Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexu...Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.展开更多
This investigation focuses on the nonlinear dynamic behaviors in the trans- verse vibration of an axiMly accelerating viscoelastic Timoshenko beam with the external harmonic excitation. The parametric excitation is ca...This investigation focuses on the nonlinear dynamic behaviors in the trans- verse vibration of an axiMly accelerating viscoelastic Timoshenko beam with the external harmonic excitation. The parametric excitation is caused by the harmonic fluctuations of the axial moving speed. An integro-partial-differential equation governing the transverse vibration of the Timoshenko beam is established. Many factors are considered, such as viscoelasticity, the finite axial support rigidity, and the longitudinally varying tension due to the axial acceleration. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the governing equation. Based on the numerical solutions, the bifurcation diagrams are presented to study the effect of the external transverse excitation. Moreover, the frequencies of the two excitations are assumed to be multiple. Further, five different tools, including the time history, the Poincaré map, and the sensitivity to initial conditions, are used to identify the motion form of the nonlinear vibration. Numerical results also show the characteristics of the quasiperiodic motion of the translating Timoshenko beam under an incommensurable re- lationship between the dual-frequency excitations.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 rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness ...A rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness of the model is verified via comparisons with the literatures and the FE models in ANSYS. The effects of the setting and pre-twisted angles on the dynamic responses of the RPICBM are analyzed. The results show that:(i) the increase in the setting or pre-twisted angle results in the increases in the first-order flapwise and torsional frequencies while the decrease in the first-order chordwise frequency under rotating conditions;(ii) a positive/negative setting angle leads to a positive/negative constant component, while a positive/negative pre-twisted angle leads to a negative/positive constant component;(iii) when the rotation speed is non-zero, the pre-twisted angle or non-zero setting angle will result in the coupled flapwise-chordwiseaxial- torsional vibration of the RPICBM under axial base excitation.展开更多
This paper focuses on a new finite-time convergence disturbance rejection control scheme design for a flexible Timoshenko manipulator subject to extraneous disturbances.To suppress the shear deformation and elastic os...This paper focuses on a new finite-time convergence disturbance rejection control scheme design for a flexible Timoshenko manipulator subject to extraneous disturbances.To suppress the shear deformation and elastic oscillation,position the manipulator in a desired angle,and ensure the finitetime convergence of disturbances,we develop three disturbance observers(DOs)and boundary controllers.Under the derived DOs-based control schemes,the controlled system is guaranteed to be uniformly bounded stable and disturbance estimation errors converge to zero in a finite time.In the end,numerical simulations are established by finite difference methods to demonstrate the effectiveness of the devised scheme by selecting appropriate parameters.展开更多
The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are...The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are studied for the response of a Timoshenko beam supported by a nonlinear foundation. The nonlinear Pasternak foundation is assumed to be cubic. Therefore, the effects of the shear deformable beams and the shear deformation of foundations are considered at the same time. The Galerkin method is utilized for discretizing the nonlinear partial dif- ferential governing equations of the forced vibration. The dynamic responses of Timoshenko beams are determined via the fourth-order Runge-Kutta method. Moreover, the effects of different truncation terms on the dynamic responses of a Timoshenko beam resting on a complex foundation are discussed. The numerical investigations shows that the dynamic response of Timoshenko beams supported by elastic foundations needs super high-order modes. Furthermore, the system parameters are compared to determine the dependence of the convergences of the Galerkin method.展开更多
The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The r...The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the eff ects of the Winkler coeffi cient, Pasternak coeffi cient and damping coeffi cient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary diff erential equations with variable coeffi cients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verifi ed through two numerical examples. The infl uences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with diff erent taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.展开更多
The optimal control of nonlinear systems has been studied for years by many researchers. However, the application of optimal control problem to nonlinear non-affine systems needs more attention. In this paper we propo...The optimal control of nonlinear systems has been studied for years by many researchers. However, the application of optimal control problem to nonlinear non-affine systems needs more attention. In this paper we propose an optimal control design technique for a class of nonlinear and control non-affine equations. The dynamic equations of a flexible shaft supported by a pair of active magnetic bearings (AMBs) are used as the nonlinear control non-affine equations. Mathematical model for the flexible beam is chosen to be the well known Timoshenko beam model, which takes rotary inertia and shear deformations into account, and it is assumed that the shaft is supported by two frictionless bearings at the ends. The effective control of such systems is extremely important for very high angular velocity shafts which are a feature of many modern machines. The control must be able to cope with unbalanced masses and hence be very robust. We shall approach the problem by discretising the Timoshenko beam model and using standard difference formulae to develop a finite-dimensional model of the system. Then we use a recently developed technique for controlling nonlinear systems by reducing the problem to a sequence of linear time-varying (LTV) systems. An optimal control designed for each approximating linear, time-varying system and recent results show that this method will converge uniformly on compact time intervals to the optimal solution.展开更多
Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanic...Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.展开更多
The excavation of foundation pit generates soil deformation around existing metro tunnel with shield driving method,which may lead to the deformation of tunnel lining.It is a challenge to evaluate the deformation of s...The excavation of foundation pit generates soil deformation around existing metro tunnel with shield driving method,which may lead to the deformation of tunnel lining.It is a challenge to evaluate the deformation of shield tunnel accurately and take measures to reduce the tunnel upward displacement as much as possible for geotechnical engineers.A new simplified analytical method is proposed to predict the longitudinal deformation of existing metro tunnel due to excavation unloading of adjacent foundation pit in this paper.Firstly,the additional stress of soils under vertical axisymmetric load in layered soil is obtained by using elastic multi-layer theory.Secondly,the metro tunnel is regarded as a Timoshenko beam supported by Winkler foundation so that the shear effect of tunnels can be taken into account.The additional stress acting on the tunnel due to excavation unloading in layered soil are compared with that in homogeneous soil.Additionally,the effectiveness of the analytical solution is verified via two actual cases.Moreover,parametric analysis is conducted to investigate the responses of the metro tunnel by considering such factors as the variation of subgrade coefficient,offset distance from the excavation center to tunnel longitudinal axis as well as equivalent shear stiffness.The proposed method can be used to provide theoretical basis for similar engineering project.展开更多
In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The init...In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.展开更多
In this paper, the partial projection finite element method is applied to the time-dependent problem-the damped vibrating Timoshenko beam model. It is proved that this method allows some new combinations of interpola...In this paper, the partial projection finite element method is applied to the time-dependent problem-the damped vibrating Timoshenko beam model. It is proved that this method allows some new combinations of interpolations for stress and displacement fields. When assuming that a smooth solution exists, we obtain optimal convergence rates with constants independent of the beam thickness. (Author abstract) 15 Refs.展开更多
The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, fl...The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.展开更多
One branch of structural health monitoring (SHM) utilizes dynamic response measurements to assess the structural integrity of civil infrastructures. In particular,modal frequency is a widely adopted indicator for stru...One branch of structural health monitoring (SHM) utilizes dynamic response measurements to assess the structural integrity of civil infrastructures. In particular,modal frequency is a widely adopted indicator for structural damage since its square is proportional to structural stiffness. However,it has been demonstrated in various SHM projects that this indicator is substantially affected by fluctuating environmental conditions. In order to provide reliable and consistent information on the health status of the monitored structures,it is necessary to develop a method to filter this interference. This study attempts to model and quantify the environmental influence on the modal frequencies of reinforced concrete buildings. Daily structural response measurements of a twenty-two story reinforced concrete building were collected and analyzed over a one-year period. The Bayesian spectral density approach was utilized to identify the modal frequencies of this building and it was clearly seen that the temperature and humidity fluctuation induced notable variations. A mathematical model was developed to quantify the environmental effects and model complexity was taken into consideration. Based on a Timoshenko beam model,the full model class was constructed and other reduced-order model class candidates were obtained. Then,the Bayesian modal class selection approach was employed to select the one with the most suitable complexity. The proposed model successfully characterizes the environmental influence on the modal frequencies. Furthermore,the estimated uncertainty of the model parameters allows for assessment of the reliability of the prediction. This study not only improves the understanding about the monitored structure,but also establishes a systematic approach for reliable health assessment of reinforced concrete buildings.展开更多
A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method.In the proposed model,the rotor was built with the ...A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method.In the proposed model,the rotor was built with the Timoshenko beam element,while the supports and bearing outer rings were modelled by the mass-centralized method.Meanwhile,the influences of the rotor’s gravity,unbalanced force and nonlinear bearing force were considered.The governing equations were solved by precise integration and the Runge-Kutta hybrid numerical algorithm.To verify the correctness of the modelling method,theoretical and experimental analysis is carried out by a rotor-bearing test platform,where the error rate between the theoretical and experimental studies is less than 10%.Besides that,the influence of the rubber damping ring on the dynamic properties of the rotor-bearing coupling system is also analyzed.The conclusions obtained are in agreement with the real-world deployment.On this basis,the bifurcation and chaos behaviors of the coupling system were carried out with rotational speed and rubber damping ring’s stiffness.The results reveal that as rotational speed increases,the system enters into chaos by routes of crisis,quasi-periodic and intermittent bifurcation.However,the paths of crisis,quasi-periodic bifurcation,and Hopf bifurcation to chaos were detected under the parameter of rubber damping ring’s stiffness.Additionally,the bearing gap affects the rotor system’s dynamic characteristics.Moreover,the excessive bearing gap will make the system’s periodic motion change into chaos,and the rubber damping ring’s stiffness has a substantial impact on the system motion.展开更多
The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the...The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.展开更多
To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy＇s higher order beam theory. The governing differentia...To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy＇s higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton＇s principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy＇s. A parametric study is also performed to investigate the effects of geometry and material parameters.展开更多
Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenk...Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions： hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.展开更多
文摘Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.
基金Project supported by the State Key Program of National Natural Science Foundation of China(No.11232009)the National Natural Science Foundation of China(Nos.11372171 and 11422214)
文摘This investigation focuses on the nonlinear dynamic behaviors in the trans- verse vibration of an axiMly accelerating viscoelastic Timoshenko beam with the external harmonic excitation. The parametric excitation is caused by the harmonic fluctuations of the axial moving speed. An integro-partial-differential equation governing the transverse vibration of the Timoshenko beam is established. Many factors are considered, such as viscoelasticity, the finite axial support rigidity, and the longitudinally varying tension due to the axial acceleration. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the governing equation. Based on the numerical solutions, the bifurcation diagrams are presented to study the effect of the external transverse excitation. Moreover, the frequencies of the two excitations are assumed to be multiple. Further, five different tools, including the time history, the Poincaré map, and the sensitivity to initial conditions, are used to identify the motion form of the nonlinear vibration. Numerical results also show the characteristics of the quasiperiodic motion of the translating Timoshenko beam under an incommensurable re- lationship between the dual-frequency excitations.
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Project supported by the National Natural Science Foundation of China(No.11772089)the Fundamental Research Funds for the Central Universities of China(Nos.N170308028 and N170306004)+1 种基金the Program for the Innovative Talents of Higher Learning Institutions of Liaoning of China(No.LR2017035)the Liaoning Revitalization Talents Program of China(No.XLYC1807008)
文摘A rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness of the model is verified via comparisons with the literatures and the FE models in ANSYS. The effects of the setting and pre-twisted angles on the dynamic responses of the RPICBM are analyzed. The results show that:(i) the increase in the setting or pre-twisted angle results in the increases in the first-order flapwise and torsional frequencies while the decrease in the first-order chordwise frequency under rotating conditions;(ii) a positive/negative setting angle leads to a positive/negative constant component, while a positive/negative pre-twisted angle leads to a negative/positive constant component;(iii) when the rotation speed is non-zero, the pre-twisted angle or non-zero setting angle will result in the coupled flapwise-chordwiseaxial- torsional vibration of the RPICBM under axial base excitation.
基金supported in part by National Natural Science Foundation of China(61803109)in part by the Innovative School Project of Education Department of Guangdong(2017KQNCX153)+3 种基金in part by the Science and Technology Planning Project of Guangzhou City(201904010494)in part by the Scientific Research Projects of Guangzhou Education Bureau(202032793)in part by the China Postdoctoral Science Foundation(2019M660463)in part by the Interdisciplinary Research Project for Young Teachers of University of Science and Technology Beijing(FRFIDRY-19-024)。
文摘This paper focuses on a new finite-time convergence disturbance rejection control scheme design for a flexible Timoshenko manipulator subject to extraneous disturbances.To suppress the shear deformation and elastic oscillation,position the manipulator in a desired angle,and ensure the finitetime convergence of disturbances,we develop three disturbance observers(DOs)and boundary controllers.Under the derived DOs-based control schemes,the controlled system is guaranteed to be uniformly bounded stable and disturbance estimation errors converge to zero in a finite time.In the end,numerical simulations are established by finite difference methods to demonstrate the effectiveness of the devised scheme by selecting appropriate parameters.
基金supported by the State Key Program of National Natural Science Foundation of China (10932006 and 11232009)Innovation Program of Shanghai Municipal Education Commission (12YZ028)
文摘The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are studied for the response of a Timoshenko beam supported by a nonlinear foundation. The nonlinear Pasternak foundation is assumed to be cubic. Therefore, the effects of the shear deformable beams and the shear deformation of foundations are considered at the same time. The Galerkin method is utilized for discretizing the nonlinear partial dif- ferential governing equations of the forced vibration. The dynamic responses of Timoshenko beams are determined via the fourth-order Runge-Kutta method. Moreover, the effects of different truncation terms on the dynamic responses of a Timoshenko beam resting on a complex foundation are discussed. The numerical investigations shows that the dynamic response of Timoshenko beams supported by elastic foundations needs super high-order modes. Furthermore, the system parameters are compared to determine the dependence of the convergences of the Galerkin method.
基金AHKJT of China under Grant Nos.1708085QE121 and 1808085ME147AHEDU of China under Grant No.TSKJ2017B13
文摘The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the eff ects of the Winkler coeffi cient, Pasternak coeffi cient and damping coeffi cient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary diff erential equations with variable coeffi cients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verifi ed through two numerical examples. The infl uences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with diff erent taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.
文摘The optimal control of nonlinear systems has been studied for years by many researchers. However, the application of optimal control problem to nonlinear non-affine systems needs more attention. In this paper we propose an optimal control design technique for a class of nonlinear and control non-affine equations. The dynamic equations of a flexible shaft supported by a pair of active magnetic bearings (AMBs) are used as the nonlinear control non-affine equations. Mathematical model for the flexible beam is chosen to be the well known Timoshenko beam model, which takes rotary inertia and shear deformations into account, and it is assumed that the shaft is supported by two frictionless bearings at the ends. The effective control of such systems is extremely important for very high angular velocity shafts which are a feature of many modern machines. The control must be able to cope with unbalanced masses and hence be very robust. We shall approach the problem by discretising the Timoshenko beam model and using standard difference formulae to develop a finite-dimensional model of the system. Then we use a recently developed technique for controlling nonlinear systems by reducing the problem to a sequence of linear time-varying (LTV) systems. An optimal control designed for each approximating linear, time-varying system and recent results show that this method will converge uniformly on compact time intervals to the optimal solution.
基金supported by the National Natural Science Foundation of China(31270989 and 11372162)the 973 Program of MOST(2010CB631005 and 2012CB934001)Tsinghua University(20121087991)
文摘Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.
基金Project(51568006)supported by the National Natural Science Foundation of ChinaProject(2018JJA160134)supported by the Natural Science Foundation of Guangxi Province,China。
文摘The excavation of foundation pit generates soil deformation around existing metro tunnel with shield driving method,which may lead to the deformation of tunnel lining.It is a challenge to evaluate the deformation of shield tunnel accurately and take measures to reduce the tunnel upward displacement as much as possible for geotechnical engineers.A new simplified analytical method is proposed to predict the longitudinal deformation of existing metro tunnel due to excavation unloading of adjacent foundation pit in this paper.Firstly,the additional stress of soils under vertical axisymmetric load in layered soil is obtained by using elastic multi-layer theory.Secondly,the metro tunnel is regarded as a Timoshenko beam supported by Winkler foundation so that the shear effect of tunnels can be taken into account.The additional stress acting on the tunnel due to excavation unloading in layered soil are compared with that in homogeneous soil.Additionally,the effectiveness of the analytical solution is verified via two actual cases.Moreover,parametric analysis is conducted to investigate the responses of the metro tunnel by considering such factors as the variation of subgrade coefficient,offset distance from the excavation center to tunnel longitudinal axis as well as equivalent shear stiffness.The proposed method can be used to provide theoretical basis for similar engineering project.
文摘In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.
文摘In this paper, the partial projection finite element method is applied to the time-dependent problem-the damped vibrating Timoshenko beam model. It is proved that this method allows some new combinations of interpolations for stress and displacement fields. When assuming that a smooth solution exists, we obtain optimal convergence rates with constants independent of the beam thickness. (Author abstract) 15 Refs.
基金Project supported by the National Natural Science Foundation of China(Nos.11672007,11402028,11322214,and 11290152)the Beijing Natural Science Foundation(No.3172003)the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education,Northeastern University(No.VCAME201601)
文摘The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.
基金Research Committee,University of Macao,China Under Grant No.RG077/07-08S/09R/YKV/FST
文摘One branch of structural health monitoring (SHM) utilizes dynamic response measurements to assess the structural integrity of civil infrastructures. In particular,modal frequency is a widely adopted indicator for structural damage since its square is proportional to structural stiffness. However,it has been demonstrated in various SHM projects that this indicator is substantially affected by fluctuating environmental conditions. In order to provide reliable and consistent information on the health status of the monitored structures,it is necessary to develop a method to filter this interference. This study attempts to model and quantify the environmental influence on the modal frequencies of reinforced concrete buildings. Daily structural response measurements of a twenty-two story reinforced concrete building were collected and analyzed over a one-year period. The Bayesian spectral density approach was utilized to identify the modal frequencies of this building and it was clearly seen that the temperature and humidity fluctuation induced notable variations. A mathematical model was developed to quantify the environmental effects and model complexity was taken into consideration. Based on a Timoshenko beam model,the full model class was constructed and other reduced-order model class candidates were obtained. Then,the Bayesian modal class selection approach was employed to select the one with the most suitable complexity. The proposed model successfully characterizes the environmental influence on the modal frequencies. Furthermore,the estimated uncertainty of the model parameters allows for assessment of the reliability of the prediction. This study not only improves the understanding about the monitored structure,but also establishes a systematic approach for reliable health assessment of reinforced concrete buildings.
基金Projects(51775277,51775265)supported by the National Natural Science Foundation of ChinaProject(190624DF01)supported by Nanjing University of Aeronautics and Astronautics Short Visiting Program,China。
文摘A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method.In the proposed model,the rotor was built with the Timoshenko beam element,while the supports and bearing outer rings were modelled by the mass-centralized method.Meanwhile,the influences of the rotor’s gravity,unbalanced force and nonlinear bearing force were considered.The governing equations were solved by precise integration and the Runge-Kutta hybrid numerical algorithm.To verify the correctness of the modelling method,theoretical and experimental analysis is carried out by a rotor-bearing test platform,where the error rate between the theoretical and experimental studies is less than 10%.Besides that,the influence of the rubber damping ring on the dynamic properties of the rotor-bearing coupling system is also analyzed.The conclusions obtained are in agreement with the real-world deployment.On this basis,the bifurcation and chaos behaviors of the coupling system were carried out with rotational speed and rubber damping ring’s stiffness.The results reveal that as rotational speed increases,the system enters into chaos by routes of crisis,quasi-periodic and intermittent bifurcation.However,the paths of crisis,quasi-periodic bifurcation,and Hopf bifurcation to chaos were detected under the parameter of rubber damping ring’s stiffness.Additionally,the bearing gap affects the rotor system’s dynamic characteristics.Moreover,the excessive bearing gap will make the system’s periodic motion change into chaos,and the rubber damping ring’s stiffness has a substantial impact on the system motion.
基金supported by the National Natural Science Foundation of China(Nos.11772182 and90816001)
文摘The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.
基金Project supported by the National High Technology Research and Development Program of China(No.2009AA032303-2)
文摘To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy＇s higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton＇s principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy＇s. A parametric study is also performed to investigate the effects of geometry and material parameters.
基金the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic
文摘Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions： hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.
