The Modeling Method for Vibration Characteristics Analysis of Composite-Laminated Rotationally Stiffened Plate

. Te composite-laminated rotationally stifened plate is widely applied in aviation, aerospace, ship, machinery, and other felds. For structural design and optimization, to investigate the vibration characteristics is important. In this paper, a modeling method of composite-laminated rotationally plate is established. Te frst-order shear deformation theory (FSDT) and the modifed Fourier series are applied to construct the admissible displacement function of the stifened plate-coupled systems. On this basis, the energy function of composite-laminated rotationally stifened plate is established. Combined with the artifcial virtual spring technology, the proposed theory could be used to analyze the vibration characteristics of composite-stifened plate-coupled systems with various classical boundary conditions or arbitrary elastic boundary conditions. Te Rayleigh–Ritz method is used to solve the energy function. Tus, the vibration characteristics of the composite-laminated rotationally stifened plate are obtained and analyzed. Te correctness of the theoretical analysis model was verifed through modal experiments. On this basis, the efect of some important parameters on the vibration characteristics of stifened plate structures is studied, such as the number, thickness, and width of the laminated stifener, varying structural parameters, and diferent boundary conditions. Tis study can provide the theoretical basis for the vibration


Introduction
Te composite-laminated rotationally stifened plate are basic structural element in aviation, aerospace, ship, machinery, and other felds, which makes it widely used in vehicle body, ship hull, and housing construction.Due to the special working environment, the plate structure is often subjected to complex dynamic loads, which causes mechanical vibration.In consequence, the vibration of composite-laminated rotationally stifened plate is inevitably generated in engineering practice.To reveal the vibration characteristics and reduce vibration and noise, it is necessary to intensively investigate the vibration characteristics of composite-laminated rotationally stifened plate.In recent years, rotary composite plate structures have been widely used in felds such as aerospace, shipbuilding, and ocean and have become an important direction for lightweight development.However, due to factors such as impurities in the environment and the imperfectness of the process during the preparation process of composite materials, defects such as pores, impurities, and fber curls will inevitably occur, which will afect the strength and stifness of the structure.Terefore, reinforcing stifeners are installed at appropriate positions of the plate structure through mechanical connections or adhesive bonding to efectively improve the overall strength and rigidity of the structure.Te addition of reinforcing stifeners results in discontinuous changes in material, mass, and damping parameters at the coupling interface between the plate and the reinforcing stifeners.Tis leads to complex changes in waveform conversion and energy loss of vibration waves at the coupling interface between the plate and the reinforcing stifeners.Terefore, it is of great signifcance to conduct theoretical modeling and vibration characteristics research on the composite-laminated rotationally stifened plate.Toshihiro et al. [1] studied the free vibration of an annular plate with radial stifeners arranged at equal angle intervals on two surfaces of the plate using the Ritz method.Te spline function was used as the allowable function for plate defection to calculate the natural frequency and mode of vibration of the plate, and the infuence of stifeners on them was analyzed.Jafari et al. [2] conducted free vibration analysis of annular plates with nonuniform eccentric stifeners and nonuniform spacing distribution stifeners.Te stifeners were treated as discrete elements using the Ritz method, and the efects of nonuniform eccentric distribution and nonuniform spacing distribution on natural frequencies were analyzed.Qin et al. [3] modeled circular stifened plates as a composite structure composed of circular plates and stifeners and proposed a meshless method for bending and free vibration analysis of circular stifened plates based on the frstorder shear deformation theory.Golmakani et al. [4,5], based on the frst-order shear deformation theory, adopted the nonlinear von Karman plate theory, combined it with the fnite element method, and studied the elastic large defection mechanical behaviour of axisymmetric composite-stifened circular and annular-laminated plates under transverse uniform load.Te reinforcing stifener is an orthotropic material with a circular shape.Te efects of plate thickness, stifener depth, thickness, and boundary conditions such as fxed support and simply supported on the nonlinear bending performance of reinforced circular and annular laminated plates were given.Ou et al. [6] used the weak form quadrature element method to analyze the nonlinear dynamic response of cylindrical composite stifened laminated plates with dome shapes.Te Newmark numerical integration method and the Newton Raphson iterative method are used to solve the nonlinear control equations.Examples including isotropicand composite-laminated plates were provided to verify the efectiveness and accuracy of the formulas.Te efects of geometric parameters, ply order, fxed, and simply supported boundary conditions on structural vibration were studied.
Peng et al. [7] proposed the meshless Galerkin method for geometrically nonlinear analysis of arbitrary polygon and circular stifened plates based on the frst-order shear deformation theory.Due to the absence of a grid in the model, reinforcing stifeners can be placed anywhere on the slab, and changing the position of the reinforcing stifeners does not require remeshing of the slab.Tis method explores the defection calculation results of polygonal and circular stifened plates under diferent boundary conditions, load forms, and reinforcement arrangement forms and proves the efectiveness of the calculation method through comparison with fnite element software.Bhimaraddi et al. [8], based on the classical plate theory, used the method of combining the annular sector plate element and the curved beam element to carry out the fnite element analysis of the orthogonal stifened annular sector plate and verifed the efectiveness of this method.Nagesh et al. [9] provided a detailed description of the noncoupled damage model based on the numerical model and used fnite element analysis to evaluate the large deformation and ductile fracture failure of laterally stifened circular plates with fxed boundary conditions under uniform pulse loads.Calin Itu et al. [10] proposed a method to improve the stifness of composite circular plates by installing radial stifeners and studied the infuence of diferent materials, thicknesses, and arrangement of stifeners on the stifness of composite circular plates under general boundary conditions using the fnite element method.Turvey et al. [11] studied the elastic-plastic large defection response of a single radial stifened circular plate by establishing an Ilyushin full-section yield model for plates and an improved Von Mises yield model for stifeners, combined with the fnite diference mesh method.Te efects of the depth of stifener, boundary conditions for simple and fxed supports, and thickness of the plate on the elastic-plastic large defection parameters are given.
In summary, although domestic and international scholars have conducted extensive research on the vibration characteristics of rotationally stifened plate and shell structures, there are relatively a few research objects focused on composite materials, especially if the reinforcing stifeners are also made of composite materials.We will only discuss a specifc metal plate structural form, and when it comes to other structural forms, we will also need to carry out tedious modeling work.Moreover, the boundary conditions are relatively single, and there is relatively little research on complex elastic boundaries.Terefore, establishing a unifed analysis model for the vibration characteristics of composite-stifened plate and shell structures with complex boundaries is of great signifcance.
In this article, a unifed analysis model for the vibration characteristics of the composite-laminated rotationally stifened plate structure is established through the frst-order shear deformation theory (FSDT) and the modifed Fourier series method [12][13][14]; specifcally, we establish a unifed analysis model of composite-laminated rotationally stifened plate structure under elastic boundary conditions.Te structure is coupled with stifened plate and stifened beam.Combined with the modifed Fourier series method and Rayleigh-Ritz method, the vibration characteristic of the model can be solved.After comparison and verifcation with the fnite element method and experiment, the efect of some important parameters on the vibration characteristics of composite-laminated rotationally stifened plate structure is studied, such as the number, thickness, and width of the stifener, varying material parameters, and diferent boundary conditions.

Establishment of a Unified Analysis Model of
Composite-Laminated Rotationally Stiffened Plate Structure the structure is an annular stifened composite plate; (2) when 0 < R 1 < R 2 , 0 < ϑ < 360 °, the structure is an annular fan-shaped stifened composite plate; (3) when R 1 � 0, ϑ � 360 °, the structure is a circular stifened composite plate; and (4) when R 1 � 0, 0 < ϑ < 360 °, the structure is fan-shaped stifened composite plate.As shown in Figure 2(a), at the edge of the compositelaminated rotationally stifened plate, general boundary conditions are defned by introducing three groups of linear springs k u , k v , k w and two groups of torsion springs K r , K θ along the u, v, w directions, respectively [15][16][17], continuous distribution of spring groups along the boundary.k u θ0 , k v θ0 , k w θ0 , K r θ0 , and K θ θ0 represent fve sets of boundary springs at the boundary θ � 0 °.Te same can be said for the boundary of the θ � ϑ, r � 0, and r � R p spring and can be represented by this method.For sector-stifened composite plates, the stifness of the boundary spring at the boundary r � 0 is set to 0; for annular stifened composite plates, the stifness of the boundary spring at the boundary θ � 0 °and θ � 360 °is set to 0; and for circular stifened composite plate, the stifness of the boundary spring at the boundary r � 0, θ � 0 °and θ � 360 °is set to 0. When rotating angle ϑ � 360 °, laminated plates (the nth laminated beam) in the composite-laminated rotationally stifened plate will produce the coupling boundary as shown in Figure 2 θc ) are uniformly arranged on the coupling boundary to realize the coupling of the composite-laminated rotationally stifened plate.Figure 2(d) shows the uniformly arranged coupling springs between the laminated plate and laminated curved beam in composite-laminated rotationally stifened plate, including three sets of linear coupling springs, i.e., k cp uc , k cp vc , and k cp wc and two sets of torsional springs, i.e., K cp xc and K cp yc .

Construction of Admissible Displacement Functions.
According to the frst-order shear deformation theory (FSDT), the displacement components U p , V p , W p and U bn , V bn , W bn at any point on the laminated plate and laminated curved beam of the composite-laminated rotationally stifened plate structure can be expressed as follows: where u p , v p , and w p represent the displacement of the middle plane in the r, θ, and z directions of the compositelaminated rotationally stifened plate, ϕ rp and ϕ θp indicate its lateral rotation in θ and r direction; u bn , v bn , and w bn indicate the displacement of the middle plane of the nth laminated curved beam in x n , θ n and z n directions, and ϕ xbn and ϕ θbn indicate its lateral rotation in θ n and x n direction.t represents a time variable.
Based on the improved Fourier series method, the admissible displacement functions of laminated plates in composite-laminated rotationally stifened plates are established.Te admissible displacement functions established in this method can ignore the infuence of the boundary conditions, and the auxiliary polynomial is introduced to eliminate the discontinuity and jump phenomenon of the displacement function at the boundary.Te specifc expression is By using the same theoretical method, the admissible displacement functions of the laminated curved beam in composite-laminated rotationally stifened plate can also be established as follows: Te displacement supplement polynomial of compositelaminated plate can be expressed as Φ M and Φ Nq (N q � 1, 2).Te displacement supplement polynomial of the Shock and Vibration 3 nth laminated curved beam can be expressed as Ψ Ω n and Ψ Θq n (Θ q � 1, 2).Professor Li [18,19] proposed the modifed Fourier series method, which expressed the displacement function of beam or plate structure in the form of traditional Fourier series plus the auxiliary function.It must be noted here that the main function of the auxiliary function is to enable the traditional Fourier series and the corresponding derivatives to avoid the occurrence of nonexistent or jumping phenomena at the boundary position.At present, the commonly used auxiliary function forms are Legendre polynomial, series polynomial, and complete series.Te auxiliary function used in this paper is the complete series method.Where A mn , B mn , C mn , D mn and E mn represent the unknown two-dimensional Fourier coefcient vectors of the admissible displacement functions of the laminated plate, these parameters can be expressed as follows: igure 1: Unifed model of composite ring fan-shaped, annular, fan-shaped, and circular stifened plates.

􏼨 􏼩
T , where A l , B l , C l , D l , and E l represent the unknown onedimensional Fourier coefcient vector of the admissible displacement functions of the laminated curved beam, which can be expressed as follows: where λ α m � mπ/α, λ β n � nπ/β, and λ α n l � lπ/α n .
Shock and Vibration 5

Stress-Strain and Displacement
Relations.According to the relevant knowledge of elastic mechanics, the normal strain and shear strain at any point on the composite-laminated plate and laminated curved beams can be defned by strain and curvature change of the middle surface as follows: in which ε According to Hooke's law, the corresponding stressstrain relationship of laminated plate and laminated curved beam at the k-th layer can be obtained as follows: where ) is the relevant stifness coefcient, which can be obtained from the following equation: In (15), T is the transformation matrix, which is defned as follows, where θ k is the included angle between the main direction and the r direction (the laminated curved beam is the x n direction), that is, the angle-ply.
) represents the material coefcient of the k layer of the laminated plate (laminated curved beam), which value can be obtained according to the engineering constant of the k layer of the laminated plate (laminated curved beam) as follows: Te forces and moments experienced by laminated plate and laminated curved beam are obtained by integrating the stresses on the plane.From one layer of laminated plate and laminated curved beam to the other layer, by integrating the thickness, we can obtain  x and Q bn θ are the resultant force of the horizontal shear force in the plane of the laminated curved beam.κ s is the shear correction coefcient [20], and N L represents the number of layers of laminated plates or laminated curved beams.Z k is the thickness coordinate value of the bottom surface of the k layer, and Z k+1 is the thickness coordinate value of the upper surface.

Energy Functions and the Solving Process.
According to the principle of energy, the energy function of the rotational stifened composite plate can be listed, and the unknown coefcient can be solved by the Rayleigh-Ritz method.Te Lagrange equation of the stifened plate can be expressed as follows: Shock and Vibration where T P and T B n represent the total kinetic energy of the laminated plate and the nth laminated curved beam of the rotational stifened plate, U P and U B n are the total potential energy of the laminated plate and the nth laminated curved beam, U P−coupling and U B n −coupling represent the coupled potential of laminated plate and nth laminated beam when ϑ � 360 °, U SP represents the potential energy of boundary spring of the laminated plate, W P&B n is the coupled potential generated when the laminated plate and the nth laminated curved beam are coupled, and W F represents the work done by simple harmonic point force F on rotational stifened composite plate.Te total kinetic energy of laminated composite plates T P and the total kinetic energy of the nth laminated beam T B n can be written as follows: in which ρ k p is the material density of the k layer of the laminated plate, and ρ k bn represents the material density of the k layer of the laminated curved beam.
Te specifc expression of the total potential energy of laminated composite plates U P and the total potential energy of nth laminated curved beam U B n are as follows: 8 Shock and Vibration In the equation, the total potential energy of the composite-laminated rotationally stifened plate U P includes tensile potential energy U stretch , bowing potential energy U bend , tensile and bowing potential energy coupling U s−b , and the expressions are also been given.
As the boundary conditions of the laminated plate in the composite-laminated rotationally stifened plate model established in this paper are determined by the boundary spring set, the potential energy of the boundary spring U SP will be generated, and its specifc expression is as follows: For composite-laminated rotationally stifened plate, the analysis of the coupled potential U P−coupling and U B n −coupling of the laminated plate and the laminated beam will be diferent due to the diferent sizes of the rotation angle ϑ; when 0 < ϑ < 360 °, since the left and right sides of the stifened plate rotation are not coupled, there is no coupled potential of the laminated plate and laminated beam, and when ϑ � 360 °, the left and right sides of the stifened plate rotation are coupled, producing coupled potential at this time.
When ϑ � 360 °, the coupled potential of the laminated plate in the stifened plate U P−coupling and the coupled potential of the nth laminated curved beam in the stifened plate U B n −coupling can be written as follows: 10 Shock and Vibration Te coupled potential generated by the coupling of the laminated plate and the nth laminated curved beam W P&B n can be expressed as follows: Te specifc expression of the work W F done by the simple harmonic point force F on the laminated plate of stifened composite plate is as follows: where f i (i � u j , v j , w j , ϕ r , ϕ θ ) is a function of external load distribution, and the location of simple harmonic point force F is (r 0 , θ 0 ).After obtaining the energy equation of the compositelaminated rotationally stifened plate, each energy equation is substituted into the Lagrange equation ( 21) and ( 22).According to the Rayleigh-Ritz method [21] in which P mn is the unknown two-dimensional Fourier coefcient matrix of the laminated plate, and Q l is the unknown one-dimensional Fourier coefcient matrix of the laminated curved beam.

Shock and Vibration
Convert equations ( 40)-( 43) into the matrix form as follows: where K P and K B n , respectively, represent the stifness matrix of the laminated plate and the nth laminated curved beam in the composite laminated rotationally stifened plate; M P and M B n are the mass matrix of the laminated plate and the nth laminated curved beam; C B n &P represents the coupling matrix between the nth laminated curved beam and the laminated plate of the stifened plate, and C P&B n � C B n &P T .When F is 0, the equations for solving the natural frequencies and modes of composite laminated rotationally stifened plates can be obtained by combining equations ( 44) and (45).At this time, direct calculation belongs to the nonlinear solution, so it needs to be converted into linear equations to be solved.Te results after conversion are as follows: Finally obtained characteristic solution ω is the natural frequency of the composite-laminated rotationally stifened plate, and the eigenvector G is its corresponding mode.By substituting the simple harmonic point force into equations ( 46)-(48), the steady-state response of the stifened plate can be obtained.

Numerical Results and Discussion
According to the unifed analysis model of compositelaminated rotationally stifened plates established in Section 2.2, this section has carried out numerical discussion and result analysis on the selected part of the calculation to further study the vibration characteristics of stifened annular sector plates, circular sector plates, annular plates, and circular plates under diferent dimensions and material parameters, mainly including the following two parts: (1) convergence and correctness verifcation and (2) the infuence of related parameters on the vibration characteristics of composite-laminated rotationally stifened plates under free vibration.Due to the lack of research on compositelaminated rotationally stifened plates, few literature parameters can be found for comparison, so the results of the calculation of this section are compared with the results of the experiment and fnite element simulation results.
Table 1 shows the material parameters used for laminated plates and laminated curved beams in this section.Te dimensionless natural frequency parameter is defned as . Te four boundary conditions of free, simply supported, fxed supported, and elastic in this section are simplifed and described as F, S, C, and E, respectively.According to the diference of stifened annular sector plate, circular sector plate, annular plate, and circular plate, there are four ways of expression: (1) stifened annular sector plate structure: FSCE indicates θ � 0, θ � ϑ, r � 0, and r � R p are free, simply supported, fxed supported, and elastic boundaries, respectively; (2) stifened circular sector plate structure: FSC indicates θ � 0, θ � ϑ, and r � R p are free, simply supported, and fxed supported boundary, respectively; (3) stifened annular plate structure: FS indicates r � 0 and r � R p are free and simply supported boundary, respectively; and (4) stifened circular plate structure: S indicates r � R p is a simply supported boundary.

Model
Validation.Tis section will verify the convergence and correctness of the model of composite-laminated rotationally stifened plate established in Section 2.2, including the convergence of natural frequency cutof value and spring stifness value.As mentioned in the previous chapters, the admissible displacement function of stifened plates in this paper is expressed as improved Fourier series expansion.Terefore, in the process of solving the energy equation including the admissible displacement function, the calculation results may not converge, so it is necessary to verify the convergence of the established analysis model.At the same time, the number of expanded items in the expansion also needs to be analyzed.Teoretically, the more terms the expansion term has, the higher the accuracy of the solution of the equation will be.However, the number of expansion terms can meet the requirement of solving accuracy.Further increasing the number of expansion terms will not signifcantly improve the solving accuracy but reduce the solving efciency.Terefore, it is necessary to confrm the cutof value of the admissible displacement function of the stifened plate.Te cutof value of the laminated plate is M p , N p , and the cutof value of the laminated curved beam is M b .
Table 2 shows the frst eight natural frequencies of stifened plates under diferent cutof values of laminated plates and laminated curved beams obtained by this method and compares them with the fnite element simulation results.Te stifened plate model is a unifed model, which can theoretically calculate the natural frequency of any size of rotational stifened composite plate; in this example, the geometric parameters of composite-laminated rotationally stifened plates have been determined as follows: R 2 � 2 m, b 1 � 0.06 m, and h 1 � 0.04 m.Te material of the laminated plate is graphite fber resin, and the angle-ply is [0 °/90 °].Te number of laminated plated curved beams is 1, located at R p /2, and the material is steel.It is not difcult to see from Table 2 that when M p × N p is 18 × 18 and M b is 50, the natural frequencies of each order basically converged.Te size and coordinate system used in the fnite element model 12 Shock and Vibration are the same as that of the model established in this paper, the mesh size is 0.02 of the global size, and the element shape of the sound cavity is AC3D20:20 node acoustic quadric hexahedron element.Te shape of the composite structural element is C3D20R: twenty-node hexahedron element.By referring to the results of fnite element simulation, the maximum error between the natural frequencies of each order and the results of fnite element simulation is 6.38%.Terefore, the cutof value is determined as M p × N p � 18 × 18 and M b � 50 in the numerical calculation later in this paper.It is not difcult to see from Table 2 that the cutof value of the laminated curved beam has little infuence on the convergence of the analytical model.Terefore, Figure 3 only shows the change curve of some order natural frequency parameter Ω of the composite-laminated rotationally stifened plate under diferent cutof values of the laminated plate (where M p � N p ).As shown in Figure 3, the natural frequency parameter Ω of the four stifened plate structures tends to be stable with the increase of the cutof value, which more intuitively refects the convergence of the present analysis model.
To have a more intuitive understanding of the modes obtained by this method, Figure 4 shows the modal shapes diagram of four types of composite-laminated rotationally stifened plate structures obtained by this method and the fnite element method (FEM).Te material parameters and dimension parameters of the four stifened plate structures are the same as those in Table 2.It can be found that the modal diagrams obtained by the present method are completely corresponding to the results of the fnite element method.
In this chapter, artifcial virtual spring technology is used to simulate the boundary conditions and coupling conditions of the model.Te change of the boundary and coupling conditions of the composite-laminated rotationally stifened plate is realized by changing the boundary and coupling spring stifness value.For boundary or coupling conditions that require rigid fxation, the spring stifness value should be set to a larger value.However, due to the limitation of the algorithm, the spring stifness value cannot be infnite in actual calculation.Terefore, to facilitate the follow-up research, it is necessary to select a reasonable spring stifness value to simulate the rigidly fxed boundary or coupling conditions and verify its convergence.Since the stifener in the composite-laminated rotationally stifened plate does not need to set the boundary conditions, the efect of boundary spring stifness value can be analyzed only for the structure of the rotationally composite plate.Te specifc scope includes linear spring k(k u , k v , k w ), torsion spring ).Taking the annular sector plate as an example, the infuence of the boundary spring stifness value is analyzed.Te change curve of the frst four frequency parameters under diferent boundary spring stifness values is shown in Figure 5. Te material parameters and size parameters of the plate structure are the same as those in Table 2.It is not difcult to see from Figure 5(a) that the stifness value k of the linear boundary spring is in the range of 10 −4 -10 1 , and the values of the frst four frequency parameters are stable and close to 0. It can be considered that the boundary condition of the annular sector plate can be regarded as a free boundary at this time.When the linear spring stifness value k is analyzed, the torsion spring stifness value K is automatically set to zero.With the increase of the linear spring stifness, the frequency parameters of the annular sector plate also continue to increase and tend to be stable in the range of 10 10 -10 16 .At this time, the boundary condition of the annular sector plate can be regarded as a simply supported boundary.In Figure 5(b), the linear boundary spring stifness value is always kept at 10 16 .With the increase of the torsion spring stifness value, the frequency parameter of the annular sector plate also continues to increase and tends to be stable in the range of 10 7 -10 16 .At this time, the boundary condition of the annular sector plate can be regarded as a fxed support boundary.According to the abovementioned conclusions, the value of boundary spring stifness under diferent boundary conditions is shown in Table 3, the classical boundary conditions of free, simply supported, clamped supported are simplifed as F, S, C respectively.Te spring stifness value can be given arbitrarily under the elastic boundary condition.Tis paper gives an example which is marked with E. In the numerical calculation in this chapter, the spring stifness values of various boundary conditions are based on those in Table 3. Te numerical calculation of the spring stifness values of various boundary conditions in this paper is based on Table 3.
Te study of the coupling condition of the interior of the plate is similar to the boundary conditions.Taking the annular plate as an example, Figure 6 shows the frequency parameter change curve of the order of the annular plate with diferent coupling spring stifness values of the interior of the plate under the two boundary conditions.Te material parameters and dimension parameters of the annular plate are the same as those in Table 2.It can be seen from Figure 6 that when the stifness value kc of the coupling spring in the plate is less than 10 4 , the frequency parameters of the corresponding part of the order remain stable and less than

Shock and Vibration 13
the frequency parameters when the stifness value kc is larger; therefore, it can be considered that the coupling spring within this range has limited infuence on the frequency parameters of the annular plate structure.With the increase of the stifness value kc of the coupling spring in the plate, the frequency parameter of the annular plate also increases.When the stifness value kc increases to 10 8 , the frequency parameters of the annular plate also tend to be stable.From the abovementioned phenomena, it can be seen that when the stifness value of the coupling spring in the plate reaches 10 8 , the rigid coupling of the coupling boundary of the rotary composite plate can be realized.To ensure the correctness of the calculation results, the stifness value kc of the coupled spring in the plate was set as 10 16 in the subsequent example.
To study the coupling conditions of plate and beam, it is necessary to take the rotational stifened composite plate as an example.Tis example selects the composite-stifened annular sector plate with the same material parameters and size parameters as those in Table 2. Figure 7 shows the frequency parameter change curve of the partial order of the stifness value of the coupling spring of diferent plates and beams under two boundary conditions.It is not difcult to see that the efect of the plate-beam coupling spring stifness value kcp on the frequency parameter Ω is similar to the interior of a plate coupling spring stifness value kc.When the stifness value kcp is less than 10 4 , the plate-beam coupling spring hardly plays the role of coupling between plates and beams in this range and has limited infuence on the frequency parameter Ω.When the stifness value kcp increases to 10 10 , the rigid coupling of the coupling boundary between the rotary composite plate and the laminated beam can be realized.In this section, the frequency response function method is used to carry out the structural modal test of a stifened circular plate with a fxed support boundary, and the test results are compared with the results of this method to verify the accuracy of the model of the stifened composite plate of revolution established by this method.Te test instruments include the LC02 force hammer, 3A105 force sensor, DH5857-1 charge adjuster, 1A116E acceleration sensor, and DH5922D dynamic signal test and analysis system.Te single point pickup method is adopted when collecting data.Te position of the sensor remains unchanged, and a force hammer is used to knock the intersection point of the grid drawn before.After hitting all the test points set, the peak position of the frequency response curve drawn is determined in the test software according to the collected data,

Shock and Vibration
Figure 9 shows the layout of the stifened circular plate structure and test equipment during the real test.Before the test, the stifened circular plate structure was divided into units.Te specifc division method was as follows: 4 equal divisions were conducted in the r direction and 12 equal divisions in the θ direction.Tere were 49 measuring points in total, and the acceleration sensor was located at 22 measuring points.Figure 10 shows the natural frequencies and modal shapes of the partial orders of stifened circular plates of Q235 steel and carbon fber composite material obtained from the test and the method in this paper.From the contents of Figure 10, when stifened circular plates are made of Q235 steel, it is not difcult to fnd that the maximum error between the test results and the calculation results of this method is 7.56%, and when stifened circular plates are made of carbon fber composite material, it is not difcult to fnd that the maximum error between the test results and the calculation results of this method is 3.18% while the experimental results correspond with the modal shapes obtained by this method, Te deviation of the abovementioned experimental results is within the acceptable range, which fully proves the correctness of the analysis model.Te experimental error is caused by many reasons.First of all, the fxed boundary conditions of the plate cannot be fully simulated by the way of clamping the foundation frame and the battens.Ten, the material parameters used in the numerical calculation of the plate deviate from the actual material parameters of the work piece, and the work piece cannot be completely ideal isotropic material.In addition, the accuracy deviation of the force sensor and acceleration sensor and the human error of the experimenter in the process of hammering will cause the error of experimental data.In the theoretical modeling in the previous section, it has been mentioned that the composite-laminated rotationally stifened plate is composed of the rotary composite plate structure and the rotary composite-laminated curved beam in the theoretical modeling in the previous section.Te infuence of the parameters of the rotationally composite plate structure on the vibration characteristics of the stifened plate structure is analyzed here.Te frequency parameters of composite-laminated rotationally stifened plates with diferent plate structure parameters are obtained by the method in this paper.Taking the stifened circular sector plate as an example, the calculation results are shown in Table 4. Te stifened plate in this example has two laminated curved beams as stifeners, of which stifener 1 is at R p /2 and stifener 2 is at R p /3. Te material of the laminated plate and laminated curved beam is glass epoxy resin, and the layer angle is the same.Te fxed geometric parameters in the calculation example are R 1 � 0 m, R 2 � 1.5 m, h p � 0.02 m, R b1 � 0.75 m, R b2 � 0.5 m, b 1 � b 2 � 0.08 m, and h 1 � h 2 � 0.05 m.It can be seen from Table 4 that the boundary conditions, angle-ply, and rotation angle all have an impact on the frequency parameter Ω of composite-laminated rotationally stifened plates.
To further analyze the infuence of boundary and parameter conditions on the vibration characteristics of composite-laminated rotationally stifened plates, the parametric study of relevant conditions is also carried out in the form of curves.Take the stifened circular sector plate in Table 4 as an example, Figure 11 shows the changing curve of the frst four frequency parameters Ω with rotation angle ϑ under diferent boundary conditions ϑ, which refects the infuence of boundary conditions and rotation angle ϑ on the frequency parameter Ω. Te angle-ply of the stifened composite circular sector plate selected in the calculation example is [90 °/0 °/90 °].It can be seen from Figure 11 that the frequency parameter Ω of stifened composite circular sector-stifened plate decreases with the increase of rotation angle ϑ, and the larger the rotation angle ϑ is, the smaller the decreasing slope of the frequency parameter Ω will be.At the same time, according to Figure 11, the rule of frequency parameter Ω of the same modal order under diferent boundary conditions can be obtained as follows: CCC > CCF > SSF > EEF.Since the spring stifness value of the fxed support boundary (C) > the spring stifness value of the simply supported boundary (S) > the spring stifness value of the elastic boundary (E) > the spring stifness value of the free boundary (F), it can be deduced that the natural frequency of the coupling system increases with the increase of the spring stifness value.
Te infuence of the parameter conditions of composite-laminated curved beams on the vibration characteristics of composite-laminated rotationally stifened plates also needs to be studied.Table 5 takes the stifened annular sector and annular plate as examples and gives the frequency parameter Ω of the frst eight orders for two kinds of composite-laminated rotationally stifened plates with diferent numbers and sizes of stifeners.Te maximum number of stifeners n is 3, where stifener 1 is at R p /2, stifener 2 is at R p /3, and stifener 3 is at 2R p /3. Te geometric parameters of the two kinds of compositelaminated rotationally stifened plates in the calculation example  5 that compared with the plate structure without stifeners, when the number of stifeners n is 1, the frequency parameter Ω of most orders of stifened annular sector plates and stifened annular plates tends to increase, and the degree of increase is related to the size parameter of stifeners, while the frequency parameter Ω will decrease with the increase of the number of stifeners n.It can be seen from Table 5 that compared with the plate structure without stifeners, the frequency

Shock and Vibration
parameter Ω of most orders of stifened annular sector plates and stifened annular plates tends to increase when the number of stifeners n is 1, and the degree of increase is related to the size parameter of stifeners, while the frequency parameter Ω will decrease with the increase of the number of stifeners n.
Figure 12 shows the modal shapes corresponding to the frst two frequency parameters of the stifened annular sector plate under diferent numbers of stifeners, which more intuitively refects the infuence of stifeners on the structure of the rotational plate.Te size and material parameters of the stifened annular sector plate are the same as those in

Shock and Vibration
Table 5; the width and thickness of the stifener are b It can be seen from Figure 12 that the vibration modal diagram at the location of the stifener is signifcantly curved, which indicates that the laminated plate and laminated curved beam structure in the model of composite laminated rotationally stifened plate are coupled.
Taking the stifened composite circular plate as an example, the variation curve of the frequency parameter Ω with the thickness h n of the laminated curved beam under the same thickness-to-width ratio is shown in Figure 13.Te number of stifeners n is 2, stifener 1 is at R p /3, and stifener 2 is at 2R p /3. Te constant geometric parameters of stifened circular plate in the calculation example are:\ R 1 � 0 m,  According to the analysis in Figure 13, the frequency parameter Ω of the same order increases with the increase of the thickness of the laminated curved beam h n under the same thickness-to-width ratio.Te variation curve of the frequency parameter Ω with the thickness h n of the laminated curved beam under the diferent thickness-towidth ratio is shown in Figure 14.Te geometric and material parameters in Figure 14 are the same as those in Figure 13, Te mode is the fourth order.It can be seen from Figure 14, the frequency parameter Ω increases with the increase of the thickness-to-width ratio; it means that as the width of the laminated curved beam increases, the frequency parameter Ω increases.To sum up, the thickness and width of the laminated curved beam are positively correlated with the frequency parameter Ω of the composite-laminated rotationally stifened plate.Continuing with the discussion on the efect of material parameters of the stifeners on the natural frequencies of composite-laminated rotationally stifened plates, the efect of

24
Shock and Vibration the layer angle of the stifeners on the frequencies is investigated.Figure 15 illustrates the efect of varying layer angle of the stifeners under diferent boundary conditions and two diferent layer schemes on the natural frequencies of composite-laminated rotationally stifened plates.In this particular case, the geometric parameters of the compositelaminated rotationally stifened plate are as follows: R 0 � 1 m, R 1 � 2 m, b 1 � 0.06 m, and h 1 � 0.04 m, with a rotation angle of 90 °.Te materials of the laminated plate and the laminated beam (stifener) are glass epoxy resin with the following material parameters: E 1 � 185 GPa, E 2 � 10.9 GPa, G 1 � G 2 � G 3 � 7.3 GPa, μ � 0.28, and ρ p � 1600 kg/m 3 .Te number of laminated beams is 1, located at R p /2. From Figure 15, it can be observed that under both layer schemes, for the SSSF and CCCF boundary conditions, the variation in the layer angle of the stifener exhibits a similar trend in afecting the natural frequencies of the composite laminated rotationally stifened plates.Within the range of 0-90 °, the natural frequencies generally increase with an increase in the layer angle before decreasing.However, for the EEEF boundary condition, under the [0/α °/0] layer scheme, except for the second order, the other natural frequencies remain relatively constant within the 0-90 °range.Under the [0/α °/0/α °] layer scheme, except for the second order, the overall trend of the other natural frequencies within the 0-90 °range is a decrease with an increase in the layer angle.Tis discrepancy is mainly due to the diferent infuences of the boundary conditions on the stifness of the structure.
Next, the efect of the anisotropy of the stifener on the frequencies of composite-laminated rotationally stifened plate is investigated.Figure 16 illustrates the efect of varying anisotropy of the stifener on the natural frequencies of composite-laminated rotationally stifened plates under diferent boundary conditions.Te anisotropy of the composite material is defned as E 1 /E 2 .In this example, the geometric parameters of the composite-laminated rotationally stifened plate are R 0 � 1 m, R 1 � 3 m, b 1 � 0.06 m, and h 1 � 0.04 m, with a rotation angle of 180 °.Te material for both the laminated plate and the laminated beam (stifener) is glass epoxy resin, with material parameters remaining unchanged except for E 1 , which varies as a parameter, consistent with Figure 15.Te number of laminated beams is 1, located at R p /4, and the layer scheme of the stifener is [0/90 °/0/90 °].From Figure 16, it can be observed that under diferent boundary conditions, within the range of 0-100, the natural frequencies of the composite-laminated rotationally stifened plate increase with an increase in the anisotropy of the stifener material.
Te stifened plate model established in this paper can not only analyze the stifened composite plate structure but also study the vibration characteristics of isotropic stifened plate structure by changing the material parameter settings.Table 6 shows the frst 8-order frequency parameter Ω of the rotationally isotropic stifened plate under diferent boundary conditions and compares it with the results of the fnite element method.Te number of stifeners n is 1, and the stifener is located at R p /2. Te invariant geometric parameters of the rotationally stifened plate in the calculation example are R 2 � 1.6 m, h p � 0.03 m, b 1 � 0.06 m, and h 1 � 0.04 m.Te plate structure and stifener material are set as isotropic material steel.As shown in Table 6, the results obtained by this method and the fnite element method are relatively close, and the error is less than 5%.
In the process of free vibration analysis, it can be found that compared with the fnite element method, the present method does not need to establish a new model when calculating examples with diferent material and size parameters but only needs to change the relevant parameters in (b) (Figure 2(c)) at the two edges of θ � 0 °and θ � 360 °.Tree sets of linear coupled springs k p uc , k p vc , k p wc (k b n uc , k b n vc , and k b n wc ) and two sets of torsional springs K p rc ,

Figure 2 :
Figure 2: Boundary spring and coupling spring of composite-laminated rotationally stifened plate.(a) Laminated plate boundary spring.(b) Laminated plate coupling spring.(c) Laminated curved beam coupling spring.(d) Coupling springs of plates and beams.
strain component on the middle plane of laminated plates, χ p r , χ p θ , and χ p rθ represent the component of curvature change on the middle plane of laminated plates, ε bn0 θ , c bn0 θx , c bn0 θz , and c bn0 xz represent the strain component on the middle plane of the nth laminated curved beam, and χ bn θ and χ bn θx represent the component of curvature change on the middle plane of a laminated curved beam.Te specifc expression is in which N p r , N p θ , and N p rθ represent the resultant force in the plane of the laminated plate, M p r , M p θ , and M p rθ represent bending and torsional moments in the plane of the laminated plate, and Q p θ and Q p r are the resultant force of the horizontal shear force of the laminated plate.N bn θ and N bn θx represent the resultant force in the plane of the laminated curved beam, M bn θ and M bn θx represent the bending and torsional moments in the plane of the laminated curved beam, and Q bn

Figure 7 :
Figure 7: Variation curve of frequency parameters Ω of composite-stifened annular sector plate under diferent values of plate-beam coupling spring stifness.(a) Boundary condition CCCC.(b) Boundary condition SSSS.
8 m, and h p � 0.035 m.Te plate structure is set as glass fber resin, the stifener material is set as graphite fber resin, and the angle-ply is set as [−45 °/0 °/45 °].It can be seen from Table

Figure 9 :
Figure 9: Layout plan of stifened circular sector plate structure and test equipment.

Figure 10 :
Figure 10: Natural frequencies and modal shapes of stifened circular plate obtained by test and this method.

Figure 11 :
Figure 11: Te change curve of frequency parameter Ω varies with rotation angle ϑ of stifened composite circular sector plate under diferent boundary conditions.(a) First order.(b) Second order.(c) Tird order.(d) Fourth order.

Figure 13 : 3 .Figure 14 :
Figure 13: Variation curve of frequency parameter Ω with the thickness of laminated curved beam for composite stifened circular plate under the same thickness to width ratio.(a) b n /h n � 0.5, (b) b n /h n � 1, (c) b n /h n � 2, and (d) b n /h n � 3.
Shock and Vibration stifened plate in the stifened plates are R 1 and R 2 , where R p � R 2 − R 1 and the thickness is h p .Te curvature radius of the laminated curved beam is R bn , width is b n , and thickness is h n .Te rotation angle of the entire stifened plate is ϑ.According to the diferent values of geometric parameters, there are the following models:(1) when annular plate, and circular plate.Te specifc model description is shown in Figure1.Te coordinates of the laminated plate are located in the coordinate system (o-z, θ, r) as shown in the fgure.Te coordinates of the nth laminated curved beam are located in the coordinate system (o n − z n , θ n , x n ) as shown in the fgure.Te inner radius and outer radius of the composite-laminated rotationally 2

Table 1 :
Material parameters used in the numerical calculations in this section for laminated plates and laminated curved beams.

Table 4 :
Frequency parameter Ω of stifened composite circular sector plate under diferent boundary and parameter conditions.