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Properties of a Semi-discrete Approximation to the Beam Equation

Properties of a Semi-discrete Approximation to the Beam Equation
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  J. Inst. Maths Applies  (1969) 5, 329-339 Properties of a Semi-discrete Approximationto the Beam Equation t ROLAND A. SWEET Department of Computer Science,Cornell University, Ithaca, New York, U.S.A. [Received 19 January 1968, and in revised form 29 October 1968]The solution of the equationw(*)Hrt+[p(*)««J«x =0, 0 <  x < L, t>0, where it is assumed that  w  and  p  are positive on the interval  [0,L],  is approximated byusing the method of straight lines. The resulting approximation is a linear system ofdifferential equations with coefficient matrix  S.  The  matrix S  is  studied under very generalboundary conditions which result in a conservative system. In all cases the matrix S iseither an oscillation matrix or possesses nearly all the properties of  an  oscillation matrix. 1.  Introduction THE STUDY  of small transverse (or lateral) vibrations of a beam with non-uniformcross-sectional area and moment of inertia is of wide interest. Of particular importancein the design of structures which possess the characteristics of a beam, for instance,chimneys, are the natural frequencies and mode shapes. The equation describing themotion is impossible to solve in most cases, hence information is obtained by numericalapproximating techniques. Linearly tapered cantilever beams have been studied byHousner & Keightley (1963).The general equation describing the free vibrations is wix)u lt +[pix)i xx \ n  =  0,  0<x<L, t>0.  (1)In the case of forced vibrations the right side of (1) is not zero, but rather, some function/of jcand  t. Throughout this paper we will assume that w(x) >  0 and  p(x)  > 0, 0 <  x  <  L. These conditions are physically meaningful and unrestrictive in vibration problems.The primary goal of this paper is the study of the coefficient matrices which arisethrough the use of the method of straight lines (a discretization of the space variableto produce a linear system of ordinary differential equations). We prove that undervery general boundary conditions the coefficient matrices possess all the propertiesof oscillation matrices (see Gantmakher & Krein, 1950). 2.  Preliminaries To solve equation (1) one must prescribe initial conditions u(x,0)  =  4>(x),  u t (x,0)  =  Mx),  (2) t This research was supported by an NDEA Fellowship at Purdue University, West Lafayette,Indiana.329   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   330 ROLAND  A.  SWEET for  0  <  x  <  L,  and  four linearly independent boundary conditions. These boundaryconditions  are  usually given  in  the form  of  two conditions  at the end  x  =  0 and two at  the end  x  =  L,  although this  is not the  only case which  can  occur  (for  example,periodic boundary conditions  can  also occur).  We  will concern ourselves, however,only with  the  former type.We shall consider only those boundary conditions which correspond to conservativeproblems. Multiplying (1) by  u t ,  integrating over the rectangle  0  ^ x ^  L, 0  ^ t ^ T, and using integration by parts twice  we  get f  r  [ L 0  =  >«„«,+(pO««r]  dx dt J  o  J o=  E(u,T)-E(u,0)+ \ T [B(u,L)-B(u,O)-]dt, J owhere1  f L £( .'i) = 2  [wuf+pull^dx and B U,X)  =  (>K XI )* r-i'«xx x<- The quantity  E(u, t)  gives  the  total energy (kinetic plus potential)  of  the system  at the time  t.  By the  conservation  of  energy  E  is  constant  for all  values  of  t.  Therefore,we must choose boundary conditions which make f[ ,) )]   o This  may be  done  by  choosing those conditions  for  which 5(II,0)  =  B(u,L)  =  0.  (3) Hence,  we  will consider only those boundary conditions which satisfy (3).Table  1  lists  the  various boundary conditions which we have selected  for  considera-tion.  We  stress that  the  definition  of  the functions  E  and  B  depend  on the  type  of boundary conditions under consideration.  The  energy function  for  boundary  con- dition 5-5,  for  instance, would  be Hence, equation (3) involves  the  new function B(u,x)  ^pu^U;,. This paper is concerned with the approximations obtained by the method  of  straightlines  (see  Berezin  &  Zhidkov, 1965).Essentially  the  method  is  concerned only with finding  the  solution  to  equation  (1) at a  finite  number of points  {xj} ^-  For  ease  let  us  assume the points are chosen  so  thatwhere  XQ  =  0  and  h  =  L\{n + 1).  Then  x n + 1  =  L.  Genin  &  Maybee  1966)  develop thismethod  by  integrating  the  equation over each interval  [x l —\h, x t +ih]  and  approxi-mating  all  integrals  not  containing  a  partial derivative  of  x  by the  midpoint rule.  Let w(xD  = w h  X*i)  =  Pi>  (' =  °>1  n+1).   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   SOLUTION  OF THE  BEAM EQUATION  331 Then,  if  we  denote  the  approximation  of  u x,, t)  by  ufj),  one has Cxt ik  fxi + i 0=  w(x)u tt dx+\ =  hwfi t  +  [(pOx]x'-xt-i*  ã  (4) If  we now  replace  all  partial derivatives  of  x  by  central difference approximations,we obtain from (4)0  =  AW|fi ( +A (5) TABLE  1 Boundary Conditions Case  Condition (at  x  =  0)  Condition  (at  x  =  L) 3  «xx(0,0  =  CP x*)x(0,')=0 4  u x (0,  t)  =  0)^(0,0  = 0  u x {L, t) = (pu^L, 5t  uJ0,t)  = 0  u x (L,t)  = 0 6t 7t8t9t — au(0,t)  = GM^CXXXO,/ —a«(0,r)  =  (pu^JP, ) cujp,t)  =  u^O.t) -bu^Tt^ujltt) (p$x(lojU f bu L,t)  =  (pu x J x (L,t)-du x (L,t)  = Uxx (L,t) t Note: we assume a > 0,  b >  0, c > 0, and  d >  0. If  an  equation  of  the type  of  (5)  is  written  for  each unknown  «, one  obtains  as the approximation,  a  linear system  of  coupled ordinary differential equations.  The  systemhas  the  formHi)+/t^SU  = F, (6) where  H  is a  diagonal matrix with positive elements,  5 is a  pentadiagonal matrix, F  is  a  vector whose components depend  on the  nature  of  the boundary conditions,and  U  is  the  vector whose  ith  component  is the  function  u£t). The precise forms  of H and S for  various boundary conditions will  be  exhibitedin  the  next section.  All the  boundary conditions  we  consider yield  an F  which  is zero, hence  we  shall solve (6)  in its  homogeneous form.  To do  this we seek  a  solutionof the formU(f)  =  V(a sin  VA  /+ P  cos  JX t),  (7) where  V  is  a  vector and  a,  /?, A,  are constants  to  be determined. Substituting (7) into  (6) and abbreviating  H^S  to just  S  we arrive  at the  condition(SV-AHV)(a  sin  Jk t+0  cos  VA  0 = 0.   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   332 ROLAND A. SWEET For this to be true for all values of /, it must be thatthat is,  k  must be an eigenvalue of H-'S and V a corresponding eigenvector. If S andH are of order  m,  then there are  m  eigenvalues  X\,X2,...,X m  and (we assume)  m  corre-sponding eigenvectors  V\,  Vz,...,V m .  Then for each  X k  and  V k  (7) is a solution of (6).Hence, U(0 = f  ot t sin  W+^cosW)^ t=i is a solution of (6), the coefficients  a k  and  p k  (k  = l,2,...,m) being determined by theinitial conditions (2), i.e.U(0) = <D, 0(0) = ¥,where <S> k  =  <l> x k )  and ^ = ^00 (fc = 0,l,2,...,n + l).Therefore, knowledge of the structure of the eigenvalues and eigenvectors of thematrix H~iS is desirable. To prepare for this let us state some definitions and mainresults of Gantmakher & Krein (1950) for the arbitrary matrix A of order  m  withelements  a,j  (1 <  i,  j  <  m). We will use the notation  d(A)  to denote the determinant of A. The minors of Aare all possible numbers ,Jl )l  ããã where 1 <  i\ < ii < ... < i p  <  m,  and 1  ^j\< ji < ... < j p  ^  m.  Also, we will denote the  pth  leading principal minor by  A p ,  i.e. Ap = A [l 2 ... Definition.  The matrix A is said to be totally non-negative (totally positive) if allits minors are non-negative (positive). Definition.  The matrix A is said to be an oscillation matrix if A is totally non-negative, and if there exists a positive integer/) such that  A*  is totally positive. Definition.  The matrix A* is defined to be the matrix of order  m  with elements ij  v /  ij* At this point let us note that the transformation of A into A* is a similarity trans-formation by the orthogonal matrix D  = diag[(—l) 1 ^—I) 2  (—1)™].Hence, A and A* have the same eigenvalues. Furthermore, if the elements of A satisfythen A* = |A|, where the elements of |A| are just  \a tJ \. THEOREM  1.  An oscillation matrix  A  has only simple, positive eigenvalues: Ai  > Xi> ... > k m > 0.   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om 
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