Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method

Free Vibration Analysis of Composite Plates with Artificial Springs by Trigonometric Ritz Method

H. Ghadirian^a*, M.R. Ghazavi^a, K. Khorshidi^b

^aFaculty of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran

^bDepartment of Mechanical Engineering, Faculty of Engineering, Arak University, 3815688349, Arak, Iran

1. 1.      Introduction   

The plate is one of the most common structural elements that are encountered by either scientific or technological interest. It’s widely utilized in aerospace, marine, mechanical, electrical, nuclear and civil engineering structures. The vibration analysis of plates is one amongst the most important issues in coming up with this sort of structure. Vibration characteristics of plates were extensively studied by other researchers. Plates with completely different shapes, boundary conditions and complicating effects were thought of and also the frequency parameters were investigated in some monographs [1, 2], normal texts [3-5] and review papers [6, 7].

In engineering applications, plates with various complications were investigated. These effects include elastically restrained boundaries, presence of elastically or rigidly connected masses, point supports, variable thickness, anisotropic material, stiffeners[8, 9], interior openings[10] and line hinged [11-13], that are used to serve different purposes required in a structure. In Refs. [13-15] general studies on the vibration of plates with point supports have been deliberated and the vibration of plates with line supports have been studied in Refs. [13-19]. For instance, a line hinge in a plate can be used to expedite folding of gates, or the opening of doors and hatches [20]. The hinge can also be used to simulate a through crack prior to the edge misalignment.

Due to its conceptual simplicity, wide flexibility, high reliability and computational efficiency, the Ritz technique has been widely utilized to resolve the vibration problem of rectangular plates. The Ritz procedure consists in approximating the normal displacement variable through a linear combination of generally assumed functions, commonly known as admissible functions, trial functions or basis functions, each satisfying at least the geometrical boundary conditions of the plate. The unknown constant factors of the combination can be obtained by the minimization of the energy functions of the system. Convergence to the exact solution is assured if the admissible functions are linearly independent and form a mathematically complete set. The chosen functions are not required to satisfy the natural boundary conditions, although, if they do, better convergence and accuracy might be achieved. Moreover, the properties of trial functions have a significant effect on computational efficiency and numerical stability of the solution [21]. The rather weak conditions imposed by the Ritz method and the great sensitivity of the related solution to the choice of admissible functions have prompted many researchers in evaluating and developing suitable ways of constructing the trial set for Kirchhoff plates with arbitrary boundary conditions and complicating effects. There is a so huge amount of literature on the topic which prevents to provide a comprehensive review of all the available approaches here. Restricting the analysis to rectangular plates, notable solutions include the use of characteristic beam functions, simple and orthogonal polynomials, static beam functions, Fourier sine and cosine series or their appropriate combinations [21, 22]. Since the initial works of Young [23], Warburton [24]and Leissa [25], admissible functions as products of vibrating beam eigenfunctions have been presented by many researchers. They work well in most conditions. However, because of the occurrence of over-restraint at free edges, the results obtained for plates involving one or more free edges are much less accurate [26]. Characteristic beam functions are also clearly dependent on the boundary conditions. There exist 21 different cases for rectangular plates by considering all possible combinations of classical edge conditions [25]. Therefore, their use involves a boring solution process since a specific set is required for each type of boundary. Finally, numerical instability happens when the common expressions for the beam mode shape functions are evaluated at high orders [27].

In this paper, a more general case of free vibration analysis of assembled plates is considered for preparing a model that can approximate vibration behaviour of a hinged rectangular plate or a uniformed rectangular plate, only with determination of various stiffness’s of translational and rotational springs that connect two plates to each other. In the process of solving the eigenvalue problem which is obtained from the variation method on the total system, the quasi analytical Ritz method is used. The trigonometric sets are used as admissible functions due to the fact that they are very effective from a computational point of view and their reliability and versatility for flexural vibration analysis of rectangular plates which may be subjected to various complicating factors. Finally some illustrating results are presented in tabular and graphical forms to validate the method.

1. 2.      The Determination of the Boundary Value Problem

Figure 1 represents an isotropic rectangular thin plate in the x-y plane. In this figure h is thickness, a and b are length and width of the plate, respectively. A slit that is parallel to the y-axis is located at x=c, as shown in Figure 1. The plate is considered to have two spans that are separated by the slit. The total domain of the plate A is divided into two sub-domains and  as is shown in Figure 1. It is seen that these two sub-domains are separated by the line  at x=c. and  are connected to each other by using linear translational and rotational springs.  and  are constants of translational and rotational springs, respectively. It can be assumed that the thickness and deflection of each sub-domain are small compared with the wave-length of flexural vibration; consequently, thin plate theory is applicable. Throughout the remainder of the paper, the counter-clockwise four-letter symbolic notation introduced by Leissa [25] is used for describing classical boundary conditions. For instance, an SFSC plate has a simply supported (zero deflection and free rotation) left edge, a free (free deflection and rotation) bottom edge, a simply supported right edge and a clamped (zero deflection and rotation) top edge, respectively.

In order to analyse the transverse displacements of the system, it can be assumed that the vertical position of the k-th (k=1, 2) plate at any time t, is defined by the function . The total kinetic energy of the system is [28]:

Where  is the mass density of the k-th plate and  such that  is constituted of both sub-domains  , and each respective boundary .

    The total potential energy due to the elastic deformation of the plate is [28]:

where  is the rigidity of the k-th plate, ν is the Poisson’s ratio, is the external load,  is the difference in the lateral displacement of the two plates along the slit,  is the difference between the normal slopes and  denotes the directional derivative of w with respect to the outward normal unit vector to the curve  and s is the coordinate along the line of slit.

    To derive the equations of motion, the extended Hamilton principle is used as:

where:

and δ is variational operator.

    Consequently, by using the Hamilton principle, the equations of motion for free vibration analysis of the coupled plates are obtained as follows:

The above equations represent the dynamical behaviour of the vibrating plates.

2.1. Classical Boundary Conditions

    In this study, plates may take any classical boundary conditions, including free, simply supported and clamped. The boundary conditions along the edges  and  are satisfied by the following relations [17]:

(a)  For a free edge:

(b)    For a simply supported edge:

(c)    For a clamped edge:

Where  is the bending moment on the edges  and , and  is the Kirchhoff equivalent force, one can write:

At :

And similarly it could be obtained for the other edges. Applying the states before boundary conditions and continuity conditions, a set of the homogeneous equations would be obtained.

2.2. Continuity conditions for connection of two regions

At the slit location (x=c), continuity conditions along the slit line can be written as:

According to the variational principle, the coefficients of   and  should be zero. Consequently, the following equations are obtained:

where all of the above equations would be evaluated at x=c. Eqs. (11) and (12) express an equal moment with the opposite signs, which are applied to the edges x=c and indicate the Kirchhoff equivalent force on the common edge between two regions.

1. 3.      Eigenvalue Problem in Ritz Method

For free vibrations of the plate, the displacements can be written as:

where ω is the circular frequency of the plate. Substituting Eq. (17) into Eqs. (1) and (2), the maximum kinetic energy  and the maximum potential energy  are obtained. For the sake of simplicity the following dimensionless parameters are used:

Therefore, it can be written:

Where  ,  and  are assumed to be constant for two areas i.e.

Due to classical boundary conditions, the functional energy of the system is expressed as:

The Ritz approximation is:

Where the superscript k denotes the k-th sub-domain,  are unknown coefficients and  and  are appropriate admissible functions satisfying at least the geometrical boundary conditions of the problem. After substituting Eqs. (19) and (20) into Eq. (22), the coefficients can be obtained by finding extremum of the functional energy as follows:

Consequently the following eigenvalue equation is obtained:

where is the dimensionless frequency.

The stiffness and mass matrices K and M are presented respectively as:

where,

In the above equations the integral statements are defined as:

where α and β denote the order of derivatives.

It is noted that  ,  and  are dimensionless coefficients. The spring stiffnesses  and  are selected either as the actual connecting stiffnesses, if flexible joints are represented, or as very high values compared with the adjoining plates, therefore approximating rigid connections. (For example, at a free edge,  and  are taken as zero, while, to approximate a hinged boundary or connection between two adjacent plates,  is given some very high value and  is taken as zero.)[21]

In the present study the admissible functions  and  are defined by means of the trigonometric set. The following trial functions which are used by Beslin and Nicolas [29] for flexural vibration of Kirchhoff plates are presented:

where the coefficients , ,  and  are listed in Table 1.

The function  is defined according to Eq. (30), where  and m are replaced by  and n, respectively. A subset of  is plotted in Figure 2 where functions of increasing order are arranged in a matrix form [21].

It is seen that the first and third functions  and  have a non-zero displacement at  and , respectively. The second and fourth trigonometric functions  and  have a free slope at the same edges at , respectively. The functions and  are arranged in a similar style for .

As it is, the first four functions to  enable one to easily satisfy any classical boundary condition by selecting a suitable combination among them. For example, the analysis of a completely free plate (FFFF) will keep all these four functions in the final sequence. If a simple support condition is imposed on the edge  (FFFS plate), the function  will be eliminated from the set. For a fully clamped plate (CCCC) all the couples of these functions both in  and in  direction will be removed. The nine combinations of classical boundary conditions are reported in Table 2 where a bullet denotes that the corresponding function must be kept in the final set. The first letter in the table refers to the edge at  or  , whereas the second letter refers to that at  or  [22].

Due to the zero determinant of the coefficient matrix in Eq. (21), the problem has a nontrivial solution. The stated determinant gives the natural frequencies of the system. It is important to note that the nontrivial solution of the system gives the mode shapes of the plate.

1. 4.      Numerical Results and Discussions

In order to examine the accuracy and applicability of the approach developed and discussed in the previous sections, numerical results were calculated for a number of different problems for which comparison values were available in the literature and also convergence studies have been carried out in the graphical and tabular form. All calculations have been performed with Poisson’s ratio .

4.1. Rectangular Plate with Different Boundary Conditions

The discussed method is applied to rectangular isotropic plates with four different combinations of classical boundary conditions. To obtain the frequency parameters corresponding to a continuous rectangular plate without slit, it is necessary that the translational and rotational stiffnesses approach infinity. For this purpose in this section dimensionless translational stiffness per unit length  is assumed to be equal to the dimensionless rotational stiffness  and it can be assumed to be equal to a high value ( ). The problems are solved for two cases which have well-known closed form solutions [25], i.e., SSSS and SFSF plates; and, last, a cantilever plate (CFFF) that there is no exact solution for this case [30]. Table 3 shows the convergence of the dimensionless frequency parameter  for the first eight modes.

Results are obtained by using a square selection method, i.e., the similar number of terms M=N is adopted in the series expansion, with no regard to symmetry. It is seen that the solutions monotonically decrease as the number of terms in the set increases. Table 3 also shows that the higher frequency convergent values to four digits are obtained with a different number of terms for each mode and case.  

Moreover, this number does not necessarily increase as the mode number increases. It is illustrated from Table 3 that the frequency parameters which are obtained by the presented method in this paper in comparison to reference [21] are higher about 0.2% at worst state. This difference is due to the discontinuity in the plate and the values of  and which are assumed to be a finite value.

4.2. Rectangular Plate with an Internal Line Hinge

The problem is the transverse vibration of two rectangular plates which hinged together along they coordinate at x=c. There are some literatures about this problem. As it was referred to previously, to approximate a hinged connection between two adjacent plates,  is given some very high value and is taken as zero. In this case some results of a convergence study of the values of the frequency   are presented in Table 4 and Table 5. The first eight values of  are presented for a square SSSS plate with an internal line hinge located at two different positions, namely  and . It can be observed that with an increase in iteration number the frequency parameters converge monotonically. From Table 4 it can be observed that the results are in a good agreement with exact frequency parameters for a SSSS plate with an internal line hinge presented in Ref. [28].

Similar results for plates with different boundary conditions, aspect ratios and position of line hinge are presented in Table 5. A convergence study is shown in Figure 3 and Figure 4 for fundamental frequency in terms of translational and rotational frequencies that vary from 0 to .

These figures are presented in two cases in terms of boundary conditions; Case 1: symmetric boundary conditions (SSSS, SFSC) and Case 2: asymmetric boundary conditions (CCFF, CCSS).

It should be noted that the first natural frequency of vibration is named as fundamental frequency and its magnitude is less than all of the other frequencies. In Figure 3 and Figure 4, the “a” and “b” letters introduce the types of boundary conditions and their indices are used to show the variation of the translational and rotational stiffnesses which are specified in each figure. As it can be seen, the fundamental frequency  oscillates in a range which is very small when only translational frequency  varies, but when both the translational and rotational stiffnesses vary, the fundamental frequency converges monotonically to a specific value. This result is due to the symmetry in boundary conditions which yields this fact that in the low values of  a good accuracy of can be obtained.

On the other hand, for asymmetric boundary conditions such as CCFF and CCSS, for the cases in which only varies or both  and  vary, the fundamental frequency parameter converges monotonically to a specific value as is shown in Figure 2.

1. 5.      Conclusions

This paper presents the formulation of an analytical model for the dynamic behaviour of rectangular isotropic plates, with an arbitrarily located slit and classical boundaries. The equations of motion and associated boundary and compatibility conditions are derived by using the extended Hamilton principle. An approach has been presented to solve the free vibration of the previously mentioned plates in a direct variational and numerical way. A not complicated, computationally efficient and accurate method has been developed for the determination of natural frequencies and modal shapes. The approach is the trigonometric Ritz method which is based on a simple, stable and computationally efficient set of admissible trigonometric functions and has been presented for free vibration analysis of rectangular Kirchhoff plates. The versatility and reliability of the present approach have been shown in various states of the slotted plate with an arbitrarily selected subset of complicating factors. Very accurate and stable solutions have been obtained for all cases with lesser computational effort in comparison with the other similar methods. Consequently, the present analysis shows that the trigonometric Ritz method is a valuable way for solving transverse free vibrations of thin rectangular plates and is easily applicable to a wide class of problems with complicating effects. To investigate the effect of the line slit and its location on the vibration behaviour, parametric studies have been performed. It is valuable to note that by using a modified version of this method, the static deflection problems and buckling can be analysed. On the other hand, this method can be easily generalized for analysing problems that include anisotropic plates.

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