Applying the Principle of Variables to Solve the Problems of Forced Vibration of the plate with three clamped and the other free with concentrated load

. In this paper, with the principle of least action with variables to solve the problems of forced vibration of the Rectangular plate with three clamped and the other free with concentrated load, and the stable solution can be worked out. We can compare the results with the literate; it also can be proved to be true. So the results by calculating not only it have important academic value, but also it can be directly referred in the actual work.


Introduction
Curved rectangular sheet has been widely used in engineering practice. when calculating the stability vibration and bending of the sheet based on sheet classical theory, there will be some errors. In the previous solution, it is difficult to find an easy displacement functions to solve it. The principle of least action mixed variables applied in article [1] requires only weak displacement, that is, displacement should meet the requirements of the strain-displacement in advance, instead of boundary conditions. This eliminates the need to make displacement hypothesis. the total potential energy of mixed variables can be set up according to the actual boundary condition of the curved rectangular sheet, thus obtained the steady-state solution of forced vibration can be obtained. This solution overcomes the classical solution of the complex calculation process and the difficulty to solve certain issues and other limitations [2][3][4][5] .

The basic equation
) ( sin sin To speed up the convergence rate and eliminate deflection and moment magnitude of the triangular series representation of the boundary that appears in the first category discontinuity, it also must points deflection surface equation for the amplitude of the triangular series and hyperbolic functions expressed mixed form.

Numerical Analysis
So we get four sets of infinite simultaneous equations (23)-(25) or (26)-(28). Remove restricted item, Solutions for the A n , B n , C m and d m . And then according to the formula (1) to (4) to obtain the moment and deflection magnitude of the amplitude. In particular, take each items of A n , B n ,C m and d m , and assuming harmonic loads concentrated on midpoint board programming calculated for different values obtained are shown in table 1 deflection magnitude of moment magnitude and fixed side edge moment , and as shown in Figure 2 and Figure  3. Discuss: (1) The issue of convergence coefficient. Since the load and structural balance in the direction axis for symmetric, so there are An=Bn, C m =d m =0(m=2,4...), C m and d m that is actually only take eight. And calculated that, An convergence of magnitude from 10 0 to 10 -2 magnitude, C m magnitude from 10 0 to converge to 10 -4 magnitude, d m magnitude from 10 -2 to 10 -5 convergence of magnitude, than the uniform load harmonic convergence is better.
(2) With regard to the distribution of M x0 . It is found, When 8 . 0 11 Z Z , M x0 larger value of the free edges will appear near the end. This suggests that, as the load frequency close to the natural frequency and a significant increase in the magnitude of the impact on its adjacent sides of the free edge of the moment.

Conclusion
(1) In this paper, mixed variables method for solving a rectangular plate with three fixed and one free vibration in any concentrated harmonic loads obtain the steadystate solution by forced vibration.
(2) Mix variables method is a simple vibration, universal, effective method to solve problem of forced vibration of bending a rectangular sheet.
(3) The results obtained by mixed variables method is correct, it may be practical engineering directly.