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On the method of collocation
ON
THE
METHOD
OF COLLOCATION. BY
EDWARD
SAIBEL,’
‘I’he term “ collocation ” is used in connection with approximations to denote the act of assigning a value, usually zero, to the error at given points or stations. The method has been applied to the problem of the approximate determination of characteristic numbers in the following way :l suppose that the homogeneous differential equation containing the parameter X together with homogeneous boundary conditions leads to characteristic numbers X1, X2, . . . for which solutions other than the trivial one, 2’= o, exist. Let this differential equation be
Iet F(X) be represented by *
y(x) =
c
k=l
bkYh(X),
Cl-‘)
where the Y’s are functions satisfying the boundary conditions and the h’s are constants. Substituting (2) into (I) leads to L { c bk Y/r(x) 1 + h+>
c &Y&c)
= E(X),
@J
where C(X) may be referred to as the error. If E(X) is made zero at as many points x1, x2, . . . as there are constants b, there results a system of linear homogeneous equations in the b’s. The vanishing of the determinant of the coefficients of the b’s, a necessary condition for a non-trivial solution, leads to the characteristic equation from which approximations to the X’s may be found. In most cases at least three or four constants must be used to get fair accuracy in the determination of the lowest characteristic number. In the Galerkin method,’ the error C(X) is distributed over thus cstent of the system by making 1 ’(‘0 ’:kociate
E(X)Y,(x)dx
= 0;
k = I, 2, . ..n.
Professor of Mechanics. 10;
EDWARD SAIREL.
108
[J.
17.I.
This also leads to a system of linear homogeneous equations in the h’s from which the characteristic equation and approximations to the X’s Grammel has proposed a converse to the method of may be found. Galerkin by starting from the homogeneous integral equation associated with the problem instead of the differential equation.2 The present aim is to propose a method which has certain advantages in many cases over collocation of the differential equation and over It is a collocation of the integral equation assoGrammel’s method. Let this equation be ciated with the problem. 1 S)Y (s)& Y(X) = x d’ m(s)@, (4) 0
where G(x, s) is the appropriate Green’s function. As before y(x) is assumed to be represented by 7%
y(x) =
c
k=l
by&).
However, since G(x, S) satisfies all of the boundary conditions of the problem, less restrictive conditions apply to the Y’s. As Grammel points out, the Y’s should at least satisfy the geometrical boundary This will be apparent in conditions in order to obtain a good result. the example shown below. The error E(X) may be found by substituting (5) into (4) leading to
c
bkYk(x) -
x J”m(s)G(x, s>[C bd’&)-Jds = E(X). 0
(6)
As before E(X) may be set equal to zero at as many points x1, ~2, . . . as there are constants b. Again this leads to a system of linear homogeneous equations in the b’s from which the characteristic equation and approximations to the X’s may be found. In some problems, particularly those which have to do with finding the natural frequencies of vibration of beams, Green’s Function need not be found or used since the integral I
s
m(s)G(x,
s) YI,(s)ds
0
may be interpreted as the deflection of the system under .a load of m(x) Yk(x) per unit length and simpler methods may be used to effect its evaluation. A simple example showing a typical application of the method is that of finding an approximate value for the lowest natural frequency Let the beam be clamped of vibration of a uniform cantilever beam. at x = o and free at x = 1. The natural boundary conditions arc y(o) = Y’(O) = o, y”(l) = y”‘(Z) = 0.
Let y be represented by bx2, a very simple function satisfying only the As a consequence geometrical constraints of the problem. bx2 -
Xmb
51
G(x,
0
s)?ds = +j.
Interpreting the integral as the deflection of the cantilever a load of intensity x2 this becomes bs2
Xmb
%c,
EI
beam under
I1”-- 1’~
^- 8
18
Setting E(I) = 0, it is found that
The esact where X is the square of the circular frequency. 12.36(EI/mZ4). Thus the error in the frequency itself is over 6y0. This error may be considerably reduced by selecting a which satisfies in addition to the geometrical requirements, For example, if in the preceding namical boundary conditions. the function y = b(6x’ - qx” + x’)
value is slightly function the dy,problem (7)
is used, then in addition to the conditions y(o) = y’(o) = o, conditions J”(I) and y”‘(1) equal to zero are also satisfied. The length of the beam is now taken as unity to simplify the numerical calculations. From area-moment or other considerations, it may easily be shown that the deflection of the cantilever beam of unit length under a load of intensity xn is given by
With the assumed deflection the integral in eq. (6) in t lw to fintl
x being measured from the fixed end. curve given by eq. (7) and interpreting light of ccl. (8), it is a simple calculation p’ =
EI 12.Q
----
ml?
The as a result of collocating the error at the free end of the beam. error in the frequency p is about 0.6 per cent. It is of interest to compare these results with those obtained through The error E(X) in the case of collocation of the differential equation.
110
I&WARD
Same..
IJ. I;. I.
the vibrating beam of uniform mass is given bY d4y -dx4
mfi2 m Y = 4x1.
Representing y by bx2 evidently leads to no result. representing y by b(6x2 - 4x3 + x4) results in b(24) -
g
b(6x2 -
4x3
+
x4)
=
On the other hand,
E(~),
where, as above, the length I of the beam has been taken as unity and is introduced into the final answer from dimensional considerations. The error at the free end of the beam being set equal to zero leads to p2 = 8EI/mZ4, an error of about 19.5 per cent. in the frequency p itself. REFERENCES. (I) FRAZER, JONES, END SKAN: “Approximations to Functions and to the .%lutions of I)ifferential Equations,” Aeronautical Research Committee Reports and Memoranda No. 1799, London, 1937. (2) C. B. BIENZENOAND R. GRAMMEL:“Technische Dynamik,” Julius Springer, Berlin, 1939. Lithoprinted by Edwards Brothers, Inc., Ann Arbor, Michigan, 1944, p. 169.
THE
METHOD
OF COLLOCATION. BY
EDWARD
SAIBEL,’
‘I’he term “ collocation ” is used in connection with approximations to denote the act of assigning a value, usually zero, to the error at given points or stations. The method has been applied to the problem of the approximate determination of characteristic numbers in the following way :l suppose that the homogeneous differential equation containing the parameter X together with homogeneous boundary conditions leads to characteristic numbers X1, X2, . . . for which solutions other than the trivial one, 2’= o, exist. Let this differential equation be
Iet F(X) be represented by *
y(x) =
c
k=l
bkYh(X),
Cl-‘)
where the Y’s are functions satisfying the boundary conditions and the h’s are constants. Substituting (2) into (I) leads to L { c bk Y/r(x) 1 + h+>
c &Y&c)
= E(X),
@J
where C(X) may be referred to as the error. If E(X) is made zero at as many points x1, x2, . . . as there are constants b, there results a system of linear homogeneous equations in the b’s. The vanishing of the determinant of the coefficients of the b’s, a necessary condition for a non-trivial solution, leads to the characteristic equation from which approximations to the X’s may be found. In most cases at least three or four constants must be used to get fair accuracy in the determination of the lowest characteristic number. In the Galerkin method,’ the error C(X) is distributed over thus cstent of the system by making 1 ’(‘0 ’:kociate
E(X)Y,(x)dx
= 0;
k = I, 2, . ..n.
Professor of Mechanics. 10;
EDWARD SAIREL.
108
[J.
17.I.
This also leads to a system of linear homogeneous equations in the h’s from which the characteristic equation and approximations to the X’s Grammel has proposed a converse to the method of may be found. Galerkin by starting from the homogeneous integral equation associated with the problem instead of the differential equation.2 The present aim is to propose a method which has certain advantages in many cases over collocation of the differential equation and over It is a collocation of the integral equation assoGrammel’s method. Let this equation be ciated with the problem. 1 S)Y (s)& Y(X) = x d’ m(s)@, (4) 0
where G(x, s) is the appropriate Green’s function. As before y(x) is assumed to be represented by 7%
y(x) =
c
k=l
by&).
However, since G(x, S) satisfies all of the boundary conditions of the problem, less restrictive conditions apply to the Y’s. As Grammel points out, the Y’s should at least satisfy the geometrical boundary This will be apparent in conditions in order to obtain a good result. the example shown below. The error E(X) may be found by substituting (5) into (4) leading to
c
bkYk(x) -
x J”m(s)G(x, s>[C bd’&)-Jds = E(X). 0
(6)
As before E(X) may be set equal to zero at as many points x1, ~2, . . . as there are constants b. Again this leads to a system of linear homogeneous equations in the b’s from which the characteristic equation and approximations to the X’s may be found. In some problems, particularly those which have to do with finding the natural frequencies of vibration of beams, Green’s Function need not be found or used since the integral I
s
m(s)G(x,
s) YI,(s)ds
0
may be interpreted as the deflection of the system under .a load of m(x) Yk(x) per unit length and simpler methods may be used to effect its evaluation. A simple example showing a typical application of the method is that of finding an approximate value for the lowest natural frequency Let the beam be clamped of vibration of a uniform cantilever beam. at x = o and free at x = 1. The natural boundary conditions arc y(o) = Y’(O) = o, y”(l) = y”‘(Z) = 0.
Let y be represented by bx2, a very simple function satisfying only the As a consequence geometrical constraints of the problem. bx2 -
Xmb
51
G(x,
0
s)?ds = +j.
Interpreting the integral as the deflection of the cantilever a load of intensity x2 this becomes bs2
Xmb
%c,
EI
beam under
I1”-- 1’~
^- 8
18
Setting E(I) = 0, it is found that
The esact where X is the square of the circular frequency. 12.36(EI/mZ4). Thus the error in the frequency itself is over 6y0. This error may be considerably reduced by selecting a which satisfies in addition to the geometrical requirements, For example, if in the preceding namical boundary conditions. the function y = b(6x’ - qx” + x’)
value is slightly function the dy,problem (7)
is used, then in addition to the conditions y(o) = y’(o) = o, conditions J”(I) and y”‘(1) equal to zero are also satisfied. The length of the beam is now taken as unity to simplify the numerical calculations. From area-moment or other considerations, it may easily be shown that the deflection of the cantilever beam of unit length under a load of intensity xn is given by
With the assumed deflection the integral in eq. (6) in t lw to fintl
x being measured from the fixed end. curve given by eq. (7) and interpreting light of ccl. (8), it is a simple calculation p’ =
EI 12.Q
----
ml?
The as a result of collocating the error at the free end of the beam. error in the frequency p is about 0.6 per cent. It is of interest to compare these results with those obtained through The error E(X) in the case of collocation of the differential equation.
110
I&WARD
Same..
IJ. I;. I.
the vibrating beam of uniform mass is given bY d4y -dx4
mfi2 m Y = 4x1.
Representing y by bx2 evidently leads to no result. representing y by b(6x2 - 4x3 + x4) results in b(24) -
g
b(6x2 -
4x3
+
x4)
=
On the other hand,
E(~),
where, as above, the length I of the beam has been taken as unity and is introduced into the final answer from dimensional considerations. The error at the free end of the beam being set equal to zero leads to p2 = 8EI/mZ4, an error of about 19.5 per cent. in the frequency p itself. REFERENCES. (I) FRAZER, JONES, END SKAN: “Approximations to Functions and to the .%lutions of I)ifferential Equations,” Aeronautical Research Committee Reports and Memoranda No. 1799, London, 1937. (2) C. B. BIENZENOAND R. GRAMMEL:“Technische Dynamik,” Julius Springer, Berlin, 1939. Lithoprinted by Edwards Brothers, Inc., Ann Arbor, Michigan, 1944, p. 169.