based on assumptions that the wave process can be simulated by summing the sinusoidal waves in different frequencies and the stress process can be simulated by summing the stress processes induced by these sinusoidal waves, the stress power spectral density(PSD) is calculated by expanding the stress processes induced by the sinusoidal waves into Fourier series and adding the amplitudes of each harmonic component with the same frequency. This analysis method can take the nonlinear relationship into consideration and the fatigue damage is then calculated based on the PSD of stress. Take an independent tank in an LNG carrier for example, the accuracy of the improved spectral analysis method is proved much better than that of the traditional spectral analysis method by comparing the calculated damage results with the results calculated by the time domain method. The proposed spectral analysis method is more accurate in calculating the fatigue damages in detail of ship liquid cargo tanks.
文摘In order to improve the office paper feeder design, and eliminate paper jam fault in running office equipment, the static deformation and dynamic response of paper were analyzed by use of the Finite Element Method (FEM). In the analysis, the three nodes mangle plate and shell element were employed, and finite element incremental formulations were derived on the basis of Updated Lagrangian (U.L) description. The newmark method was used to analyze the transient response of paper. All the results calculated in this article coincide with those by experiments.
文摘This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.
基金the National Natural Science Foundation of China(Grant no.51175017)the Innovation Foundation of BUAA for Ph.D.Graduates(Grant no.YWF-12-RBYJ-008)。
文摘To effectively select random variable in nonlinear dynamic reliability analysis,the extremum selection method(ESM)is proposed.Firstly,the basic idea was introduced and the mathematical model was established for the ESM.The nonlinear dynamic reliability analysis of turbine blade radial deformation was taken as an example to verify the ESM.The results show that the analysis precision of the ESM is 99.972%,which is almost kept consistent with that of the Monte Carlo method;moreover,the computing time of the ESM is shorter than that of the traditional method.Hence,it is demonstrated that the ESM is able to save calculation time and improve the computational efficiency while keeping the calculation precision for nonlinear dynamic reliability analysis.The present study provides a method to enhance the nonlinear dynamic reliability analysis in selecting the random variables and offers a way to design structure and machine in future work.
文摘This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing polynomial approximation interval methods. Interval arith- metic using the Chebyshev basis and interval arithmetic using the general form modified affine basis for polynomials are developed to obtain tighter bounds for interval computation. To further reduce the overestimation caused by the ＂wrap- ping effect＂ of interval arithmetic, the derivative information of dynamic responses is used to achieve exact solutions when the dynamic responses are monotonic with respect to all the uncertain variables. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effective- ness of the proposed hybrid interval method, in particular, its ability to effectively control the overestimation for specific timepoints.
文摘The analysis of cable structures is one of the most challenging problems for civil and mechanical engineers.Because they have highly nonlinear behavior,it is difficult to find solutions to these problems.Thus far,different assumptions and methods have been proposed to solve such structures.The dynamic relaxation method(DRM)is an explicit procedure for analyzing these types of structures.To utilize this scheme,investigators have suggested various stiffness matrices for a cable element.In this study,the efficiency and suitability of six well-known proposed matrices are assessed using the DRM.To achieve this goal,16 numerical examples and two criteria,namely,the number of iterations and the analysis time,are employed.Based on a comprehensive comparison,the methods are ranked according to the two criteria.The numerical findings clearly reveal the best techniques.Moreover,a variety of benchmark problems are suggested by the authors for future studies of cable structures.
基金supported by the National Key Research and Development Program of China(Grant No.2023YFB4502500)the National Natural Science Foundation of China(Grant No.12404564)Anhui Province Science and Technology Innovation(Grant No.202423s06050001)。
文摘Nonlinear partial differential equations(PDEs)are crucial for modeling complex fluid dynamics and are foundational to many computational fluid dynamics(CFD)applications.However,solving these nonlinear PDEs is challenging due to the vast computational resources they demand,highlighting the pressing need for more efficient computational methods.Quantum computing offers a promising but technically challenging approach to solving nonlinear PDEs.Recently,Liao[arXiv:2406.15821]proposed a framework that leverages quantum computing to accelerate the solution of nonlinear PDEs based on the homotopy analysis method(HAM),a semi-analytical technique that transforms nonlinear PDEs into a series of linear PDEs.However,the no-cloning theorem in quantum computing poses a major limitation,where directly applying quantum simulation to each HAM step results in exponential complexity growth with the HAM truncation order.This study introduces a“quantum-compatible linearization”approach that maps the whole HAM process into a system of linear PDEs,allowing for a one-time solution using established quantum PDE solvers.Our method preserves the exponential speedup of quantum linear PDE solvers while ensuring that computational complexity increases only polynomially with the HAM truncation order.We demonstrate the efficacy of our approach by applying it to the Burgers'equation and the Korteweg-de Vries(KdV)equation.Our approach provides a novel pathway for transforming nonlinear PDEs into linear PDEs,with potential applications to fluid dynamics.This work thus lays the foundation for developing quantum algorithms capable of solving the Navier-Stokes equations,ultimately offering a promising route to accelerate their solutions using quantum computing.
基金supports from the National Natural Science Foundation of China(Grant Nos.52478305 and 52308483)Postgraduate Research&Practice Innovation Program of Jiangsu Province(No.KYCX22_0213).
文摘Quasi-static testing is the primary seismic research method employed.The method proposed in this study utilizes the neural network(NN)algorithm for restoring force identification to extend the hysteretic performance of nonlinear complex components obtained from quasi-static tests shared or performed at a lower cost to the time history analysis of the seismic response of the entire structure.This approach enables accurate analysis of the seismic performance of the structure under real earthquake ground motions at a relatively low experimental costs.At the level of restoring force model recognition,the eight-path hysteresis model recognition theory and the corresponding complete set of input and output variables in the NN algorithm are proposed.The NN restoring force model was established using input and output parameters that characterize hysteresis state features,with a two-hidden-layer NN architecture.The case study results indicate that the prediction results of the NN restoring force model align well with the target values when trained on samples obtained under both seismic and quasi-static loading conditions.At the level of the nonlinear dynamic analysis of structures,the hybrid analysis method of structural seismic response based on NN restoring force model is proposed.In this method,the potentially severe nonlinear and elastic parts of the structure are divided into several NN substructures and principal numerical substructure,respectively.The pseudo-static test data of nonlinear regions were used to train the proposed NN restoring force model to identify the restoring force of NN substructures in the same region under time-history dynamic analysis.The platform was built to complete the data interaction between several NN substructures and principal numerical substructures,and a precise integration method was used to program the dynamic equation solving module,gradually completing dynamic response analysis of the entire structure.A multi-degree-offreedom nonlinear frame case study indicate that the proposed method has good accuracy and can effectively analyze the structural nonlinear seismic response.
文摘The endurance time (ET) method is a time history based dynamic analysis in which structures are subjected to gradually intensifying excitations and their performances are judged based on their responses at various excitation levels. Using this method, the computational effort required for estimating probable seismic demand parameters can be reduced by an order of magnitude. Calculation of the maximum displacement or target displacement is a basic requirement for estimating performance based on structural design. The purpose of this paper is to compare the results of the nonlinear ET method with the nonlinear static pushover (NSP) method of FEMA 356 by evaluating performances and target displacements of steel frames. This study will lead to a deeper insight into the capabilities and limitations of the ET method. The results are further compared with those of the standard nonlinear response history analysis. We conclude that results from the ET analysis are in proper agreement with those from standard procedures.
文摘For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .
基金Science Council,Taipei 106-08,Chinese Taipei,under Grant No. NSC-99-2221-E-027-029
文摘The performance of a classical damping matrix, constructed either from the use of initial structural properties or current structural properties, in the step-by-step solution of a nonlinear multiple degree of freedom （MDOF） system is analytically evaluated. The analytical results are confirmed by numerical examples. Consequently, some conclusions are drawn from these analytical results that might be considered as rough guidelines for practical applications. It is found that a classical damping matrix constructed from initial structural properties is adequate for practical applications, since it has approximately the same damping effect as obtained by current structural properties and is more efficient in terms of computing.
