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**Mohamed A. El-Sayad ^{*} and Ahmed M. Farag**

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt

- *Corresponding Author:
- Mohamed A. El-Sayad

Department of Engineering Mathematics and Physics

Faculty of Engineering, Alexandria University, Alexandria, Egypt

**E-mail:**[email protected]

**Received Date:** April 17, 2012; **Accepted Date:** May 19, 2012; **Published**** Date:** May 22, 2012

**Citation:** El-Sayad MA, Farag AM (2012) Numerical Solution of Vibrating Double and Triple-Panel Stepped Thickness Plates. J Appl Computat Math 1:110. doi: 10.4172/2168-9679.1000110

**Copyright:** © 2012 El-Sayad MA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

The main objective of the present paper is to achieve a modified numerical method for investigating the vibration characteristics of the stepped thickness plate with many types of boundary conditions surrounding certain number of panels. The presented technique relies on dividing the entire plate into several regions of uniform thickness separated by sudden steps. Each region is divided to number of strips which are assembled and solved numerically by the Finite Strip-Transition Matrix method FSTM. A convenient basic function is applied to reduce the partial differential equation of motion of plate inside a single region into an ordinary differential one. Step continuity conditions are applied to achieve the final solution of plate. Regional rigidities of plates and mass per unit area are changed due to the change of plate thickness from a region to another. Consequently, new straining actions are occurred and then compatibility conditions become necessary to modify the nodal vector at each step. Various types of restrained boundary conditions against rotation are included in the present paper. The validity of present method is checked and the accuracy of the results is compared with those available in literature showing a good agreement.

Numerical; Transition; Stepped; Panel; Vibration; Plate

Regional dimension-less plate displacement in the domain of region

a,b Plate dimension in ζ ,η directions respectively

x, y Plate coordinates

t Time

ζ ,η Plate dimension-less coordinates;

Aspect ratio

h_{i} Regional thickness of plate

ρ Plate density

M Number of panel

k Parameter of homogenous sub-grade

Plate mass per unit area

Regional flexural rigidity of plate

Normalized parameter of homogenous sub-grade

k Parameter of homogenous sub-grade

ω Natural frequency

Natural frequency parameter

Integral values

Panel thickness ratio

Regional nodal vector

Regional transition matrix

Nodal vectors of strip K_{j}

Φ Restrained cofficient of rotation and translation at ξ = 0,1

Restrained cofficient of rotation and translation at η = 0,1

Step thickness ratio

Poisson’s ratio

A certain structural optimization of a panel may be achieved by possessing a suitable variation of thickness of plate structure. The importance of types of structures increases when the aero-space and underground structures are considered. Survey of the literature on the flexural vibration of thin plate reveals that the work on this topic has been mainly confined to plates with uniform thicknesses. A relatively few studies have been published on the free vibration of isotropic plates of stepped thicknesses. During the past few decades, many researchers were devoted to mathematical modeling and numerical solution for static elastic multi-structure problems. Since vibration analysis of elastic structures plays important roles in engineering applications, this paper is concerned with Finite Strip-Transition Matrix method (FSTM) for vibration analysis of paneled stepped thickness plate.

Although, Chopra [1] has attempted an exact solution for a simply supported stepped thickness plate with two panels, Warburton [2] pointed out that continuity conditions used by Chopra [1] were incorrect. He presented a modified analytical technique for two paneled stepped plate with different properties of orthotropic. Sakata [3] proposed an approximate formula for estimating the fundamental nature frequency of an isotropic plate with stepped thickness from the natural frequencies of the isotropic plate reduced from the orthotropic one. Recently, Farag [4] applied a closed form solution for vibrating surfaces of partially restrained and clamped double-panel plates via a power matrix exponential method. Xiang and Wang [5] also studied the exact vibration solutions of stepped rectangular plates. Xiang et al. [6,7] extended these studies in cooperation with others to include the case of stepped rectangular mindlin plate and the case of stepped circular mindlin plate. Gorman and Singha [8] outlined the vibration analysis of stepped cantilever plate using a supper position method. A numerical approach for rectangular stepped plate with sides restrained against rotation has been proposed by Laura and Filipich [9]. They ensured that vibration of plate with stepped thicknesses was studied in a very few number of literature. Filipich et al. [10] used a simple polynomial coordinate function which identically satisfied the restrained boundary conditions of plate with discontinuous thickness. Modal study via a discrete numerical approach, such as finite element method by Mukherjee and Mukhopadhyay [11], or finite strip method by Cheung and Li [12,13] introduces solution accuracy depending upon the suggested element size. Vibrations of plates with variable thickness were studied by Zanzi and Laura [14], Gutierrez et al. [15], Laura and Gutierrez [16] via different methods. Kaabi and Aksu [17,18] applied a modified method to examine the dynamic behavior of rectangular plates with bilinear variation of thickness. Cheung and Li [19] achieved a simple finite strip method to analyze the hunched continuous bridges. Vibration of orthotropic rectangular plate with free edges in the case of discontinuously varying thickness was discussed by Laura et al. [20]. They extended their studies to include two other boundary conditions different from Laura et al. [21]. Farag [22] applied Finite Strip Transition Matrix method to solve the free and forced vibration problems of uniform thickness plate. In extended work, Farag [23] analyzed the stepped thickness plate analytically by means of the matrix exponential method. Recently, a semi and fully discrete finite element methods for investigating vibration analysis of elastic plate-plate structures are proposed by Junjiam et al. [24]. Semie [25] explained the numerical modeling of thin plates using the finite element method. Sanches et al. [26] studied the dynamic stationary response of reinforced plate by the boundary element method.

The present paper offers a modified numerical method relying on reducing the partial differential equation of motion into an ordinary differential one. A convenient basic function is used to obtain the later equation which is solved by Finite Strip Transition Matrix Method (FSTM) inside each panel of plate. Due to the sudden change of thickness between two adjacent panels, continuity conditions must be satisfied to generate the final solution of the entire plate. The known basic function is expressed in a variable parallel to the steps. The reduced differential equation of rectangular stepped plate is still carrying the other unknown variable in the other plate direction. The restrained boundary conditions are included and employed to derive the solution of the particular cases of clamped and simply supported edges.

The isotropic rectangular plates with elastically restrained boundary conditions in the two directions ζ and η of plates are considered here. The plate thickness shown in **Figure 1** is a uniform in ζ -direction, while it changes suddenly in η -direction at a step S_{i} from h_{i} to h_{i+1} .

Referring to the domain of the isotropic region R_{i} , the regional dimensionless partial differential equation of motion of plate vibration is:

(1)

Where

The different symbols are denoted in the nomenclatures part where the suffix i means that the mentioned magnitudes are calculated inside the region R_{i}.

The partial differential equation (1) is solved numerically under the proposed boundary conditions at (ζ = 0,1) and (η = 0,1) by using the Technique of Finite Strip-Transition Matrix FSTM. In this Technique, the investigated plate is divided into three stepped panels which are separated by a number M-1 of steps. Each individual panel is divided to a number N of finite strips separated by a number N+1 of nodal lines as in **Figure 1**. The basic function of strip [23], is applied to reduce the partial differential equation of motion into an ordinary differential one. The reduced differential equation is solved numerically by the FSTM as an initial value problem under the proposed initial boundary conditions at (η = 0) . The initial nodal conditions of each strip are applied to derive the strip end nodal conditions which are used as initial nodal conditions to the next strip. All nodal straining actions of each strip are obtained until the end of the current panel is reached. Because the intermediate end of a current panel is sudden step, the initial nodal conditions of the next panel must be modified by satisfying the compatibility conditions at this step. The FSTM is applied for all strips of all panels until the final boundary end of plate is reached. Satisfying the boundary conditions at (η = 1) , one can obtain the final solution of the equation motion of plate.

**Reduction technique**

The regional displacement of the stepped plate is:

(2)

Where is known regional shape function satisfying the plate boundary conditions [23] at (ζ = 0,1) and is unknown regional longitudinal function to be determined at (η = 0,1) .

Eq. (2) is applied to reduce the partial differentia Eq. (1) Inside the Region R_{i }i = 1,2,3; into [4]:

(3)

Where

, , ,

Consequently, the equation of motion for any panel becomes:

(4)

where:

is the panel thickness ratio at region R_{i }i = 1,2,3;

The ordinary differential equation (4) is transformed within a region R_{i} to a linear system of differential equations expressed as [22]:

(5)

Where

(6)

And

(7)

The general solution of the ordinary differential equation is:

(8)

Where δ_{i} is the width of strip κ_{j} bounded by two nodal lines K_{j} ,K_{j-1} inside region R_{i} .

The regional Transition Matrix [Y_{K}]_{i} is calculated according to the Runge Kutta method such as:

(9)

Where I is the unit matrix?

**Continuity conditions**

The Magnitudes of displacement W slope Moment M_{V} and shear Q_{v} must satisfy the continuity conditions at the sudden step S_{i} so that:

(10)

Where is step thickness ratio at step S_{i} ;i=1,2 and ν_{x} is Poisson’s ratio.

A right step vector of the sudden step S_{i} is achieved by updating the step left vector due to Eq. (10).

**Boundary and initial conditions**

Applying the proposed boundary conditions at η = 0 , one can reduce the four initial unknowns to only two unknowns. Consequently, by satisfying the boundary conditions of plate at η = 1 , two characteristic equations for the plate vibration are established. The natural frequency parameters of plate are the Eigen values of the characteristic matrix of these equations. The corresponding Eigen vectors create the mode shapes. Boundary conditions at the edges η = 0,1 are considered for various types of edges [22], such as simply supported S, clamped C, and free F, elastically restrained against rotation E_{R} . The restrained coefficients of rotation Φ η = 0,1 are usually applied to vary from 0 to ∞.

The initial conditions are expressed as initial values which are the components of initial vector {V_{m}}_{o} of the first region R1. This initial vector {V_{m}}_{o} can be derived according to the known boundary conditions at the first nodal edge η = 0 [23].

Different cases of plates composed of different panels with unequal thicknesses and panel widths are investigated by the present technique. This study takes in account the variation of the aspect ratios β and various magnitudes of panel thickness ratio α_{i }i = 1,2,3; and the coefficients of partially restrained boundary conditions. The natural frequency parameter λ_{mn} is calculated for every case where Poisson’s ratio is taken as ν = 0.3 .

The natural frequencey coefficients based on the present technique are compared as shown in **Table 1** with those obtained by Xiang et al. [5] using an exact solution and using closed form solution [4].

SSSS; | CSCS; | CCCC; | ||||||

Present | Xiang et al. [5] | Farag [4] | Present | Xiang et al. [5] | Farag [4] | Present | ||

s11 |
02.9015 | 0 2.9015 | 02.9015 | 04.1711 | 04.1711 | 04.1711 | 05.3028 | |

1/2 | s12 |
07.1157 | 07.1156 | 07.1156 | 09.9048 | 0 9.9047 | 09.9047 | 10.5159 |

s13 | 13.7849 | 13.7850 | 13. 7848 | 18. 0455 | 18.0450 | 18. 0453 | 18.6238 | |

s11 | 02.4470 | 2.4471 | 02.4470 | 03.5609 | 03.5610 | 03.5609 | 04.4631 | |

2/3 | s12 | 06.2230 | 6.2229 | 06.2229 | 08.7200 | 08.7199 | 08.7199 | 09.2528 |

s13 | 11.9125 | 11.9480 | 11.9124 | 15.6217 | 15.6210 | 15.6215 | 16.0596 |

**Table 1:** Comparesons of the natural frequency coefficient σ_{mn} for square Plates SSSS and CSCS.

The comparisons are available only for the case of double panel square plate with panel width ratio as shown in **Table 1**. Calculations are carried out in two cases of boundary conditions of square plates SSSS and CSCS where S, C mention to the simply supported and clamped edges respectively. The general method of restrained boundary conditions is applied when , to posses the case of full simply supported plate SSSS. Another case of plate CSCS is obtained where and . The third case is full clamped plate CCCC where and . The results are obtained for panel thickness ratios equal to 0.5 and 2/3. The comparisons show excellent agreement.

The natural frequencey parameters λ_{mn} in the first six modes are obtained in **Table 2** for the cases of 3-panels square full clamped plates CCCC with panel width ratio . The results are calculated under the variation of α_{2} from 1.0 to 0.1 where . The results show that the natural frequency parametrer inreases by decreasing α_{2} .

Natural frequencey parameters λ_{mn} |
||||||

α_{2},α_{3} |
α_{11} |
α_{12} |
α_{13} |
α_{14} |
α_{15} |
α_{16} |

1.0, 1.0 | 35.9195 | 73.5929 | 132.3785 | 211.2511 | 309.0456 | 416.1822 |

0.9, 1.0 | 37.9563 | 77.3438 | 138.6326 | 222.3554 | 327.3623 | 437.8916 |

0.8, 1.0 | 40.5830 | 81.3422 | 145.6952 | 236.6180 | 348.0211 | 459.2436 |

0.7, 1.0 | 43.9586 | 85.5344 | 154.3948 | 236.6180 | 370.3955 | 480.2570 |

0.6, 1.0 | 46.3017 | 71.6753 | 102.2939 | 255.4175 | 332.9891 | 385.8950 |

0.5, 1.0 | 54.0737 | 95.5105 | 184.6845 | 267.9179 | 410.2973 | 539.8316 |

0.4, 1.0 | 62.4249 | 104.1121 | 215.7646 | 343.2175 | 425.5430 | 616.2196 |

0.3, 1.0 | 76.9445 | 122.0380 | 268.1892 | 366.2887 | 447.2614 | 819.1406 |

0.2, 1.0 | 109.3091 | 168.1862 | 333.4251 | 376.4200 | 543.4720 | 1265.0045 |

0.1, 1.0 | 215.0652 | 311.9320 | 359.3840 | 410.0752 | 1080.9598 | 2590.7068 |

**Table 2:** The natural frequency parameter λ_{mn} for 3-panels square clamped plate CCCC.

**Table 3** shows the natural frequencey parameters λ_{mn} for the cases of 3- panels square full plates CERCER with two opposite edges clamped and other edges ellastically restrained against rotation with restrained coefficient varying from 0 to ∞ . The results are obtained for cases of panel width ratio and α_{2} varying from 1.0 to 0.25, where . The natural frequencey parameter is obtained for the first three modes. The results show that the natural frequency parametrer decreases by increasing the coefficient of restrained .

Restraint coefficients against rotation | |||||||

α_{2},α_{3} |
λ_{mn} |
0.0 | 0.02 | 0.2 | 2.0 | 200 | |

1.0, 1.0 | α_{11} |
35.9195 | 34.7141 | 31.1493 | 29.1644 | 28.8413 | 28.8379 |

α_{11} |
73.5929 | 69.7890 | 50.1225 | 55.6106 | 54.9253 | 54.9181 | |

α_{13} |
132.3787 | 125.0569 | 109.4910 | 103.5695 | 102.7365 | 102.7279 | |

0.75, 1.0 | α_{11} |
42.1654 | 40.8311 | 37.1138 | 35.1818 | 34.8760 | 34.7828 |

α_{11} |
83.4136 | 78.8536 | 67.9529 | 63.1631 | 62.4507 | 62.4432 | |

α_{13} |
149.7636 | 141.3653 | 124.3771 | 118.1902 | 117.3307 | 117.3219 | |

0. 5, 1.0 | α_{11} |
54.0736 | 52.5033 | 48.5825 | 46.7700 | 46.4956 | 46.4927 |

α_{11} |
95.5105 | 90.5687 | 79.5519 | 57.0258 | 74.3679 | 74.3610 | |

α_{13} |
184.8631 | 175.7379 | 157.7919 | 151.4550 | 150.5831 | 150.7542 | |

0.25, 1.0 | α_{11} |
84.6291 | 88.3274 | 58.8198 | 84.7820 | 84.6307 | 84.6291 |

α_{11} |
123.5494 | 135.5536 | 127.3214 | 124.0274 | 123.5543 | 123.5494 | |

α_{13} |
233.6159 | 282.3131 | 246.3862 | 235.1273 | 233.6312 | 233.6164 |

**Table 3:** The natural frequency parameter λ_{mn} for 3-panels square clamped restrained plate CE_{R}CE_{R}.

This case is carried out for 3-panel square plate, SE_{R}SE_{R}, simply supported at two opposite edges and partially restrained against rotation at the other edges (**Table 4**). The results are obtained for panel width ratios and panel thickness ratio 1 :α_{2} : 1 where α_{2} varies from 1.0 to 0.25. **Table 4** shows that the natural frequency parameter increases by decreasing the restrained coefficient . For the double panel isotropic, square, clamped supported plate at all edges CCCC the relationships of the variation of natural frequency parameters due to the change of thickness ratio α_{2} are plotted in **figures** **2-a,b,c**. For the panel width ratio , the natural frequency parameters λ_{mn} are obtained for the first three modes where m =1, 2 or 3 and n = 1,2 or 3.

Restraint coefficients against rotation | |||||||

α_{2},α_{3} |
λ_{mn} |
0.0 | 0.02 | 0.2 | 2.0 | 200 | |

1.0, 1.0 | α_{11} |
35.9195 | 34.7141 | 31.1493 | 29.1644 | 28.8413 | 28.8379 |

α_{11} |
73.5929 | 69.7890 | 50.1225 | 55.6106 | 54.9253 | 54.9181 | |

α_{13} |
132.3787 | 125.0569 | 109.4910 | 103.5695 | 102.7365 | 102.7279 | |

0.75, 1.0 | α_{11} |
42.1654 | 40.8311 | 37.1138 | 35.1818 | 34.8760 | 34.7828 |

α_{11} |
83.4136 | 78.8536 | 67.9529 | 63.1631 | 62.4507 | 62.4432 | |

α_{13} |
149.7636 | 141.3653 | 124.3771 | 118.1902 | 117.3307 | 117.3219 | |

0. 5, 1.0 | α_{11} |
54.0736 | 52.5033 | 48.5825 | 46.7700 | 46.4956 | 46.4927 |

α_{11} |
95.5105 | 90.5687 | 79.5519 | 57.0258 | 74.3679 | 74.3610 | |

α_{13} |
184.8631 | 175.7379 | 157.7919 | 151.4550 | 150.5831 | 150.7542 | |

0.25, 1.0 | α_{11} |
84.6291 | 88.3274 | 58.8198 | 84.7820 | 84.6307 | 84.6291 |

α_{11} |
123.5494 | 135.5536 | 127.3214 | 124.0274 | 123.5543 | 123.5494 | |

α_{13} |
233.6159 | 282.3131 | 246.3862 | 235.1273 | 233.6312 | 233.6164 |

**Table 4:** The natural frequency parameter λ_{mn} for 3-panels square simply supported- restrained plate SERSER.

Similarly, the case of square isotropic stepped plate CSCS with two opposite edges clamped and other edges simply supported is carried out as shown in **figures 3-a,b,c**. The frequency parameters λ_{mn} are recorded for the case of panel width ratio when the thickness ratio α_{2} changes from 0.1 to 1.0. The results show that the natural frequency parameters increase by decreasing the thickness ratio α_{2} for all modes.

The case of full clamped stepped rectangular plate CCCC with thickness ratio α_{2}=.5, panel width ratio and aspect ratio β of the intire plate varying from 1 to 2 is investigated as shown in **figures 4-a,b,c**. The resuls show that the parameter λ_{mn} increases by increasing the aspect ratio β for all recorded modes.

The finite strip transition matrix method FSTM described here involvoes a numerical solution of stepped paneled plate with classical and restrained boundary conditions. This method is a combination between the strip and transition matrix method to solve the vibration problem of stepped plates as an initial value proplem. Transition matrix method is a semi analytical method relying on estmiating the numerical solution of the intial value problem by means of Range Kutta method. The plate domain is divided into paneled regoins consisting of strips bounded by nodal lines. Each strip is governed by the transition matrix formula which transite from one strip to another via nodal vectors until the final edge is reached. Several cases of double and triple panel plates are investigated for the variation of thickness ratio, aspect ratio, panel width ratio and boundary conditions. To show the accuracy of the present method, the results have been summerized and compared with those obtained by other methods. A good exteremely agreement of results is found for all compared cases.

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