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The present work is concerned with dynamic characteristics of beam-stiffened rectangular plate by an improved Fourier series method (IFSM), including mobility characteristics, structural intensity, and transient response. The artificial coupling spring technology is introduced to establish the clamped or elastic connections at the interface between the plate and beams. According to IFSM, the displacement field of the plate and the stiffening beams are expressed as a combination of the Fourier cosine series and its auxiliary functions. Then, the Rayleigh–Ritz method is applied to solve the unknown Fourier coefficients, which determines the dynamic characteristics of the coupled structure. The Newmark method is adopted to obtain the transient response of the coupled structure, where the Rayleigh damping is taken into consideration. The rapid convergence of the current method is shown, and good agreement between the predicted results and FEM results is also revealed. On this basis, the effects of the factors related to the stiffening beam (including the length, orientations, and arrangement spacing of beams) and elastic parameters, as well as damping coefficients on the dynamic characteristics of the stiffened plate are investigated.

The stiffened plate component can be regarded as a coupling structure that is composed of a plate and several beams. In practical engineering, the connection between the plate and beam does not only involve the classical coupling, but the elastic coupling is also frequently encountered. In addition, due to the complexity of the actual working conditions, the stiffened plates will be often subjected to the elastic boundary restrictions. A good understanding of the dynamics of the stiffened plate with general boundary restrains can provide direct benefits to the structural design of complex systems. However, there are only a few literatures on the dynamic analysis of the rectangular plate with some beams of arbitrary lengths and orientations. The current work is to present the dynamic analysis of the beam-stiffened rectangular plate with general boundary restraints.

In the past decades, many scholars have made a lot of efforts on free vibration characteristics of the stiffened plate. Mukherjee and Mukhopadhyay [

Through the abovementioned review of literatures, the predecessors have conducted in-depth research on the free vibration of the stiffened plate. Nevertheless, many scholars are not satisfied with research in the field of free vibration. Harik and Salamoun [

Through the review of the abovementioned literature, it can be found that the existing researches are limited to the dynamic of beam-stiffened plate with classical boundary conditions (like clamped, simply supported, and free). Only the free vibration analysis related to elastic boundary conditions and nonfixed connections between a plate and beams is carried out in Xu’s et al. [

The geometry of a typical beam-stiffened plate and the coordinate system of the coupling structure are depicted in Figure _{b} and _{b}, respectively, which is in the local coordinate system (_{u}, and _{pb1}, _{pb2}, and _{pb3}) and three types of rotational springs (_{pb1}, _{pb2}, and _{pb3}). By setting stiffness values of coupling springs, different coupling relationships between the beams and plate can be achieved, including various elastic coupling and clamped coupling. It should be noted that the coupling between two intersecting beams is ignored in this paper. From Figure _{0}, _{0}).

Schematic diagram of the coupling systems. (a) Geometry of rectangular plate reinforced by arbitrarily orientated beams. (b) Schematic of an arbitrarily placed beam.

In this study, both the in-plane displacement components (_{p}) of the plate is composed of the out-of-plane strain potential energy (

Similarly, the kinetic energy (_{p}) expression of the plate can be written as

In equations (_{p}, _{p},

Since the boundary conditions of the plate are realized by the artificial virtual spring technology, the elastic potential energy (_{sp}) stored in the boundary springs here should be taken into the potential energy of plate. _{sp} can be expressed as

In this paper, four degrees of freedom for a single beam are considered, which are two bending displacement components (_{b}) and torsional displacement component (_{bi}) of

The kinetic energy of

In equations (_{bi}. In addition, Young’s modulus, shear modulus, mass density, and cross-sectional area of _{bi}, _{bi}, _{bi}, respectively.

Since the stiffening beam and plate is connected by a coupling spring, the coupling potential (

It should be noted that six degrees of freedom between the beam and the plate are considered here. As shown in equation (_{pb1}, _{pb2}, and _{pb3}) are used for linear displacement constraints and three sets of rotational springs (_{pb1}, _{pb2}, and _{pb3}) are used for angular displacement constraints. Besides,

In the present research, the admissible displacement functions of a plate and a beam are established by IFSM, which has been successfully applied to solve the vibration of beams [_{ml}, _{ln}, _{ml}, _{ln}, _{mp}, and _{pn} (

Similarly, the displacement field functions of the

In the above equations (_{mi}, _{mi}, _{mi}, and _{mi} represent the unknown coefficient of

For the calculation of the steady-state response for the beam-stiffened plate under external excitation, it is necessary to consider the work done by the external force. In terms of the coupling structure shown in Figure _{exc} represents the work done by the external excitation force acting on the plate:

In the above equations (_{u},

Substituting equations (

For the external excitation force with any frequency, the unknown series expansion coefficient related to displacement functions of the coupled structure can be directly calculated from equation (

Substituting equation (_{p}) and structural loss factor (

The structure mobility can quantitatively describe the law of power flow in the structure. Therefore, understanding the admittance characteristics of the stiffened plate is of great significance for structural vibration control. The structure mobility can be calculated by the following equation:_{nk} represents the transfer mobility from point _{k} is utilized to express the velocity at point _{n}. The mobilities of the drive point

The vibration energy of the structure can be described by the structural intensity vector, which helps the designers understand the energy distribution of the structure. The definition of structural intensity can be found in references [_{x} (_{y} (

In equation (

In the above equations (

The structural intensity associated with the in-plane vibration components in equations (

Substituting equations (

The mass matrix

Generally, damping constants have frequency-varying characteristics. Therefore, it is difficult to accurately define the damping matrix in numerical simulation. For the sake of simplicity, only Rayleigh damping of the coupling structure is considered in this study, namely,

In order to solve the transient response of the system, the Newmark method is adopted, which is an extensive implicit algorithm. Its integration method is as follows:

In equations (30)∼(31), when the velocity function and the displacement function are expanded by Taylor series, the expansion is retained to the second derivative and equations (

Combining equations (

The accuracy and stability of the Newmark method depends on

In this section, a series of numerical results for dynamic behaviors of the stiffened plate with general boundary restraints are carried out based on the theoretical model established in Section _{p} = _{bi} = 2.07 GPa, ^{3},

In addition, for the sake of simplicity, the symbols of

At

^{2}:

^{5}:

^{7}:

^{55}:

^{77}:

Another special note is the type and magnitude of the external load used in this paper. The excitation forces applied in the current research are point excitation and uniform pressure in the normal direction of the stiffened plate, which are denoted by the symbol of

External load types for the stiffened plates: (a) point force, (b) surface force, and (c) rectangular pluse.

In this section, the stiffened rectangular plate with _{b}/_{b}/

Convergence of velocity response for rectangular plate with one central

Convergence of velocity response for rectangular plate with one central

Next, a square plate with one beam located at its diagonal line is utilized to implement a comparative study between the current method and FEM result (_{Ansys15.0)}. It is clarified that the configurations of the reinforcing component are _{1} = _{b}_{b}_{0} = _{0} = 0, and

Comparison of velocity response for square plate with one beam located at its diagonal line under unit force (

Comparison of velocity response for square plate with one beam located at its diagonal line under local uniform pressure (

In this part, the effect of the stiffening beam with different lengths on mobility characteristics of the reinforced plate subjected to _{1} = 0 m, 0.2 m, 0.5 m, and 1 m. Particularly, except for the length of the stiffening beam and the cross-section parameter _{b}_{1} = 0.5 m or _{1} = 1 m, the peak of their mobility curves will shift to the high frequency and magnitude of some peaks will appear to decrease in the range of 200–800 Hz, especiall_{1} = 1 m, which shows that the rib increases the ratio of stiffness to mass of the plate in high frequency. The above phenomena also reveals that in structural vibration control, reinforcement helps to reduce the response peak of the plate in the midhigh-frequency range and changes the structural resonance frequency, but has a little effect on the response in low-frequency domain.

Mobility characteristics for square plate with one beam of different lengths located at its diagonal line under unit force (

Figure ^{2}^{2}^{2}^{5}^{2}^{5}, ^{5}^{5}^{5}^{5}, and ^{5}^{5}). The points C and D are selected as the observed points, and a local uniform pressure _{0}, _{0}, _{1} = _{2} = _{b1}/_{b2}/_{b1}/_{b2}/^{2}^{2}^{5}^{5}, the resonant peak of the structural mobility response moves to the high-frequency direction and the peak value of the response curves will show a downward trend, especially under ^{5}^{5}. Therefore, it can be known that the frequency position corresponding to the vibration formant can be adjusted by modifying the boundary constraint spring stiffness to achieve structural vibration control. Another interesting finding is that the number of formants of the mobility responses increases when the stiffened plate is subjected to ^{2}^{5}^{2}^{5} and ^{5}^{5} boundary conditions.

Mobility characteristics of diagonally stiffened square plate subjected to local uniform pressure with various elastic boundary conditions (

In order to investigate the effect of coupling spring stiffnesses on the structural mobility characteristics, a stiffened rectangular plate with ^{2}^{2}^{2}^{2} case is presented in Figure _{1} = 2 m in the _{2} = _{3} = _{4} = 1 m in the _{b1}/_{b2} = _{b3} = _{b4} = _{b1}, _{b1}/_{b2} = _{b3} = _{b4} = _{b1}. And point A and point B are selected as the drive point and transfer point, respectively. In addition, it should be noted that the coupling springs (including _{pb1}, _{pb2}, _{pb3}, _{pb1}, _{pb2}, and _{pb3}) shown in Section _{b}. The curves in Figure

Mobility characteristics of beam-reinforced rectangular plate with different coupling spring stiffnesses (

In the analytical model constructed in this paper, structural damping is introduced by means of complex Young’s modulus. Damping effects on the mobility characteristics of a rectangular plate stiffened by three ^{5}^{5}^{5}^{5} boundary restrains and measured points follow the same values of the cases studied in Figure

Damping effects on the mobility characteristics of a rectangular plates stiffened by three

In the previous section, the mobility characteristics of the stiffened plates are clearly described. Undoubtedly, these numerical results provide reference data for further research. However, for the specific structural design, it is extremely necessary to understand the distribution and transmission of vibration energy in the structural system. To this end, the following analysis of structural intensity will be performed to show the flow strength and direction of the vibrational energy in the structure.

Firstly, structural intensity of a rectangular plate attached by beams with different configurations under two-point excitation is carried out, where four symmetrical boundary conditions are employed, namely, ^{77}^{77}^{77}^{77}, and ^{55}^{55}^{55}^{55}. The geometric parameters of the beams (see Figure _{1} = _{2} = _{3} = _{b1}_{b2} = _{b3} = _{b1}, and _{b1} = _{b2} = _{b3} = _{0} in the _{0} = 0. Therefore, the coordinates _{0} in the ^{nd} frequency of the rectangular plate with ^{5}^{5}^{5}^{5} and ^{7}^{7}^{7}^{7}). The above phenomenon may be because reinforcements hinder energy flow on the plate, resulting in energy accumulation in some areas. By further observing, it is found that in the case of symmetrical boundary cases and symmetrical point forces, there is always at least a section where the vibrational energy cancels each other out, which is called the power-insulation section. Furthermore, the number of such power-insulation section can be increased by setting an appropriate reinforcement spacing. Also, it can be intuitively found that the vibration energy is transmitted from the excitation to the periphery of the plate, but the force source is not always output source of the vibration energy (see Figure

Structural intensity of a rectangular plate attached by beams with different configurations under two-point excitation: (a) CSCS; (b) SFSF; (c) ^{77}^{77}^{77}^{77}; (d) ^{55}^{55}^{55}^{55}.

Next, structural intensity of an _{1} = 1, _{b1}/_{b1}/^{nd} frequency and 4^{th} frequency for the square plate. From Figure

Structural intensity of an ^{nd} frequency (122.44 Hz); (b) 4^{th} frequency (195.90 Hz).

This section is concerned with the transient vibration analysis of the reinforced plate under point force and local uniform pressure. The convergence and accuracy of transient responses for the plate with beams attached are first validated. For a validation case, transient responses of a coupling structure identical to the reinforced rectangular plate used in Figures _{0} = 0.2 s shown in Figure

Convergence of transient responses for rectangular plate with one central

Convergence of transient responses for rectangular plate with one central

Comparison of transient responses for rectangular plate with one central

Next, transient responses of an SSSS square plate stiffened by a beam with various orientations are presented in Figure

Transient response of an

As mentioned in Section _{1} = _{2} = _{b1}_{b2} = _{b1}, and _{b1} = _{b2} =

Effect of Rayleigh damping coefficient on transient response of stiffened square plate with

In the last transient study, transient responses of a rectangular plate with three ^{12}; for instance, while the elastic parameters ^{12}, the stiffness values of _{u}, _{,} and ^{12}. Figure ^{5} to 10^{11}, the amplitude and phase of the lateral displacement response will change significantly. Specifically, as the stiffness value increases, the oscillation period of the response curve will be shortened and the vibration amplitude will decrease. When the stiffness value of spring ^{11}, the transient displacement response remains unchanged and is close to the response under CCCC case. For a rotating spring ^{3} to 10^{8}, but its effect is not as obvious as that of the transverse spring _{u} and

Effect of the boundary spring stiffness on transient response of stiffened plate under local uniform pressure (_{u}. (b) Linear spring:

An analysis model is established for investigating the linear dynamic analysis of rectangular plates reinforced by beams of any lengths and orientations. Employing IFSM, displacement functions of the plate and reinforcements are obtained, respectively. Based on Rayleigh–Ritz method, dynamic response of the coupling structure is obtained, in which various boundary restrains and beam-plate coupling relationships are achieved using the artificial spring technology. Several numerical analyses verify the effectiveness of the current method, and a series of novel numerical results are also given. Besides, it should also be noted that even though only the linear dynamic performances of the beam-stiffened plate are reported here, the presented methods can also be extended to perform other dynamics analysis related to the stiffened plate, such as the vibration control of the stiffened plate in electrothermal-magnetic field and vibration analysis of reinforced coupling plates.

In addition, there are some conclusions that are stated as follows: (1) The mobility characteristics of the beam-stiffened plate are not only related to the length of the beams, but also to coupling spring stiffness and structural damping, especially in the high-frequency range, which provides a design basis for structural vibration control. (2) The arrangement spacing and orientation of stiffening beams have a significant effect on the structural intensity distribution of the plate. Moreover, the structural intensity distribution of the stiffened plates is sensitive to the frequency of the excitation force and boundary conditions, which give a reference for discriminating the position of the load and the transmission path of the vibration energy. (3) Reducing the distance between the measured point and the reinforcing beams or increasing the spring stiffness (_{u} and

All the underlying data related to this article are available upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (grant no. 51905511).