Study on the Influence of the Internal Elastic Constraint Stiffness on the Vib-Acoustic Performance of a Coupled Plate-Cylindrical Shell System

. Regarding the vib-acoustic performance of internal coupled structures, basically all relevant studies have been confned to the classic form of connections. In this investigation, the boundary and internal elastic constraint matrices of the coupled structure are established, the vibration and sound calculation model of the coupled structure is then established, and numerical analysis is performed to show the efect of constraint stifness on the vibration and acoustical performance of a coupled plate-cylindrical shell system. Te results show that impedance matching between the structures is improved with the increase of the elastic connection stifness, which is conducive to the vibration energy propagation. Moreover, the supporting coupling stifness between elastic coupled structures plays an important role in vibration energy transfers.


Introduction
Te dynamic behavior of diferent structures and a series of problems caused by them have been the focus of research [1][2][3][4][5]. It is well known that cylindrical shells separated by longitudinal bottom plates are important simplifed structures in ocean engineering, aerospace engineering, and other felds. Terefore, it is necessary to further investigate the vibration and acoustic properties of the plate-cylindrical shell coupling system.
Although there are many studies on the coupled platecylindrical shell systems, the internal connections of these structures are usually assumed to be rigid and fxed. Missaoui and Cheng combined the artifcial spring technique with the integral mode method to build a numerical model to discuss the sound performance of the shell system [6]. To better understand the internal physical mechanisms and ofer advice on noise and vibration management, Li et al. studied the acoustic propagation property interaction between a cylindrical shell with compartments and a constrained acoustic enclosure [7]. Wang et al. studied the power fow characteristics of a complicated plate-cylindrical shell system by using the receptance substructure approach, and the theoretical description of the mode shape function was employed to represent the receiving function of each substructure [8]. Zhao et al. established the receptivity expression of plates and shells by using the conventional plate theory and Loew's shell theory. Based on the geometric compatibility constraints and force balancing, they were able to develop the frequency equation for shell and platecoupled systems and analyze the vibration transmission properties of cylindrical shells [9]. Te researchers Clot et al. developed a cogent model for a double-deck tunnel. Te model considered the pipe model to describe the coupling system. Te computational fndings indicate that there is a discernible distinction between the two types of tunnels, with the double-layer tunnel exhibiting a greater capacity for the passage of radiant power [10]. By applying the reciprocity theorem, Zou et al. was able to determine the characteristic of underwater acoustic radiation that is produced by an infnitely long structure consisting of an inner plate and a cylindrical shell [11]. Deng et al. conducted their research on a composite cylinder shell that had an interior thin plate and a number of acoustic black holes inserted within the plate. Tey solved the structural modal parameters by employing the Gaussian expansion approach, and they decreased the order of the model by utilizing the modal truncation technique [12]. In order to study the dynamic characteristics of cylindrical shells that have an internal fexural foor structure, Tian et al. developed a formula that combined analytical and numerical methods. Te entirety of the construction may be broken down into three distinct parts: the cylinder shell, the axisymmetric ring plate, and the nonaxisymmetric foor plate [13]. Taking into account the displacement continuity requirement, Zhao et al. derived the coupling controlling equation of a spinning constructed cylinder of shell plates by using the Donnell shell theory, the Kirchhof plate theory, and the Lagrange equation. Free vibration results for the combined cylinder shell and plate structure are calculated by using the assumption mode approach [14]. For a simply supported shell system, Lee et al. were able to determine the system's natural frequency as well as its mode function by employing the Rayleigh-Ritz method, which is predicated on the principle of energy. In addition, they were able to determine the dynamics behavior of a cylindrical shell with an inner plate by employing the tolerance method [15].
According to an analysis of the relevant published research, earliest investigations on plate-cylindrical shell coupling architectures were restricted to clamped or hinging systems.
Regarding the vib-acoustic performance of the coupled structure, basically all the studies have been confned to the classic form of connections, such as welding or hinging. However, there are numerous diferent instances of elastically connected structures in the actual application of engineering, for instance, space vehicles, building structures, and ship hulls, as shown in Figure 1. According to the knowledge of the author, few works have been reported on the vibrational and acoustic performance of elastically linked structures. In addition, the particulars of the vibrational and acoustic behavior, as well as the processes that underlie this behavior in such elastically linked materials, are not yet completely known. Te primary objective of the research is to contribute in bridging this research gap. In the future, when a better knowledge of the interactions between elastically coupled plate-cylindrical shell systems is achieved, it will be possible to gain insight into how elastically coupled structures may be utilized in the control of vibration and noise in real settings.
Tere are four sections in this paper. Te frst part of the paper is the introduction. Te corresponding matrix representation is provided in Section 2. In Section 3, the impacts of the internal constraint stifness on the vib-acoustic performance of structures are discussed. Te main results of this paper are summarized in the fnal section.

Modelling of the Shell Element.
A point (x, y, z) within the element is made to have a displacement that takes the following form [16]: In the above formula, the displacement of the corresponding points is represented by u, v, and w; meanwhile, the midplane's rotations are represented by θ x and θ y , respectively.
According to the Mindlin bending theory, the shell's strain components are as follows [17]: In this study, the displacement feld is expressed in terms of the nodal variables in a four-node isoparametric shell element [17,18].
where U 0 and U 1 can be written as follows: Ten, equation (3) can be expressed as as follows: Te stifness matrix of the plate structure can be written as follows [17,18]: where [B p ] is

Modelling of the Beam Element.
Te Timoshenko beam element is a two-node beam element that was developed using the Timoshenko beam theory and taking into account transverse shear deformation. Each node contains three displacement degrees u, v, and w and three rotation angles θ x , θ y , and θ z . It is an element of the displacement degree of freedom w and the angle degree of freedom θ independently interpolated. It belongs to the C 0 type element, which is represented by the following interpolation: where w is the displacement of the ith node, θ is the angle of the ith node, n is the number of nodes, and N is the Lagrange interpolation function. Similar to the shell element, the beam element can be expressed as follows: where k e s represents the element's overall stifness, k e s represents its shear stifness, and k e b represents its bending stifness.

Elastic Boundary Stifness and Internal Constraint
Stifness. Te elastic coupling model of the coupled stifened plate-cylindrical shell system is considered, as shown in Figure 2. Te boundary around the stifened plate is elastically connected with the closed cylindrical shell; that is, a pair of boundaries is elastically connected with the circular end plate and another pair of boundaries is elastically connected with the cylindrical shell. Tese pairs are expressed by four kinds of independent spring constraints (k cx , k cy , k cz , and k sx ), while the elastic constraints on the boundary of the cylindrical shell are expressed by k sx , k sθ , k sw , and k sr . Te cylindrical shell is modelled using the cylindrical coordinate system, while the stifening plate is simulated using the Cartesian coordinate system. Furthermore, u, v, and w represent the axial, tangential, and radial displacements, respectively. R, t s , and L s are the cylindrical shell's radius, thickness, and length, respectively. Te elastic boundary of the cylindrical shell is described by the translation stifness and rotational stifness, and its strain energy is obtained as follows [19]: Te above equation can be further written as follows: where K b e can be written in detail as follows: Shock and Vibration where Te total stifness matrix of the elastic boundary can be expressed as follows: Similarly, the internal constraint connection stifness matrix of the coupled system [K pb ] can be easily obtained, and the elastic boundary stifness matrix and the coupling structure connection stifness matrix are superimposed in the same coordinate system.

Modal Analysis.
Ignoring the infuence of the damping junction, the motion equation for the coupled system's modal analysis is expressed as follows: Assuming harmonic vibrations, U { } � U { }e iω n t and then where [M] is the global mass matrix and [K] is the global stifness matrix. Tis is a standard eigenvalue problem that is solvable for eigenvalues and eigenvectors. where and λ � (1/ω 2 n ). [18,20]. Te equation of motion of an elastic structure under a time-harmonic load can be written as follows:

Acoustic Radiation Model
where v n is the normal velocity vector of the elastic structure and [Z] is the acoustic impedance matrix.
Terefore, the displacement vector of the elastic structure is After solving the normal velocity v n , the acoustic radiation power of the elastic structure can be calculated as follows: In the above equation, an asterisk represents a complex conjugate.

Numerical Simulation Results
Generally, the vibration frequency of an elastic structure mainly depends on the material properties and the geometric size. Regarding the coupled system, it is also related to the relative position and connection form of each structure. In this section, a closed cylindrical shell containing a stifened plate is taken as an example (for clear display, the seal plate is not drawn). As shown in Figure 1, ψ is the starting position of the stifened plate, and L p is the length of the stifened plate. Its dimensions in the width direction are related to the coupling position. Te thickness of the coupled system remain the same, that is, t p � t s � t e , (the subscript s represents the cylindrical shell, p represents the stifened plate, and e represents the end plate). Te material of the structure is the same, and the stifened plate structure is the same as the stifened plate structure on the upper section. Te T profle is arranged along the length direction of the cylindrical shell, and the L profle is arranged along the width direction of the stifened plate. Te fuid medium is air. Te reference value of the velocity level is 1 m 2 /s 2 , and the reference value of the sound power level is 10 − 12 w.
Te accuracy of the computation model must be confrmed. First, taking the natural frequencies of the cylindrical shell under rigid fxed boundary conditions as an example, all the stifness coefcients at the edges of the cylinder are infnite. In this example, the value is 10 15 . Te dimension proportion relation of the cylindrical shell is t s /R � 0.002, L s /R � 20, and the natural frequencies parameter is Ω � ωR ��������� � ρ(1 − v 2 )/E. Te calculation results are compared to those found in the published literature (Table 1). Te numerical results are basically consistent with those in the literature.
Next, under elastic boundary conditions, this model is used to validate the natural frequency of cylindrical shell structures. One side is rigid and fxed, and the other side is elastically supported. Only the radial stifness coefcient k r changes, and the other elastic stifness coefcients are set to zero, where k � k r /K and K is the internal rigidity of a cylindrical shell. Te cylindrical shell dimensions are L s � 1.25 m, R � 0.25 m, t s � 0.008 m, ρ � 7800 kg/m 3 , E � 2.1e11 N/m 2 , and ] � 0.3. Table 2 displays a comparison between the computation fndings and the literature. Te numerical outcomes are essentially in line with those reported in the literature.
Finally, the vibration frequency of the elastic coupled plate-cylindrical shell system is compared to verify the accuracy of the internal connection stifness. As an illustration, the elastic coupled plate-cylindrical shell structure is considered with simply supported constraints. Moreover, the coefcients of the elastic coupling stifness between a fat plate and a cylinder are k cx � k cy � k cz � k cr � 10 6 . Te coupling position is ψ � 115 o , with L s � L p � 1.27 m, t s � t p � 0.00508 m, ρ � 7500 kg/m 3 , E � 2.1e11 N/m 2 , and ] � 0.3. Table 3 presents the frst 10 natural frequencies, and it is evident that the numerical outcomes mostly agree with those reported in the literature.
Te stifened plate and the cylindrical shell in the coupling structure have the following measurements: L s � L p � 1 m, t p � t s � t e � 0.002 m, and R � 0.18 m. Te structural material is steel, with properties of ρ � 7800 kg/m 3 , E � 2.06e11 N/m 2 , ] � 0.3, and ψ � 110 o .
Te T-profle size is 2 × 15/2 × 6 (unit: mm), the L-profle size is 10 × 5 × 2 (unit: mm), the boundary stifness coefcients of the cylindrical shell are k sx � k sθ � k sw � k sr � 10 11 , and the connection stifness coefcients of the coupled structures are k cx � k cy � k cz � k cr � 10 11 . Figures 3(a)-3(f ) depict the frst six modes of the elastically coupled plate-cylindrical shell system (part of the structure is hidden for the convenience of observation, as shown below).
According to the amplitude ratio of each part of the coupling structure, the modes in the fgure can be roughly divided into two types. One mode is the stifness control of a single structure. In Figures 3(a), 3(b), and 3(e), the modal amplitude of the stifened plate is much larger than those of the cylindrical shell and the circular plate at both ends. In this instance, the modal amplitude of the stifened platecylindrical shell coupling structure is controlled by the stifened plate structure because the stifened plate has a larger side length. Compared to the cylinder and circular plate, its bending stifness is lower. Te stifened plate in this scenario receives boundary stifness from the circular plate and the cylindrical shells at each end, and the circular plate and the cylindrical shell at both ends possess enough rigidity to control the stifened plate's border movement.
In Figures 3(c) and 3(d), the modal amplitudes of the circular plates at both ends are much larger than those of the cylindrical shell and the stifened plates. Te modes of the coupling structure are controlled by the circular plates at both ends. Te others are the coupled modes of multiple structural stifness controls. Figure 3(f ) shows the coupled modes of the structure. Te amplitudes of each structural Shock and Vibration  Table 2: Natural frequencies for a cylindrical shell with the clamped-elastic condition.    Shock and Vibration mode in the coupled structure are equivalent or nonnegligible. Te infuence of the internal connection stifness of the elastic coupled stifened plate-cylindrical shell system on these two modes is further explained below. Only the elastic connection stifness is changed. It is assumed that the stifness coefcients in all directions increase in proportion from the free end to the rigid fxed end. Taking the antisymmetric coupling mode as an example, this mode is mainly the elastic coupling deformation between the stifened plate and the cylindrical shell, and the circular plate just slightly deforms at each end.

Modal orders
Te variation curve of modal frequencies as a function of the stifness coefcient is shown in Figure 4. Te fgure demonstrates that when the rigidity coefcient k is from 0 to 10 2 , the modal frequencies of this coupled structure change a little because the stifness coefcients in all directions are small at this time, the cylindrical shell and the stifened plate cannot be efectively connected, and the modes of the coupled structure are mainly controlled by the cylindrical shell. Stifened plates have insignifcant modal amplitudes, as shown in Figure 5(a).
When the stifness coefcient connection k ranges from 10 2 to 10 10 , the vibration frequencies increase rapidly with increasing stifness coefcients (Figure 4), the stifened plate has a greater rise in its modal amplitude, and the stifened plate and the cylindrical shell exhibit coupled vibration as a whole, as shown in Figures 5(b) and 5(c). At this stage, the coupling modes of the structure are sensitive to the stifness coefcient. At this stage, the infuence of the constraint structure dynamic performance has practical signifcance to the actual structure. For example, the actual structural connection constraints are often complicated. Moreover, the connection stifness between the ship deck and the side shell plate is neither freely supported nor rigidly fxed but between a free support and rigid fxation, and the constraint between them, which is often called an elastic constraint, can be considered as k � 10 8 (Figure 1).
When the stifness coefcient connection k ranges from 10 10 to 10 15 , the coupling modal frequencies of the structure are basically unchanged at this stage with the increase in the stifness coefcient ( Figure 4). Because the stifness coefcient is sufciently large at this time, the connection among the stifened plate, the circular plate, and the cylindrical shell at both ends is equivalent to rigid fxation, and the connection between each structure no longer has relative translation or rotation, which is equivalent to the welding situation in the actual structure. As shown in Figure 5(d), an organic overall structure has been formed.
Te other type of mode is the single structural mode; that is, with an increasing connection stifness, the vibration mode of the coupled system is always a single structural mode. Taking the frst mode as an example, as shown in Figure 6, the vibration mode of the coupled structure resembles that of the stifened plate with the increase in the connection stifness in each direction. When the connection stifness between the coupling structures is large, the vibration frequency of the structure is 159 Hz, which is signifcantly diferent from that of the stifened plates when the boundary stifness around the stifened plates is fxed (d) (e) (f ) Figure 3: (a-f ) Te frst six modes of the elastic coupled plate-cylindrical shell system. (289 Hz). Te reason is that the cylindrical shell and the round plates at both ends of the structure are not sufciently "rigidly fxed" around the stifened plates, and the boundary around the stifened plates can still translate and rotate in a limited range.
Ten, it is assumed that the point excitation force acts on the midpoint of the internal plate at the cylindrical shell, and when the elastic connection stifness of the coupling structure is between 10 5 and 10 10 . Figures 7 and 8 depict the mean square velocity (MSV) and radiated sound power   Shock and Vibration (RSP) curves, respectively. Te results show that the coupling stifness has a substantial impact on the acoustic and vibrational forces of the coupled structure. First, when the connection stifness coefcient increases, the number of resonant frequencies of the connected structure reduces as the connection stifness increases, and the self-vibration modes of each structure gradually become the coupled vibration modes of the structure. Second, when the connection stifness increases, the peaks of the curves obviously increase. Tis is because the impedance matching between the structures is improved with an increasing connection stifness, which is conducive to vibration energy propagation. Tird, the curves of the structure both move towards high frequencies. Tis is because the increase in the stifness of the connection leads to an increase in the frequency of the structure.
To explore the impact of the coupling spring stifness on the acoustic and vibration performance of a coupled structure separately, the stifness value of one coupling spring was modifed in turn, while the stifness of the other coupling spring was set to 1e5. Te MSV curve and the RSP curve of the elastic coupled structural system are shown in Figures 9 and 10. While the coupling stifness in the direction of k cy has a signifcant impact on the vibration and sound radiation of the coupled structure, the coupling stifnesses in the other directions have a much less impact on the structural system's ability to dampen vibration and sound. Tat is, the supporting coupling stifness between the elastic plate and the cylindrical shell structure plays a major role in the vibration energy transfer. In fact, the bending vibration transfer of the inner plate is what truly constitutes the vibration coupling transfer in the y direction of the structure. Terefore, in the actual structure, it is recommended that the support's strength and stifness be decreased to lessen the vibration sound radiation produced by the structure. One way to do this would be to install rubber gaskets.

Summary
In this research, we build a calculation model of a coupled plate-cylindrical shell system to investigate the efect of internal constraint stifness on this performance. Te following conclusions are the most important: (1) According to the amplitude ratio of each part of the coupled structures, the modes can be roughly divided into two types. One mode is single-structure stifness control, and the other type is the coupled mode of multiple-structure stifness control. (2) Te coupled modes are closely related to the connection stifness, while the single structural modes are basically independent of the connection stifness. (3) When the connection stifness increases, the MSP curve and the RSP curve of the coupled structure approach a high frequency and the resonance peak value increases, while the number of resonant frequencies decreases. (4) Te supporting coupling stifness between the elastic coupled structures plays an important role in the vibration energy transfer.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that there are no conficts of interest.