This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Ber...This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams.The proposed algorithm,named as two-parameter multiple shooting method,is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions.The impact of the axial force and additional point masses is also taken into account.Due to the fact that the method is based directly on the fourth-order ordinary differential equation,the structures do not have to be divided into many small elements to obtain an accurate enough solution,even though the geometry is very complex.To verify the proposed method,three different examples are considered,i.e.,a three-segment non-prismatic beam,a prismatic column subject to non-uniformly distributed compressive loads,and a two-segment beam with an additional point mass.Numerical analyses are carried out with the software MATHEMATICA.The results are compared with the solutions computed by the commercial finite element program SOFiSTiK.Good agreement is achieved,which confirms the correctness and high effectiveness of the formulated algorithm.展开更多
The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and...The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.展开更多
The inflation tests of rubbery membranes have been widely employed as an efficient method to characterize the stress response as biaxial loading states.However,most of the previous theoretical works have employed clas...The inflation tests of rubbery membranes have been widely employed as an efficient method to characterize the stress response as biaxial loading states.However,most of the previous theoretical works have employed classic hyperelastic models to analyze the deformation behaviors of inflated membranes.The classic models have been demonstrated to lack the ability to capturing the biaxial deformation of rubbers.To address this issue,we have combined the analytical method and the finite element simulation to investigate the deformation response of soft membranes with different constitutive relationships.For the analytical method,the governing ordinary differential equations have been set up for the boundary value problem of inflation tests and further solved using the shooting method.The analytical results are consistent with those obtained from finite element simulation.The results show that the deformation belongs to the unequal biaxial condition rather than the equi-biaxial state unless a neo-Hookean model is adopted.We also perform a parameter study using the extended eight-chain model,which shows that a change in different parameters affects the mechanical response of inflation tests variously.This work may shed light on the future experimental characterization of soft materials using inflation experiments.展开更多
This paper proposes a novel impulsive thrust strategy guided by optimal continuous thrust strategy to address two-player orbital pursuit-evasion game under impulsive thrust control.The strategy seeks to enhance the in...This paper proposes a novel impulsive thrust strategy guided by optimal continuous thrust strategy to address two-player orbital pursuit-evasion game under impulsive thrust control.The strategy seeks to enhance the interpretability of impulsive thrust strategy by integrating it within the framework of differential game in traditional continuous systems.First,this paper introduces an impulse-like constraint,with periodical changes in thrust amplitude,to characterize the impulsive thrust control.Then,the game with the impulse-like constraint is converted into the two-point boundary value problem,which is solved by the combined shooting and deep learning method proposed in this paper.Deep learning and numerical optimization are employed to obtain the guesses for unknown terminal adjoint variables and the game terminal time.Subsequently,the accurate values are solved by the shooting method to yield the optimal continuous thrust strategy with the impulse-like constraint.Finally,the shooting method is iteratively employed at each impulse decision moment to derive the impulsive thrust strategy guided by the optimal continuous thrust strategy.Numerical examples demonstrate the convergence of the combined shooting and deep learning method,even if the strongly nonlinear impulse-like constraint is introduced.The effect of the impulsive thrust strategy guided by the optimal continuous thrust strategy is also discussed.展开更多
An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the appr...An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the approximate solution obtained by former researchers.展开更多
In order to improve the safety and reliability of a launch vehicle, we present a reconfigurable guidance method based on online trajectory optimization for thrust decrease at the ascending stage of the launch vehicle ...In order to improve the safety and reliability of a launch vehicle, we present a reconfigurable guidance method based on online trajectory optimization for thrust decrease at the ascending stage of the launch vehicle in this paper. We used a multiple shooting method to convert the optimal control problem into a nonlinear programming(NLP) problem. Finally, we used the interior point method to solve the NLP problem. The simulation results show that the method can effectively adapt to thrust decreases by 5% and 10%. This reconfigurable guidance method based on online planning is proposed to realize launch missions under the condition of power descent, thus verifying the feasibility of the method.展开更多
In space,surface tension plays an important role and liquid behaviour is much different from that on the ground.The static capillary surfaces in the annular space between two coaxial cones under microgravity are studi...In space,surface tension plays an important role and liquid behaviour is much different from that on the ground.The static capillary surfaces in the annular space between two coaxial cones under microgravity are studied in this paper.Theoretical expressions of the capillary surfaces are derived and a procedure is developed to predict the capillary surfaces based on the expressions.By considering various liquid contact angles,liquid volumes,and container geometries,numerical simulation with the volume of fluid method is carried out and microgravity experiments in Beijing Drop Tower are performed.The numerical and experimental results are in good agreement with theoretical predictions.Furthermore,capillary surfaces in an annulus with constant cross-section and in a spherical tank with a central column are also discussed.z3 will decrease obviously with the increase of the liquid contact angle.The theoretical models and findings will be great helpful for liquid management in space and the evaluation of propellant residue.展开更多
Magnetic monopoles stand for the static solution arising from a(1 + 3)–dimensional theory describing the interaction between a real scalar triplet and non–Abelian gauge field. In this paper, we obtain a two–point b...Magnetic monopoles stand for the static solution arising from a(1 + 3)–dimensional theory describing the interaction between a real scalar triplet and non–Abelian gauge field. In this paper, we obtain a two–point boundary value problem of a first–order ordinary differential equations from the self–dual monopole model. Then we establish the existence and uniqueness theorem for the problem by using a dynamical shooting method, we also obtain sharp asymptotic estimates for the solutions at infinity.展开更多
This paper proposes a new dynamic model for the study of friction-induced self-excited vibration of a disc brake system,where the pad’s motions in both radial and circumferential/tangential directions are included,wh...This paper proposes a new dynamic model for the study of friction-induced self-excited vibration of a disc brake system,where the pad’s motions in both radial and circumferential/tangential directions are included,which is in stark contrast to the previous studies that normally consider the pad’s motion in the tangential/circumferential direction only.The non-smooth dynamics of the system including three different states of motion,i.e.,stick,slip and separation,is investigated.Both the linear stability analysis and the transient dynamic analysis are performed.The numerical results in the linear stability analysis indicate that the inclusion of pad’s radial motion in the present brake model significantly expands the ranges of operating parameters for dynamic instability than the brake model with only circumferential/tangential motion for the pad.For the transient dynamic analysis,two different methods,i.e.,the time integration method and the shooting method,are employed for the calculation of steady-state response.The accuracy and efficiency of the shooting method are subsequently examined.The numerical results show rich bifurcation behaviours of the steady-state response in the present brake model with the variations of brake pressure N_(0) and disc speedΩ,and that k_(ir)(the stiffness of the inclined spring in the radial direction)is a key parameter for controlling the occurrence of chaotic vibration in the system.展开更多
This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering the...This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.展开更多
This paper is concerned with optimal motion planning for vibration reducing of free-floating flexible redundant manipulators. Firstly, dynamic model of the system is established based on Lagrange method, and the motio...This paper is concerned with optimal motion planning for vibration reducing of free-floating flexible redundant manipulators. Firstly, dynamic model of the system is established based on Lagrange method, and the motion planning model for vibration reducing is proposed. Secondly, a hybrid optimization approach employing Gauss pseudospectral method （GPM） and direct shooting method （DSM）, is proposed to solve the motion planning problem. In this approach, the motion planning problem is transformed into a non-linear parameter optimization problem using GPM, and genetic algorithm （GA） is employed to locate the approximate solution. Subsequently, an optimization model is formulated based on DSM, and sequential quadratic programming （SQP） algorithm is used to obtain the accurate solution, with the approximate solution as an initial reference solution. Finally, several numerical simulations are investigated, and the global vibration or residual vibration of flexible link is obviously reduced by the joint trajectory which is obtained by the hybrid optimization approach. The numerical simulation results indicate that the approach is effective and stable to the motion planning problem of vibration reducing.展开更多
The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by con...The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.展开更多
A nonlinear dynamic model of a simple nonholonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton＇s Variatio...A nonlinear dynamic model of a simple nonholonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton＇s Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon （i.e. catastrophe phenomenon） is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is nonzero. The results obtained by two methods are consistent.展开更多
Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equati...Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.展开更多
This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The non-linear govern...This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The non-linear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance （10^-5）. The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.展开更多
The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary dif...The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.展开更多
Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc len...Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc length s(x) of axial line and the longitudinal displacement u(x) are taken as the basic unknown functions. This is a two point boundary value problem of first order ordinary differential equations with strong non_linearity. By using shooting method and analytical continuation, the nonlinear boundary value problems are numerically solved. The thermal post_buckled states of the rods with transversely simply supported and clamped ends are obtained respectively and the corresponding numerical data tables and characteristic curves are also given.展开更多
The indirect method for the continuous low-thrust near minimum cumulative longitude orbit transfer problem is addressed.The movement of the satellite is described by the Gauss equation using the modified equinoctial e...The indirect method for the continuous low-thrust near minimum cumulative longitude orbit transfer problem is addressed.The movement of the satellite is described by the Gauss equation using the modified equinoctial elements and replacing time as the system independent variable by the cumulative longitude.The maximum principle is adapted to design the optimal control in order to minimize the final cumulative longitude, and the twopoint-boundary-value problem is derived from the orbit transfer problem.The single shooting method is applied in a numerical experiment, and the simulations demonstrate that the orbit transfer mission is fulfilled and the product of the maximal thrust and the minimum cumulative longitude is near constant.展开更多
In this article, using the contraction mapping principle and the shooting method, the authors obtain the existence and uniqueness of the local solution and the global solution to a class of quasilinear elliptic system...In this article, using the contraction mapping principle and the shooting method, the authors obtain the existence and uniqueness of the local solution and the global solution to a class of quasilinear elliptic systems with p-Laplacian as its principal. They also obtain the continuous dependence of the solutions on the boundary data.展开更多
Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transve...Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.展开更多
文摘This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved.A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams.The proposed algorithm,named as two-parameter multiple shooting method,is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions.The impact of the axial force and additional point masses is also taken into account.Due to the fact that the method is based directly on the fourth-order ordinary differential equation,the structures do not have to be divided into many small elements to obtain an accurate enough solution,even though the geometry is very complex.To verify the proposed method,three different examples are considered,i.e.,a three-segment non-prismatic beam,a prismatic column subject to non-uniformly distributed compressive loads,and a two-segment beam with an additional point mass.Numerical analyses are carried out with the software MATHEMATICA.The results are compared with the solutions computed by the commercial finite element program SOFiSTiK.Good agreement is achieved,which confirms the correctness and high effectiveness of the formulated algorithm.
文摘The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.
基金supported by the National Natural Science Foundation of China(Grant Nos.12211530061 and 12321002)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LD22A020001)the 111 Project(Grant No.B21034).
文摘The inflation tests of rubbery membranes have been widely employed as an efficient method to characterize the stress response as biaxial loading states.However,most of the previous theoretical works have employed classic hyperelastic models to analyze the deformation behaviors of inflated membranes.The classic models have been demonstrated to lack the ability to capturing the biaxial deformation of rubbers.To address this issue,we have combined the analytical method and the finite element simulation to investigate the deformation response of soft membranes with different constitutive relationships.For the analytical method,the governing ordinary differential equations have been set up for the boundary value problem of inflation tests and further solved using the shooting method.The analytical results are consistent with those obtained from finite element simulation.The results show that the deformation belongs to the unequal biaxial condition rather than the equi-biaxial state unless a neo-Hookean model is adopted.We also perform a parameter study using the extended eight-chain model,which shows that a change in different parameters affects the mechanical response of inflation tests variously.This work may shed light on the future experimental characterization of soft materials using inflation experiments.
基金funded by the National Natural Science Foundation of China(No.U21B6001)。
文摘This paper proposes a novel impulsive thrust strategy guided by optimal continuous thrust strategy to address two-player orbital pursuit-evasion game under impulsive thrust control.The strategy seeks to enhance the interpretability of impulsive thrust strategy by integrating it within the framework of differential game in traditional continuous systems.First,this paper introduces an impulse-like constraint,with periodical changes in thrust amplitude,to characterize the impulsive thrust control.Then,the game with the impulse-like constraint is converted into the two-point boundary value problem,which is solved by the combined shooting and deep learning method proposed in this paper.Deep learning and numerical optimization are employed to obtain the guesses for unknown terminal adjoint variables and the game terminal time.Subsequently,the accurate values are solved by the shooting method to yield the optimal continuous thrust strategy with the impulse-like constraint.Finally,the shooting method is iteratively employed at each impulse decision moment to derive the impulsive thrust strategy guided by the optimal continuous thrust strategy.Numerical examples demonstrate the convergence of the combined shooting and deep learning method,even if the strongly nonlinear impulse-like constraint is introduced.The effect of the impulsive thrust strategy guided by the optimal continuous thrust strategy is also discussed.
基金The work is financially supported by the National Natural Science Foundations of China (No.50476083).
文摘An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the approximate solution obtained by former researchers.
文摘In order to improve the safety and reliability of a launch vehicle, we present a reconfigurable guidance method based on online trajectory optimization for thrust decrease at the ascending stage of the launch vehicle in this paper. We used a multiple shooting method to convert the optimal control problem into a nonlinear programming(NLP) problem. Finally, we used the interior point method to solve the NLP problem. The simulation results show that the method can effectively adapt to thrust decreases by 5% and 10%. This reconfigurable guidance method based on online planning is proposed to realize launch missions under the condition of power descent, thus verifying the feasibility of the method.
基金supported by the China Manned Space Engineering Program(Fluid Physics Experimental Rack and the Priority Research Program of Space Station)the Natural Science Foundation Project(Grant No.12032020).
文摘In space,surface tension plays an important role and liquid behaviour is much different from that on the ground.The static capillary surfaces in the annular space between two coaxial cones under microgravity are studied in this paper.Theoretical expressions of the capillary surfaces are derived and a procedure is developed to predict the capillary surfaces based on the expressions.By considering various liquid contact angles,liquid volumes,and container geometries,numerical simulation with the volume of fluid method is carried out and microgravity experiments in Beijing Drop Tower are performed.The numerical and experimental results are in good agreement with theoretical predictions.Furthermore,capillary surfaces in an annulus with constant cross-section and in a spherical tank with a central column are also discussed.z3 will decrease obviously with the increase of the liquid contact angle.The theoretical models and findings will be great helpful for liquid management in space and the evaluation of propellant residue.
文摘Magnetic monopoles stand for the static solution arising from a(1 + 3)–dimensional theory describing the interaction between a real scalar triplet and non–Abelian gauge field. In this paper, we obtain a two–point boundary value problem of a first–order ordinary differential equations from the self–dual monopole model. Then we establish the existence and uniqueness theorem for the problem by using a dynamical shooting method, we also obtain sharp asymptotic estimates for the solutions at infinity.
基金supported by the Hong Kong Innovation and Technology Commission(Project No.MRP/030/21 under PiH/026/23)The Chinese University of Hong Kong(Project ID:3134167)the Research Grants Council(Project No.CUHK14211823)of Hong Kong Special Administrative Region,China.
文摘This paper proposes a new dynamic model for the study of friction-induced self-excited vibration of a disc brake system,where the pad’s motions in both radial and circumferential/tangential directions are included,which is in stark contrast to the previous studies that normally consider the pad’s motion in the tangential/circumferential direction only.The non-smooth dynamics of the system including three different states of motion,i.e.,stick,slip and separation,is investigated.Both the linear stability analysis and the transient dynamic analysis are performed.The numerical results in the linear stability analysis indicate that the inclusion of pad’s radial motion in the present brake model significantly expands the ranges of operating parameters for dynamic instability than the brake model with only circumferential/tangential motion for the pad.For the transient dynamic analysis,two different methods,i.e.,the time integration method and the shooting method,are employed for the calculation of steady-state response.The accuracy and efficiency of the shooting method are subsequently examined.The numerical results show rich bifurcation behaviours of the steady-state response in the present brake model with the variations of brake pressure N_(0) and disc speedΩ,and that k_(ir)(the stiffness of the inclined spring in the radial direction)is a key parameter for controlling the occurrence of chaotic vibration in the system.
文摘This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.
基金National Natural Science Foundation of China （10902121）
文摘This paper is concerned with optimal motion planning for vibration reducing of free-floating flexible redundant manipulators. Firstly, dynamic model of the system is established based on Lagrange method, and the motion planning model for vibration reducing is proposed. Secondly, a hybrid optimization approach employing Gauss pseudospectral method （GPM） and direct shooting method （DSM）, is proposed to solve the motion planning problem. In this approach, the motion planning problem is transformed into a non-linear parameter optimization problem using GPM, and genetic algorithm （GA） is employed to locate the approximate solution. Subsequently, an optimization model is formulated based on DSM, and sequential quadratic programming （SQP） algorithm is used to obtain the accurate solution, with the approximate solution as an initial reference solution. Finally, several numerical simulations are investigated, and the global vibration or residual vibration of flexible link is obviously reduced by the joint trajectory which is obtained by the hybrid optimization approach. The numerical simulation results indicate that the approach is effective and stable to the motion planning problem of vibration reducing.
基金Project supported by the National Natural Science Foundation of China(No.11272278)
文摘The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.
基金The project supported by the National Natural Science Foundation of China （10272002） and the Doctoral Program from the Ministry of Education of China （20020001032） The English text was polished by Yunming Chen.
文摘A nonlinear dynamic model of a simple nonholonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton＇s Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon （i.e. catastrophe phenomenon） is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is nonzero. The results obtained by two methods are consistent.
基金supported by the National Natural Science Foundation of China(No.11272278)
文摘Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.
文摘This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The non-linear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance （10^-5）. The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.
文摘The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.
文摘Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc length s(x) of axial line and the longitudinal displacement u(x) are taken as the basic unknown functions. This is a two point boundary value problem of first order ordinary differential equations with strong non_linearity. By using shooting method and analytical continuation, the nonlinear boundary value problems are numerically solved. The thermal post_buckled states of the rods with transversely simply supported and clamped ends are obtained respectively and the corresponding numerical data tables and characteristic curves are also given.
基金supported by the National Natural Science Foundation of China (10832006 60874011)
文摘The indirect method for the continuous low-thrust near minimum cumulative longitude orbit transfer problem is addressed.The movement of the satellite is described by the Gauss equation using the modified equinoctial elements and replacing time as the system independent variable by the cumulative longitude.The maximum principle is adapted to design the optimal control in order to minimize the final cumulative longitude, and the twopoint-boundary-value problem is derived from the orbit transfer problem.The single shooting method is applied in a numerical experiment, and the simulations demonstrate that the orbit transfer mission is fulfilled and the product of the maximal thrust and the minimum cumulative longitude is near constant.
文摘In this article, using the contraction mapping principle and the shooting method, the authors obtain the existence and uniqueness of the local solution and the global solution to a class of quasilinear elliptic systems with p-Laplacian as its principal. They also obtain the continuous dependence of the solutions on the boundary data.
基金Project supported by the National Natural Science Foundation of China (No.10472039)
文摘Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.
