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An extended transfer matrix-finite element method for free vibration of plates
Journal ofSound and Vibration (1980) 70(2), 205-211
AN EXTENDED
TRANSFER
ELEMENT
MATRIX-FINITE
METHOD
FOR FREE VIBRATION
OF PLATES
S. SANKAR AND S. V. HOA Department of Mechanical Engineering, Concordia University,Montreal, Canada (Received 28 November 1978, and in revised form 22 November 1979)
A combination of extended transfer matrix and finite element methods is proposed for obtaining vibration frequencies of structures. This method yields the value of the frequency once a trial value is assumed. By using this technique, the number of nodes required in the regular finite element method is reduced and therefore a smaller computer can be used. Besides, no plotting of the values of the determinants corresponding to each assumed frequency is necessary. A worked example is given for the case of vibration of a cantilever plate. The results show fast convergence from the assumed value to the true natural frequency. 1. INTRODUCTION
In vibration analysis of structures, exact solutions for the natural frequencies are possible only for a limited set of simple structures and boundary conditions. Approximate numerical methods are therefore important for the analysis of more complex systems. The different numerical methods available include the Holzer-Myklestad type transfer matrix method. This method is successful for systems described by a single space variable such as beams and shafts [ 11. For more complicated structures, the finite element method has proved to be powerful and versatile. However, the disadvantage of the finite element method is that for some systems large matrices are produced which require large computers to handle them. In order to reduce the size of the matrices some techniques [2-4] have been proposed which consist of keeping the important degrees of freedom and suppressing the less important ones. Which degrees of freedom are to be retained depends on judgment and on the physical system. However, this approach may lead to considerable inaccuracy if the wrong degrees of freedom are suppressed. Recently, Dokainish [5] suggested a method in which the finite element technique is combined with the transfer matrix approach for obtaining frequencies of vibration of thin plates and shells. In his method the size of the stiffness and mass matrices is equal to the number of degrees of freedom for one strip, as compared to the total number of degrees of freedom for the entire structure in the finite element representation. Hence his approach certainly has an advantage over the regular finite element approach in reducing the number of degrees of freedom of the structure to be considered and, consequently, the computer memory requirement. However, the method has a drawback: it requires calculations at a significant number of frequencies and interpolation must be used if even only a few of the lowest natural frequencies are to be determined. To overcome this drawback of the Dokainish method, an extended transfer matrix approach may be used, as described in this paper. In this method, an extended transfer 205 @ 1980 Academic Press Inc. (London) Limited 0022-460X/80/100205+07 $02.00/O
206
S. SANKAR
AND S. V. HOA
matrix relating the respective state vectors, consisting of state variables (displacements and forces) and their derivatives with respect to frequency, at the two boundaries is formulated by using the regular transfer matrix method [6,7]. This extended transfer matrix relation is then used in the determination of natural frequencies via a Newton-Raphson iterative technique. The proposed method gives a quadratic convergence to a natural frequency from within a range on either side of the true natural frequency and hence allows a greater degree of error in the selection of the trial frequency. 2. THEORY
2.1. EXTENDED TRANSFER MATRIX-FINITE ELEMENT METHOD Figure 1 shows the plate or shell structure divided into m strips and each strip subdivided into finite elements. Edge BE is the left section of strip i + 1 and the right section of strip i. There are a total of 2n nodes on strip i with n nodes on the left section AD and n nodes on the right section BE.
Y
i-I
X
lb)
(a)
Figure 1. Subdivision of structure into strips and finite elements. (a) Structure divided into m strips; (b) strip i divided into finite elements.
To calculate the natural frequencies of the given structure using the extended transfer matrix-finite element method, it is required to formulate first the transfer matrix relation between the state variables (displacement and force) at the two ends of the structure. Proceeding as in reference [5], one obtains (a list of nomenclature is given in the Appendix)
where the global transfer matrix is
WI = ~TlmlYm-I.. . PII.
(2)
In equation (2), the [T]i’S are the local matrices relating the state variables at the left and right sections of each strip. In general,
Wli=[;:
;;I;
EXTENDED
TRANSFER
MATRIX-FINITE
ELEMENT
207
where
To formulate the extended transfer matrix relation, one differentiates, equation (1) with respect to w and combines it with the result:
(5)
where the dot represents extended transfer matrix
the differentiation
The matrix [P] can be partitioned
with respect to w and [P] is the global
and thus equation (5) can be written as
For a structure with the left edge clamped and the right edge free, the boundary conditions are (8) Substituting equations (8) into equation (7) gives iURhn {tiR)m =
=
[B*I{&h,
[F*lUG + [B*l{&
-~-FR),
=
[D*IFL~,
(90)
{-fiR>m
=
[H*IUCh+ [~*l{&h.
(%d)
From equations (8) and (9b), a non-trivial solution exists only if the determinant of [D*] vanishes: that is, the natural frequencies are determined from the roots of the polynomial A*(o) = det [D*(O)] = 0.
(10)
Instead of resorting to a trial and error search procedure to solve equation (lo), one can adopt the Newton-Raphson iteration technique. The recurrence relation between the trial frequencies based on the Newton-Raphson method is %ew = Wtrial-
det [D*] (det [DJ +det [D2] +. * . + det [D,])’
(11)
where p is the order of the matrix [D*]. In equation (11) the determinants are evaluated at w = utrial and the coefficients of the matrices [DJ, [D2], . . . , [D,] are identical to those of matrix [D*] except for the following coefficients which are related to the coefficients of
S. SANKAR AND S. V. HOA
208 matrix [d*]: Dr(I, 1) = D*(I, l),
D,(I,2)=d”(l,2),
Z=l,p.
. . .,D,(I,p)=D)*(Z,p),
(12) Suppose that det [D*] does not vanish for a chosen value of the trial frequency; then there is a residual of {-F’}m. Hence {-FR}~ is a function of the trial frequency w. Now differentiating equation (9b) with respect to w and comparing with equation (9d) gives [H*] = d[D* J/dw.
(‘13)
Utilizing the relationship (13) in equation (12) yields DlV, 1) = H*(z, I),
D2K
2)
=
H*V,
21,
. . . , &V,
PI = H*K
PI,
Z=l,p. (14)
Hence
by formulating an extended transfer matrix the matrices [D*], [DJ, [DJ, are known and these can be directly used in equation (11) to calculate the natural frequencies systematically. The number of steps for convergence when using this method depends on the closeness of the initial trial frequency to the true natural frequency. However, within the vicinity of a root, convergence is quadratic as compared to superlinearly convergent, as in the case, for example, for the inverse interpolation method. Since the Newton-Raphson iteration technique requires a derivative of the function at each step, the computation time doubles per step. However this increase in computational time per step is offset by the fewer number of steps for the same final accuracy. An additional advantage of the NewtonRaphson method is that it is a single point method requiring only one initial trial value. Further it has a known sufficient condition for convergence [8] given by ]utriar- utrue] 6 d/(n - l), where d is the separation between the true natural frequency under consideration and its nearest neighbouring natural frequency, and II is the degree of the polynomial d(w) under consideration.
[&I, . . . , [D,]
3. EXAMPLE:
PLATE
VIBRATION
PROBLEM
To illustrate the accuracy of the method, natural frequencies have been calculated for a cantilevered square plate clamped along one edge and free along the opposite edge. A schematic of a cantilevered plate and a plate element in shown in Figure 2.
4 2, w (0)
(b)
Figure 2. (a) Cantilever plate; (b) plate element nomenclature.
EXTENDED For
out-of-plane
TRANSFER
MATRIX-FINITE
bending, the displacement
ELEMENT
209
is assumed to be of the following form:
w=al+a2~+a3q+a4~2+a25577+a6q2+a753+a85277+a~Sr12+a~~773 f~115317 where
5 = x/a,
T-, =
f@2&?3,
(15)
yf b and al--al2 are unknown coefficients. The bending strain energy U
of an isotropic plate of uniform flexural rigidity D is
U=;j$$)2+($)z+2~($)($)+2(l-I_)(&)2] 0
dxdy,
(16)
0
where D = Eh 3/12(1 - v’). With rotary inertia neglected, the kinetic energy for the plate is a
K = ;phw2
a
w* dx dy.
II
(17)
0 0 The finite element method used was that of Dawe [9], together with the stiffness and mass matrices which he derived. The cantilever plate was divided into three strips, each strip consisting of three elements. True natural frequencies were obtained after only a few iterations. Table 1 shows a comparison of natural frequency values obtained by finite element methods [9] and by using the method of this paper. A list of trial frequency values assumed for different modes and the true natural frequencies obtained at the termination of the iterative scheme outlined in this paper are also given in Table 1. In this method outlined the trial and error TABLE
1
Natural frequencies of a square plate, convergence factor = 0.0001 Extended
Mode number
Finite element solution [9] (rad/s)
1
5.888773432
E-6
2
3.563182946
E-5
3
2.3025581569
4
3.532463628
E -4 E-4
transfer matrix-finite
Starting trial frequency (rad/s) 1.0
E-21 1.91 E-S
element
method
Value of natural frequency after convergence (rad/s) 5.888818699113 E-6 5.8888290064 E-6
1.92 E-5
3.5631696279 E-5
1.1 E-4
3.5631754145
1.2 E-4 2.8 E-4
2.3025750415 E-4 2.30258301878 E-4
2.9 E-4 4.1 E-4
3.5324978908 E-4 3.532493072 E-8
E-5
procedure is completely eliminated. Instead one can select a trial frequency arbitrarily, and the method then provides a systematic, efficient and accurate procedure for determining the nearest natural frequency in a limited number of steps. Hence, by using different trial frequencies all the required natural frequencies can be determined. Table 2 shows the smallest and largest values of trial frequencies that one can select for each mode, the calculated natural frequency, the number of iterative steps and the computer processing time for each case. From the above results, it can be concluded that the method allows one
210
S. SANKAR AND S. V. HOA TABLE 2 Lowest
(a) and largest (b) trial frequencies
and convergence results
(4
Mode number 1
2 3 4
Lowest trial frequency (rad/s)
Calculated natural frequency (rad/s)
Number of iterations
Computer processing time (s)
1.0 E-21 1.01 E-5 5.01 E-5 1.01 E-4
5.88818699113 E-6 3.5631696279 E-5 2.3025750415 E-4 3.5324978908 E-4
6 7 5 7
24.187 27,804 20.015 27.532
(b)
Mode number
Largest trial frequency (rad/s)
Calculated natural frequency (rad/s)
Number of iterations
Computer processing time (s)
1 2 3 4
0.99 E-5 4.99 E-5 0.99 E-4 4.5 E-4
5.8888290064 E-6 3.5631754145 E-5 2.30258301878 E-4 3.532493072 E-4
7 7 4 8
27.905 28.022 15.675 31.888
a very great deal of latitude in the selection of the trial frequency of each mode. However, if the assumed value is too far from the frequency in question, the results would converge to the next frequency. For the case of the fundamental frequency, the trial frequency can be lo-l4 times smaller than the real frequency and convergence still occurs in only a few steps of iteration. 4. CONCLUSION
A combination of an extended transfer matrix method with the finite element method has been proposed for obtaining natural frequencies of structures. Results for vibration frequencies of a cantilevered plate show that the method enables the user to successfully calculate the frequency from one assumed value of frequency. The method has the same advantage as the method proposed by Dokainish of reducing the size of the matrix relative to that for the regular finite element method; it also has an additional advantage in that one does not need to calculate so many values of the determinants and plot them versus assumed values of the frequencies. Other advantages of the method presented are that it allows a greater degree of error in the selection of trial frequencies and gives quadratic convergence in calculating natural frequencies. ACKNOWLEDGMENT The financial assistance from the National Research Council of Canada, Grants No. A0413 and A2685 and from the FCAC Grant No. 042-110 are gratefully appreciated.
REFERENCES 1. J. M. PRENTIS and F. A. LECKIE 1963 Mechanical Methods. London: Longman.
Vibrations: An Introduction to Matrix
EXTENDED
TRANSFER
MATRIX-FINITE
ELEMENT
211
2. J. G. GUYAN 1965 American Institute of Aeronautics and Astronautics Journal 3,380. Reduction of stiffness and mass matrices. 3. R. G. ANDERSON, B. M. IRONS and 0. ZIENKIEWICZ 1968 Journal of Solids and Structures 4, Vibration and stability of plates using finite elements. 1031-1055. 4. J. M. RAMSDEN and J. R. STOKER 1969 International Journal of Numerical Methods in Engineering 1, 333-349.Mass condensation: a semiautomatic method for reducing the size of vibration problems. 5. M. A. DOKAINISH 1972 Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 94, 526-530. A new approach for plate vibrations: combination of transfer matrix and finite element technique. 6. B. DAWSON and M. DAVIES 1974 International Journal for Numerical Methods in Engineering 8, 11 l-l 17. An improved transfer matrix procedure. 7. S. SANKAR 1977 The Shock and Vibration Bulletin, 47, part 4. Extended transfer matrix method for free vibration of shells of revolution. 8. P. LANCASTER 1964 Mathematical Gazette 48, 291-295. Convergence of the NewtonRaphson method of arbitrary polynomials. 9. D. J. DAWE 1965 Journal of Mechanical Engineering Science 7,28-32. Finite element approach to plate vibration problems. APPENDIX: a al-a12 A*
NOMENCLATURE
dimension of finite element unknown coefficients partitioned matrix dimension of finite element partitioned matrix partitioned matrix Eh3/12(1
- v’)
partitioned matrix modified matrices Young’s modulus partitioned matrix force vectors on left and right sections of strip i, respectively partitioned matrix thickness of plate partitioned matrix kinetic energy stiffness matrix of strip i number of the last strip mass matrix of strip i a matrix defined in equation (5)
[Kli- w*[Mlz
a matrix defined by equations (2) and (3) derivative of [ Tli with respect to frequency displacements on the left and right sections of strip i,
strain energy displacement space variables x/a ylb frequency @ad/s) Poisson’s ratio material density
respectively
AN EXTENDED
TRANSFER
ELEMENT
MATRIX-FINITE
METHOD
FOR FREE VIBRATION
OF PLATES
S. SANKAR AND S. V. HOA Department of Mechanical Engineering, Concordia University,Montreal, Canada (Received 28 November 1978, and in revised form 22 November 1979)
A combination of extended transfer matrix and finite element methods is proposed for obtaining vibration frequencies of structures. This method yields the value of the frequency once a trial value is assumed. By using this technique, the number of nodes required in the regular finite element method is reduced and therefore a smaller computer can be used. Besides, no plotting of the values of the determinants corresponding to each assumed frequency is necessary. A worked example is given for the case of vibration of a cantilever plate. The results show fast convergence from the assumed value to the true natural frequency. 1. INTRODUCTION
In vibration analysis of structures, exact solutions for the natural frequencies are possible only for a limited set of simple structures and boundary conditions. Approximate numerical methods are therefore important for the analysis of more complex systems. The different numerical methods available include the Holzer-Myklestad type transfer matrix method. This method is successful for systems described by a single space variable such as beams and shafts [ 11. For more complicated structures, the finite element method has proved to be powerful and versatile. However, the disadvantage of the finite element method is that for some systems large matrices are produced which require large computers to handle them. In order to reduce the size of the matrices some techniques [2-4] have been proposed which consist of keeping the important degrees of freedom and suppressing the less important ones. Which degrees of freedom are to be retained depends on judgment and on the physical system. However, this approach may lead to considerable inaccuracy if the wrong degrees of freedom are suppressed. Recently, Dokainish [5] suggested a method in which the finite element technique is combined with the transfer matrix approach for obtaining frequencies of vibration of thin plates and shells. In his method the size of the stiffness and mass matrices is equal to the number of degrees of freedom for one strip, as compared to the total number of degrees of freedom for the entire structure in the finite element representation. Hence his approach certainly has an advantage over the regular finite element approach in reducing the number of degrees of freedom of the structure to be considered and, consequently, the computer memory requirement. However, the method has a drawback: it requires calculations at a significant number of frequencies and interpolation must be used if even only a few of the lowest natural frequencies are to be determined. To overcome this drawback of the Dokainish method, an extended transfer matrix approach may be used, as described in this paper. In this method, an extended transfer 205 @ 1980 Academic Press Inc. (London) Limited 0022-460X/80/100205+07 $02.00/O
206
S. SANKAR
AND S. V. HOA
matrix relating the respective state vectors, consisting of state variables (displacements and forces) and their derivatives with respect to frequency, at the two boundaries is formulated by using the regular transfer matrix method [6,7]. This extended transfer matrix relation is then used in the determination of natural frequencies via a Newton-Raphson iterative technique. The proposed method gives a quadratic convergence to a natural frequency from within a range on either side of the true natural frequency and hence allows a greater degree of error in the selection of the trial frequency. 2. THEORY
2.1. EXTENDED TRANSFER MATRIX-FINITE ELEMENT METHOD Figure 1 shows the plate or shell structure divided into m strips and each strip subdivided into finite elements. Edge BE is the left section of strip i + 1 and the right section of strip i. There are a total of 2n nodes on strip i with n nodes on the left section AD and n nodes on the right section BE.
Y
i-I
X
lb)
(a)
Figure 1. Subdivision of structure into strips and finite elements. (a) Structure divided into m strips; (b) strip i divided into finite elements.
To calculate the natural frequencies of the given structure using the extended transfer matrix-finite element method, it is required to formulate first the transfer matrix relation between the state variables (displacement and force) at the two ends of the structure. Proceeding as in reference [5], one obtains (a list of nomenclature is given in the Appendix)
where the global transfer matrix is
WI = ~TlmlYm-I.. . PII.
(2)
In equation (2), the [T]i’S are the local matrices relating the state variables at the left and right sections of each strip. In general,
Wli=[;:
;;I;
EXTENDED
TRANSFER
MATRIX-FINITE
ELEMENT
207
where
To formulate the extended transfer matrix relation, one differentiates, equation (1) with respect to w and combines it with the result:
(5)
where the dot represents extended transfer matrix
the differentiation
The matrix [P] can be partitioned
with respect to w and [P] is the global
and thus equation (5) can be written as
For a structure with the left edge clamped and the right edge free, the boundary conditions are (8) Substituting equations (8) into equation (7) gives iURhn {tiR)m =
=
[B*I{&h,
[F*lUG + [B*l{&
-~-FR),
=
[D*IFL~,
(90)
{-fiR>m
=
[H*IUCh+ [~*l{&h.
(%d)
From equations (8) and (9b), a non-trivial solution exists only if the determinant of [D*] vanishes: that is, the natural frequencies are determined from the roots of the polynomial A*(o) = det [D*(O)] = 0.
(10)
Instead of resorting to a trial and error search procedure to solve equation (lo), one can adopt the Newton-Raphson iteration technique. The recurrence relation between the trial frequencies based on the Newton-Raphson method is %ew = Wtrial-
det [D*] (det [DJ +det [D2] +. * . + det [D,])’
(11)
where p is the order of the matrix [D*]. In equation (11) the determinants are evaluated at w = utrial and the coefficients of the matrices [DJ, [D2], . . . , [D,] are identical to those of matrix [D*] except for the following coefficients which are related to the coefficients of
S. SANKAR AND S. V. HOA
208 matrix [d*]: Dr(I, 1) = D*(I, l),
D,(I,2)=d”(l,2),
Z=l,p.
. . .,D,(I,p)=D)*(Z,p),
(12) Suppose that det [D*] does not vanish for a chosen value of the trial frequency; then there is a residual of {-F’}m. Hence {-FR}~ is a function of the trial frequency w. Now differentiating equation (9b) with respect to w and comparing with equation (9d) gives [H*] = d[D* J/dw.
(‘13)
Utilizing the relationship (13) in equation (12) yields DlV, 1) = H*(z, I),
D2K
2)
=
H*V,
21,
. . . , &V,
PI = H*K
PI,
Z=l,p. (14)
Hence
by formulating an extended transfer matrix the matrices [D*], [DJ, [DJ, are known and these can be directly used in equation (11) to calculate the natural frequencies systematically. The number of steps for convergence when using this method depends on the closeness of the initial trial frequency to the true natural frequency. However, within the vicinity of a root, convergence is quadratic as compared to superlinearly convergent, as in the case, for example, for the inverse interpolation method. Since the Newton-Raphson iteration technique requires a derivative of the function at each step, the computation time doubles per step. However this increase in computational time per step is offset by the fewer number of steps for the same final accuracy. An additional advantage of the NewtonRaphson method is that it is a single point method requiring only one initial trial value. Further it has a known sufficient condition for convergence [8] given by ]utriar- utrue] 6 d/(n - l), where d is the separation between the true natural frequency under consideration and its nearest neighbouring natural frequency, and II is the degree of the polynomial d(w) under consideration.
[&I, . . . , [D,]
3. EXAMPLE:
PLATE
VIBRATION
PROBLEM
To illustrate the accuracy of the method, natural frequencies have been calculated for a cantilevered square plate clamped along one edge and free along the opposite edge. A schematic of a cantilevered plate and a plate element in shown in Figure 2.
4 2, w (0)
(b)
Figure 2. (a) Cantilever plate; (b) plate element nomenclature.
EXTENDED For
out-of-plane
TRANSFER
MATRIX-FINITE
bending, the displacement
ELEMENT
209
is assumed to be of the following form:
w=al+a2~+a3q+a4~2+a25577+a6q2+a753+a85277+a~Sr12+a~~773 f~115317 where
5 = x/a,
T-, =
f@2&?3,
(15)
yf b and al--al2 are unknown coefficients. The bending strain energy U
of an isotropic plate of uniform flexural rigidity D is
U=;j$$)2+($)z+2~($)($)+2(l-I_)(&)2] 0
dxdy,
(16)
0
where D = Eh 3/12(1 - v’). With rotary inertia neglected, the kinetic energy for the plate is a
K = ;phw2
a
w* dx dy.
II
(17)
0 0 The finite element method used was that of Dawe [9], together with the stiffness and mass matrices which he derived. The cantilever plate was divided into three strips, each strip consisting of three elements. True natural frequencies were obtained after only a few iterations. Table 1 shows a comparison of natural frequency values obtained by finite element methods [9] and by using the method of this paper. A list of trial frequency values assumed for different modes and the true natural frequencies obtained at the termination of the iterative scheme outlined in this paper are also given in Table 1. In this method outlined the trial and error TABLE
1
Natural frequencies of a square plate, convergence factor = 0.0001 Extended
Mode number
Finite element solution [9] (rad/s)
1
5.888773432
E-6
2
3.563182946
E-5
3
2.3025581569
4
3.532463628
E -4 E-4
transfer matrix-finite
Starting trial frequency (rad/s) 1.0
E-21 1.91 E-S
element
method
Value of natural frequency after convergence (rad/s) 5.888818699113 E-6 5.8888290064 E-6
1.92 E-5
3.5631696279 E-5
1.1 E-4
3.5631754145
1.2 E-4 2.8 E-4
2.3025750415 E-4 2.30258301878 E-4
2.9 E-4 4.1 E-4
3.5324978908 E-4 3.532493072 E-8
E-5
procedure is completely eliminated. Instead one can select a trial frequency arbitrarily, and the method then provides a systematic, efficient and accurate procedure for determining the nearest natural frequency in a limited number of steps. Hence, by using different trial frequencies all the required natural frequencies can be determined. Table 2 shows the smallest and largest values of trial frequencies that one can select for each mode, the calculated natural frequency, the number of iterative steps and the computer processing time for each case. From the above results, it can be concluded that the method allows one
210
S. SANKAR AND S. V. HOA TABLE 2 Lowest
(a) and largest (b) trial frequencies
and convergence results
(4
Mode number 1
2 3 4
Lowest trial frequency (rad/s)
Calculated natural frequency (rad/s)
Number of iterations
Computer processing time (s)
1.0 E-21 1.01 E-5 5.01 E-5 1.01 E-4
5.88818699113 E-6 3.5631696279 E-5 2.3025750415 E-4 3.5324978908 E-4
6 7 5 7
24.187 27,804 20.015 27.532
(b)
Mode number
Largest trial frequency (rad/s)
Calculated natural frequency (rad/s)
Number of iterations
Computer processing time (s)
1 2 3 4
0.99 E-5 4.99 E-5 0.99 E-4 4.5 E-4
5.8888290064 E-6 3.5631754145 E-5 2.30258301878 E-4 3.532493072 E-4
7 7 4 8
27.905 28.022 15.675 31.888
a very great deal of latitude in the selection of the trial frequency of each mode. However, if the assumed value is too far from the frequency in question, the results would converge to the next frequency. For the case of the fundamental frequency, the trial frequency can be lo-l4 times smaller than the real frequency and convergence still occurs in only a few steps of iteration. 4. CONCLUSION
A combination of an extended transfer matrix method with the finite element method has been proposed for obtaining natural frequencies of structures. Results for vibration frequencies of a cantilevered plate show that the method enables the user to successfully calculate the frequency from one assumed value of frequency. The method has the same advantage as the method proposed by Dokainish of reducing the size of the matrix relative to that for the regular finite element method; it also has an additional advantage in that one does not need to calculate so many values of the determinants and plot them versus assumed values of the frequencies. Other advantages of the method presented are that it allows a greater degree of error in the selection of trial frequencies and gives quadratic convergence in calculating natural frequencies. ACKNOWLEDGMENT The financial assistance from the National Research Council of Canada, Grants No. A0413 and A2685 and from the FCAC Grant No. 042-110 are gratefully appreciated.
REFERENCES 1. J. M. PRENTIS and F. A. LECKIE 1963 Mechanical Methods. London: Longman.
Vibrations: An Introduction to Matrix
EXTENDED
TRANSFER
MATRIX-FINITE
ELEMENT
211
2. J. G. GUYAN 1965 American Institute of Aeronautics and Astronautics Journal 3,380. Reduction of stiffness and mass matrices. 3. R. G. ANDERSON, B. M. IRONS and 0. ZIENKIEWICZ 1968 Journal of Solids and Structures 4, Vibration and stability of plates using finite elements. 1031-1055. 4. J. M. RAMSDEN and J. R. STOKER 1969 International Journal of Numerical Methods in Engineering 1, 333-349.Mass condensation: a semiautomatic method for reducing the size of vibration problems. 5. M. A. DOKAINISH 1972 Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 94, 526-530. A new approach for plate vibrations: combination of transfer matrix and finite element technique. 6. B. DAWSON and M. DAVIES 1974 International Journal for Numerical Methods in Engineering 8, 11 l-l 17. An improved transfer matrix procedure. 7. S. SANKAR 1977 The Shock and Vibration Bulletin, 47, part 4. Extended transfer matrix method for free vibration of shells of revolution. 8. P. LANCASTER 1964 Mathematical Gazette 48, 291-295. Convergence of the NewtonRaphson method of arbitrary polynomials. 9. D. J. DAWE 1965 Journal of Mechanical Engineering Science 7,28-32. Finite element approach to plate vibration problems. APPENDIX: a al-a12 A*
NOMENCLATURE
dimension of finite element unknown coefficients partitioned matrix dimension of finite element partitioned matrix partitioned matrix Eh3/12(1
- v’)
partitioned matrix modified matrices Young’s modulus partitioned matrix force vectors on left and right sections of strip i, respectively partitioned matrix thickness of plate partitioned matrix kinetic energy stiffness matrix of strip i number of the last strip mass matrix of strip i a matrix defined in equation (5)
[Kli- w*[Mlz
a matrix defined by equations (2) and (3) derivative of [ Tli with respect to frequency displacements on the left and right sections of strip i,
strain energy displacement space variables x/a ylb frequency @ad/s) Poisson’s ratio material density
respectively