According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of th...According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.展开更多
The relativistic external shock model of gamma-ray burst （GRB） afterglows has been estab- lished with five free parameters, i.e., the total kinetic energy E, the equipartition parameters for electrons ee and for the...The relativistic external shock model of gamma-ray burst （GRB） afterglows has been estab- lished with five free parameters, i.e., the total kinetic energy E, the equipartition parameters for electrons ee and for the magnetic field eB, the number density of the environment n and the index of the power- law distribution of shocked electrons p. A lot of modified models have been constructed to consider the variety of GRB afterglows, such as： the wind medium environment by letting n change with radius, the energy injection model by letting kinetic energy change with time and so on. In this paper, by as- suming all four parameters （except p） change with time, we obtain a set of formulas for the dynamics and radiation, which can be used as a reference for modeling GRB afterglows. Some interesting results are obtained. For example, in some spectral segments, the radiated flux density does not depend on the number density or the profile of the environment. As an application, through modeling the afterglow of GRB 060607A, we find that it can be interpreted in the framework of the time dependent parameter model within a reasonable range.展开更多
基金Project supported by the National Natural Science Foundation of China (No.19872076)
文摘According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.
基金supported by the National Basic Research Program of China(973 Program,Grant 2014CB845800)the National Natural Science Foundation of China(Grant Nos.11773010,U1431124 and U1738132)
文摘The relativistic external shock model of gamma-ray burst （GRB） afterglows has been estab- lished with five free parameters, i.e., the total kinetic energy E, the equipartition parameters for electrons ee and for the magnetic field eB, the number density of the environment n and the index of the power- law distribution of shocked electrons p. A lot of modified models have been constructed to consider the variety of GRB afterglows, such as： the wind medium environment by letting n change with radius, the energy injection model by letting kinetic energy change with time and so on. In this paper, by as- suming all four parameters （except p） change with time, we obtain a set of formulas for the dynamics and radiation, which can be used as a reference for modeling GRB afterglows. Some interesting results are obtained. For example, in some spectral segments, the radiated flux density does not depend on the number density or the profile of the environment. As an application, through modeling the afterglow of GRB 060607A, we find that it can be interpreted in the framework of the time dependent parameter model within a reasonable range.
