Phân tích động lực học phi tuyến tấm FGM và vỏ trụ tròn sandwich FGM chứa đầy chất lỏng tt t

MINISTRY OF EDUCATION AND TRAINING MINISTRY OF DEFENSE ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY LE XUAN DOAN NONLINEAR DYNAMIC ANALYSIS OF FGM PLATES AND SANDWICH-FGM CYLINDRICAL SHELL FILLED WITH FLUID Specialization: Mechanical engineering Code: 52 01 01 SUMMARY OF TECHNICAL DOCTORAL THESIS HA NOI – 2020 This Thesis was completed at: Academy of Military Science and Technology, Ministry of Defense Scientific Supervisors: Assoc Prof Dr Khuc Van Phu Assoc Prof Dr Nguyen Minh Tuan Reviewer 1: Prof Dr Hoang Xuan Luong Military Technical Academy Reviewer 2: Prof Dr Tran Ich Thinh Hanoi University of Science and Technology Reviewer 3: Assoc Prof Dr Tran Ngoc Thanh Academy of Military Science and Technology This thesis will be defended in front of Doctor Evaluating Committee at Academy of Military Science and Technology At: , … date ….month … year 2020 This thesis may be found at: - Library of Academy of Military Science and Technology - The Vietnam National Library LIST OF SCIENTIFIC PUBLICATIONS Khuc Van Phu, Le Xuan Doan, (2017), “Analysis of nonlinear dynamic response of corrugated FGM-Sandwich plates” Tuyển tập cơng trình Hội nghị học tồn quốc lần thứ X, Hà Nơi, 8-9/12/2017, pp 862-869 Khuc Van Phu, Le Xuan Doan, (2018), “Nonlinear vibration of trapezoidal corrugated FGM-Sandwich plates” Tuyển tập cơng trình Hội nghị Khoa học toàn quốc Cơ học Vật rắn lần thứ XIV, Đại học Trần Đại Nghĩa, TP Hồ Chí Minh, 19-20/7/2018 Tập II, pp 470-477 Phu Van Khuc, Bich Dao Huy, Doan Xuan Le, (2017), “Analysis of nonlinear thermal dynamic responses of sandwich functionally graded cylindrical shells containing fluid” Journal of Sandwich Structures & Materials, Vol 21 (6), pp 1953-1974 doi:10.1177/1099636217737235 Phu Van Khuc, Bich Dao Huy, Doan Xuan Le, (2019) “Nonlinear thermal vibration and dynamic buckling of eccentrically stiffened sandwich-FGM cylindrical shells containing fluid” Journal of Reinforced Plastics and Composites, Vol 38(6), pp 253-266, Khuc Van Phu, Le Xuan Doan, (2019) “Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells” Vietnam Journal of Mechanics, 2(41), pp 179 – 192 doi: 10.15625/0866-7136/13306 Khuc Van Phu, Nguyen Minh Tuan, Dao Huy Bich, Le Xuan Doan, (2019), “Investigation of nonlinear dynamic responses of Sandwich-FGM cylindrical shells containing fluid resting on elastic foundations in thermal environment”.Journal of Military Science and Technology, Special Issue, No.60A, pp 95-107 Khuc Van Phu, Le Xuan Doan and Nguyen Van Thanh, (2019), “Nonlinear Dynamic Analysis for Rectangular FGM Plates with Variable Thickness Subjected to Mechanical Load” VNU Journal of Science: Mathematics – Physics, Vol 35, No pp 30-45 doi:10.25073/25881124/vnumap.4363 INTRODUCTION The necessity of thesis FGM plate structures with special shapes (corrugated and variable thickness) and FGM cylindrical shell containing fluid are widely used in many industries such as: aerospace, national defense and civil industries etc However, results of research on these structures are incomplete, especially the nonlinear dynamics problem is still new and not much research results has been published To improve further the theory and meet the requirement of applying these structures in practice, First of all, there should be theoretical research Therefore, the topic “Nonlinear dynamic analysis of FGM plates and sandwich-FGM cylindrical shell filled with fluid” has scientific and practical significance The research purpose of the thesis This thesis focuses on solving nonlinear dynamics problems of FGM plate with special shaped and sandwich-FGM cylindrical shell filled with fluid subjected to mechanical load in the thermal environment Object, scope and content of the study - Object of the study: corrugated FGM plate, variable thickness FGM plates and eccentrically stiffened sandwich-FGM cylinder shells filled with fluid, surrounded by elastic foundations, subjected to thermal-mechanical load - Scope of the study: Studied on vibration and geometrical nonlinear dynamic stability of structures, materials work in elastic range, material properties are independent of temperature - Research content: + Studied on nonlinear vibration and nonlinear stability of corrugated FGM plate and variable thickness FGM plate subjected to mechanical load + Investigated nonlinear vibration of eccentrically stiffened sandwich functionally graded cylindrical shells filled with fluid, surrounded by elastic foundations under mechanical load in thermal environment + Studied on nonlinear dynamic buckling of eccentrically stiffened sandwich-FGM cylindrical shells filled with fluid, surrounded by elastic foundations and subjected to mechanical load in thermal environment according to Budiansky-Roth standards Research Methods By using analytical method based on classical shell theory, Lekhnitsky’s smeared stiffener technique and Xia’s proposition to establish the system of motion equations of structures Nonlinear dynamic responses of structures are received by using Galerkin method and the fouth-oder Runge-Kutta method The critical dynamic loads are determined according to the Budiansky – Roth dynamic stability standard Scientific and practical meanings Scientific meanings: Adding the further theory and methods of studied on nonlinear dynamics of FGM plate and sandwich-FGM cylindrical shells Practical meanings: Research results can be used as reference and direction for manufacture and use these structures in practice Contents of the Thesis The thesis includes: beginning, chapters, and conclusion as follows: Beginning: presents the urgency, objectives, subjects, scope and research methods of the thesis Chapter 1: Overview of studies on dynamics of FGM plate and shell structures Chapter 2: Nonlinear dynamics analysis of FGM plates Chapter 3: Studied on nonlinear vibration of eccentrically stiffened sandwich-FGM cylindrical shells containing fluid Chapter 4: Nonlinear dynamic stability of stiffened sandwich-FGM cylindrical shells filled with fluid Conclusions CHAPTER OVERVIEW OF STUDIES ON DYNAMICS OF FGM PLATE AND SHELL STRUCTURES Overview of FGM material and main research directions for FGM plate and shell structure, analysis several domestic and foreign studies published on this structure From there, find out problems which need to be further researched and developed that the thesis needs to focus on as following: - Investigate nonlinear vibration of FGM plate with special shapes (corrugated and variable thickness) based on classical plate theory, Galerkin method and the fourth-oder Runge-kutta method Study on nonlinear dynamic stability of FGM variable thickness plate according to BudianskyRoth standards - Study on nonlinear vibration of eccentrically stiffened sandwich FGM cylindrical shells filled with fluid, surrounded by elastic foundations subjected to mechanical load in thermal environment Examine the effect of fluid, temperature, materials and geometric parameters on nonlinear dynamic responses of the shell - Analyze nonlinear dynamic stability of eccentrically stiffened sandwich-FGM cylindrical shells filled with fluid and surrounded by elastic foundations subjected to mechanical load in thermal environment using Budiansky-Roth standards From the obtained results, give somes scientific and practical meaningful comments, It can be used as a reference for the design, manufacture and apply these structures in practice CHAPTER 2: NONLINEAR DYNAMICS ANALYSIS OF FGM PLATES Chapter of the thesis focuses on nonlinear dynamics analysis of two special-shaped FGM plate structures (corrugated and variable thickness) Results of this study are presented in papers 1, and 2.1 Basic equations Displacement field: u ( x, y, z , t )  u0 ( x, y,0, t )  z. x ( x, y,0, t )  u  v( x, y, z, t )  v0 ( x, y,0, t )  z. y ( x, y,0, t )   w( x, y, z , t )  w0 ( x, y,0, t ) (2.1) Strain-displacement relations:     0  z.k with:     ,  ,    0N x y (2.2)        ;   T xy 0L 2    w   w  w w   ,      ,  x   y  x y      0N 0L  u v u v   0, 0,  0 x   x y y T ; k  kx , k y , kxy  T T ; T  2w 2w 2w    ,  , 2  y xy   x Stress-displacement relations:    Q   T  (2.3) Internal force components: N   A    M  B B  0    D  k  (2.7) The equations of motion: N x N xy  2u   1 ; x y t N xy x N y  y  1 2v ; t  M xy  M y 2 M x 2 w 2 w 2 w 2 w w 2   N x  N xy  N y  q(t)  1  21 2 xy xy t x y x y t (2.9) 2.2 Nonlinear vibration of corrugated FGM-Sandwich plates subjected to mechanical load 2.2.1 Corrugated FGM Sandwich plate and basic equations a) Round corrugation b) Trapezoidal corrugated Figure 2.2 Corrugated FGM Sandwich plate Basic equations - Force resultants and moment resultants  v0  w    u0  w   N x  A11         A12   ;  x  x    y  y    v0  w    u0  w    u0 v0 w w  ; N y  A12         ; N xy  A66     A22   x x y   y  y  y    x  x   M x   D11 2w x  D12 2w y ; M y   D12 2w x  D22 2w y ; M xy  2 D66 (2.14) 2w xy - Differential equations of motion of corrugated sandwich-FGM plate: A11   2v0 w  w   2u0  2u0  u0 w  w w  w         A A A A A    11 66 12 66  66 x y x x x y t  xy y xy  (2.15)   2u0 w  w   2v0  2v0  2v0 w  w w  w A A A A         22 66 22 66 y x y y y x t  xy x xy   A12  A66   (2.16) 4w 4w  w  w  w   w  w   D11   D12  D66  2  D22  A11    A12    x x y y x  x  x  y  2 u  w v  w  w w w  w  w   w  w  2 A66  A12    A22    A11  A12  xy x y y  x  y  y  x x y x 2 2 u  w u  w v  w v  w w w A66  A12  A22  A66  q( t )  1  21 y xy x y y y x xy t t (2.17) 2.2.2 Solution method The plate is simply supported, thus the boundary conditions are: w=0; v=0; Mx=0 at x=0 x=a w=0; u=0; My=0 at y=0 y=b Displacement field of plate satisfying boundary conditions can be chosen as: u  U mncos m x n y m x n y m x n y sin ; v  Vmn sin ; w  Wmn sin sin cos a b a b a b Applying Galerkin method and Volmir's assumptions we obtain: a11U mn  a12Vmn  a13Wmn  0; a21U mn  a22Vmn  a23Wmn  0; (2.19) a31Wmn  a32Wmn  a33U mnWmn  a34VmnWmn+ 4mn abq d Wmn dW =1  21 mn 2 dt mn dt From Eqs (2.19), we obtain: d 2Wmn dWmn 4  ab Wmn  .Wmn  m n q  2  mn   dt 1mn dt (2.20) - Linear free vibration analysis The equation of linear free vibration of plate can be given as: d 2Wmn dWmn Wmn   2  mn dt dt Natural frequencies of corrugated FGM-Sandwich plate: mn 4   mn  mn  m   n         D D D D 2  11   22    12 66   1 ab b 4 m n 1  a        a31 (2.23) - Nonlinear vibration analysis The nonlinear frequency-amplitude relationship can be determined by harmonic equilibrium method and given as: 2  2    1   A2  mn   - Nonlinear responses of FGM-Sandwich plate: Assume that the dynamic load acting on plate is q(t)=Q0SinΩt Then the equation (2.20) can be rewritten as: d 2Wmn dWmn 4 m n ab       W W Q0 sin t    mn  mn mn  dt dt 1mn (2.27) Nonlinear responses of struture are received from Eq (2.27) by using Runger-Kutta method 2.2.3 Some results of nonlinear vibration analysis of corrugated Sandwich FGM plates Consider a rectangular corrugated FGM-Sandwich plate simply supported Pressure load acting on plate is q(t)=Q0SinΩt and uniformly distributed on surface of plate * Natural frequencies of corrugated FGM-Sandwich plate Natural frequencies of considered plate are determined from (2.23) and shown in Table 2.2 Table 2.2 Effect of mode (m, n) on natural frequencies (s-1) a=0,9m; b=1,5m, h=0.002m,f=0,01m, c=0,06m,h t =h b =0,2h, k=1 (m, n) (1, 1) (3, 1) (5, 1) (1, 3) (1, 5) (1, 7) ω mn 959,3 1020,2 1346,8 8622,4 23950 46941 * Nonlinear response of corrugated Sandwich-FGM plate -4 x 10 k=1, m=n=1,q = 300*sin(200*t) W(m) -1 -2 a=0,9 m; b=1,5 m, h=0.002m, k=1, (m, n) = (1, 1), q=300 sin(200t) -3 -4 a=0,9 m; b=1,5 m, r=0,03m ; d=0.005m 0.05 0.1 0.15 0.2 t(s) 0.25 0.3 0.35 0.4 b) Trapezoida corrugated a) Round corrugation Figure 2.4 Nonlinear response of sandwich-FGM corrugated plate 2.3.4.2 Vibration results - Natural-vibration frequency of variable plate: Table 2.5 Effect of vibration mode on natural frequency (1/s) (m, n) ω0 (1, 1) 291,70 a=1,5m, b=0,8m, h =0.008m, h =0.005m, k=1 (1, 3) (3, 1) (5, 1) (3, 3) 2165,4 805,1 1713,4 2747,1 - Nonlinear dynamic response of variable thickness plate Fig 2.13 shows nonlinear dynamic response of variable thickness plate q0=400sin800t a=1,5m; b=0,8m, h1=0.008m, h0=0.005m, k=1, (m, n) = (1, 1), Fig 2.13 Dynamic responses of variable thickness plates - Effect of geometric factors on dynamic response of plate b=0,8m, h1=0.008m, h0=0.005m, k=1, (m, n) = (1, 1), q0=400sin800t Fig 2.15 Effect of ratio a/b on dynamic response of plate b=0,8m; a=1,5m; h1=0.008m, k=1, (m, n) = (1, 1), q=400sin800t Fig 2.16 Effect of ratio h0/h1 on dynamic response of plate Comment: Results shows that, dynamic responses amplitude of plate increases when increasing ratio a/b, that means the stiffness of plate decreases Dynamic response amplitude decrease when ratio h0/h1 increase That means, stiffness of plate increase when h0 increase and reaches the maximum value when h0=h1 10 2.3.4.3 Nonlinear dynamic stability analysis of variable thickness plate - Effect of geometric parameters b= 0,8m, h1=0.008m, h0=0.005m,k=1, (m, n) = (1, 1), q0=0, c1=c2=1e8 b= 0,8m, a=1,5m, b=0.8m,k=1,q0=0, (m, n) = (1, 1), c1=c2=1e8 Fig 2.21 Effect of ratio h0/h1 on Fig 2.20 Effect of ratio a/b on dynamic response of plate dynamic response of plate Table 2.10 Critical load of plate with various of a/b ratio(MPa) a/b 1,25 1,50 2,00 3,00 h =0.008m, h =0.005m, c =c =1e8 Pa/s; m=1; n=1 k=1 k=2 k=3 k=5 21.31 17.24 16.09 15.46 19.67 16.14 14.96 14.52 18.39 15.14 14.08 13.61 17.27 14.40 13.61 13.23 Comment: the load-bearing capacity of plate will decrease if size of plate increase Table 2.11 Critical load of plate with various of h0/h1 ratio(MPa) h /h 0,5 0,75 b=0.8m, a=1,5m, c =c =1e8 Pa/s; m=1; n=1 k=1 k=2 k=3 k=5 15.93 13.21 12.49 12.11 21.26 17.42 16.27 15.52 26.67 21.70 20.08 19.23 Comment: when ratio h0/h1 increases, the plate will work more stability 2.4 Conclusions of chapter Differential equations of motion of FGM plate with special shapes (corrugated and variable thickness) has been established Investigate nonlinear vibrations of special shapes FGM plate Effect of materials and geometric parameters on nonlinear response of plate are also examined Studied on nonlinear dynamic stability of variable thickness FGM plate according to Budiasky-Roth standard Effect of materials and geometric parameters on response and critical load of plate are investigated Results of research are published in papers 1, and 11 CHAPTER STUDIED ON NONLINEAR VIBRATION OF ECCENTRICALLY STIFFENED SANDWICH-FGM CYLINDRICAL SHELLS CONTAINING FLUID 3.1 Configuration of sandwich-FGM cylindrical shells containing fluid Fig 3.1 Configuration of stiffened sandwich-FGM cylinder shell filled with fluid 3.2 Basic equations - Displacement field w  u ( x, y, z , t )  u0 ( x, y,0, t )  z x ( x, y,0, t )  w u  v( x, y, z, t )  v0 ( x, y,0, t )  z ( x, y,0, t ) y  w x y z t w x y t ( , , , ) ( , ,0, )     (3.3) - Strain-displacement relations:     0  z.k ;  0   x0 ,  y0 , xy0    L    N  (3.4) In which:   0L T T   w 2  w 2 w w    u0 v0 w u0 v0   0N  , ;   , ,  ,       y  x y  ; x x      x y R y         (3.5) T  2 w 2 w 2 w  k   x2 ,  y ,  xy    Thermal deformation at the point (x,y,z): {εT }=εTx ,εTy ,0 =αx (z),α y (z),0 ΔT={αT (z)}ΔT với ΔT=T2-T1 T T (3.6) - Stress-displacement relations:    Q   T  (3.8) 12 With stiffener:  xs  Em x  Em mT ; ys  Em y  Em mT (3.9) - The force resultants and moment resultants: Applying Lekhnitsky’s smeared stiffener technique, we obtain: Em Ax   0  A12 B11 C1 B12  A11  s x   Em Ay   x    *   Nx   A  a a 0   A B B C 12 22 12 22  Ny      y0   **  sy       a a  0 0 A66 B66   xy0     N xy     M       E I  *  x   B11 C1 D11  m x   k x   b **b  B12 D12 sx My     k y   b  b   M xy    k    E I m y     B   xy   0   B C D D 12 22 12 22 sy    0 0 B66 D66  (3.11) 3.3 Motion equations of a stiffened sandwich-FGM cylindrical shell filled with fluid and surrounded by elastic foundations N xy N y  N x N xy  2u  2v   1 ;   1 ;  y t x y t  x 2   M  M xy  M y 2w 2w 2w N y x 2        K1w  N N N  x xy y xy y x xy y R  x   2w 2w  2w w   K     q  pL  1  1 , y  t t   x (3.15) pL- Dynamic fluid pressure acting on the shell and expressed as: pL    L  L t (3.17) This fluid pressure is determined and expressed as:  L 2w pL    L  mL t t (3.24) N xy N y  N x N xy   0;   0;  y x y  2x  M xy  M y 2w 2w 2w N y  M x       N N N 2  x xy y 2 2         x x y y x x y y R  2  w  w  w w         K w K q m        1 L 2      x y t t    (3.26) Substituting (3.24) into (3.15) then applying Volmir’s assumption we obtain: The first two equation of (3.26) are satisfied identically when introducing the stress function F: Nx  2 F 2F 2F   N ; ; N  xy y xy y x (3.27) 13 Substituting (3.5), (3.14) and (3.27) into (3.7) and the third equation of (3.26) we obtain governing equations used to investigate nonlinear dynamics of stiffened sandwich-FGM cylindrical shell filled with fluid: 4 4F 4F 4w * * *  F *  w * * * 2 A A A B B B B            66 12 22 21 11 22 66 x x 2y 22 y x x 2y 4w 2w  2w  2w 2w  B12*      y R x  xy  x y A11* (3.28)  1  mL  tw2  21 wt  D11* xw4   D12*  D21*  4D66*  x2wy  D22* yw4 4  F  B*  F  B*  F   F   F  w   F  w * *   B11*  B22  B66  x2y 21 x4 12 y R x2 y x2 xy xy 2     F2  w2  K1w  K   w2   w2   q x y  x y  (3.29) 3.4 Solution method Suppose that the shell is simply supported and under an axial compression N01   ph and external pressure load q(t) The boundary conditions are: w  0, M x  0, N x  N01, N xy  at x=0 and x=L Satisfying boundary condition, deflection w of shell can be chosen as: w  f (t ).sin  x.sin  y (3.30) Stress function F can be found in the form: F  F1 cos 2 x  F2 cos 2 y  F3 sin  x sin  y  N 01 y2  2  x2 A*    * ft2   ft   N o1 12*    A11  A11 2 (3.34) Putting Eq(3.30) and Eq (3.34) into (3.29) then applying Galerkin procedure yields:  4 4  A12*  1  mL  ft   21 f  H f  H f  H3 ft   mn R  A* N01     mn q (3.35)  11  t  2 t  3.4.1 Nonlinear Vibration analysis of cylindrical shell filled with fluid Suppose that shell subjected to axial compression N01= const and exciting force in the form q  Q sin t , Eq (3.35) can be rewritten as:  H  H1 H 2 1 4  A12* f t   f t   f t   f t   f (t )  N      01 H3 mn RH  A11*  1  mL   1  mL   H  4 Q sin t  mn  1  mL  (3.37) Solve Eq (3.37) by using fourth-order Runge–Kutta method we obtain nonlinear response of Sandwich-FGM cylindrical shell filled with fluid 14 3.4.2 Natural frequency of cylindrical shell filled with fluid: - Natural frequency of shell: In the case of linear free-vibrations, Eq (3.37) can be rewritten as: 1 df (t) d f (t) H3   f (t)  1  mL dt 1  mL dt (3.38) Natural frequencies of the shell can be drawn from Eq (3.38): mn  H3  1  mL  (3.40) - The frequency–amplitude relationship of nonlinear vibration The nonlinear frequency-amplitude relationship of nonlinear vibration can be determined by harmonic equilibrium method and given as: 2    A*   21   4  H A  H1 A2  (3.43)   12* N01     Q   1     H3 H3    1  mL  mn  1  mL  mn Amn2  R  A11   In which:    mn , mn - Natural frequency of shell 3.5 Numerical results 3.5.1 Validation The natural frequencies of fluid-free cylindrical shell in thesis will be compared with the publication of Loy et al [26] and Shen [58] for FGM shell made of Stainless steel and Nickel Table 3.1 Comparison of free-vibration frequencies of FGM cylinders (Hz) Source m=1, n=7, h=0.05m, L/R=20, R/h=20, T=300K k=0 k=0.5 k=1 k=2 k=5 k=15 Loy [26] 580.78 570.25 565.46 560.93 556.45 553.37 Shen [58] 585.788 575.266 570.47 565.93 561.399 558.27 Thesis 591.591 579.151 573.33 567.74 562.20 558.55 As can be seen, the obtained results are closed to each other Thus, the results of the thesis are reliable 3.5.2 Nonlinear Vibration analysis of FGM cylindrical shell filled with fluid Consider a stiffened circular cylindrical shell made of Sandwich-FGM filled with fluid and surrounded by an elastic medium subjected to thermalmechanical load (Fig 3.1) 15 3.5.2.1 Natural frequency of cylindrical shell Table 3.2 Natural-vibration frequencies of cylindrical shell (s-1) m=3, k=1, h=0.01m, h c =0.2h, h m =0.2h, L=20R, R=20h, T  5000 C; K1  2,5e8( N / m3 ); K2  5e5( N / m) n=1 n=2 n=4 n=5 n=6 n=3 No fluid 9417.61 4284.97 2018.42 3765.67 7488.24 10199.8 Full-filled fluid 1596.28 719.664 333.663 608.11 1170.62 1525.97 Comment: natural frequencies of the Sandwich-FGM shell full-filled fluid are lower than those of the shell without fluid The lowest fundamental frequency of the considered shell corresponds to mode (m, n)=(3, 3) 3.5.2.2 Nonlinear dynamic response of cylindrical shells Nonlinear dynamic response of fluid-filled and filled-free sandwichFGM cylindrical shells are shown in Fig 3.2 -7 x 10 Full-filled fluid No fluid f(m) -2 -4 m=3,k=1, h=0.01m, h c =0.2h, h m =0.2h, L=20R, R=20h, T  5000 C; K1  2,5e8( N / m3 ); K2  5e5( N / m) -6 -8 0.02 0.04 0.06 0.08 0.1 t(s) 0.12 0.14 0.16 0.18 0.2 Fig 3.2 Nonlinear response of fluid-filled and filled-free shell Comment: From the graph, it can be seen that the response amplitude of full-filled fluid cylindrical shells are larger than those of the fluid-free cylindrical shells It also shows that the bound of dynamic response amplitude of full-filled fluid cylindrical shell changes according to sine shape law, while the bound of dynamic response amplitude of fluidfree shell changes as a straight line b Effect of geometric parameters Effect of geometric parameters on nonlinear dynamic responses of the fluid-filled sandwich-FGM cylindrical shell are illustrated in Fig 3.5 and Fig 3.6 16 1-L/R=15; 2-L/R=17; 3-L/R=20 1-h=0.01; 2-h=0.015; 3-h=0.02 3 m=3; n=3; k=1; R/h=20, L/R=20; K1=2.5e8; K2=5e5; sy=L/50; sx=πR/25, ΔT=5000C; q=200sin500t m=3, n=3, k=1, R/h=20, h=0.01m ; K1=2.5e8; K2=5e5; sy=L/50; sx=πR/25 q=300sin100t, ΔT=500oC Fig 3.5 Effect of thickness of shell Fig 3.6 Effect of L/R ratio on dynamic responses of shell dynamic responses of shell Comment: The thickness of the shell increases, the amplitude of nonlinear dynamic response of shell decreases, it mean the stiffness of shell increases Fig 3.6 shows that, the ratio L/R increases, it means the length of shell increases, the dynamic response amplitude of full filled fluid cylindrical shell increases It mean, the longer shells work weaker than shorter ones c Effect of stiffners and elastic foundations -7 -8 x 10 x 10 1- Stiffener; 2- No Stiffener q=300sin100t 4 f(m) f(m) 1- Foundation; 2- No foundation 0 -2 -2 -4 -4 -6 -6 m=3; n=3; k=1; R/h=20, L/R=20; ΔT=5000C; h=0.01; K1=2.5e8; K2=5e5; sy=L/50; Sx=πR/25 -8 0.05 0.1 0.15 0.2 t(s) 0.25 0.3 0.35 -8 0.4 m=3; n=3; k=1; R/h=20, L/R=20; h=0.01; K1=2.5e8; K2=5e5; sy=L/50; Sx=πR/25, ΔT=5000C; q=300sin500t 0.01 0.02 0.03 0.04 t(s) 0.05 0.06 0.07 0.08 Fig 3.8 Effect of elastic Fig 3.7 Effect of stiffners on foundations on responses of shell dynamic responses of shell Comment: Effect of stiffners and elastic foundations dynamic responses of shell are shown in Fig 3.7 and Fig 3.8 When sandwich-FGM cylindrical shell is stiffened by stiffeners, the stiffness of shell increases; therefore the vibration amplitude of cylindrical shells decreases (Fig 3.7) Nonlinear response amplitudes of shell decrease when the shell is surrounded by an elastic foundation (Fig 3.8) It means, elastic foundations prevent the vibration of shell 17 e Characteristics of nonlinear vibration of cylindrical shells Characteristics of nonlinear vibration of cylindrical shells filled with fluid depending on relationship of natural frequencies of shell and exciting frequencies When exciting frequencies is equal to the natural-frequency then the resonance phenomenon occurs (Fig 3.11) when the frequencies of exciting force close to the natural-vibration frequencies of cylindrical shell The phenomenon harmonic beat occurs (Fig 3.13) q = 200sin333.t q = 200sin353.t q = 200sin370.t m=n=3, k=1, R/h=20, L/R=20, h=0.01, K1=2.5e8; K2=5e5; sy=L/50; sx=πR/25 ΔT=5000C; m=n=3, k=1, R/h=20, L/R=20, h=0.01, K1=2.5e8; K2=5e5; sy=L/50; sx=πR/25 Fig 3.12 The phenomenon harmonic beat When excited frequency is far from natural frequency of the shell, the relationship between the velocity and the deflection of shell are shown in Fig 3.15 When the frequency of exciting force is much far from the natural frequency of the shell, chaos fluctuations occurs (Fig 3.16) Fig 3.11 The resonance phenomenon Ω=1600 rad/s Ω=4200 rad/s m=n=3, k=1, R/h=20, L/R=20, h=0.01 ΔT=500oC, p=100N/m2 ωmn=333 (1/s) m=n=3, k=1, R/h=20, L/R=20, h=0.01 ΔT=500oC, ωmn=333 (1/s) Fig 3.16 df/dt-f when excited frequency is much far from natural frequency Fig 3.15 df/dt-f when excited frequency is far from natural frequency 18 3.6 Conclusion of chapter Chapter of the thesis based on classical shell theory, using Galerkin method and the fourth-order Runge-Kutta method to analyze nonlinear vibration of stiffened sandwich-FGM cylindrical filled with fluid in elastic foundations Some achieved results: + Establish differential equations of motion of stiffenened SandwichFGM cylindrical shell containing fluid in elastic mediums + Analyze nonlinear vibration of structure, effects of fluids, materials and other factors on dynamics responses of cylindrical shell are also examined Results show that, the fluid remarkably influenced on nonlinear dynamic responses of cylindrical shells It increases the nonlinear dynamic response amplitudes and reduces natural frequencies of shell + The resonance phenomenon and harmonic beat occurs when the frequency of the excitation force is equal and close to the natural frequency of shell When excitation frequency is much greater than natural frequency of shell , chaos fluctuations occur CHAPTER 4: NONLINEAR DYNAMIC STABILITY OF STIFFENED SANDWICH-FGM CYLINDRICAL SHELLS FILLED WITH FLUID Equation Chapter (Next) Section 4.1 Configuration of cylindrical shells Consider cylindrical shell with configuration is shown in Fig 3.1 4.2 Differential equations of motion of FGM cylindrical shell containing fluid From results of Chapter 3, the basic equations to investigate dynamic stability of cylindrical shell filled with fluid are: 4 4 F 4 F 4w * * *  F *  w * * * A A A B B B B        2     66 12 22 21 11 22 66 x x y y x x y 2 4w 2w  2w  2w 2w  B12*      y R x  xy  x y (4.1) 4 2w w 4w *  w * * * *  w *  F  1  mL   1  D11   D12  D21  D66  2  D22  B21 t t x x y y x 4 2 2  F  F  F  F  w  F  w * *   B11*  B22  B66  x2y  B12* y  R x  y x  xy xy  2w 2w  2 F 2w   K1w  K     q x y y   x (4.2) A11* 19 4.3 Solution method Consider a cylinder shell simply supported at both edges, subjected to a preloaded axial compression N01   ph and external pressure q(t) Basic differential equations of structure:  4 4  A12*  1  mL  ft   21 f  H f  H f  H ft   mn R  A* N01     mn q (4.3)  11  t  2 t  Investigate dynamic stability of structures by solving equation (4.3) according to two following cases: Case 1: Cylinder shell subjected to linear axial compression load interm of time N01   ph with p  c1t ( c1 -loading speed), and external pressure q=0 Case 2: Cylinder shell subjected to uniform axial compression load N01  const and external pressure in terms of time q  c2t ( c2 - loading speed) 4.4 Numerical results and discussion 4.4.1 Validation Dynamic critical stress of fluid-free FGM cylindrical shell in the thesis will be compared with the publication of Huang Han [65] Table 4.1 Comparison of critical stress of FGM cylindrical shell (MPa) Source L/R=2; R/h=500; c=100 MPa/s k=0.2 k=1.0 k=5.0 Huang & Han [65] 194.94 (2,11) 169.94 (2,11) 150.25 (2,11) Thesis 193.914 (1,9) 168.685 (1,9) 149.167 (1,9) Tables 4.1 shows that, the comparison obtain a good agreement with above publication, therefor, results of this thesis are reliable and can be used to investigate dynamics stability of sandwich-FGM cylindrical shell filled with fluid 4.4.2 Numerical results Case 1: Cylinder shell subjected to linear axial compression load interm of time N01   ph with p  c1t ( c1 -loading speed), and external pressure q=0 - Effects of fluid and stiffeners Effects of fluid and stiffeners on nonlinear dynamic response and critical load of the shell are shown in Fig 4.4, Fig 4.5 and Table 4.2 We 20 can see that fluid and stiffeners remarkably influenced on nonlinear dynamic stability of shells It increases critical load of the shell So, fluid and stiffeners increase the loadbearing capacity of the shell 1- No fluid; 2- Full filled fluid 1- Stiffener; 2- Unstiffener 2 Fig 4.4 Effect of fluid on nonlinear response of shell Fig 4.5 Effect of stiffener on nonlinear response of shell Table 4.2 Effect of fluid on the critical load of cylindrical shell.(GPa) m=n=3; R/h=20, L/R=20; h=0.01; ΔT=50 C; K =2.5e8; K =5e5; s y =L/50; s x =πR/25, c =1e12 Full filled fluid No fluid h=0.01 h=0.015 h=0.02 h=0.01 h=0.015 h=0.02 k=0 30.3 37.8 44.6 12.2 13.3 16.1 k=0.5 29.2 36.9 43.8 11.2 12.5 15.7 k=1 28.8 36.5 43.4 10.5 11.9 15.5 k=3 28.5 36.2 43.0 9.8 11.5 15.0 - Effects of structural of material and elastic foundation + Effects of structural of material and elastic foundations on nonlinear dynamic response of cylindrical shell are shown in Fig 4.7 and Fig 4.8 From the graph, with the same geometry dimensions, sandwich- FGM cylindrical shell will work more stability than FGM one + If cylindrical shell is surrounded by an elastic foundation, the critical force of shell will increase That mean, the elastic foundation enhances the stability of a cylindrical shell when it is compressed 21 1- Foundation; 2- No foundation 1- Sandwich-FGM; 2- FGM k=1; R/h=20, L/R=20; h=0.01m; K1=2.5e8; ΔT=500C; K2=5e5; Sy=L/50; Sx=πR/25, c1=1e11 m=n=3; R/h=20,L/R=20; k=1;h=0.01m;K1=2.5e8; ΔT=500C; K2=5e5;Sy=L/50; Sx=πR/25, c1=1e11 2 1 Fig 4.7 Nonlinear response of Fig 4.8 Effect of foundation on FGM and Sandwich- FGM shell nonlinear response of shell Case 2: Cylinder shell subjected to uniform axial compression load N01  const and external pressure in terms of time q  c2t ( c2 - loading speed) - Effects of fluid and stiffeners on nonlinear responses of structures 1- No fluid; 2- Full filled fluid 1- Stiffener; 2- No Stiffener 2 (m, n)= (3,3); k=1; R/h=20; L/R=20; h=0.01m; K1=2.5e8; K2=5e5; N01=1e3; c2=1e8; ΔT=5000C (m,n)=(3,3); k=1; R/h=20; L/R=20; h=0.01m; K1=2.5e8; K2=5e5;sy=L/50; sx=πR/25; N01=1e3; c2=1e10; ΔT=5000c Fig 4.10 Effect of fluid on Hình 4.11 Effect of stiffener on nonlinear response of shell nonlinear response of shell Table 4.8 Effect of fluid on the critical load of cylindrical shell.(MPa) (m=n=3; k=1; R/h=20; L/R=20;h=0.01;ΔT=500 C; K =2.5e8; K =5e5; s y =L/50; s x =πR/25; c =1e10; Full filled fluid No fluid h=0.01 h=0.015 h=0.02 h=0.01 h=0.015 h=0.02 k=0 53,32 72,65 86,67 24,27 32,01 42,07 k=0.5 52.41 69.38 86.12 14.83 20.60 26.41 22 k=1 51.10 70.82 83.66 14.01 19.62 24.50 k=3 50.65 70.30 82.57 13.26 18.73 24.30 Results in Fig 4.10 and Table 4.8 shows that, critical load of full filled fluid shell (pcr=51,10 MPa) is 3.5 times higher than those of fluid-free shell (pcr=14,1 MPa) Therefor, fluid increases the stability of cylindrical shells - Effects of structural of material and elastic foundation Comment: Effects of structural of material and elastic foundations on nonlinear dynamic response of cylindrical shell are shown in Fig 4.13 and Fig 4.14 Similar to case 1, the reseach results show that, with the same geometry dimensions and working conditions, Sandwich-FGM cylindrical shell will work more stability than FGM ones Fig 4.14 shows that, the critical force of the shell surrounded by an elastic foundation is greater than those of shell without elastic foundation Thus, the shell surrounded by an elastic foundation works more stability than without elastic foundation one 1- FGM 2- Sandwich-FGM 2 Sóng biên hình m=n=3;R/h=20, L/R=20; h=0.01m; K1=2.5e8; K2=5e5; N01=1e3; ΔT=5000C; Sy=L/50; Sx=πR/25, c2=1e9 m=n=3; k=1; R/h=20, L/R=20; h=0.01m; K1=2.5e8; K2=5e5; N01=1e3; ΔT=5000C; Sy=L/50; Sx=πR/25, c2=1e9 Fig 4.13 Nonlinear response of FGM Fig 4.14 Effect of foundation on and Sandwich- FGM shell nonlinear response of shell 4.5 Conclusion of chapter Base on the basic equations established in chapter 3, Chapter analyzed dynamic stability of stiffened sandwich-FGM cylindrical shell filled with fluid, in an elastic foundations, taking into account the effect of temperature From the nonlinear dynamic response of the cylindrical shell, the critical dynamic load of structure are determined according to BudianskyRoth stability criterion Investigated effects of fluid, material, Geometric parameters and load on nonlinear dynamic response and critical dynamic load of sandwich-FGM cylindrical shell filled with fluid in thermal environment 23 CONCLUSION The main results of the thesis: - Based on classical plate theory and Xia’s proposition to study nonlinear vibration of specially shaped FGM plate (Corrugated and variable thickness) using Galerkin method and the four-order Runge-Kutta method - Analyzed nonlinear dynamic stability, determined critical dynamic load of variable thickness FGM plate according to Budiansky-Roth standard - Established basic differential equations of motion and studied on nonlinear vibration of stiffened Sandwich-FGM cylindrical shell containing fluid surrounded by an elastic mediums subjected to thermal-mechanical load - Investigated nonlinear dynamic stability, determined critical dynamic load of stiffened Sandwich-FGM cylindrical shell containing fluid surrounded by an elastic mediums according to Budiansky-Roth standard Main contributions - Examined nonlinear dynamic responses of FGM plate with special shaped Studied on nonlinear dynamic stability, determined critical dynamic load of variable thickness FGM plate according to Budiansky-Roth standard - Established basic equation and investigate nonlinear dynamic response of stiffened Sandwich- FGM cylinder shell containing fluid, in elastic foundations subjected to thermal-mechanical load - Analyzed nonlinear dynamic stability and determined the critical dynamic load of stiffened sandwich-FGM cylindrical shell filled with fluid, in elastic mediums according to Budiansky-Roth dynamic stability standard Further research directions - Study on nonlinear vibration and dynamics stability of FGM cylindrical shell containing flow fluid - Investigate nonlinear vibration and stability of FGM cylindrical shell containing fluid with material properties depending on temperature - Analyze nonlinear dynamics of variable thickness FGM cylindrical fluid-filled and fluid-free subjected to different load in thermal environment 24 ...  Thermal deformation at the point (x,y,z): {? ?T }=εTx ,εTy ,0 =αx (z),α y (z),0 ? ?T= {? ?T (z)}? ?T với ? ?T= T2 -T1 T T (3.6) - Stress-displacement relations:    Q   ? ?T  (3.8) 12 With... x y t t    (3.26) Substituting (3.24) into (3.15) then applying Volmir’s assumption we obtain: The first two equation of (3.26) are satisfied identically when introducing the stress function... responses of structures are received by using Galerkin method and the fouth-oder Runge-Kutta method The critical dynamic loads are determined according to the Budiansky – Roth dynamic stability standard