Original Article Effect of plate s parameters on vibration of isotropic tapered rectangular plate ith different boundar conditions Journal of Lo Frequenc Noise, Vibration and Active Control 216, Vol. 35(2) 139 151! The Author(s) 216 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: 1.1177/26392316644134 lfn.sagepub.com Anupam Khanna 1 and Ashish Singhal 2 Abstract A mathematical model is presented to analse the vibration of a tapered isotropic rectangular plate under thermal condition. Tapering in the thickness of rectangular plate is considered bi-parabolic. Here, temperature variation is assumed bi-linearl, i.e. temperature varies linearl in the -direction and linearl in the -direction. First to modes of frequenc of rectangular plate are calculated for five boundar conditions such as C-C-C-C, SS-SS-SS-SS, C-SS-SS-SS, C-SS-C-SS and C-C-C-SS, here C and SS stand for clamped boundar and simpl supported boundar, respectivel. The plate is considered homogeneous and made up of a visco-elastic isotropic material. Raleigh Ritz technique is applied to get the first to modes of frequencies at different values of plate s parameters. Numeric results are presented in tabulated and graphical forms. Keords Isotropic tapered plate, clamped plate, simpl supported plate, thermal gradient, aspect ratio Introduction In recent ears, tapered plates are being increasingl used in modern engineering structures or space vehicles due to their ide technical importance. Consideration of tapered plates not onl helps to reduce the eight of structural elements but also improves the utilisation of the material. Almost ever structure or vehicle orks under the influence of elevated temperature field due to hich a heat flu is created and it directl affects the efficienc of a vehicle or structure. In the available literature, man authors have discussed the vibration of plates having one or to directional tapering ith one-dimensional thermal effect but little ork has been done in the field of to-dimensional thermal effects. Therefore, the authors of this paper discuss vibration problem of visco-elastic rectangular plate ith biparabolic thickness variation along ith bi-linear thermal condition. A surve of research papers, monographs and books published in the last si decades is given as follos. A monograph on vibration of plates ith different shapes and boundar conditions is given b Leissa. 1 Leissa 2 studied the effect of non-homogeneit on the free vibrations of rectangular plate of various combinations of clamped, simpl supported, and free boundar conditions. Jain and Soni 3 discussed the free vibrations of rectangular plate ith parabolic varing thickness using classic theor of plate. In this paper, to parallel edges of the plate are considered as simpl supported and different boundar conditions are considered for the remaining to edges. Tomar and Gupta 4 studied the effect of thermal gradient on the free vibrations of an orthotropic rectangular plate ith bi-linearl thickness variation for various boundar conditions. Leissa 5 analsed the effect of thermal 1 Department of Mathematics, D.A.V. College-Sadhaura, Yamuna Nagar, Harana, India 2 Department of Mathematics, Northern India Engineering College, Delhi, India Corresponding author: Anupam Khanna, Department of Mathematics, D.A.V. College-Sadhaura, 13324 Yamuna Nagar, Harana, India. Email: rajieanupam@gmail.com
14 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) gradient on the vibration of parallelogram plate ith bi-directional thickness variation in both directions for various combinations of boundar conditions. Qiu et al. 6 presented some eperiments on the active control of circular disk vibration. Lal 7 studied transverse vibrations of an orthotropic non-uniform plate ith continuousl varing densit. To parallel edges are considered simpl supported and different combinations of boundar conditions are considered for the other to parallel edges. Li 8 has given a vibrational analsis of rectangular plate ith general elastic boundar support. Leissa 9 discussed the historical bases of Raleigh and Ritz s methods to minimise the frequenc of the vibrations of different structures of beams, bars, plates, etc. Gupta and Kumar 1 studied the thermal effect on vibrations of non-homogeneous rectangular plate ith bi-linearl varing thickness variations. Lal and Dhanpati 11 analsed the free transverse vibrations of non-homogeneous plate of varing thickness. Hota et al. 12 analsed the free vibration of perforated plates. Gupta and Kumar 13 analsed the effect of thermall induced vibration of orthotropic trapezoidal plate ith linearl varing thickness. Shooshtari and Razavi 14 orked on non-linear vibration of laminated fiber-reinforced rectangular plates. Abu Bakar et al. 15 analsed aismmetric vibration of circular plate ith attached annular piezoceramic plate. Khanna and Arora 16 analsed the effect of sinusoidal varing thickness on the vibrations of non-homogeneous parallelogram plate ith bi-linearl temperature variation. The also discussed the effect of non-homogeneit of the material. Khanna and Kaur 17 studied the effect of eponentiall varing thickness, temperature, and Poisson ratio on the free vibration of visco-elastic rectangular plate. The main endeavour of the present investigation is to stud the effects of varing plate s parameters, i.e. thermal gradient (due to temperature variation), taper constants (due to thickness variation) and aspect ratio (ratio of length and breadth of the plate) on the vibration of visco-elastic isotropic rectangular plate. Here, the first to modes of frequenc of the vibration of isotropic rectangular plate are calculated ith the help of Raleigh Ritz technique for five different boundar conditions. Results are given in the form of tables and graphs. Analsis of equation of motion The differential equation describing the motion of a visco-elastic isotropic rectangular plate ma be ritten as 18 @ 2 M @ 2 þ 2 @2 M @@ þ @2 M @ 2 ¼ h @2 @t 2 ð1þ here and are the coordinates of the plate geometr, M and M are the bending moments, M is the tisting moment per unit length of plate, is the mass per unit volume, h is the thickness of plate and is the displacement at time t. The epressions for M,M and M are given b 19 @ 2 M ¼ ~DD 1 @ 2 þ @2 @ 2 @ 2, M ¼ ~DD 1 @ 2 þ @2 @ 2 here ~D is visco-elastic operator. Here, D 1 is the fleural rigidit of the plate s material and it is epressed as 2 and M ¼ ~DD 1 ð1 Þ @2 @@ Eh 3 D 1 ¼ 12ð1 2 Þ Substituting the values of M,M and M in equation (1), one gets @ ed 4 D 1 @ 4 þ 2 @4 @ 2 @ 2 þ @4 @ 4 þ 2 @D 1 @ 3 @ @ 3 þ 2 @3 @@ 2 þ 2 @D 1 @ 3 @ @ 3 þ 2 @3 @@ 2 þ @2 D 1 @ 2 @ 2 @ 2 þ @2 @ 2 þ @2 D 1 @ 2 @ 2 @ 2 þ @2 @ 2 þ 2ð1 Þ @2 D 1 @ 2 þ h @2 @@ @@ @t 2 ¼ ð2þ ð3þ
Khanna and Singhal 141 Using variable separation method, deflection ma be considered as the product of to functions as 21 ð,, tþ ¼ð, Þ:TðtÞ ð4þ here (, ) is the deflection function in and and T(t) is a time function for the vibration of rectangular plate. Substituting equation (4) into equation (3), one obtains þ @2 D 1 @ 2 @ 4 D 1 @ 4 þ 2 @4 @ 2 @ 2 þ @4 @ 4 þ 2 @D 1 @ 3 @ @ 3 þ 2 @3 @@ 2 þ 2 @D 1 @ 3 @ @ 3 þ 2 @3 @@ 2 @ 2 @ 2 þ @2 @ 2 þ @2 D 1 @ 2 @ 2 @ 2 þ @2 @ 2 þ 2ð1 Þ @2 D 1 @ 2 =h ¼ @2 T=@t 2 ð5þ @@ @@ edt Equating both sides of equation (5) to a constant p 2, one obtains @ 4 @ 4 þ 2 @4 @ 2 @ 2 þ @4 @ 4 þ 2 @D 1 @ þ @2 D 1 @ 2 @ 2 @ 2 þ @2 @ 2 þ @2 D 1 @ 2 D 1 @ 3 @ 3 þ 2 @3 @@ 2 @ 2 @ 2 þ @2 @ 2 þ 2 @D 1 @ 3 @ @ 3 þ 2 @3 @@ 2 þ 2ð1 Þ @2 D 1 @ 2 p 2 h ¼ @@ @@ ð6þ and @ 2 T @t 2 þ p2 ~DT ¼ ð7þ Equations (6) and (7) are differential equations of motion and time function for visco-elastic rectangular plate. Since the research area of vibration of plates is too ide to discuss at once, the authors proceed ith a fe limitations. Limitation 1 Thickness of the rectangular plate is considered non-uniform, i.e. thickness of the plate varies bi-parabolic 18 as (shon in Figure 1) 2 2 h ¼ h 1 þ 1 1 þ 2 a 2 b 2 ð8þ Figure 1. Rectangular Plate ith 2-directional thickness variation.
142 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) here a and b are the length and breadth of rectangular plate, respectivel, 1 and 2 are the taper parameters in the -direction and -direction, respectivel, and h is the thickness of the plate at ¼ ¼. Limitations 2 The structures of high-speed space vehicles, i.e. supersonic flights, missiles, etc. are subjected to high surface temperature and large thermal gradient. These conditions can seriousl affect or damage the structure of these vehicles. Variations in the structure s characteristics as a result of thermal effect can be observed b the changes in frequenc of vibration. Therefore, the authors assumed bi-linear temperature variations as 22 ¼ 1 1 ð9þ a b here denotes the temperature ecess above the reference temperature at an point on the plate and denotes the temperature ecess above the reference temperature at ¼ ¼. The temperature dependence of the modulus of elasticit for most of the engineering materials can be epressed as follos 2 E ¼ E ð1 Þ ð1þ here E is value of the Young s modulus at reference temperature, i.e. ¼ and is slope of variation of E and. Using equation (9) in equation (1), one obtains h E ¼ E 1 1 ð11þ a b here ¼ ( <1) is the thermal gradient. Using equations (8) and (11) in equation (2), one gets E 1 1 a 1 3 3 b h 3 1 þ 2 1 a 1 þ 2 2 D 1 ¼ 2 b 2 12ð1 2 Þ ð12þ Limitation 3 In order to fulfil the practical aspects for using in structures or vehicles, the authors discuss five different boundar conditions for the rectangular plate, i.e. C-C-C-C, SS-SS-SS-SS, C-SS-SS-SS, C-SS-C-SS and C-C-C-SS 23,24 as follos: (i) hen the boundar of plate is C-C-C-C, the boundar conditions are ¼ @ @ ¼ at ¼, a and ¼ @ ¼ at ¼, b @ To satisf these boundar conditions, the to-term deflection function is defined as ¼ 2 2 2 2 h 1 1 A 1 þ A 2 1 (ii) hen the boundar of the plate is SS-SS-SS-SS, the boundar conditions are ¼ @2 @ 2 ¼ at ¼, a and ¼ @2 ¼ at ¼, b @2
Khanna and Singhal 143 To satisf these boundar conditions, the to-term deflection function is defined as h ¼ 1 h A 1 þ A 2 1 (iii) hen the boundar of the plate is C-SS-SS-SS, the boundar conditions are ¼ @ @ ¼ at ¼, ¼ @2 @ 2 ¼ at ¼ a and ¼ @2 ¼ at ¼, b @2 To satisf these boundar conditions, the to-term deflection function is defined as ¼ 2 1 1 h A 1 þ A 2 1 (iv) hen the boundar of the plate is C-SS-C-SS, the boundar conditions are ¼ @ @ ¼ at ¼, a and ¼ @2 ¼ at ¼, b @2 To satisf these boundar conditions, the to-term deflection function is defined as ¼ 2 1 2 h 1 A 1 þ A 2 1 (v) hen the boundar of plate is C-C-C-SS, the boundar conditions are ¼ @ @ ¼ at ¼, a ; ¼ @ @ ¼ at ¼ and ¼ @2 @ 2 ¼ at ¼ b To satisf these boundar conditions, the to-term deflection function is defined as ¼ 2 2 2 h 1 1 A 1 þ A 2 1 Here, A 1 and A 2 are the arbitrar constants occurred due to the first to modes of vibration. Solution of frequenc equation Raleigh Ritz technique is applied to solve the frequenc equation. In this method, one requires that the maimum strain energ (S E ) must be equal to the maimum kinetic energ (K E ). So it is necessar for the problem under consideration that 25 ðs E K E Þ¼ ð13þ Here K E ¼ 1 2 p2 Z a Z b h 2 dd and S E ¼ 1 2 Z a Z b " 2 2 D 1 @2 @ 2 þ @2 @ 2 þ2 @2 @ 2 @ 2 @ 2 þ 2ð1 Þ @ 2 2 # dd @@
144 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) Assuming the non-dimensional variables as X ¼ a, Y ¼ a, ¼ a, h ¼ h a ð14þ Using equation (14), the kinetic energ (K E ) and strain energ (S E ) become K E ¼ 1 Z 1 Z b=a 2 p2 h a 5 ð1 þ 1 X 2 a 2 Þ 1 þ 2 b 2 Y2 2 dydx ð15þ and S E ¼ E h 3 Z 1 Z b=a n a3 24ð1 2 1 ð1 XÞ 1 a oð1 Þ b Y þ 1 X 2 Þ 3 a 2 3 1 þ 2 b 2 Y2 " 2 2 @2 @X 2 þ @2 @Y 2 þ2 @2 @ 2 @ 2 2 # @X 2 þ 2ð1 Þ dydx @Y2 @X@Y ð16þ Using the modified values of kinetic and strain energ in equation (13), one gets S E l2 K E ¼ ð17þ here K E ¼ Z 1 Z b=a ð1 þ 1 X 2 a 2 Þ 1 þ 2 b 2 Y2 2 dydx and Z 1 Z b=a n S E ¼ 1 ð1 XÞ 1 a oð1 b Y þ 1 X 2 Þ 3 a 2 3 1 þ 2 b 2 Y2 " 2 2 @2 @X 2 þ @2 @Y 2 þ2 @2 @ 2 @ 2 2 # @X 2 þ 2ð1 Þ dydx @Y2 @X@Y Here, l 2 ¼ 12p2 a 2 ð1 2 Þ is the frequenc parameter. E h 2 Equation (17) consists of to unknon constants, i.e. A 1 and A 2 arising due to the substitution of corresponding to the different boundar conditions. These to constants are to be determined as follos Simplifing equation (18), one gets @ S E @A l2 K E ¼, n ¼ 1, 2 ð18þ n bn 1 A 1 þ bn 2 A 2 ¼, n ¼ 1, 2 ð19þ here b n1 and b n2 include parametric constants, i.e. taper constants, thermal gradient, aspect ratio and frequenc parameter. For a non-trivial solution, the determinant of the coefficient must be zero. So, one gets the frequenc equation as b 11 b 12 b 21 b 22 ¼ ð2þ
Khanna and Singhal 145 Table 1. Frequenc versus thermal gradient at fied a/b ¼ 1.5. 1 ¼ 2 ¼. 1 ¼ 2 ¼.4 1 ¼ 2 ¼.8 B.C s Mode I Mode II Mode I Mode II Mode I Mode II C-C-C-C 256.2 64.83 339.26 86.1 45.47 113.58 SS-SS-SS-SS 256.92 34.25 351.22 44.99 48.69 59.51 C-SS-SS-SS 26.97 39.75 378.4 55.56 535.3 76.38 C-SS-C-SS 288.81 41.94 392.21 54.96 532.36 72.17 C-C-C-SS 357.65 55.24 554.18 74.97 794.72 1.11.4 C-C-C-C 243.5 61.5 327.12 82.9 438.91 11.24 SS-SS-SS-SS 243.73 32.5 34.3 43.27 471.12 57.76 C-SS-SS-SS 252.6 38.37 371.31 54.9 528.79 74.79 C-SS-C-SS 273.99 39.79 379.64 52.84 521.15 7. C-C-C-SS 353.7 53.12 549.95 72.71 789.85 97.65.8 C-C-C-C 229.15 57.98 314.53 79.54 427.5 16.76 SS-SS-SS-SS 229.79 3.64 329.1 41.47 461.36 55.95 C-SS-SS-SS 243.95 36.92 364.9 52.57 522.2 73.16 C-SS-C-SS 258.32 37.51 366.64 5.63 59.69 67.76 C-C-C-SS 349.71 5.91 545.69 7.37 784.95 95.13 Table 2. Frequenc versus taper constants at fied a/b ¼ 1.5, ¼.2. 1.4.8 2 B.C s Mode I Mode II Mode I Mode II Mode I Mode II C-C-C-C 249.71 63.18 281.18 71.64 316.63 81.12 SS-SS-SS-SS 25.41 33.39 284.62 38.36 324.28 44.1 C-SS-SS-SS 256.82 39.7 314.26 47.77 377.57 57.5 C-SS-C-SS 281.5 4.88 314.83 47.1 351.36 54.33 C-C-C-SS 355.68 54.19 396.42 61.5 44.64 69.75.4 C-C-C-C 296.47 74.78 333.24 84.52 374.53 95.37 SS-SS-SS-SS 35.39 38.37 345.8 44.14 392.38 5.83 C-SS-SS-SS 39.34 44.76 374.87 54.83 447.23 65.61 C-SS-C-SS 345.64 46.82 385.98 53.91 43.3 62.14 C-C-C-SS 495.89 65.1 552.7 73.85 612.77 83.72.8 C-C-C-C 353.25 88.23 396.43 99.47 444.73 111.92 SS-SS-SS-SS 372.85 44.9 42.89 5.82 475.93 58.64 C-SS-SS-SS 373.74 51.34 448.93 63.4 532.6 75.59 C-SS-C-SS 424.47 53.6 473.46 61.69 526.78 71.1 C-C-C-SS 641.91 76.87 714.27 87.21 792.29 98.89 ith the help of equation (22), one can obtain a quadratic equation in l 2 from hich the to values of frequenc parameter for both the modes of vibration can be evaluated easil. Results and discussion Computations ere made for the first to modes of frequenc at different values of thermal gradient (), taper constants ( 1 and 2 ) and aspect ratio (a/b) for the five different boundar conditions. In calculations, Poisson ratio is considered as constant, i.e..345, and thickness of the plate h at X ¼ Y ¼ is taken as.1 m.
146 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) Table 3. Frequenc versus aspect ratio. ¼ 1 ¼ 2 ¼. ¼ 1 ¼ 2 ¼.4 ¼ 1 ¼ 2 ¼.8 a/b B.C s Mode I Mode II Mode I Mode II Mode I Mode II.5 C-C-C-C 14.94 26.14 134.25 33.54 175.58 43.17 SS-SS-SS-SS 15.91 13.17 14.28 16.45 19.8 2.96 C-SS-SS-SS 135.3 2. 21.61 26.14 33.96 34.11 C-SS-C-SS 86.9 25.38 112.18 32.69 148.62 42.51 C-C-C-SS 85.71 26.19 118.62 35.48 163.45 47.51 1 C-C-C-C 15.1 38.35 191.34 49.6 219.3 63.21 SS-SS-SS-SS 149.67 21.8 198.25 26.84 23.95 35.3 C-SS-SS-SS 169.42 27.19 253.4 36.7 34.74 49.35 C-SS-C-SS 151.36 3.95 197.75 39.6 228.63 51.5 C-C-C-SS 176.67 35.78 266.2 47.93 318.97 63.76 1.5 C-C-C-C 256.2 64.82 327.12 82.9 383.36 16.76 SS-SS-SS-SS 256.92 34.25 34.3 43.27 396.35 55.95 C-SS-SS-SS 26.97 39.75 371.31 54.9 453.81 73.16 C-SS-C-SS 288.81 41.94 379.64 52.84 452.66 67.76 C-C-C-SS 357.65 55.24 549.95 72.71 691.74 95.13 Figure 2. Frequenc (Mode 1) versus thermal gradient. Frequenc for the first to modes of vibration ere calculated for increasing values of thermal gradient at various combinations of taper parameters and fied value of aspect ratio, and the results are shon in Table 1. It is evident that the frequenc for both the modes of vibration of plate ith different boundar conditions decrease continuousl as thermal gradient () increases corresponding to each paired value of 1 and 2. It is interesting to note that the first mode of frequenc is maimum for C-C-C-SS plate but minimum for C-C-C-C plate. For second mode, frequenc is maimum for C-C-C-C plate, hile minimum for SS-SS-SS-SS. Table 2 shos the frequenc for the first to modes of vibration for various values of taper parameters at a fied aspect ratio and thermal gradient. The authors conclude that the frequenc continuousl increases as one of
Khanna and Singhal 147 Figure 3. Frequenc (Mode 2) versus thermal gradient. Figure 4. Frequenc (Mode 1) versus taper constant ( 1 ). the taper parameter increases for a fied value of another. It is clear from Table 2 that the frequenc increases faster hen 2 increases from. to.8 as compared to increasing 1. Frequenc for the first to modes of vibration at different values of aspect ratio for the fied values of plate s parameters are calculated and presented in Table 3. It is noted that the frequencies for both the modes of vibration increase rapidl as the aspect ratio (a/b) increases from.5 to 1.5 for the five different boundar conditions at each paired value of a, 1 and 2. For better understanding of the results in terms of boundar conditions, the results in Tables 1 to 3 are presented in a graphical form. It is clear from Figures 2 and 3 that the frequenc for both the modes of vibration decreases continuousl at fied values of taper constant and the aspect ratio for different boundar conditions.
148 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) Figure 5. Frequenc (Mode 2) versus taper constant ( 1 ). Figure 6. Frequenc (Mode 1) versus taper constant ( 2 ). One can clearl observe from Figures 4 to 7 that the frequenc increases continuousl for the first to modes of vibrations as the taper constants ( 1 and 2 ) increase for the fied values of the thermal gradient and aspect ratio. The authors conclude from the graphs in Figures 8 and 9 that the frequenc for both the modes of vibration increases ver fast as the aspect ratio (a/b) increases from.5 to 1.5 for all the five combinations of boundar conditions. Comparison and conclusions Table 4 shos a comparison of the results of frequenc for both the modes of vibration for C-C-C-C boundar condition onl ith Khanna and Singhal 22 at the corresponding values of the plate s parameters. Here, the authors
Khanna and Singhal 149 Figure 7. Frequenc (Mode 2) versus taper constant ( 2 ). Figure 8. Frequenc (Mode 1) versus aspect ratio (a/b). found that the frequenc for the both modes of vibration is less than that in the present paper (shon in italic) than 22 (shon in bold) at the corresponding values of the plate s parameters. It shos that the frequenc of the rectangular plate changes ith the change in tapering of the plate. The results, in light of the comparison, can thus be summarised in the folloing points: 1. Frequenc can be controlled b appropriate tapering, i.e. b assuming suitable values of taper parameters. Desired values of frequenc can be obtained b choosing appropriate taper parameters.
15 Journal of Lo Frequenc Noise, Vibration and Active Control 35(2) Figure 9. Frequenc (Mode 2) versus aspect ratio (a/b). Table 4. Frequenc of present paper versus 15 at a/b ¼ 1.5. 1 ¼ 2 ¼.4 1 ¼ 2 ¼.8 Mode 1 Mode 2 Mode 1 Mode 2.2 333.24 37.94 84.52 93.77 444.73 529.98 111.92 133.59 2. Boundar conditions directl affect the vibrational characteristics of the rectangular plate. 3. First mode of frequenc is maimum at C-C-C-SS boundar condition but minimum at C-C-C-C boundar condition. But second mode is maimum at C-C-C-C boundar condition and minimum at SS-SS-SS-SS boundar condition. 4. ith increasing values of thermal gradient, frequenc decreases for both the modes of vibration of plate. It implies that temperature variations also directl affect the vibration of plate. 5. Vibration can be controlled effectivel b bi-parabolic thickness variation (present paper) as compared to the bi-linear thickness variation. 22 Therefore, scientists and design engineers ho are orking in designing space vehicles or structures are advised to analse the findings of the present stud in a practical manner to make more efficient and authentic mechanical structures and safe designs of vehicles. Declaration of conflicting interests The author(s) declared no potential conflicts of interest ith respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. References 1. Leissa A. Vibration of plates, NASA SP-16 (1969), Scientific and Technical Information Division, National Aeronautics and Space Administration (NASA).
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