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Free vibration analysis of mindlin plates with parabolically varying thickness
0045s7949/89 $3.00+ 0.00 PergamonPressplc
Compurrrs & SrrucfuresVol. 33. No. 6, pp. 1417-1421,1989
Printedin GreatBritain.
FREE VIBRATION ANALYSIS OF MINDLIN PLATES WITH PARABOLICALLY VARYING THICKNESS S. A. AL-KAABI and G. AKSU Department of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia (Received 4 January 1989)
Abstract-The free transverse vibration characteristics of rectangular plates, including the effect of transverse shear deformation and rotary inertia, are presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions am presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.
1. INTRODUCTION
The energy approach based on the finite difference technique has been applied to examine the free vibration characteristics of Mindlin plates with parabolically varying thickness. In the aeronautical field, there is a growing interest in plates of variable thickness owing to their utility in aircraft wings. It appears that most of the previous investigations have been conflned to the solutions based on the classical thin plate theory. Jain and Soni [l] have applied the Frobenius method to study the free transverse vibrations of rectangular plates of parabolically varying thickness on the basis of classical theory of plates. Later, Dovganich and Korol[2] investigated the dynamic behaviour of rectangular thin plates by using the conjugation conditions sweep method. Prasad et al. [3] presented the effect of transverse shear and rotary inertia on vibrations of an infinite strip of parabolically varying thickness. To the best of the authors’ knowledge, Mikami and Yoshimura [4] were the first to include the effect of both transverse shear deformation and rotary inertia in the study of the free vibration of plates with variable thickness. They applied the collocation method to examine the free vibration characteristics of Mindlin plates with linearly varying thickness. Recently, Aksu and Al-Kaabi [5] developed a method based on the variational procedure in conjunction with the finite difference technique to study Mindlin plates with linearly varying thickness. The applicability of the proposed method has been demonstrated by studying the problem for various values of relative thickness ratio and boundary conditions. The objective of the work reported herein is to apply a method in conjunction with the finite difference technique to analyse the dynamic behaviour of rectangular plates with parabolically varying thick-
ness including both transverse shear and rotary inertia effects, Natural frequencies and corresponding mode shapes are calculated and new findings are assessed by comparing the results with solutions based on thin plate theory.
2. PLATE
CONFIGURATION
AND ANALYSIS
Figure 1 shows a rectangular plate of parabolically varying thickness h in one direction and sides of length a and b respectively. The displacements u, v and w, with respect to position coordinates x, y and z, may be fully described by the components iit, #b, and &., where I is the lateral displacement of the midsurface of the plate and +,, #+. are the rotations made by the normal about the y and x axes, respectively. The strain and kinetic energy equations in Mindlin’s theory [6] based on the following nondimensional parameters
a*(&+y 0 1417
(1)
S. A. AL-KAABIand G. AKW
1418
x
/--
___---
The total energy is minimized with respect to discretized displacement and rotational components, and the natural frequencies and mode shapes are determined as solutions of the linear algebraic eigenvalue problem
___ ___-----b
7 Y
&Z
[Al(x) = J(x) a+
tFig. I. Geometry and dimensions for a rectangular plate of
parabolic thickness variation.
where 1 = (1 - v2)phiw2/Eu. The stress boundary conditions are satisfied automatically by the minimization process and only the geometric boundary conditions are considered in the analysis. A more comprehensive formulation for this section can be found in Aksu and Al-Kaabi [S].
can be rewritten, by taking into account the variable
rigidity due to the variation follows:
3. RESULTSAND
It appears that no available results for rectangular plates of parabolically varying thickness exist except for the results based on the classical thin plate theory. Therefore, at first, the convergence of the present method is tested by varying the pO(=&in) ratio and running a comparison with the classical thin plate solution given by Dovganich and Korol[2]. In [1,2], a non-dimensional parameter
+*v(%)(%)+(2)
n =
+(d,+z)2}]di drl 2
R = Jlu/j;.
h(x,y) =
In order to test the existing solutions based on the thin plate theory, the simply supported plate with a taper ratio c = 0.5 and aspect ratio y = 1 is analysed using an (8 x 8) nodal set. The value of shear factor adopted is I[‘/ 12. The predictions for the fundamental frequency including the result obtained by Dovganich and Korol[2] are shown in Fig. 2. As can be seen from the convergence curve, the results predicted by the present method approach the thin plate solution
and the non-dimensional
is given as
w - cr2)
(8)
(2)
did?.
The thickness function h&y)
(7)
be related to 1 defined in the present study by
( )I aw’ at
12( 1 - v2)a2po2/Eh;
is used to express the frequency parameters which can
+k(l ;v’“{(&+!!K)
a*
DISCUSSlON
of plate thickness, as
u=&jjO’[${($
+E
(6)
(4)
17.0 r
form as 166
where c is the taper ratio and h, is the thickness at (i = 0, rj = 0). The energy integrals are re-expressed by dividing the integrand into four subfunctions, and the interlacing grid technique is used to express each integral as the summation of the rectangular subdomains. The discrete displacement and rotational components are transformed to @Ji, r~1, @,,K, ~1 and WK, ~1 by assuming harmonic motion to eliminate the time dependence. Then partial derivatives appearing in the functionals are replaced by finite difference equations with equal intervals in both the x and y directions.
-
Present
---Thin 150
1 IO
I loo
methcd
plate [2] I 1000
I loo00
I/B0
Fig. 2. Variation of fundamental frequency with B,, for an S-S-S-S plate of parabolic thickness variation (b/a = 1, c = +0.5).
1419
Vibration analysis of Mindlin plates Table 1. Frequency parameter n for an S-!&S-S plate of parabolic thickness, c = 0.5, v = 0.3, k = 7?/12, y = b/a = 1 Present method Bo=
Thin plate solutions
r21
Frobenius 111
1
16.22
2
40.43
1
40.42
2
64.81
1
79.37
3
80.58
3
104.82
2
106.14
Mode m n
Conju~tion
0.05
0.1
0.2
-
16.0354 (-1.1%)
15.6187 (-3.7%)
14.3366 (11.6%)
-
40.1730 (-0.6%)
38.2274 (-5.4%)
32.8668 (- 18.7%)
40.42
40.2726 (-0.3%)
38.3575 (-5.1%)
33.0161 (- 18.3%)
64.79
62.4862 (-3.6%)
58.0670 (- 10.4%)
47.3972 (-26.9%)
-
81.1027 (-2.2%)
73.9977 (-6.8%)
57.9855t (-26.9%)
-
82.4029 (-2.2%)
74.6958 (-7.3%)
58.02333 (-28%)
101.2181 (-3.4%)
90.3494 (-13.8%)
68.5184 (-34.6%)
103.1738 (-2.8%)
91.5523 (- 13.7%)
68.8516 (-35.1%)
104.78
-
t, SThese values have been found to correspond to diagonal and circular nodal shapes respectively.
Fig. 3. The mode shapes for the fourth mode.
Fig. 4. The mode shapes for the fifth mode.
S. A. AL-KAABIand G. AKSU
1420
Table 2. Fundament~
frequency parameter a2 for S-S-S-S plates of parabolic thickness, v = 0.39, b/a = I
Thin plate
80
VI
0.01
0.05
0.1
-0.1
416.04
412.11 (-0.9%)
400.80 (-3.7%)
370.62 (- 10.9%)
292.23 (-29.8%)
-0.3
470.32
465.37 (- 1.1%)
450.60 (-4.2%)
412.14 (- 12.4%)
317.31 (-32.5%)
-0.5
526.82
520.79 (- 1.14%)
501.68 (-4.8%)
453.42 (- 13.9%)
340.85 (-35.3%)
-0.7
585.69
578.34 (- 1.25%)
554.01 (- 5.4%)
494.45 (- 15.6%)
362.96 (- 38.0%)
C
as j10 decreases. It can also be noticed that the convergence curve almost approaches the thin plate solution with the /3,,ratio ~0.01. The discrepancy is 0.06%, which indicates the validity of the present method. The frequency parameters including the results obtained by Jain and Soni [l] and Dovganich and Korol[2] are given in Table 1. A simply supported square plate with the taper constant c = 0.5 is considered in the analysis. By comparing with thin plate solutions obtained by Dovganich and Korol[2], it is observed that the discrepancy in general increases towards the higher modes and more effectively with higher /30 ratios. These observations are due to the well known effects of transverse shear deformation and rotary inertia. It is found that the modes corresponding to (m = 3, n = 1) and (m = 1, n = 3) are
replaced by diagonal and circular nodal mode shapes due to the effect of increase in the relative thickness ratio B,,. Various mode shapes for the simply supported plate with ,!$,= 0.1 are presented in Figs 3 and 4. The nodal lines corresponding to the direction of thickness variation are shifted towards the direction of lesser thickness. The effect of the taper constant with various values of /lo is studied by considering only the fundamental frequency. A simply supported rectangular plate with the taper constant having various values, i.e. -0.1, -0.3, -0.5 and -0.7, is considered in the analysis. For these values of c, the thickness increases in the q direction, The predicted results are compared with the thin plate solutions obtained by Jain and Soni [1] in Table 2. As expected, the discrepancy increases as the taper constant increases. This result is due to the
Table 3. Frequency parameter G for an S-XX-C r zn.5
plate of parabolic thickness, b/a = 1, Present method ts, =
Thin plate Mode m
n
0.2
Conjugation
Frobenius
PI
[[I
0.05
0.1
0.2
-
23.4165 (+2.5%)
22.3470 (-2.2%)
19.3231 (-15.4%)
44.9303 (+0.7%)
42.2354 (-5.3%)
35.3497 (-20.7%)
59.0382 (+6.9%)
53.2764 (-3.5%)
40.7480 (-26.2%)
77.1873 (+0.89%)
69.0414 (-9.8%)
52.5038 (-31.4%)
22.84 44.60 55.21 76.51
44.91 77.45
83.29
-
85.9973 (-3.3%)
77.4207 (-7.0%)
59.4812 (-28.6%)
103.49
-
118.3075 (- 14.3%)
97.5266 (-5.8%)
65.7449 (-36.5%)
115.06
-
113.677 f- 1.2%)
98.7828 (-14.1%)
71.9770 (-37.4%)
132.5603 (-6.1%)
109.1830 (- 12.6%)
74.2929 (-40.6%)
124.97
128.30
Vibration analysis of Mindlin plates
effects of transverse shear deformation and rotary inertia, since the plate becomes thicker as the taper constant increases. Also, this effect is more marked as the /I0 ratio increases. We conclude the study of the present problem by analysing an S-C-S-C supported plate (clamped at both edges 9 = 0 and q = 1). The taper ratio is shown as 0.5 and the first eight modes are considered for various values of the &, ratio. The numerical results predicted by the present method, as well as the thin plate solutions obtained by Jain and Soni [l] and Dovganich and Korol[2], are given in Table 3. Due to the existence of clamped edges, some of the predicted results have positive discrepancy. This is due to the strain energy over-estimation of the clamped and simply supported edges. Moreover, the strain energy over-estimation due to the simply supported edges is especially pronounced in the case of rigid boundaries.
i.e.. clammed boundaries.
It can also
be noticed that the discrebancy in general increases towards the higher modes due to the error of the finite difference
technique.
This can be noticed more clearly
for higher values of the POratio where the effects of transverse shear deformation and rotary inertia are more pronounced. 4. CONCLUSION
It has been demonstrated that the enerav armroach based on finite differences is highly effe&e for the analysis of vibration of rectangufar Mindlin plates with various boundary conditions. The significance of the present approach lies in its simplicity and
straightforwardness.
1421 Moreover,
the present
method
may be extended, without special treatment, to vibration problems of Mindlin plates with various types of thickness variation. The natural frequencies and mode shapes have been determined and the effect of the thickness parameter and taper thickness ratio on vibration have been clarified. The accuracy of predicted results is expected to be reduced as the relative thickness ratio increases, since the finite difference technique is based on a flexible model assumption. It has been shown that thin plate theory cannot present accurate values for the higher modes and even lower modes of moderate thickness or relatively high taper ratio. REFERENCES
1. R. K. Jain and S. R. Soni, Free vibrations of rectangular plates of parabolic varying thickness. Indian J. Pure appl. Math. 4, 261-277 $973).
2. M. I. Dovganich and I. Yu. Korol, Method of investigating free vibrations of rectangular plates of variable thickness. J. Soviet appt. Mech. 14, 173-178 (1978). 3. C. Prasad, R. K. Jain and S. R. Soni, Effect of
transverse shear and rotary inertia on vibrations of an infinite strip of variable thickness. J. phys. Sot. Japan 33, 115&l 159 (1972). 4. T.‘Mikami and-J. Yoshimura, Application of the collo-
cation method to vibration analysis of rectangular Mindlin plates. Cornput. Struct. 18, 425-431 (1984). 5. G. Aksu and S. A. Al-Kaabi. Free vibration analvsis of
Mindlin plates with linearly varying thickness. J. kound V&r. 119-2, 189-205 (1987). 6. R. D. Mindlin, Influence of rotary inertia and shear on
flexural motions of isotropic elastic plates. J. appt. Mech. 18, 31-38 (1951).
Compurrrs & SrrucfuresVol. 33. No. 6, pp. 1417-1421,1989
Printedin GreatBritain.
FREE VIBRATION ANALYSIS OF MINDLIN PLATES WITH PARABOLICALLY VARYING THICKNESS S. A. AL-KAABI and G. AKSU Department of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia (Received 4 January 1989)
Abstract-The free transverse vibration characteristics of rectangular plates, including the effect of transverse shear deformation and rotary inertia, are presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions am presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.
1. INTRODUCTION
The energy approach based on the finite difference technique has been applied to examine the free vibration characteristics of Mindlin plates with parabolically varying thickness. In the aeronautical field, there is a growing interest in plates of variable thickness owing to their utility in aircraft wings. It appears that most of the previous investigations have been conflned to the solutions based on the classical thin plate theory. Jain and Soni [l] have applied the Frobenius method to study the free transverse vibrations of rectangular plates of parabolically varying thickness on the basis of classical theory of plates. Later, Dovganich and Korol[2] investigated the dynamic behaviour of rectangular thin plates by using the conjugation conditions sweep method. Prasad et al. [3] presented the effect of transverse shear and rotary inertia on vibrations of an infinite strip of parabolically varying thickness. To the best of the authors’ knowledge, Mikami and Yoshimura [4] were the first to include the effect of both transverse shear deformation and rotary inertia in the study of the free vibration of plates with variable thickness. They applied the collocation method to examine the free vibration characteristics of Mindlin plates with linearly varying thickness. Recently, Aksu and Al-Kaabi [5] developed a method based on the variational procedure in conjunction with the finite difference technique to study Mindlin plates with linearly varying thickness. The applicability of the proposed method has been demonstrated by studying the problem for various values of relative thickness ratio and boundary conditions. The objective of the work reported herein is to apply a method in conjunction with the finite difference technique to analyse the dynamic behaviour of rectangular plates with parabolically varying thick-
ness including both transverse shear and rotary inertia effects, Natural frequencies and corresponding mode shapes are calculated and new findings are assessed by comparing the results with solutions based on thin plate theory.
2. PLATE
CONFIGURATION
AND ANALYSIS
Figure 1 shows a rectangular plate of parabolically varying thickness h in one direction and sides of length a and b respectively. The displacements u, v and w, with respect to position coordinates x, y and z, may be fully described by the components iit, #b, and &., where I is the lateral displacement of the midsurface of the plate and +,, #+. are the rotations made by the normal about the y and x axes, respectively. The strain and kinetic energy equations in Mindlin’s theory [6] based on the following nondimensional parameters
a*(&+y 0 1417
(1)
S. A. AL-KAABIand G. AKW
1418
x
/--
___---
The total energy is minimized with respect to discretized displacement and rotational components, and the natural frequencies and mode shapes are determined as solutions of the linear algebraic eigenvalue problem
___ ___-----b
7 Y
&Z
[Al(x) = J(x) a+
tFig. I. Geometry and dimensions for a rectangular plate of
parabolic thickness variation.
where 1 = (1 - v2)phiw2/Eu. The stress boundary conditions are satisfied automatically by the minimization process and only the geometric boundary conditions are considered in the analysis. A more comprehensive formulation for this section can be found in Aksu and Al-Kaabi [S].
can be rewritten, by taking into account the variable
rigidity due to the variation follows:
3. RESULTSAND
It appears that no available results for rectangular plates of parabolically varying thickness exist except for the results based on the classical thin plate theory. Therefore, at first, the convergence of the present method is tested by varying the pO(=&in) ratio and running a comparison with the classical thin plate solution given by Dovganich and Korol[2]. In [1,2], a non-dimensional parameter
+*v(%)(%)+(2)
n =
+(d,+z)2}]di drl 2
R = Jlu/j;.
h(x,y) =
In order to test the existing solutions based on the thin plate theory, the simply supported plate with a taper ratio c = 0.5 and aspect ratio y = 1 is analysed using an (8 x 8) nodal set. The value of shear factor adopted is I[‘/ 12. The predictions for the fundamental frequency including the result obtained by Dovganich and Korol[2] are shown in Fig. 2. As can be seen from the convergence curve, the results predicted by the present method approach the thin plate solution
and the non-dimensional
is given as
w - cr2)
(8)
(2)
did?.
The thickness function h&y)
(7)
be related to 1 defined in the present study by
( )I aw’ at
12( 1 - v2)a2po2/Eh;
is used to express the frequency parameters which can
+k(l ;v’“{(&+!!K)
a*
DISCUSSlON
of plate thickness, as
u=&jjO’[${($
+E
(6)
(4)
17.0 r
form as 166
where c is the taper ratio and h, is the thickness at (i = 0, rj = 0). The energy integrals are re-expressed by dividing the integrand into four subfunctions, and the interlacing grid technique is used to express each integral as the summation of the rectangular subdomains. The discrete displacement and rotational components are transformed to @Ji, r~1, @,,K, ~1 and WK, ~1 by assuming harmonic motion to eliminate the time dependence. Then partial derivatives appearing in the functionals are replaced by finite difference equations with equal intervals in both the x and y directions.
-
Present
---Thin 150
1 IO
I loo
methcd
plate [2] I 1000
I loo00
I/B0
Fig. 2. Variation of fundamental frequency with B,, for an S-S-S-S plate of parabolic thickness variation (b/a = 1, c = +0.5).
1419
Vibration analysis of Mindlin plates Table 1. Frequency parameter n for an S-!&S-S plate of parabolic thickness, c = 0.5, v = 0.3, k = 7?/12, y = b/a = 1 Present method Bo=
Thin plate solutions
r21
Frobenius 111
1
16.22
2
40.43
1
40.42
2
64.81
1
79.37
3
80.58
3
104.82
2
106.14
Mode m n
Conju~tion
0.05
0.1
0.2
-
16.0354 (-1.1%)
15.6187 (-3.7%)
14.3366 (11.6%)
-
40.1730 (-0.6%)
38.2274 (-5.4%)
32.8668 (- 18.7%)
40.42
40.2726 (-0.3%)
38.3575 (-5.1%)
33.0161 (- 18.3%)
64.79
62.4862 (-3.6%)
58.0670 (- 10.4%)
47.3972 (-26.9%)
-
81.1027 (-2.2%)
73.9977 (-6.8%)
57.9855t (-26.9%)
-
82.4029 (-2.2%)
74.6958 (-7.3%)
58.02333 (-28%)
101.2181 (-3.4%)
90.3494 (-13.8%)
68.5184 (-34.6%)
103.1738 (-2.8%)
91.5523 (- 13.7%)
68.8516 (-35.1%)
104.78
-
t, SThese values have been found to correspond to diagonal and circular nodal shapes respectively.
Fig. 3. The mode shapes for the fourth mode.
Fig. 4. The mode shapes for the fifth mode.
S. A. AL-KAABIand G. AKSU
1420
Table 2. Fundament~
frequency parameter a2 for S-S-S-S plates of parabolic thickness, v = 0.39, b/a = I
Thin plate
80
VI
0.01
0.05
0.1
-0.1
416.04
412.11 (-0.9%)
400.80 (-3.7%)
370.62 (- 10.9%)
292.23 (-29.8%)
-0.3
470.32
465.37 (- 1.1%)
450.60 (-4.2%)
412.14 (- 12.4%)
317.31 (-32.5%)
-0.5
526.82
520.79 (- 1.14%)
501.68 (-4.8%)
453.42 (- 13.9%)
340.85 (-35.3%)
-0.7
585.69
578.34 (- 1.25%)
554.01 (- 5.4%)
494.45 (- 15.6%)
362.96 (- 38.0%)
C
as j10 decreases. It can also be noticed that the convergence curve almost approaches the thin plate solution with the /3,,ratio ~0.01. The discrepancy is 0.06%, which indicates the validity of the present method. The frequency parameters including the results obtained by Jain and Soni [l] and Dovganich and Korol[2] are given in Table 1. A simply supported square plate with the taper constant c = 0.5 is considered in the analysis. By comparing with thin plate solutions obtained by Dovganich and Korol[2], it is observed that the discrepancy in general increases towards the higher modes and more effectively with higher /30 ratios. These observations are due to the well known effects of transverse shear deformation and rotary inertia. It is found that the modes corresponding to (m = 3, n = 1) and (m = 1, n = 3) are
replaced by diagonal and circular nodal mode shapes due to the effect of increase in the relative thickness ratio B,,. Various mode shapes for the simply supported plate with ,!$,= 0.1 are presented in Figs 3 and 4. The nodal lines corresponding to the direction of thickness variation are shifted towards the direction of lesser thickness. The effect of the taper constant with various values of /lo is studied by considering only the fundamental frequency. A simply supported rectangular plate with the taper constant having various values, i.e. -0.1, -0.3, -0.5 and -0.7, is considered in the analysis. For these values of c, the thickness increases in the q direction, The predicted results are compared with the thin plate solutions obtained by Jain and Soni [1] in Table 2. As expected, the discrepancy increases as the taper constant increases. This result is due to the
Table 3. Frequency parameter G for an S-XX-C r zn.5
plate of parabolic thickness, b/a = 1, Present method ts, =
Thin plate Mode m
n
0.2
Conjugation
Frobenius
PI
[[I
0.05
0.1
0.2
-
23.4165 (+2.5%)
22.3470 (-2.2%)
19.3231 (-15.4%)
44.9303 (+0.7%)
42.2354 (-5.3%)
35.3497 (-20.7%)
59.0382 (+6.9%)
53.2764 (-3.5%)
40.7480 (-26.2%)
77.1873 (+0.89%)
69.0414 (-9.8%)
52.5038 (-31.4%)
22.84 44.60 55.21 76.51
44.91 77.45
83.29
-
85.9973 (-3.3%)
77.4207 (-7.0%)
59.4812 (-28.6%)
103.49
-
118.3075 (- 14.3%)
97.5266 (-5.8%)
65.7449 (-36.5%)
115.06
-
113.677 f- 1.2%)
98.7828 (-14.1%)
71.9770 (-37.4%)
132.5603 (-6.1%)
109.1830 (- 12.6%)
74.2929 (-40.6%)
124.97
128.30
Vibration analysis of Mindlin plates
effects of transverse shear deformation and rotary inertia, since the plate becomes thicker as the taper constant increases. Also, this effect is more marked as the /I0 ratio increases. We conclude the study of the present problem by analysing an S-C-S-C supported plate (clamped at both edges 9 = 0 and q = 1). The taper ratio is shown as 0.5 and the first eight modes are considered for various values of the &, ratio. The numerical results predicted by the present method, as well as the thin plate solutions obtained by Jain and Soni [l] and Dovganich and Korol[2], are given in Table 3. Due to the existence of clamped edges, some of the predicted results have positive discrepancy. This is due to the strain energy over-estimation of the clamped and simply supported edges. Moreover, the strain energy over-estimation due to the simply supported edges is especially pronounced in the case of rigid boundaries.
i.e.. clammed boundaries.
It can also
be noticed that the discrebancy in general increases towards the higher modes due to the error of the finite difference
technique.
This can be noticed more clearly
for higher values of the POratio where the effects of transverse shear deformation and rotary inertia are more pronounced. 4. CONCLUSION
It has been demonstrated that the enerav armroach based on finite differences is highly effe&e for the analysis of vibration of rectangufar Mindlin plates with various boundary conditions. The significance of the present approach lies in its simplicity and
straightforwardness.
1421 Moreover,
the present
method
may be extended, without special treatment, to vibration problems of Mindlin plates with various types of thickness variation. The natural frequencies and mode shapes have been determined and the effect of the thickness parameter and taper thickness ratio on vibration have been clarified. The accuracy of predicted results is expected to be reduced as the relative thickness ratio increases, since the finite difference technique is based on a flexible model assumption. It has been shown that thin plate theory cannot present accurate values for the higher modes and even lower modes of moderate thickness or relatively high taper ratio. REFERENCES
1. R. K. Jain and S. R. Soni, Free vibrations of rectangular plates of parabolic varying thickness. Indian J. Pure appl. Math. 4, 261-277 $973).
2. M. I. Dovganich and I. Yu. Korol, Method of investigating free vibrations of rectangular plates of variable thickness. J. Soviet appt. Mech. 14, 173-178 (1978). 3. C. Prasad, R. K. Jain and S. R. Soni, Effect of
transverse shear and rotary inertia on vibrations of an infinite strip of variable thickness. J. phys. Sot. Japan 33, 115&l 159 (1972). 4. T.‘Mikami and-J. Yoshimura, Application of the collo-
cation method to vibration analysis of rectangular Mindlin plates. Cornput. Struct. 18, 425-431 (1984). 5. G. Aksu and S. A. Al-Kaabi. Free vibration analvsis of
Mindlin plates with linearly varying thickness. J. kound V&r. 119-2, 189-205 (1987). 6. R. D. Mindlin, Influence of rotary inertia and shear on
flexural motions of isotropic elastic plates. J. appt. Mech. 18, 31-38 (1951).