THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM

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THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM

Co. H. Tran - Phong . T . Ngo

Faculty of Mathematics, University of Natural Sciences - VNU-HCM

coth123@math.com & cohtran@math.com ntphong_6@yahoo.com

Copyright 2007

August 16 2007

NOTE:

This worksheet demonstrates Maple's capabilities in researching the numerical and graphical solution of the variable boundary problem of a thick-walled cylinder of material enclosed in a thin metallic shell .

All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .

Abstract :

The relations between stress and strain in linear viscoelastic theory are discussed from the viewpoint of application to problems of stress analysis. This consideration includes some important differences from the estimation of linear viscoelastic laws for the representation of material properties , and the integral operators expressing the creep function or relaxation function can be applied . By using of the differential operator for the relation between stress and strain it is usually most convenient to solve some problems which have the variable boundary .

REFERENCES

[1] ю.н. РАБОТНОВ , Пoлзyчесть элементов конструкций , издательство “наука” , MOCKBA 1966

[2] Lee E .H ., Radok J.R.M., Woodward W.B., Stress Analysis for Linear Viscoelastic Materials , Trans. Soc. Rheol., 3, 41-59 (1959)

Disclaimer: While every effort has been made to validate the solutions in this worksheet, the author is not responsible for any errors contained and are not liable for any damages resulting from the use of this material.

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The Solution Of A Variable Boundary Problem

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Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

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 Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
 Pure mathematics is, in its way, the poetry of logical ideas. 

 Albert Einstein .

THE RELAXATION FUNCTION PROBLEM .

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The Relaxation function problem of an orthotropic cylinder .

Co. H. Tran. Faculty of Mathematics, University of Natural Sciences - VNU-HCMcoth123@math.com & cohtran@math.com
Copyright 2007
June 06 2007

NOTE:This worksheet demonstrates Maple's capabilities in researching the numerical and graphical solution of the relaxation function problem of an orthotropic cylinder .
All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .

Use Maple 10


Abstract
The worksheet presents some thoughts about the plane strain problem of the viscous orthotropic composite materials cylinder under internal and external pressure with
respect to using the direct method . To compute the interior stress , from the elastic solution we use the correspondence principle and the inverse Laplace transform .

1. Analysis of the composite orthotropic cylinder :


We examine an orthotropic viscoelastic composite material cylinder which has the horizontal section within limit of 2 circles : r = a , r = b ( a <>

2. Direct method : The direct method is an approximate inversion technic based on the direct relation between the time dependence and the transformed solution . If the plot of the viscoelastic solution has small curvature when plotted with variables logt then : (1) where C is Euler's constant .. (1) is exact if , is proportional to logt . (1) can be rewritten : (2) Note that (2) is used when , has small curvature with respect to logt . From the correspondence principle we obtain the viscoelastic solution . (3) (4) (5) (6) The operator moduli : (7) We consider the relaxation test , in which , is a constant at t = 0 (8) . We have , , (9) By the similar way , we find out : (10) Assume that the relaxation moduli have power form : (11) where are constants . By applying the Laplace transfom for (11) , we obtain the operator moduli : (12) with the values of Gamma function : ; (13)

3. Parameters - The Numerical and Graphical Solution : >
restart;cycrstrecom:=proc(T,Gamma1,c1,P1,Q1,M1,d1) global P,Q,sigmaat1,sigmaat2,sigmabt2,sigmabt1,sigmaatisotropic,sigmabtisotropic ; local To,E,E1,M,d,j,Gamma,Gamma_form,gamma;with(inttrans):with(plottools):with(plots):print(" PARAMETERS DEFINITION : ");print( T=To,gamma=Gamma1,c=c1);;;print(" REPRESENTATION OF STRESS : ");;;;;print(" LAPLACE TRANSFORM OF MODULI : ");;;;;Gamma_form:=sqrt(E1[theta]/E1[r]);print(" EXPRESSION OF : ",gamma=Gamma_form;;print(" SUBSTITUTE ",c=c1 ,p =1/(2*t),gamma=Gamma) ;;;;print(" CHANGE THE PRESENTATION OF TIME INTO LOG(t/To) ");;;print(" OUTPUT DATA ");;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;M:=M1;;;d:=d1;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;printf(" s=log(t/To) sigma[Theta](a)(s)/P \n\n");
>
for j from 0 to M do printf("%10.1f %10.4f \n", -d*(10-j), subs(s=-d*(10-j),sigmaat2)) ; end do;
>
;;;;;;;;;;;;;;;;;;;;;;;;;;;;for j from 1 to M do printf("%10.1f %10.4f \n", d*j, subs(s=d*j,sigmaat2)) ; end do;
>
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;print(" NUMERICAL AND GRAPHICAL SOLUTION ");;printf("\n%s"," KET THUC BAI TOAN ONG TRU COMPOSITE DAN NHOT TRUC HUONG BANG PHUONG PHAP TRUC TIEP "); ;plot([sigmaat2,sigmaat2,sigmaatisotropic],s=-10..30,y=0.85..5.2,color=[grey,black,black],style=[line,point,point],thickness=1,symbol=[cross,diamond,cross],linestyle=1,axes=boxed,labels=["logt/To","sigma(a,t)/P"],legend=[`sigma(a,t)/P`,`sigma(a,t)/P`,`Isotropic solution`],title="Numerical solution");;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
>
end:>
cycrstrecom(1, .83, 1/2, 1, 0, 30, 1);


s=log(t/To) sigma[Theta](a)(s)/P -10.0 1.4286 -9.0 1.4286 -8.0 1.4287 -7.0 1.4289 -6.0 1.4294 -5.0 1.4305 -4.0 1.4335 -3.0 1.4409 -2.0 1.4595 -1.0 1.5056 0.0 1.6182 1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725 6.0 4.8419 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.0 2.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 1.0 1.8804 2.0 2.4264 3.0 3.3300 4.0 4.3240 5.0 4.8725 6.0 4.8419 7.0 4.5127 8.0 4.1110 9.0 3.7260 10.0 3.3828 11.0 3.0857 12.0 2.8323 13.0 2.6184 14.0 2.4396 15.0 2.2913 16.0 2.1691 17.0 2.0691 18.0 1.9877 19.0 1.9217 20.0 1.8684 21.0 1.8255 22.0 1.7910 23.0 1.7634 24.0 1.7413 25.0 1.7237 26.0 1.7096 27.0 1.6984 28.0 1.6894 29.0 1.6823 30.0 1.6766

KET THUC BAI TOAN ONG TRU COMPOSITE DAN NHOT TRUC HUONG BANG PHUONG PHAP TRUC TIEP

>

REFERENCES

[1] Ngo Thanh Phong , Nguyen Thoi Trung , Nguyen Dình Hien , Ap dung
phap gan dung bien doi Laplace nguoc de giai bai toan bien dang phang trong
lieu composite dan nhot truc huong , Tap chí phat trien KHCN , tap 7 , so 4 &
in Vietnamese ) , 2002 .

[2] R.A. Schapery , Stress Analysis of Viscoelastic Composite Materials ,
Edited by G.P.Sendeckyj ,Academic Press , Newyork –London , 1971 .


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The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.

View details online at the link : http://www.maplesoft.com/applications/app_center_view.aspx?AID=2136

The Relaxation function problem of an orthotropic cylinder .

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License. -------------------------------------------------------------------------------------------
Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas. 

Albert Einstein . 

THE SOKOLOV'S METHOD .

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.

THE  AVERAGE APPROXIMATING METHOD ON FUNCTIONAL ADJUSTMENT  QUANTITY   FOR SOLVING

The Volterra Integral Equation II 

                                                                                                                                                                                                                                              

by Co.H Tran , University of Natural Sciences  , HCMC  Vietnam - 

             Institute of Applied Mechanics , HCMC  -  coth123@math.com   &  coth123@yahoo.com        

                                                       Copyright  2004

                                                   Sat , November 06  2004  

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** Abstract  : Solving the Volterra's  integral equation II  with applying the Neumann series and the average approximating method on functional adjustment quantity . 

** Subjects: Viscoelasticity Mechanics , The Integral equation  . 

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Copyright

Co.H Tran --

.  The  Average Approximating Method on Functional Adjustment Quantity ( Sokolov's method ) All rights reserved.  No copying or transmitting of this material is allowed without the prior written permission of Co.H Tran 

The  Average Approximating Method on Functional Adjustment Quantity ( Sokolov's method ) 


In consideration of  The Volterra Integral Equation II  ( second kind ) , we find the explicit expression for the resolvent kernel Gamma ( t , t )  in the general form :
                                         v   = ( 1 + lambda K* ) u 
 here  lambda : arbitrary parameter . The solution of  u  can be represented with the Neumann series  :    .
The resolvent operator  Gamma*   is determined by a Neumann series :   , then the kernel     . The convergence of this series  must be investigated  in  a connection with the Neumann series .
The average approximating method on the functional adjustment quantity ( Sokolov's method ) makes  increasing  for  the rate of convergence of this series . 
From the first approximation of the solution u , we find the adjustment quantity for the next and so on .    
We consider the following equation : 
             ( 1 )
the first approximation  :      ( 2 )  by choosing the initial adjustment quantity :          ( 3 )
From  ( 2 )   and  ( 3 )  we obtain  :     ( 4 )      with       ( 5 )
the n-th  approximation  :       ( 6 )      and the adjustment quantity of the n-th order can then be written  as :
   ( 7 )   here   ( 8 )   . From ( 6 ) , ( 7 ) and ( 8 )  we have  :       ( 9 )   
Denoting the formulas ( 6 ) to ( 9 )  can be carried out by the computer  programming language. We can show that the convergence condition of this method is     ( 10 )  here   : the project-operator from the Banach's space B into  its space  Bo (  the solution  u   B  )   

                               Sokolov's method  

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The Average Approximating Method On Functional Adjustment Quantity For Solving The Volterra Integral Equation II

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License. 
------------------------------------------------------------------------------------------- 
Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas.

 Albert Einstein . 

Demis Roussos

DEMIS  ROUSSOS  

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Artemios (Demis) Ventouris Roussos (Hy Lạp: Ντέμης Ρούσος) là ca sĩ người Hy Lạp và người biểu diễn quốc tế độc tấu trong năm suốt thập kỷ từ 1960 và 1970 sau khi đã tham gia ban nhạc Rock tiến bộ Aphrodite's Child  trong đó có cả nghệ sĩ  huyền thoại Vangelis nổi tiếng . Demis Roussos đã bán được hơn 40 triệu album trên toàn thế giới.

Roussos được sinh ra và lớn lên ở Alexandria, Ai Cập ngày 15 tháng sáu 1946 trong một gia đình mà cả cha George (Kỹ sư Yorgos Roussos) và mẹ Olga có nguồn gốc Hy Lạp. Cha mẹ của ông đã khánh kiệt trong cuộc khủng hoảng kênh đào Suez và do đó họ quyết định chuyển đến Hy Lạp.

Sau khi định cư ở Hy Lạp, Roussos tham gia vào một loạt các nhóm nhạc bắt đầu với The Idols khi ông 17 tuổi, khi ông gặp Evangelos Papathanassiou (sau này được biết đến như Vangelis), và Sideras Loukas, các thành viên tương lai trong nhóm nhạc Aphrodite's Child 1967-1972. Sau này, ông tham gia ban We Five (không phải là nhóm rock dân gian San Francisco, California ) ,  một ban nhạc khác cũng đã biểu diễn rất thành công ở Hy Lạp.

Ông hướng đến đối tượng khán giả lớn hơn trong năm 1968 khi gia nhập ban nhạc rock Aphrodite's Child , gồm có Vangelis và Sideras, ban đầu họ là ca sĩ nhưng sau đó cũng chơi guitar bass, và đạt được thành công về âm nhạc thương mại rất lớn ở Pháp và châu Âu vào những năm 1968 cho mãi đến năm 1972.  Chính với phong cách giọng hát Opera đặc trưng của ông đã thúc đẩy các ban nhạc trở nên có danh tiếng tầm cỡ quốc tế, đặc biệt là album cuối cùng của họ 666, đã trở thành một tiêu biểu cho phong cách Rock tiến bộ mang tính chất cổ điển .


Sau khi ban nhạc Aphrodite's Child giải tán,, Roussos tiếp tục ghi âm không thường xuyên với cựu thành viên ban nhạc Vangelis. Năm 1970 phát hành hai Sex Power (mặc dù album này cũng đã được từ chối ghi là của Aphrodite's Child ), cũng ghi âm các album Magic 1977 với nhau. Sự hợp tác thành công nhất của họ là "Race to End", thể hiện sự thích ứng giọng hát của Roussos với  các chủ đề âm nhạc từ giai đoạn chiến thắng giải Oscar với phim Chariots of Fire, và khả năng hát bằng tiếng Tây Ban Nha trong tác phẩm "Tu Libertad", đồng thời Roussos cũng là khách mời trong soundtrack trong bộ phim hành động  Blade Runner (1982). Bài hát được mang tên " Truyền thuyết của tương lai". ("Tales of the Future".)


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KHẢO SÁT HÀM MẬT ĐỘ PHỔ PHƯƠNG TRÌNH DUFFING BẰNG PHƯƠNG PHÁP TUYẾN TÍNH HÓA TƯƠNG ĐƯƠNG

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Investigation of the Power Spectral Density of Duffing's Equation By Equivalent Linearization Method

Co. H. Tran.
Faculty of Mathematics, University of Natural Sciences - VNU-HCM

Abstract

We consider the non-linear random vibration model demonstrated by the Duffing's differential equation :

(*)

The stationary random process is f( t) which is satisfied <> = 0
with the spectral density function Sf ( w ) . To find the solution Sx ( w ) of (*) we use the equivalent linearization method .

1. Model Definition

The non-linear random vibration model includes the mass (m) - dashpot (c) -spring (k)
( fig.1 ) . This model moves on the rough surface which is described by the random variable y(s) with the constant velocity v . If we have the relation s = vt and the mass m is also influenced under the non-linear stimulating force , then the vibration differential equation of the mass m can be rewritten as :

( 1.0 )
( fig . 1)

2. The equivalent linearization method

The conditions of the stationary solution and equivalent approximation : ( 2.1 )
The linear operator : ( 2 2 )
Substitute D = iy1 into (2.2) we obtain the frequency response :
( 2.3 )
The impulse response : ( 2.4 )
The power spectral density :
( 2. 5 )
Assuming S f ( y1 ) = So : const ( white-noise) then we have :
( 2. 6 )
By altering : and choosing S f ( y1 ) = So = 1 ( to simplify the next algorithm ) , we take into account the integral expression :
( 2.7 )
The function h(z) :
: ( 2.14 )
Calculation in details : eq:=subs(psi=1,mu=0.1,beta=0.2,S[0]=1,Gamma=omega[0]^2+Delta,eqndelta);eq:=subs(omega[0]=0.5,eq);
> nodelta:=solve(eq,Delta);

The Duffing's equation can be approximated in the linear form with the values of nodelta : ( 2.15 )
The investigation on components of the Duffing's differential equation will be calculated by other methods of linear random vibration , and we can obtain the corresponding approximate values in the meaning of minimum variance .

3. Parameters - Solution of the equivalent differential equation
The graph of Duffing's differential equation ( non-linear random ) :
The comparison of two graphical solutions : non-linear and equivalent-linearization .

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.


See more details at :
http://www.maplesoft.com/applications/app_center_view.aspx?AID=1993

Equivalent Linearization Method

Publish at Calameo or read more publications
Author : Co Hong Tran MMPC-VN.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.





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 Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic. 
Pure mathematics is, in its way, the poetry of logical ideas. 

Albert Einstein .