With the nonlinear material constitutive relation was considered,the single-freedom vibration control equation of nonlinear material cable under concentrated load was presented. And then the vibration displacement of cable was analyzed by Fourier-series,the formula for its vibration was deduced with L-P method,finally the vibrational peculiarity between nonlinear material and linear material was compared.

In process of finding the Fourier-series expansion of the function, usually get some different result,for example the sine-series and the cosine-series. To choice proper examples and explain them exactly in teaching,it will play important role on the students comprehend the meaning of this type of topic and the not uniqueness of Fourier series of the function.

The generalized Fourier-series method was used to derive the impact responses formula of an unrestrained planar frame structure when subjected to an impact of a moving rigid-body. By using these formula,the analytic solutions of dynamic responses of the contact-impact system can be obtained.

The Fourier-series method is adopted to analyze the responses of an unrestrained bar under a rigid axial impact. The rigid response of the bar is separated directly from its total response of the bar and the relationship and affecting factors between the rigid displacement and elastic displacement of the bar is investigated.

For simplicity, however, all numerial results presented and discussed throughout paper are dealing with a single harmonic of such Fourier-series expansions.

This is achieved by suitably expanding the lateral surface boundary conditions in the form of appropriate Fourier-series.

The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it.

The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule.

Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines.

Using the x,y,p coordinates,and neglecting the small terms in thesteady state vorticity equation, we have(?)taking two layer baroclinic model,as Sawyer and Bushby,we let(?)in whichA(p)=(p_0+p_1-2p)/(p_0-p_1),(3)where p_0=1000mb and p_1=200mb.Introducing (2) to (1) and integrating from p=p_0 to p=p_1 under thefollowing bundary conditions(?)We obtain(?)where H=(RT_0)/g is the height of the homogeneous atmosphere,and η=η(x,y) is the topography of the earth's surface.For the distribution of vertical velocity,we...

Using the x,y,p coordinates,and neglecting the small terms in thesteady state vorticity equation, we have(?)taking two layer baroclinic model,as Sawyer and Bushby,we let(?)in whichA(p)=(p_0+p_1-2p)/(p_0-p_1),(3)where p_0=1000mb and p_1=200mb.Introducing (2) to (1) and integrating from p=p_0 to p=p_1 under thefollowing bundary conditions(?)We obtain(?)where H=(RT_0)/g is the height of the homogeneous atmosphere,and η=η(x,y) is the topography of the earth's surface.For the distribution of vertical velocity,we assume ω(x,y,p)=B(p) (?)(x,y)+C(p) ω_0(x,y),(5)whereB(p)=1-A(p)~2,C(p)=(p-p_1)/(p_0-p_1).(6)Introducing (2) and (5) to (1),multiplying A(p),and then integratingit with respect to p from p_0 to p_1,we find(?)Using again the relations of (2) and (5) and the geostrophic approximationin (Ⅰ)(1),taking steady state and then eliminating (?) from it with (10),we have(?)whereC_1=[R/(gc_p)]ln[p_0/(?)],M=[(4f~2)/(p_0-p_1)][1/(C_2RΓ_p)],C_2=(1/2)[(p_0+3p_1)/(p_0-p_1)]-[(4p_0p_1)/(p_0-p_1)~2]ln[(2p_0)/(p_0+p_1)].The motion is considered as consisting of small perturbations superim-posed upon a basic zonal current (?)+A(p)U_T,and taking geostrophic app-roximation,wo obtained the two required equations:(?)in which β is the variation of Coriolis parameter with latitude.We used the equation (9) and (10) to a rectangular reigon with a mar-ginal cyclicity and with a length equal to that of the latitud circle at 45°and take 1 day as the time unit and the radius of 45° latitude circle aslength unit.By means of Fourier analysis,the solution may be written inthe following form(?)where the topographical influence function Φ_0 and the heating influencefunction Φ_h are respectively defined by the Fourier series (?)As the equation was linearzed, we can put Q_m=0 to find the topographiceffect, and put η=0 to find the effect of heat sources and sinks;whenboth Q_m≠0 and η≠0,we can find the combined influences of heat sour-ces and sinks and topography.For case of winter,we take (?)=15m/s and U_t=13m/s.In the unitsdescribed above,we find (?)=2.87×10~(-1),β=2π,M=231,β/(?)=21.9.We have constructed separately the flow patterns respectively due tothe thermal effect,the topographical effect and the combined effect of themon the westerlies.By examining these patterns we may state the followingpoints on the formation of mean troughs and ridges and jet stream:(1)For the case Q_m=0.There is a trough to the downstream ofmountains and a ridge over the mountains created by the forced ascent ofthe westerly current over the topography.The Tibetan Platean is importanton the formation of the Asiatic mean troughs and ridges.(fig.1) Consi-dering the combined effect of the mountains of Rocky and Greenland,theposition and intensity of the computed mean trough and ridge (fig.6) iswell in agreement with the observation.(fig.4,10) However the effect ofthe forced ascent of the westerly current over the mountains does notgive a jet stream.(2)For the case η=0.Through dynamic processes the large-scaleheat sources and sinks show even larger influences than that of the moun-tains on the midtropospheric flow.The 500 mb trough is located west ofthe heat source,and the ridge west of the heat sink.The jet stream isformed at the south-western part of the heat source.The perturbationpattern of heating and cooling (fig.2,7) is in better agreement with theobserved one than that due to topography.(3)The flow pattern obtained by considering both heat sources andsinks and topography (fig.3,9) is in good agreement with the observedones.The agreement with observation is much better than any one of thefactors considered alone.

Denote by σ_n~α(f; x) the n-th Cesàro mean of order a for the Fourier series of f(x) and write R_N(f,α;x)=sum from n=N to ∞ |σ_n~α(f;x)-σ_(n+1)~α(f;x)|. The object of this note is to estimate R_N(f,α;x), the main results obtained can be stated as follows: Theorem 1. If (x)∈Lip 1, then the relation holds uniformly in x. Theorem 2. If f (x)∈Lip 1, then for 0<α≤1, the relation R_N(f,α;x)=0((log N)/N~α) holds uniformly in x. Moreover, for almost all x, there give Theorem 3. If f (x)∈kw~((r))(r<1), then there holds...

Denote by σ_n~α(f; x) the n-th Cesàro mean of order a for the Fourier series of f(x) and write R_N(f,α;x)=sum from n=N to ∞ |σ_n~α(f;x)-σ_(n+1)~α(f;x)|. The object of this note is to estimate R_N(f,α;x), the main results obtained can be stated as follows: Theorem 1. If (x)∈Lip 1, then the relation holds uniformly in x. Theorem 2. If f (x)∈Lip 1, then for 0<α≤1, the relation R_N(f,α;x)=0((log N)/N~α) holds uniformly in x. Moreover, for almost all x, there give Theorem 3. If f (x)∈kw~((r))(r<1), then there holds uniformly good, for almost every x, there holds R_N(f,α;x)=O(1/N~r),α=1 or α≥2. Furthermore these results can not be improved without changing the conditions.

§1.导言设f(x)～1/2α_0+sum from n=1 to ∞(α_ncos nx++b_nsin nx),帕蒂于[1]中证明了: 定理A.设f(x)是一个周期2π的可积周期函数。{λ_n}是一个凸的数列,它满足∑n~(-1)λ_n<∞。则当x_0是f(x)的勒贝格点时,级数1/2α_0λ_0+sum from n=1 to ∞λ_n(α_ncos nx_0+b_nsin nx_0)是

The shallow elliptical shell has different curvature in two principal directions, and is more complicate in analysis than spherical shell. Practical methods are available to compute the maximum bending moments and membrane direct forces when the shell is uniformly loaded. But from the viewpoint of shell design, the effective method to determine maximum twisting moments and membrane shear forces about the corners of shell are urgently needed. In the present paper, using the technique of substituting a double...

The shallow elliptical shell has different curvature in two principal directions, and is more complicate in analysis than spherical shell. Practical methods are available to compute the maximum bending moments and membrane direct forces when the shell is uniformly loaded. But from the viewpoint of shell design, the effective method to determine maximum twisting moments and membrane shear forces about the corners of shell are urgently needed. In the present paper, using the technique of substituting a double Fourier series by a double Fourier integral, simple formulas for the twisting moments and for the membrane shear forces are obtained readily. The accuracy of formulas and the rapid convergency of the series involved are fully discussed and proved to be satisfied.