Journa of American Science, ;8(3) Comparison of wo Kinds of Functionay Graded Conica Shes with Various Gradient Index for Vibration Anaysis Amirhossein Nezhadi *, Rosan Abdu Rahman, Amran Ayob Facuty of Mechanica Engineering, Universiti eknoogi Maaysia (UM), 83 Skudai, Johor, Maaysia * E-mai: a_h_nezhadi@yahoo.com Abstract: In this paper, a study on the effects of the FGM configuration is taken into account by studying the frequencies of two FG conica shes. ype I FG conica she has auminum on its inner surface and amina on its outer surface and ype II FG cyindrica she has amina on its inner surface and auminum on its outer surface. he study is done based on Rayeigh-Ritz method. he objective is to study the effects of configurations of the constituent materias on the frequencies. he properties are graded in the thickness direction according to the gradient index distribution. he anaysis is carried out with strains-dispacement reations are given by Soede (98). he governing equations are obtained using energy functiona with the Rayeigh-Ritz method. Resuts are presented on the frequency characteristics and the infuences of constituent various voume fractions for ype I and II FG conica shes. the boundary conditions are simpy supported. [Amirhossein Nezhadi, Rosan Abdu Rahman, Amran Ayob. Comparison of wo Kinds of Functionay Graded Conica Shes with Various Gradient Index for Vibration Anaysis. ;8(3):65-657]. (ISSN: 545-3).. 87 Keywords: Functionay Graded Materias, Conica she, Rayeigh-Ritz Method, Energy Functiona, Vibration. Introduction Conica shes have been widey utiized in a variety of engineering fieds as important structura components due to their specia geometric shapes, especiay in marine industries and aerospace. Considerabe investigations have been conducted to examine the dynamic responses of such structures. Leissa (993) provided an earier survey on the free vibration of conica shes, the effects of semi-vertex anges and different boundary conditions on the frequency characteristics of conica shes were investigated. he vibration anaysis of shaow conica shes by a goba Ritz formuation based on the energy principe is done by Liew and Lim 994. Next, a formuation for the free vibration of moderatey thick conica she panes based on shear deformabe theory was aso presented by them (995). He et a. (), Ng et a. () and Liew et a. (4) examined the finite eement anaysis of she and she panes subjected to vibration. he generaized differentia quadrature method was empoyed to study the free vibration of composite aminated conica shes by Shu (996), and the vibration characteristics of open conicay curved, isotropic she panes using a h p version of finite eement method was investigated by Barde et a. (998). A new kind of composite materias are known as functionay graded materias (FGM) that are formed by mixing two or more different materias according to a pre-determined formua that depends on the voume fractions of constituents. Such materias possess smooth and continuous materia properties, which make them more suitabe in engineering appications. Much effort has been used to various structura anayses of functionay graded structures, such as static, therma stresses, vibration anayses and bucking. Noda (999) presented a review on therma stresses in functionay graded materias, the therma stresses on the functionay graded pates and the therma stress intensity factor in the functionay graded pates with crack were discussed. he stresses and strains in a functionay thick-waed tube under uniform therma oading was examined by Fukui et a. (993), and studied the effects of FGM materias on the parametric resonance of pate structures was studied by Ng et a. (). he therma stress behavior of functionay graded hoow circuar cyinders was investigated by Liew et a. (3), and the static and dynamic response of functionay graded pates in terms of the combination of the first-order shear deformation pate theory and the von Kármán strains was examined by Praveen and Reddy (998). A finite eement formuation for the active contro of functionay graded pates with integrated piezoeectric actuators and sensors was provided by He et a. (), and the vibration anaysis of variabe thickness annuar functionay graded pates was conducted by Efraim and Eisenberger (7). A few pubications on the anaysis of functionay graded conica shes have been reported in iterature. he stabiity of truncated conica shes of functionay 65
Journa of American Science, ;8(3) graded materia subjected to externa pressure was investigated by Sofiyev (4). ornabene (9) was cacuated free vibration anaysis of moderatey thick functionay graded conica, cyindrica she and annuar pate structures with a four-parameter power-aw distribution. he formuation was based on generaized differentia quadrature method and the first order shear deformation theory. ornbene et a. (9) aso examined the same structures with two different power-aw distributions. he appications of functionay graded conica she can be very extensive. Due to their high strength and resistance to temperature change, the functionay graded conica she can be appied to miitary aircraft propusion system, fuseage structures of civi airiners, and other machine parts. In this study, Rayeigh-Ritz method is used to comparison natura frequencies of two kinds of functionay graded conica shes with various gradient indexes.. Functionay gradient materias In which for functionay graded materias with two constituent materias Poisson ratio υ is assumed to be constant through the thickness, whereas the variations through the thickness of Young s moduus and the mass density per unit voume ρ(η) can be written as (Matsunaga, 9) E η = E E E (.5 η h )p () ρ(η) = ρ ρ ρ (.5 η h )p () Where η is the thickness coordinate (-h/ η h/), and p is the gradient index. he materia properties vary continuousy from materia at the inner surface of the conica she to materia at the outer surface of the conica she. 3. Equation of motion of FG conica she A thin and FG conica she with constant thickness is assumed. Fig. shows the schematic diagram of the conica she. he two boundaries of the conica she are simpy supported (S-S). he corresponding curviinear surface coordinates O ζη and Cartesian coordinates O xyz are aso shown in Fig.. he curviinear surface coordinates are imited to be orthogona ones which coincide with the ines of principa curvature of the neutra surface. For conica shes, the ines of principa curvature of the neutra surface are the circes (ζ-axis) and parae meridians (-axis).. Fig.. he schematic diagram of a FGM conica she (a) he geometry and the curviinear surface and Cartesian Coordinate Systems; (b) the infinitesima she eement and the corresponding stresses. For a thin conica she, pane stress condition is assumed and the constitutive reation is given by ζ ζ ζ ζ 3 = ( μ ) μ ( μ ) μ ( μ ) ( μ ) ( μ) ( μ) ε ε ε ε 3 (3) where ε ij (i, j =,, 3) are the strains and σ ij (i, j =,, 3) are the stresses in which, and 3 coincide with the, ζ and η directions and is the Young s moduus and μ is the Poisson s ratio. Where σ and σ are the norma stresses acting in the and ζ directions, σ and σ 3 are the shear stresses in the curviinear coordinate O ζη as shown in Fig. b. o determine the equation of motion of the conica she, the Lagrangian function with the Rayeigh-Ritz method wi be used. he Lagrangian function is written by (Soede, 4) t δ U dt = (4) t Where the kinetic energy, U strain energy and W work, t and t are the integration time imits, δ() denotes the first variation. he strain energy and kinetic energy and virtua work of a conica she can be written as 65
Journa of American Science, ;8(3) = h/ L ρ(η) u v w h/ L t t t sinα d d ζ d η (5) h/ L U = (ζ ε ζ ε ζ ε h/ L ζ 3 ε 3 ) sinα d d ζ d η (6) For simpy supported conica she, the boundary conditions at both ends can be written as v = w = N = M = (7) at = and = woud be considered. he dispacement fieds which satisfy these boundary conditions can be written as u, ζ, t = cos iπ cos jζ p(t) (8) v, ζ, t = sin iπ sin jζ r(t) (9) w, ζ, t = sin iπ cos jζ s(t) () i =,,, m ; j =,,, n, where i and j denote the wave numbers in the meridiona and circumferentia directions and p, r, s are the generaized coordinates or moda coordinates. Substituting Eqs. (5) and (6) in terms of the dispacement fieds into Eq. (4) and fufiing the variation operation in terms of p, r and s. hey can be obtained as M t d X dt K tx = () where M t the generaized mass matrix, K t the stiffness matrix, X the generaized coordinate matrix and written by M K K K 3 M t = M K t = K K 4 K 5 M 3 K 3 K 5 K 6 X = [p r s ] () where M, M and M 3 are the moda mass matrices and K, K,, K 6 are the moda stiffness matrices which are given in Appendix A. A soution of Eq. () is in the form X t = X e λt (3) where λ is the characteristic vaues or the eigenvaue and X is the eigenvector. Substituting Eq. (3) into the homogeneous differentia equation of Eq. () eads to the foowing standard eigenvaue probem: M t λ K t X = (4) From which the eignvaues and eignvectors can be obtinethed. he imaginary parts of the eigenvaues are the natura frequencies of the FG conica she. 4. Resuts and discussions he resuts for Meta are compared with the open iterature in abe. In the numerica cacuations, the non-dimensiona frequency parameter is defined as ((Lam and Li, 999) ; (Liew et a., )) f = ω α ρ m (μ ) E m (5) where ω is the natura frequency of the conica she in radians per second. he materia properties used in the present study is: Meta (Auminium, A): E M = 7 GPa, ρ M = 7 kg/m 3, μ =.3 Ceramic (Amina, A O 3 ): E C = 38 GPa, ρ C = 38 kg/m 3, μ =.3 he variation through the thickness of Young s moduus and mass density per unit voume ρ(η) are the same as Eqs. () and (). he structura parameters are h =.4 m, h/a =., (L L ) sin α /a =.5. For meta, the frequency parameters computed by Eq. (5) are isted in abe. Aso the corresponding resuts by ((Lam and Li, 999) ; (Irie et a., 984)) are isted in abe. he frequencies of two FG conica shes with two different FGM configurations are studied: ype I FG conica she and ype II FG conica she. ype I FG conica she has auminum on its inner surface and amina on its outer surface and ype II FG conica she has amina on its inner surface and auminum on its outer surface. abes, 3 and 4 show the variations of the natura frequencies (Hz) with the circumferentia wave numbers n for a ype I FG conica she. he coumns P C and P M show the natura frequencies for a ceramic conica she and a meta conica she, respectivey. he effects of changing gradient index (P) can be seen from tabes. As P increased, the natura frequencies decreased. When P is sma, the natura frequencies approached those of P C and when P is arge they approached those of P M. Hence, the natura frequencies for P > fe between those of P C and P M for a given circumferentia wave number n. 653
Journa of American Science, ;8(3) abe. Comparisons of frequency parameter f for the conica she with S-S boundaries (m =, Meta). α = 3 α = 45 α = 6 n Irie et a Lam and (984) Li (999) Present.79.84.8437 3.784.7376.7463 4.635.636.6494 5.553.558.559 6.4949.495.579 7.4653.466.4779 8.4654.466.469 9.489.496.4938.6879.7655.7644 3.6973.7.78 4.6664.6739.67467 5.634.633.63364 6.63.635.649 7.598.59.593 8.599.6.645 9.657.673.669.577.6348.6343 3.6.638.636 4.654.645.6459 5.677.6.68 6.659.67.67 7.6343.635.63479 8.665.666.6655 9.784.7.7873 abe. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype I FG conica she α = 3 n P C P= P=3 P=8 P=3 P M 3847 393 7 338 79 955 356 97 469 38 9 787 3 393 563 7 883 675 57 4 677 3 877 633 453 36 5 33 98 63 48 7 85 6 88 74 454 88 5 6 7 963 58 358 94 998 8 957 554 343 3 4 994 9 57 6 4 34 73 45 47 74 5 433 9 4 abe 3. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype I FG conica she α = 45 n P C P= P=3 P=8 P=3 P M 336 737 3 4 8 686 387 68 3 947 734 6 3 37 474 84 64 58 4 84 35 96 78 54 43 5 64 5 837 63 456 343 6 53 37 746 566 4 8 7 473 977 7 547 387 57 8 54 98 73 578 47 7 9 65 46 779 658 49 39 799 7 896 784 68 4 abe 4. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype I FG Conica She α = 6 n P C P= P=3 P=8 P=3 P M 679 75 859 657 48 36 645 4 833 638 464 344 3 6 99 799 65 445 3 4 563 57 768 598 43 3 5 549 3 75 597 43 95 6 574 36 76 6 456 38 7 647 78 84 676 57 345 8 774 6 883 764 589 4 9 955 89 887 7 5 389 456 5 4 84 6 abes 5, 6 and 7 show the aterations of the natura frequencies (Hz) with the circumferentia wave numbers n for a ype II FG conica she. abe 5. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype II FG conica she α = 3 n P C P=5 P=8 P=5 P=3 P M 3847 3779 375 366 3558 955 356 3453 343 3344 35 787 3 393 336 99 938 855 57 4 677 64 583 536 463 36 5 33 79 4 98 33 85 6 88 33 994 95 89 6 7 963 899 857 83 754 998 8 957 879 83 783 7 994 9 57 964 97 85 783 45 47 35 67 4 98 4. 654
Journa of American Science, ;8(3) he infuence of the gradient index (P) on the natura frequencies is the opposite of a ype I FG conica she. Unike a ype I FG conica she where the natura frequencies decreased with P, the natura frequencies for a ype II FG conica she increased with P. hus the infuence of the gradient index for a ype II FG conica she is different from a ype I FG conica she. abe 6. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype II FG conica she α = 45 n P C P=5 P=8 P=5 P=3 P M 336 348 397 338 348 686 387 3 37 33 96 6 3 37 94 89 836 753 58 4 84 746 697 643 564 43 5 64 57 5 468 39.3 343 6 53 444 39 337 6.6 8 7 473 384 37 7 93.5 57 8 54 4 337 74 94.3 7 9 65 495 4 35 66. 39 799 66 577 5 45. 4 change continuousy from auminum on its inner surface to amina on its outer surface. he other is named as a ype II FG cyindrica she and has properties that change continuousy from amina on its inner surface to auminum on its outer surface. he anaysis was done by Rayeigh-Ritz method. For vaidation, the resuts are compared with those in the iterature and have found to be accurate. he infuence of the gradient index (P) on the frequencies for ypes I and II FG conica shes has been found to be different. For the ype I FG conica shes, the natura frequencies decreased when (P) increased, and for the ype II FG conica shes, the natura frequencies increased when P increased. In ypes I and II FG conica shes, the natura frequencies for a vaues of P are between those for auminum and amina conica shes. herefore, the gradient index and the configurations of the constituent materias affect the natura frequencies. Appendix A. he expressions of the moda mass, moda stiffness and forcing matrices in Eqs. () are given by abe 7. Variation of natura frequencies (Hz) against circumferentia wave number n (m=). ype II FG conica she α = 6 n P C P=5 P=8 P= 5 P=3 P M 679 63 55.4 496 48 36 645 566.9 54. 458 38 344 3 6 59. 464.9 48 33 3 4 563 476. 49.3 36 83 3 5 549 454.4 393.9 333 54 95 6 574 468.9 43.3 339 56 38 7 647 59.8 457.5 388 3 345 8 774 64. 56.9 486 39 4 9 955 86.9 77.6 634 53 5 389 3. 9. 89 79 6 5. Concusions A study on the vibration of functionay graded (FG) conica shes made of auminum and amina has been presented. he study was done for two kinds of functionay graded conica shes where the configurations of the constituent materias in the functionay graded conica shes are different. ype I FG conica she has auminum on its inner surface and amina on its outer surface and ype II FG conica she has amina on its inner surface and auminum on its outer surface. One is named as a ype I FG conica she and has properties that M = sinα M = sinα M 3 = sinα K = sin α μ μu sin α h/ UU ρ(η)dηddζ h/ h/ )E η dηddζ K = μ h/ ( h/ h/ VV ρ(η)dηddζ WW ρ(η)dηddζ UU μ U G η dηddζ ζ ζ h/ (U V ζ μ V ζ h/ ( V ζ ζ V h/ )E η dηddζ )E η dηddζ ( μ) μ ( ( μ )tanα ) V η dηddζ h/ ζ (U) V η ( μ )tanα ζ dηddζ ( μ)tanα ζ ( V ) η dηddζ h/ V dηddζ ( μ)tanα ζ h/ 655
Journa of American Science, ;8(3) h/ K 3 = COS α μ (UW h/ h/ μ W )E η dηddζ sinα W ( μ ) ( ) dηddζ μ sinα ( W ( μ )sin α h/ ζ μ sinα ( W )dηddζ ( μ ) h/ U ( W μ sinα ζ ) dηddζ sinα U ( W ( μ ) h/ ) dηddζ μ sinα (U)( W ( μ ) ) dηddζ ( W μ sinα ζ )( ζ ) h/ ( μ sinα ζ )( W ζ ) K 4 = μ sin α η V V tan α ζ ζ sinα ( V h/ ( V V ζ ζ h/ 3) E η dηddζ h/ V V VV V V V )G η dηddζ sin α h/ h/ VV η V tan α V 4 VV V 3 V G η dηddζ ( V ( μ )sinα tanα ζ )( V ζ ) cos α ( μ) 3 cos α ( μ) 3 cos α ( μ) 4 cos α ( μ) K 5 = h/ h/ h/ h/ h/ h/ ) dηddζ dηddζ dηddζ ( V )( V ) dηddζ ( V )(V ) dηddζ V ( V ) V (V ) h/ V W μ tan α ζ h/ V W ( (μ )tan α ζ h/ V W sin α ζ tan α V 3 W ζ 3) dηddζ ( V W ζ h/ dηddζ dηddζ E η dηddζ V V dηddζ μ V W ζ V W V W ζ ζ )G η ζ η dηddζ ( μ )tan α V ζ (W ) h/ dηddζ V W ( μ )sin α ζ ( ζ ) h/ dηddζ V μ ζ ( W ) dηddζ h/ μ V W μ ζ ( ) dηddζ h/ V W μ ( ζ ) dηddζ h/ V μ sin α ( W ζ ) h/ dηddζ V ( W μ ζ ) dηddζ h/ V ( W μ ζ ) dηddζ K 6 = sin α (μ ) h/ h/ μ W W μ W W ( μ sin α sin α ( W W W W ) dηddζ W W ζ ζ 3 μ W W W ζ μ W W W ζ ζ dηddζ sin α μ tan α 4 sin α W ζ h/ W W h/ ( W ζ h/ W ζ )G(η) h/ h/ cosα μ η dηddζ ( W ζ ) W h/ h/ h/ W E η dηddζ W ζ ) W W W W W ζ ζ ζ ζ ζ μ sinα tanα dηddζ ( W ) W dηddζ μ cosα ( W μ ) W dηddζ h/ W ( W μ sinα tanα ζ ) h/ dηddζ cos α W ( W μ ) dηddζ μ cos α μ h/ Acknowedgements: W ( W ) dηddζ (6) Authors are gratefu to the Universiti eknoogi Maaysia (Johor Bahru) for Internationa Doctora Feowship (IDF) to carry out this research. 656
Journa of American Science, ;8(3) Corresponding Author: Amirhossein Nezhadi Facuty of Mechanica Engineering Universiti eknoogi Maaysia (UM) 83 UM Skudai, Johor, Maaysia. E-mai: a_h_nezhadi@yahoo.com References:. Leissa A. Vibration of shes. he Acoustic Society of America, 993.. Liew KM, Lim CW. Vibratory characteristics of cantievered rectanguar shaow shes of variabe thickness. AIAA J 994; 3:387 96. 3. Liew KM, Lim CW. Vibration of perforated douby-curved shaow shes with rounded corners. Int J Soids Struct 994; 3:59 36. 4. Lim CW, Liew KM. Vibratory behaviour of shaow conica shes by a goba Ritz formuation. Eng Struct 995; 7:63 9. 5. Lim CW, Liew KM. A higher order theory for vibration of shear deformabe cyindrica shaow shes. Int J Mech Sci 995; 37:77 95. 6. He XQ, Liew KM, Ng Y, Sivashanker SA. FEM mode for the active contro of curved FGM shes using piezoeectric sensor/actuator ayers. Int J Numer Methods Eng ; 54:853 7. 7. Ng Y, He XQ, Liew KM. Finite eement modeing of active contro of functionay graded shes in frequency domain via piezoeectric sensors and actuators. Comput Mech;8: 9. 8. Liew KM, He XQ, Kitipornchai S. Finite eement method for the feedback contro of FGM shes in the frequency domain via piezoeectric sensors and actuators. Comput Methods App Mech Eng 4;93:57 73. 9. Shu C. Free vibration anaysis of composite aminated conica shes by generaized differentia quadrature. J Sound Vib 996; 94:587 64.. Barde NS, Dunsdon JM, Langey RS. Free vibration of thin, isotropic, open, conica panes. J Sound Vib 998; 7:97 3.. Noda N. herma stresses in functionay graded materias. J herma Stress 999; :477 5.. Fukui Y, Yamanaka N, Wakashima K. he stresses and strains in a thick-waed tube for functionay graded under uniform therma oading. Int J Jpn Soc Mech Eng Ser A 993 ; 36:56 6. 3. Ng Y, Lam KY, Liew KM. Effects of FGM materias on the parametric resonance of pate structures. Comput Methods App Mech Eng ;9:953 6. 4. Liew KM, Kitipornchai S, Zhang XZ, Lim CW. Anaysis of the therma stress behavior of functionay graded hoow circuar cyinders. Int J Soids Struct 3; 4:355 8. 5. Praveen GN, Reddy JN. Noninear transient thermoeastic anaysis of functionay graded ceramic meta pates. Int J Soids Struct 998; 33:4457 76. 6. He XQ, Ng Y, Sivashanker S, Liew KM. Active contro of FGM pates with integrated piezoeectric sensors and actuators. Int J Soids Struct ; 38:64 55. 7. Efraim E, Eisenberger M. Exact vibration anaysis of variabe thickness thick annuar isotropic and FGM pates. J Sound Vib 7; 99:7 38. 8. Sofiyev AH. he stabiity of functionay graded truncated conica shes subjected to aperiodic impusive oading. Int J Soids Struct 4; 4:34 4. 9. Sofiyev AH. he vibration and stabiity behavior of freey supported FGM conica shes subjected to externa pressure. Compos Struct 9; 89:356 66.. ornabene F. Free vibration anaysis of functionay graded conica, cyindrica she and annuar pate structures with a four-parameter power-aw distribution. Comput Methods App Mech Eng 9; 98:9 35.. ornabene F, Vioa E, Inman DJ. -D differentia quadrature soution for vibration anaysis of functionay graded conica, cyindrica she and annuar pate structures. J Sound Vib 9; 38:59 9.. Matsunaga H. Free vibration and stabiity of functionay graded circuar cyindrica she according to a D higher -order shear deformation theory, Compos. Struct. 9; (88) 59 3. 3. Soede W. Vibrations of Shes and Pates, third ed. Marce Dekker Inc., New York, USA, 4. 4. Lam KY and Li H. Infuence of boundary conditions on the frequency characteristics of a rotating truncated circuar conica she, J. Sound and Vib. 999, (3) 7 95. 5. Liew KM and He XQ and Ng Y and Kitipornchai S. Active contro of FGM she subjected to a temperature gradient via piezoeectric sensor/actuator patches. Int. J. Numer. Meth. Engng. ; (55) 653 668. 6. Irie and Yamada G and anaka K. Natura frequencies of truncated conica shes, J. Sound Vib. 984; (9) 447 53. 3// 657