This paper presents the differential biharmonic equation of thin plates through which, the vibration mode shapes for a rectangular thin plate simply supported on contour were obtained. Also, the first four vibration mode and the first four natural frequencies of this rectangular thin plate of steel, were obtained. Using MATLAB software, the vibration mode shapes were graphically represented

Cornel Hatiegan

Eugen Raduca

Gilbert-Rainer Gillich

Lenuta Cindea

Marian-Dumitru Nedeloni

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  127    Cornel HaŃiegan, Gilbert Rainer Gillich, Eugen Răduca, Marian Dumitru Nedeloni, LenuŃa Cîndea Equation  of  Motion  and  Determining  the  Vibration Mode  Shapes  of  a  Rectangular  Thin  Plate  Simply Supported on Contour Using MATLAB This paper presents the differential biharmonic equation of thin plates through which, the vibration mode shapes for a rectangular thin plate simply supported on contour were obtained. Also, the first four vibra tion mode and the first four natural frequencies of this rectangular thin plate  of  steel,  were  obtained.  Using  MATLAB  software,  the  vibration mode shapes were graphically represented. Keywords: Equation of motion, vibration mode shapes, MATLAB  1. Introduction Various construction contain in their structure, plates of various shapes and sizes which means that plates have a great importance in mechanical engineering.  In the study of plates, there are two well known manners of scientific approach:  A.  The first approach is based on the development in a series of the functions of stress and strain, on the Z coordinate, as we know it from Cauchy. By keeping a minimum possible number of terms into these series, Sophie Germaine equation was obtained [1]. Using this approach, Navier solved the problems of bending and stability for a rectangular plate with all edges simply supported. The analysis of plate behavior in general, was made by Poisson,  who  introduced  a  further  condition  for  imposing  the  boundary conditions on contour. B.  The second approach that led to a technical theory of plates is owed to Kirchhoff.  He  introduced  the  hypothesis  of  the  straight  normal.  The Mindlin Reissner theory generalizes Kirchhoff theory, which can obviously be applied to plates of any thickness [2], [3]. In the present paper, for the study of plane plates, in accordance with [4], [5], [6], the following hypotheses are used:  ANALELE UNIVERSITĂłII   “EFTIMIE MURGU” RE IłA ANUL XX, NR. 1, 2013, ISSN 1453 - 7397    128 a.  the plate is made of a continuous, homogeneous and isotropic material; b.  for moderate loads, the plate is elastic and Hooke law applies; c.  the deformations are negligible compared to the unit; d.  the displacements are small compared to the thickness of the plate and the equilibrium can be written on the undeformed shape; e.  when  the  plate  is  deformed,  there  are  no  linear  deformations  and displacements on the mid surface.   2. Formulation of problem and equation of motion It is consider a homogeneous and isotropic plate of a constant thickness h, with a uniformly distributed mass and subjected to loads in the direction normal to its plane, with the mid surface in the XOY plane. The deformed shape of the mid surface Z(x,y) satisfies the Sophie Germaine differential biharmonic equation of the plane plate with a constant thickness as follows [5], [7], [8], [9], [10]:  ( ),,Dy x pyZy xZ2xZ442 2444=¶¶+¶ ¶¶+¶¶   (1) where:  ( ) u -=1 12EhD3  is  the  bending  rigidity  of  the  plate,  Z=Z(x,y)  is  the  mid surface displacement of the plate on the normal direction, h is the thickness of the plate andu  is the Poisson coefficient. When the plate undergoes vibrations on the Z direction, the  ) , , ( t y x W  dis placements are added to the static arrows Z(x,y), whereas the momentum per sur face unit 22tWh¶¶r -  and the perturbation forces  ) , , ( t y x p0 are added to the static load.  These displacements verify the differential equation of the 4th order:  ( ),, ,Dt y x ptWDhyWy xW2xW 022442 2444=¶¶ r+¶¶+¶ ¶¶+¶¶   (2) where: Dh 2 4 × rw = l .  (2’)  In the case of a rectangular plate simply supported on two parallel edges: x=0 and x=a, the function  ) , ( y x Zmn has the form:   129 , sin ) ( ) , (  × p ×=ax ny Y y x Z m mn   (3) where: * N n Î . If the plate is simply supported on the edges: x=0 and x=a, the arrow and the bending moment parallel to the edge are zero and satisfy in this case the limit conditions [5], [7 10]:  ( ) ( ) ( ) ( ) ... , , , , sin , ; , 3 2 1 n 0 x y Y y a Z 0 y 0 Z m = = p = =   (4)    . 0yZxZa x0 x 2222= ¶¶+¶¶==   (5)  Introducing the function (3) in the differential equation of the 4th order (2), it is obtain for the function  ) (y Ym a linear differential equation of the 4th order with constant coefficients:   . 0 YandyY dan2dyY d 444 42224 244= l -p+ ×p-   (6)  From the characteristic equation:  , 0anran2 r444222 24 = l -p+p-   (7) it is obtain the solution:  .222 22anr l ±p=   (8) Using the notations: , ,22 22 222 22 2anan p+ l = bp- l = a   (9)  the solutions as the following form are obtained:  2 2 r b = , with  b ± = 2 1 r ,  and 2 2 r a = , with a ± = i r 4 3, , because: 22 22an p< l . The general solution to the differential equation (6) is a function of the follow ing form:   130  ( ) .y i4y i3y2y1y r4y r3y r2y r1 e C e C e C e C e C e C e C e C y Y4 3 2 1 × a - × a × b - × b × × × × + + + = + + + =   (10)  The  constants  4 3 2 1 C C C C , , , are  determined  from  the  boundary  conditions that have to be satisfied on the other two edges parallel to the OX axis.  If the plate is simply supported on the edges: y=0 and y=b, for which the ar rows and the bending moment parallel to the edge are null, the boundary condi tions are:   0 b Y 0 Y 0 b x Z 0 x Z m m = = ⇒ = = ) ( ) ( ) , ( ) , (   (11)    . 0yZxZb y0 y2222= ¶¶+¶¶==  (12)  Plugging this condition into (3) and using the Euler relations, it will be:   . sin ) ( ) ( cos' ' 0ax my Y y Yax mamb y0 ym 22 2= × p ×× + ×  × p ××p ×-== (13)  Then:  , ) ( ) (' ' ' ' 0 b Y 0 Y m m = =      (14)  this means that the boundary conditions are satisfied. Introducing the  conditions (11) and (14) into the equation (10), the linear homogeneous system with the unknowns  4 3 2 1 C C C C , , ,  can be obtained: = a - a - b + b= a - a - b + b= + + += + + +× a - × a × b - × b× a - × a × b - × b0 e C e C e C e C0 C C C C0 e C e C e C e C0 C C C Cb i42 b i32 b 22b 2124232221b i4b i3b2b14 3 2 1  (15)  We get the constants  4 3 2 1 C C C C , , ,  from the boundary conditions that have to be satisfied on the other two edges, parallel to OX axis.  If, having the conditions that the system (15) also admits solutions other than the banal solution, the vibration mode equation can be obtained:   131 0 b b sh i 4e e e ee e e e e ee e e ee e e e1 1 1 12 2 2b i b i b b 2 2 2b b b i b b b i 2 2 2b i 2 b i 2 b 2 b 22 2 2 2b i b i b b= × a × × b × × b + a =- - b + a == - - - b + a =a - a - b ba - a - b b× a - × a × b - × b× b - × b × a - × b - × b × a× a - × a × b - × b× a - × a × b - × b) sin( ) ( ) () )( ( ) ()] ( ) ( [ ) (    (16)  By solving the vibration mode equation (16) it can be obtained:  . ) sin( ) ( ) ( 0 b b sh i 42 2 2 = × a × × b × × b + a   (17)  From the (17) equation it can see that:  ( ) 0 b = × a sin , which means that: * , N m m b m Î p × = × a  and:  . 0 = b  (18)  From  the  expression  (18)  can  be  observed  that  for  every  natural  different from zero value of m, there is a infinity of values of the  b m b argument, so:  .22 22mbm p= a   (19)  Introducing the value of 2m a  into (9), according to [5], [7], [8], [9], [10], the characteristic equation of the form can be obtained:  . , ;* N n manbman2222222 22mn2mn Î + p =p+ b = l   (20)  Put (2’) into (20), the natural pulsation of motion that has the form can be ob tained: .hDanbmhD22222 2mn mn × r + p =× rl = w   (21)  From (21) it can notice that there is a double infinity of natural pulsations with which the plate can perform free harmonic vibrations.     132 For  the  integration  constants  4 3 2 1 C C C C , , ,   results  that  0 C C C 3 2 1 = = =  and 0 C4 ¹ . Then the solution to the differential equation (6) has the form:                                        , sin ) (by mC y Y 4 m× p ×=   (22)  and the functions of  movement Zmn(x,y) have the form:  ( ) , sin sin ,by max nC y x Z 4 mn× p × × p ×=   (23)  than  1 C4 = , because the values are normalized. Once determined the self pulsations and the self functions, it can be write the expression of displacement in vibration in a self mode:  ( ) ( ) ( ), sin , , , mn mn mn mn t y x Z t y x w q + w =   (24)  and the resulting motion is expressed as a superposition of modes:  ( ) ( ) ( ). sin , , ,,... , ,... ,mn mn3 1 n 3 1 mmn t y x Z t y x w q + w = ∑ ∑¥=¥=  (25)  The fundamental node of vibration of the rectangular plate corresponds to the values m=1 and n=1, being characterized by the self pulsation  11 w and by the self form  11 Z , expressed by the particular relations:  hDb1a12 2211 × r+ p = w     (26) and: . sin sinbyaxn m16Z2 11× p × pp × ×=   (26’)   3. Simulation and the numerical results obtained For a rectangular plate made of steel with the edges a=1000 mm and b=500 mm, with the Young module:11 10 2,1 E × = N/m2, the Poisson coefficient  3 0, = u  and the density  7850 = r  kg/m3, though Microsoft Excel with the equation:     133 ,hDanbmhD22222 2mn mn × r + p =× rl = w   (27)  the first four vibration mode and natural frequencies are calculated in table 1, cal culated values according with [11], where n  i m are mode numbers:  Table 1. Angular frequency [rad/s]  Natural frequencies [Hz] n m=1  m=2  m=3  m=4  m=1  m=2  m=3  m=4 1  149.607  239.372  388.979  598.43  23.822  38.116  61.939  95.291 2  508.665  598.43  748.037  957.487  80.997  95.291  119.114  152.466 3  1107.09  1196.86  1346.47  1555.92  176.288  190.582  214.406  247.758 4  1944.9  2034.66  2184.27  2393.72  309.697  323.990  347.813  381.165  With MATLAB [12], it was determined their vibration mode shapes for the rec tangular plate simply supported on the edges, mode shape presented in Figures 1 ÷ 16, Where it was used the step of 0,005 and the relationship:  ( ) . sin sin ,by max nC y x Z 4 mn× p × × p ×=   (28)   Figure 1. Vibration mode shapes, with n=1 and m=1.   134  Figure 2. Vibration mode shapes, with n=2 and m=2.   Figure 3. Vibration mode shapes, with n=3 and m=3.   Figure 4. Vibration mode shapes, with n=4 and m=4.   135     Fig. 5 Vibration mode, with n=1 and m=2.  Fig. 6 Vibration mode, with n=2 and m=1.       Fig. 7 Vibration mode, with n=1 and m=3.  Fig. 8 Vibration mode, with n=3 and m=1.       Fig. 9 Vibration mode, with n=1 and m=4.  Fig. 10 Vibration mode, with n=4 and m=1.    136    Fig. 11 Vibration mode, with n=2 and m=3.  Fig. 12 Vibration mode, with n=3 and m=2.       Fig. 13 Vibration mode, with n=2 and m=4.  Fig. 14 Vibration mode, with n=4 and m=2.       Fig. 15 Vibration mode, with n=3 and m=4.  Fig. 16 Vibration mode, with n=4 and m=3.       137 4. Conclusion The following conclusions can be made: ·  it is observed from the mathematical model that the solution of the Y direction, can be achieved only if the solution of the characteristic equation is complex; ·  in Table 1 and 2, it can be observed that the fundamental vibration mode,  has  the  lowest  value  of  their  mode  shapes,  respectively  the fundamental natural frequencies has the lower of their calculated natural frequencies; ·  in Figures 2÷16, it is observed that the vibration mode shapes of the  rectangular  plate  are  damped  harmonic,  periodic  harmonic  and  with sudden variations; ·  the results can be used to validate finite element models. Acknowledgements The authors gratefully acknowledge the support of the Managing Authority for Sectoral Operational Programme for Human Resources Development (MASOPHRD), within the Romanian Ministry of Labour, Family and Equal Opportunities by co financing  the  project  “Investment  in  Research innovation development  for  the future (DocInvest)” ID 76813. References [1]  Wu  J.H.,  Liu  A.Q.,  Chen  H.L.,  Exact  Solutions  for  Free Vibration Analysis of Rectangular Plates Using Bessel Functions, Journal of Applied Mechanics, vol. 74, 2007, 1247 1251.  [2]  Xing Y., Liu B., Closed form solutions for free vibrations of rectangular Mindlin plates, Acta Mech Sin, vol. 25, 2009, 689 698.  [3]  Hashemi S.H., Arsanjani M., Exact characteristic equations for some of classical  boundary  conditions  of  vibrating  moderately  thick  rectangular plates, International J. of Solids and Structures, vol. 42, 2005, 819 853.  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[11]  Leissa  A.W,  Vibration  of  plates,  Scientific  and  Technical  Information Division. Office of Technology Utilization, National Aeronautics And Space Administration, Washington DC., 1969.  [12]  Predoi  M.V.,  VibraŃii  mecanice.  Modele   i  aplicaŃii  în  MATLAB (Mechanical  vibration.  Models  and  Applications  in  MATLAB),  Editura Matrix Rom, Bucure ti, 2011.  Addresses: ·  PhD. stud. Phys. Cornel HaŃiegan, “Eftimie Murgu” University of Re iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, Re iŃa, c.hatiegan@uem.ro ·  Prof. PhD. Eng. Gilbert Rainer Gillich, “Eftimie Murgu” University of Re  iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, Re iŃa, gr.gillich@uem.ro ·  Prof. PhD. Eng. Eugen Răduca, “Eftimie Murgu” University of Re iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, Re iŃa, e.raduca@uem.ro ·  Asist. PhD. Eng. Marian Dumitru Nedeloni, “Eftimie Murgu” University of Re iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, m.nedeloni@uem.ro ·  Lect. PhD. Eng. LenuŃa Cîndea, “Eftimie Murgu” University of Re iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, Re iŃa, l.cindea@uem.ro 

Equation of Motion and Determining the Vibration Mode Shapes of a Rectangular Thin Plate Simply Supported on Contour Using MATLAB

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