This study investigates the nonlinear dynamic properties of rotating functionally graded sandwich rectangular plates in a thermal environment.The nonlinear vibration equations for a rotating metal-ceramic functionally...This study investigates the nonlinear dynamic properties of rotating functionally graded sandwich rectangular plates in a thermal environment.The nonlinear vibration equations for a rotating metal-ceramic functionally graded sandwich rectangular plate in a thermal environment are derived using classical thin plate theory and Hamilton’s principle,considering geometric nonlinearity,temperature-dependent material properties,and power law distribution of components through the thickness.With cantilever boundary conditions,the flexural nonlinear differential equations of the rectangular sandwich plate are obtained via the Galerkin method.Since the natural vibration differential equations exhibit nonlinear characteristics,the multiscale method is employed to derive the expression for nonlinear natural frequency.An example analysis reveals how the natural frequency of a functionally graded sandwich rectangular plate varies with rotational speed and temperature.Results show that the nonlinear/linear frequency ratio increases with rotational angular velocity Ω and thickness-to-length ratio h/a,follows a cosine-like periodic pattern with the setting angle,and shows a sharp decrease followed by a rapid increase with increasing width-to-length ratio b/a.The derived analytical solutions for nonlinear frequency provide valuable insights for assessing the dynamic characteristics of functionally graded structures.展开更多
In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates w...In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.展开更多
The natural frequencies, complex modes and critical speeds of an axially moving rectangular plate, which is partially immersed in a fluid and subjected to a pretension, are investigated. The effects of free surface wa...The natural frequencies, complex modes and critical speeds of an axially moving rectangular plate, which is partially immersed in a fluid and subjected to a pretension, are investigated. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The subsection functions are used to describe the discontinuous characteristics of the system due to partial immersion. The classical thin plate theory is adopted to formulate the equations of motion of a vibrating plate. The velocity potential and Bernoulli＇s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be equivalent to the added mass on the plate. The effects of distance ratio, moving speed, immersed-depth ratio, boundary conditions, stiffness ratio and aspect ratio of the plate as well as the fluid-plate density ratios on the free vibrations of the moving plate-fluid system are investigated.展开更多
In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection bein...In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.展开更多
This paper investituites the problems of lateral buckling of rectangular plates. In the text we discuss the minimum critical load of the lateral buckling occurring on under a concentrated force, uniformly distributed ...This paper investituites the problems of lateral buckling of rectangular plates. In the text we discuss the minimum critical load of the lateral buckling occurring on under a concentrated force, uniformly distributed load and the concentrated couples, respectively. The energy method is used in this article.展开更多
In this paper, on the basis of von Karman large deflection equations and its double trigonometric series solution, we present a simple, fast and effective iteration algorithm for solving simply-supported rectangular p...In this paper, on the basis of von Karman large deflection equations and its double trigonometric series solution, we present a simple, fast and effective iteration algorithm for solving simply-supported rectangular plate subjected to biaxial compression.展开更多
In this paper, an exact solution for an uniformly loaded rectangular plate with two adjacent edges clamped, one edge simply supported and the other edge free, was given by using the concept of generalized simply suppo...In this paper, an exact solution for an uniformly loaded rectangular plate with two adjacent edges clamped, one edge simply supported and the other edge free, was given by using the concept of generalized simply supported edges and superposition method. The numerical results were given for the deflections along the free edge and bending moments along the clamped edges of a square plate.展开更多
This paper studies transverse vibration of rectangular plates with two opposite edges simply supperted other two edges arbitrarily supported and free edges elaslically supported at points,A highly accurate solution is...This paper studies transverse vibration of rectangular plates with two opposite edges simply supperted other two edges arbitrarily supported and free edges elaslically supported at points,A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular platen elaslically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourter series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given.Inherent frequencies and mode shape with any accutacy can be gained.展开更多
In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundam...In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.展开更多
This paper discusses by energy theorem the methodof approximate computation for the lowest eigenfrequencies of rechmguhir plates,on which there are symmetrical concentrated masses,supported at corner points,In the cas...This paper discusses by energy theorem the methodof approximate computation for the lowest eigenfrequencies of rechmguhir plates,on which there are symmetrical concentrated masses,supported at corner points,In the case of seseral concentrated masses,by using the prineiple of superposition we mayfiml the reduneed coefficients of masses comveniently.llence we can louain the lowest eigenfrequencies of thin plates.In the paper a good mamy mmerical caleuhting eximples are inustrated展开更多
The present paper investigates several problems for unsymmetrically lateral instability of rectangular plates by the energy method. In the text we discuss the minimum critical load of rectangular plates which possess ...The present paper investigates several problems for unsymmetrically lateral instability of rectangular plates by the energy method. In the text we discuss the minimum critical load of rectangular plates which possess the unsymmetrical supporters and to which the lateral buckling occurs unsymmetrically under a concentrated force, uniformly distributed load and the concentrated couples respectively.展开更多
All possible exact solutions are successfully obtained in terms of 10 sets of distinct eigensolutions for the free in-plane vibration of isotropic rectangular plates. The plates have simply supported condition at two ...All possible exact solutions are successfully obtained in terms of 10 sets of distinct eigensolutions for the free in-plane vibration of isotropic rectangular plates. The plates have simply supported condition at two opposite edges and any combination of classical boundary conditions at the other two edges. The exact solutions are validated through both mathematical proof and comparisons with the solutions of differential quadrature method. Some unusual phenomena are revealed in free in-plane vibrations of rectangular plates due to one of the eigenvalues being zero. This work constitutes an improved version of very recent corresponding work by the same authors lint. J. Mech. Sci., 2009, 51： 246-255]. Both the solution forms and solving procedures in the previous work are substantially simplified. Some new results are also given, which are useful for validation purpose in future.展开更多
This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the...This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.展开更多
A method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges. By introdu...A method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges. By introducing two displacement functions and two stress functions, two independent state equations were constructed based on the three-dimensional elasticity equations for transverse isotropy. The original differential equations are thus decoupled with the order reduced that will facilitate obtaining solutions of various problems. For the simply supported rectangular plate, two relations between the state variables at the tap and bottom surfaces were established. In particular, for the free vibration ( stability) problem, it is found that there exist two independent classes: One corresponds to the pure in-plane vibration (stability) and the other to the general bending vibration (stability). Numerical examples are finally presented and the effects of same parameters are discussed.展开更多
Under internal blast loading,the response of a beam or plate is highly correlated with the phenomenon of saturated impulse,which governs the deflection of the structure.This paper aims to investigate the phenomenon of...Under internal blast loading,the response of a beam or plate is highly correlated with the phenomenon of saturated impulse,which governs the deflection of the structure.This paper aims to investigate the phenomenon of saturated impulse for fully clamped rectangular plates subjected to internal blast loading.Based on the rigid,perfectly plastic assumption,the relationship between saturation duration and saturation deflection is derived.Influences of the peak shock wave,the duration of shock wave and the peak quasi-static pressure loading on saturation duration and saturation deflection are discussed.It is found that there is a critical duration for the internal blast impulse to reach saturation,and beyond this duration,the deflection of plate will no longer increase as the loading increases further.The saturation deflection and saturation duration both exhibit regular variation patterns with the changes of the dimensionless peak shock wave,the duration of shock wave and the peak quasi-static pressure loading.展开更多
On the basis of Karman's theory of thin plates with large deflection, the Boltzmann law on linear viscoelastic materials and the mathematical model of dynamic analysis on viscoelastic thin plates, a set of nonline...On the basis of Karman's theory of thin plates with large deflection, the Boltzmann law on linear viscoelastic materials and the mathematical model of dynamic analysis on viscoelastic thin plates, a set of nonlinear integro partial differential equations is first presented by means of a structural function introduced in this paper. Then, by using the Galerkin technique in spatial field and a backward difference scheme in temporal field, the set of nonlinear integro partial differential equations reduces to a system of nonlinear algebraic equations. After solving the algebraic equations, the buckling behavior and multiple equilibrium states can be obtained.展开更多
The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for t...The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.展开更多
Full-range analysis for the buckling and post buck ling at rectangular plates under in-plane compression has been made by perturbation technique which takes deflection as its perturbation parameter.In this paper the e...Full-range analysis for the buckling and post buck ling at rectangular plates under in-plane compression has been made by perturbation technique which takes deflection as its perturbation parameter.In this paper the effects of initial geometric imperfection on the postbuc kling behavior of plates have been discussed. It is seen that the effect of initial imperfection on the inelastic postbuckling oj plates is sensitive. By comparison, it is found that the theoretical results of this paper are in good agreement with experiments.展开更多
The nonlinear dynamic behaviors of viscoelastic rectangular plates including the damage effects under the action of a transverse periodic load were studied. Using the von Karman equations, Boltzmann superposition prin...The nonlinear dynamic behaviors of viscoelastic rectangular plates including the damage effects under the action of a transverse periodic load were studied. Using the von Karman equations, Boltzmann superposition principle and continuum damage mechanics, the nonlinear dynamic equations in terms of the mid_plane displacements for the viscoelastic thin plates with damage effect were derived. By adopting the finite difference method and Newmark method, these equations were solved. The results were compared with the available data. In the numerical calculations, the effects of the external loading parameters and geometric dimensions of the plate on the nonlinear dynamic responses of the plate were discussed. Research results show that the nonlinear dynamic response of the structure will change remarkably when the damage effect is considered.展开更多
In this paper,applying the method of the reciprocal theorem,we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic...In this paper,applying the method of the reciprocal theorem,we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates,the figures and tables of number value of bending moment and the deflection amplitudes as well.展开更多
基金supported by the National Natural Science Foundation of China(No.11772090).
文摘This study investigates the nonlinear dynamic properties of rotating functionally graded sandwich rectangular plates in a thermal environment.The nonlinear vibration equations for a rotating metal-ceramic functionally graded sandwich rectangular plate in a thermal environment are derived using classical thin plate theory and Hamilton’s principle,considering geometric nonlinearity,temperature-dependent material properties,and power law distribution of components through the thickness.With cantilever boundary conditions,the flexural nonlinear differential equations of the rectangular sandwich plate are obtained via the Galerkin method.Since the natural vibration differential equations exhibit nonlinear characteristics,the multiscale method is employed to derive the expression for nonlinear natural frequency.An example analysis reveals how the natural frequency of a functionally graded sandwich rectangular plate varies with rotational speed and temperature.Results show that the nonlinear/linear frequency ratio increases with rotational angular velocity Ω and thickness-to-length ratio h/a,follows a cosine-like periodic pattern with the setting angle,and shows a sharp decrease followed by a rapid increase with increasing width-to-length ratio b/a.The derived analytical solutions for nonlinear frequency provide valuable insights for assessing the dynamic characteristics of functionally graded structures.
文摘In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.
基金Project supported by the National Natural Science Foundation of China(Nos.11302046 and 11172063)
文摘The natural frequencies, complex modes and critical speeds of an axially moving rectangular plate, which is partially immersed in a fluid and subjected to a pretension, are investigated. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The subsection functions are used to describe the discontinuous characteristics of the system due to partial immersion. The classical thin plate theory is adopted to formulate the equations of motion of a vibrating plate. The velocity potential and Bernoulli＇s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be equivalent to the added mass on the plate. The effects of distance ratio, moving speed, immersed-depth ratio, boundary conditions, stiffness ratio and aspect ratio of the plate as well as the fluid-plate density ratios on the free vibrations of the moving plate-fluid system are investigated.
文摘In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.
文摘This paper investituites the problems of lateral buckling of rectangular plates. In the text we discuss the minimum critical load of the lateral buckling occurring on under a concentrated force, uniformly distributed load and the concentrated couples, respectively. The energy method is used in this article.
文摘In this paper, on the basis of von Karman large deflection equations and its double trigonometric series solution, we present a simple, fast and effective iteration algorithm for solving simply-supported rectangular plate subjected to biaxial compression.
文摘In this paper, an exact solution for an uniformly loaded rectangular plate with two adjacent edges clamped, one edge simply supported and the other edge free, was given by using the concept of generalized simply supported edges and superposition method. The numerical results were given for the deflections along the free edge and bending moments along the clamped edges of a square plate.
文摘This paper studies transverse vibration of rectangular plates with two opposite edges simply supperted other two edges arbitrarily supported and free edges elaslically supported at points,A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular platen elaslically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourter series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given.Inherent frequencies and mode shape with any accutacy can be gained.
文摘In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.
文摘This paper discusses by energy theorem the methodof approximate computation for the lowest eigenfrequencies of rechmguhir plates,on which there are symmetrical concentrated masses,supported at corner points,In the case of seseral concentrated masses,by using the prineiple of superposition we mayfiml the reduneed coefficients of masses comveniently.llence we can louain the lowest eigenfrequencies of thin plates.In the paper a good mamy mmerical caleuhting eximples are inustrated
文摘The present paper investigates several problems for unsymmetrically lateral instability of rectangular plates by the energy method. In the text we discuss the minimum critical load of rectangular plates which possess the unsymmetrical supporters and to which the lateral buckling occurs unsymmetrically under a concentrated force, uniformly distributed load and the concentrated couples respectively.
基金supported by the China Postdoctoral Science Foundation (No. 20100470179)
文摘All possible exact solutions are successfully obtained in terms of 10 sets of distinct eigensolutions for the free in-plane vibration of isotropic rectangular plates. The plates have simply supported condition at two opposite edges and any combination of classical boundary conditions at the other two edges. The exact solutions are validated through both mathematical proof and comparisons with the solutions of differential quadrature method. Some unusual phenomena are revealed in free in-plane vibrations of rectangular plates due to one of the eigenvalues being zero. This work constitutes an improved version of very recent corresponding work by the same authors lint. J. Mech. Sci., 2009, 51： 246-255]. Both the solution forms and solving procedures in the previous work are substantially simplified. Some new results are also given, which are useful for validation purpose in future.
文摘This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.
文摘A method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges. By introducing two displacement functions and two stress functions, two independent state equations were constructed based on the three-dimensional elasticity equations for transverse isotropy. The original differential equations are thus decoupled with the order reduced that will facilitate obtaining solutions of various problems. For the simply supported rectangular plate, two relations between the state variables at the tap and bottom surfaces were established. In particular, for the free vibration ( stability) problem, it is found that there exist two independent classes: One corresponds to the pure in-plane vibration (stability) and the other to the general bending vibration (stability). Numerical examples are finally presented and the effects of same parameters are discussed.
基金The authors would like to thank the support from the National Natural Science Foundation of China under Grant No.11802030.
文摘Under internal blast loading,the response of a beam or plate is highly correlated with the phenomenon of saturated impulse,which governs the deflection of the structure.This paper aims to investigate the phenomenon of saturated impulse for fully clamped rectangular plates subjected to internal blast loading.Based on the rigid,perfectly plastic assumption,the relationship between saturation duration and saturation deflection is derived.Influences of the peak shock wave,the duration of shock wave and the peak quasi-static pressure loading on saturation duration and saturation deflection are discussed.It is found that there is a critical duration for the internal blast impulse to reach saturation,and beyond this duration,the deflection of plate will no longer increase as the loading increases further.The saturation deflection and saturation duration both exhibit regular variation patterns with the changes of the dimensionless peak shock wave,the duration of shock wave and the peak quasi-static pressure loading.
文摘On the basis of Karman's theory of thin plates with large deflection, the Boltzmann law on linear viscoelastic materials and the mathematical model of dynamic analysis on viscoelastic thin plates, a set of nonlinear integro partial differential equations is first presented by means of a structural function introduced in this paper. Then, by using the Galerkin technique in spatial field and a backward difference scheme in temporal field, the set of nonlinear integro partial differential equations reduces to a system of nonlinear algebraic equations. After solving the algebraic equations, the buckling behavior and multiple equilibrium states can be obtained.
基金Project supported by the National Natural Science Foundation of China (Grant No.10872163)the Natural Science Foundation of Education Department of Shaanxi Province (Grant No.08JK394)
文摘The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.
文摘Full-range analysis for the buckling and post buck ling at rectangular plates under in-plane compression has been made by perturbation technique which takes deflection as its perturbation parameter.In this paper the effects of initial geometric imperfection on the postbuc kling behavior of plates have been discussed. It is seen that the effect of initial imperfection on the inelastic postbuckling oj plates is sensitive. By comparison, it is found that the theoretical results of this paper are in good agreement with experiments.
基金ProjectsupportedbytheNationalNaturalScienceFoundationofChina (No .1 0 2 72 0 2 4)
文摘The nonlinear dynamic behaviors of viscoelastic rectangular plates including the damage effects under the action of a transverse periodic load were studied. Using the von Karman equations, Boltzmann superposition principle and continuum damage mechanics, the nonlinear dynamic equations in terms of the mid_plane displacements for the viscoelastic thin plates with damage effect were derived. By adopting the finite difference method and Newmark method, these equations were solved. The results were compared with the available data. In the numerical calculations, the effects of the external loading parameters and geometric dimensions of the plate on the nonlinear dynamic responses of the plate were discussed. Research results show that the nonlinear dynamic response of the structure will change remarkably when the damage effect is considered.
文摘In this paper,applying the method of the reciprocal theorem,we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates,the figures and tables of number value of bending moment and the deflection amplitudes as well.
