Beam Approximation for Dynamic Analysis of Launch Vehicles Modelled as Stiffened Cylindrical Shells

A beam approximation method for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells is proposed. Firstly, an initial beam model of the stiffened cylindrical shell is established based on the cross-sectional area equivalence principle that represents the shell skin and its longitudinal ribs as a beam with annular cross-section, and the circumferential ribs as lumped masses at the nodes of the beam elements. Then, a fine finite element model (FE model) of the stiffened cylindrical shell is constructed and a modal analysis is carried out. Finally, the initial beam model is improved through model updating against the natural frequencies and mode shapes of the fine FE model of the shell. To facilitate the comparison between the mode shapes of the fine FE model of the stiffened shell and the equivalent beam model, a weighted nodal displacement coupling relationship is introduced. To prevent the design parameters used in model updating from converging to incorrect values, a pre-model updating procedure is added before the proper model updating. The results of two examples demonstrate that the beam approximation method presented in this paper can build equivalent beam models of stiffened cylindrical shells which can reflect the global longitudinal, lateral and torsional vibration characteristics very well in terms of the natural frequencies.


Introduction
A stiffened cylindrical shell consists of skins and ribs with high specific stiffness and specific strength and is widely used as the main load-bearing body of a launch vehicle [Davila, Bisagni and Rose (2015); Hao, Wang, Tian et al. (2016) ;Hilburger, Lindell, Waters et al. (2017); Wang, Tian, Hao et al. (2016)]. To minimise the scale of computational models in load calculation, control, dynamic analysis, stability analysis, etc., launch vehicles are often reduced to equivalent simple models [Grimes, Mc Tigue, Riley et al. (1970) ;Nurre, Ryan, Scofield et al. (1984); Pinson (1970); Zipfel (2014)] which should reflect their global longitudinal, lateral and torsional vibration characteristics. The local vibration characteristics are usually considered in a simple manner in the above analysis.
Since the slenderness ratios of launch vehicles are large, the stiffened cylindrical shells in launch vehicles are usually modelled as beams or spring-mass oscillators [Zipfel (2014)], and the on-board equipment is simplified as lumped masses. The key to a successful reduced model of a launch vehicle is the appropriate treatment of the stiffened cylindrical shell. There is a strong motivation to model the stiffened shell as a beam which can accurately reflect the global longitudinal, lateral and torsional vibration characteristics. In order to predict the mechanical behaviour of a large complex structure accurately, it is necessary to establish a large-scale fine finite element model (FE model). This is often the case in static analysis as it does not take much time to do this. In dynamic analysis, however, a large-scale fine FE model presents a significant impact on computation time, particularly when optimization, control, or other theoretical work is also required. So reducing the scale of the model without losing the required accuracy is a way forward. In recent decades, model reduction theory [Afonso, Lyra, Albuquerque et al. (2010); Benfield and Hruda (1971); Box and Wilson (1951); Cortex and Vapnik (1995); Craig and Bampton (1968) ;Friswell, Garvey and Penny (1995); Guyan (1965); Hajela and Berke (1992); Hou (1969); Kaintura, Spina, Couckuyt et al. (2017); Koutsovasilis and Beitelschmidt (2008); Kuhar and Stahle (1974); Matheron (1963) ;Samuel, Ferranti, Knockaert et al. (2016); Wilson (1974); Wilson, Yuan and Dickens (1982)] has been developed and widely used. It has become an important part of the dynamic analysis of large and complex structures. One methodology is to reduce the mass, stiffness and damping matrices of the FE model by mapping the nodal displacements of the original model to the vector space of the reduction basis. The common methods in this category includes Guyan method [Guyan (1965)], Kuhar method [Kuhar and Stahle (1974)], improved reduction system method [Friswell, Garvey and Penny (1995)], Ritz method [Wilson, Yuan and Dickens (1982)], modal synthesis method [Benfield and Hruda (1971); Craig and Bampton (1968); Hou (1969)], etc. These methods can significantly reduce the scale of a FE model. The accuracy of the reduced model depends on the composition of the reduced basis. Another methodology is the proxy or surrogate model technique, which uses a mathematical expression with a small amount of computation to represent the original model for specific analysis. The commonly used proxy models are the response surface model [Box and Wilson (1951)], Kriging model [Kaintura, Spina, Couckuyt et al. (2017); Matheron (1963)], artificial neural network model [Hajela and Berke (1992)], support vector machine model [Cortex and Vapnik (1995)], etc. Proxy models are established by repeated reanalysis of the original model on a series of sample points and the established reduced model usually lacks physical connections with the original physical parameters. The accuracy of a proxy model depends on the number of sample points and their distribution. It is usually very time-consuming to establish a proxy model of a complex structure with high nonlinearity. To reduce the scale of the computational models of stiffened cylindrical shells, Cheng and his group [Cai, Xu and Cheng (2014) ;Cheng, Cai and Xu (2013)] developed a homogenization method and applied it to a three-dimensional periodic structure. Firstly, the complex three-dimensional periodic structure was divided into numerous typical unit cells. Then each typical unit cell was simplified to an equivalent structure with similar stiffness characteristics according to the mechanical properties of each cell. Finally, all the unit cells are replaced by those simple structures. And Hao et al. [Hao, Wang, Tian et al. (2017); Wang, Tian, Hao et al. (2016); Wang, Tian, Zheng et al. (2017)] developed an efficient equivalent analysis model of stiffened shells to facilitate the post-buckling optimization based on the numerical implementation of asymptotic homogenization method. This kind of methods is suitable for periodic structures with the same unit cells. The difficulty lies in the equivalence of the unit cells. The accuracy is dependent on the understanding of the mechanical properties of the typical unit cells. The method proposed by Zheng et al. [Wang, Cheng and Li (2013)] divided a FE model of a stiffened cylindrical shell into several regions and assumed that each region was a rigid body. Then, a transformation relationship between the local motion and the global motion was constructed to reduce the FE model. To improve the accuracy of the reduced model, a deformation correction was introduced. Cheng et al. [Cheng and Wang (2014)] extended Zheng and Ding's method and reduced the stiffened cylindrical shell to a super-beam. To improve the accuracy of the super-beam model, three shear coefficients were introduced. This method maintains a good conversion relationship between the super-beam model and the original FE model. The accuracy of the super-beam is good. However, as the superbeam model lacks an obvious physical meaning, it is difficult to establish directly an intuitive finite beam element model for the shell. On the basis of this method, Wang et al. Li, Wang and Cheng (2017)] established a reduced-order model for the structural dynamic analysis of complicated beam-type structure by using a novel reduction basis along with the polynomial interpolation function. The basic idea was to convert the displacements of finite element model nodes in each cross section to a small set of nodes with a few generalized degrees of freedom. Li et al. [Li, Hao, Wang et al. (2019)] presented a model-reduction method based on the proper orthogonal decomposition technique. The basic idea was to extract the principal component of correlation matrix assembled by the nodal displacement field of full-order models subjected to different crosssectional loads as the transformation matrix. Pan et al. [Pan, Wang, Ma et al. (2014)] compared the simplified models obtained from several equivalence principles, and proposed a cross-sectional area equivalence principle. After studying the cause for the incorrect increase in torsional stiffness, they obtained a correction coefficient to modify the torsional stiffness of the simplified model established through a cross-sectional area equivalence principle. That method can directly establish an equivalent beam model according to the geometry of the original structure, but it is difficult to take into account the influences of the circumferential ribs and the internal components of the structure. A good accuracy can be obtained only if the cross-sectional moment of inertia of the equivalent beam is close to that of the original structure. This paper proposes a new beam approximation method for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells. The flow chart of the method is shown in Fig. 1. An initial beam model of the stiffened cylindrical shell is built based on a crosssectional area equivalence principle and improved through model updating against the modal analysis results of the shell. First, an improvement has been made to the crosssectional area equivalence principle, the circumferential ribs are considered as lumped masses at the nodes of the beam elements, and so the mass distribution of the initial beam model is much closer to that of the shell along the longitudinal direction. Then, model updating, generally, which is implemented to minimise the difference between model predictions and corresponding test results [Friswell and Mottershead (1995); Mottershead and Friswell (1993); Mottershead, Link and Friswell (2011)], is utilized to improve the accuracy of the equivalent beam model in this paper, and the iterative model updating method using modal data is selected. During the model updating process, because of the great difference between a beam and a stiffened cylindrical shell, two problems arise: a) the comparison of mode shapes is difficult; b) the design parameters used in model updating are prone to converging to incorrect values. By introducing a weighted nodal displacement coupling relationship, problem a) has been solved. The mode shapes of stiffened cylindrical shell are transformed into the displacements of a series of nodes located on the longitudinal axis of the shell. For problem b), a pre-model updating procedure is added before the proper model updating. The contents of this paper are arranged as follows: The cross-sectional area equivalence principle, the weighted nodal displacement coupling relationship, and the iterative model updating method using modal data are established or discussed in the second to fourth sections. In the fifth Section, an outline of modelling and updating the initial beam is presented. Two examples are given in the sixth section. And some conclusions are drawn in the last Section.
2 Cross-sectional area equivalence principle A cross-sectional area equivalence principle [Pan, Wang, Ma et al. (2014)] is commonly used to build simple equivalent models of launch vehicles for load calculation, control, dynamic analysis, etc. It is easy to establish an equivalent beam model with annular cross-section according to the geometry of the stiffened cylindrical shell through this principle. The core idea of the principle lies in the equivalence of the longitudinal ribs. In launch vehicles, the number of longitudinal ribs is usually 40 to 120, which can be regarded as uniformly distributed. The longitudinal ribs influence the bending, torsional and longitudinal stiffness of the stiffened cylindrical shell through their contributions to the moments of area of the shell. The effects of circumferential ribs are usually not considered. The shape and eccentricity of the longitudinal rib has negligible influence on the equivalence accuracy of this principle [Pan, Wang, Ma et al. (2014)] because the diameter of the stiffened cylindrical shell is much larger than the cross-section size of the ribs. For convenience, it is assumed that all the longitudinal ribs are identical. The crosssection shape of the ribs is circular with a radius rib , and the axis of the ribs locates in  [Pan, Wang, Ma et al. (2014)]. The cross-sectional area, rib , the second moment of area, rib , and the polar moment of area, rib , relative to its centre of the longitudinal rib can be easily written as: The cross-section area of the stiffened cylindrical shell, 0 , can be written as: where skin is the average radius of the stiffened cylindrical shell, skin is the thickness of the shell skin, is the cross-section area of each longitudinal rib, is the number of the longitudinal ribs. The thickness of the equivalent annular cross-section beam, , can be calculated by where rib is the thickness contributed by longitudinal ribs. The area moment of inertia, , and the polar moment of area, , of the equivalent beam can be easily written as: and = 2 = 2 skin �4 skin 2 + 2 � ≈ 2 skin 3 (7) In Eqs. (6) and (7), the second-order small quantity 2 has been neglected. The area moment of inertia, 0 , of the stiffened cylindrical shell can be written as: The polar moment of area of the stiffened cylindrical shell, 0 , is not twice as much as its area moment of inertia [Pan, Wang, Ma et al. (2014)], 0 . And the contribution from the longitudinal ribs to the torsional stiffness of the stiffened cylindrical shell is far less than that from the shell skin, the polar moment of area of the shell, 0 , can be written approximately as: In Eqs. (8) and (9), the second-order small quantity skin 2 has been neglected.
From Eqs. (6) and (8) Since skin is far larger than rib in launch vehicles, 0 equals to 1 approximately, i.e., the area moment of inertia of the equivalent beam is equal to that of the stiffened cylindrical shell approximately.
From Eqs. (7) and (9), 0 can be calculated Since rib skin cannot be neglected, the polar moment of area of the equivalent beam is much different from that of the stiffened cylindrical shell, a correction of the polar moment of area of the equivalent beam is needed [Pan, Wang, Ma et al. (2014)], and from Eq. (11), the correction factor, tor , is given by The final polar moment of area of the equivalent beam is * = tor = 2 skin skin �4 skin 2 + 2 � ≈ 2 skin 3 skin where, , , are weighting coefficients for each degree of freedom of each node, usually taken as 1. Then, assuming that there are no relative motions between the nodes, a rigid region is formed. According to the rigid body motion relationship, the rigid-body displacement of node , * = � * , * , * � T , can be written as: , , and are the angles of rotation about -, -and -axis of the rigid region. They can be determined by minimizing the modulus, | |, of vector = � , , � T of the error functions of * and using the least-squares method Finally, the displacement vector of node c can be determined The specific steps of using this relationship to transform the displacements of the nodes in the fine FEM model of a stiffened cylindrical shell to the node displacements of the equivalent beam model are introduced in Section 5.2 in detail.

Iterative model updating method using modal data
Modal data include natural frequencies and mode shapes of a structure, which are the common kinds of targets used in model updating. The iterative model updating method using modal data [Friswell and Mottershead (1995)] needs to select some physical and geometric parameters of the structure as design parameters and analyse their sensitivities. Normally, the natural frequencies and mode shapes vary with the design parameters nonlinearly, so it is necessary to linearize them by truncating the first-order Taylor expansion [Mottershead, Link and Friswell (2011)]. Using obj to represent the vector consisting of the interested natural frequencies and mode shapes, to represent the vector consisting of the corresponding finite element analysis results, to represent the vector consisting of the design parameters, the relationship between them is defined as where, subscript represents the th iteration, is sensitivity matrix consisting of the first derivatives of the natural frequencies, ( = 1,2, ⋯ , ), and the mode shapes, , with respect to the design parameters, ( = 1,2, ⋯ ).
For an eigenvalue problem = 2 (20) where and are the mass matrix and stiffness matrix of the structure. The expression of the first derivative of the natural frequency, , is [Friswell and Mottershead (1995) The complete eigenvector derivative is given by a linear combination of all eigenvectors of the structure [Friswell and Mottershead (1995) the eigenvector, , follows mass normalization equation T = 1 (23) By differentiating Eq. (23) with respect to , and combining with Eq. (22), the participation factor, , is obtained A problem arises in the computation of and a solution is given by Nelson [Friswell and Mottershead (1995)].
Eqs. (21) and (22) are the theoretical formulas for calculating the sensitivities of natural frequencies and mode shapes, and in the practical operation, they are usually calculated approximately by using numerical differentiation method. By rearranging Eq. (19), the explicit iteration updating relationship of design parameters is obtained where + is the inverse sensitivity matrix. Since the system of linear algebraic equations in Eq. (19) is often over-determined or under-determined, it cannot be inverted directly.

Beam approximation process
In this section, the beam approximation process for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells is presented in detail. Firstly, an initial beam model is established based on the cross-sectional area equivalence principle. Then, a fine FE model of the stiffened cylindrical shell is established and a modal analysis is carried out. Some natural frequencies and mode shapes are selected as the model updating targets, and the mode shapes are translated by using the weighted nodal displacement coupling relationship. Finally, the initial beam model is improved through model updating against the natural frequencies and mode shapes of the fine FE model.

Initial equivalent beam model
The stiffened cylindrical shell mainly consists of skin, longitudinal ribs and circumferential ribs. Based on the cross-sectional area equivalence principle, the skin and the longitudinal ribs are represented as a beam model with annular cross-section located on the axis of the stiffened cylindrical shell and the circumferential ribs are ignored. In this paper, for the purpose of ensuring that the mass distribution of the beam model is the same as that of the stiffened cylindrical shell along the longitudinal direction, the circumferential ribs are represented as lumped masses applied to the nodes of the beam model. The specific steps are as follows: a) According to the geometry of the stiffened cylindrical shell, the cross-section area of each longitudinal rib is calculated, and then the thickness of the annular crosssection beam is obtained from Eqs. (4) and (5). b) Taking the diameter of the shell skin as the diameter, the beam model with annular cross-section is established. The longitudinal locations of the beam nodes correspond to those of the circumferential ribs of the shell. c) According to the material properties of the stiffened cylindrical shell, the mass of each circumferential rib is calculated and applied to the corresponding node of the beam model as a lumped mass. The material properties of the initial equivalent beam model are set in step b) according to those of the stiffened cylindrical shell. If the materials of the skin and the longitudinal ribs are different, the material properties of the beam model need to be estimated. It has to be noted that the mass distribution of the initial beam model should be as close as possible to the longitudinal mass distribution of the stiffened cylindrical shell. Other material attributes can be roughly selected.

Model updating targets
The fine FE model of the stiffened cylindrical shell is established, and modified if there are experimental data. Then, a modal analysis of the fine FE model is implemented. Some natural frequencies and mode shapes are selected as the model updating targets. The natural frequencies can be directly used to improve the equivalent beam model. While the mode shapes cannot, because of the nodes of the fine FE model of the shell and the equivalent beam model are not one-to-one correspondence. The weighted nodal displacement coupling relationship is used to transform the nodal displacements between the two models. For a mode shape, the specific steps are as follows: a) The nodes on the plane located the circumferential rib are selected and the node of equivalent beam model corresponding to the circumferential rib is selected. b) According to Eqs. (14) to (18), the displacements of the nodes on the plane are represented by the displacement of the node of the beam model. c) Steps a) and b) are repeated several times until the displacements of the nodes on the planes located the circumferential ribs are all represented by the displacements of the nodes of the beam model.

Model updating
Since the mass distribution of the beam model has been settled in Section 5.1, it is considered to be exact. A problem may occur during model updating, because of the great difference between the fine FE model and the equivalent beam model. If all the design parameters are directly used for model updating at the same time, they can easily converge to incorrect values, for example, the Young's modulus, , may become negative. And it is hard to give the exact ranges of the design parameters directly. Having a pre-model updating procedure before the proper model updating to shift design parameters close to the convergent values can ease this problem. In pre-model updating, based on the sensitivity analysis and practical experiences, the area moments of inertia, 1 , and, 2 , in two directions, the polar moment of area, p , and Young's modulus, , of the beam element are selected as design parameters. There are no range limitations of the design parameters, and, generally, they are converged to reasonable values. Usually, using these four design parameters the bending and torsional modes of the equivalent beam model can be corrected. The axial modes are sensitive to the cross-sectional area, , and Young's modulus, , of the beam, and these two parameters are closely related to the mass distribution and the bending modes of the beam model. So, in order to improve the axial modes, some linear spring elements are introduced between each pair of adjacent nodes of the equivalent beam model, and their stiffness, , is selected as design parameter. All the design parameters of each element in the beam model are same. Those five design parameters are determined by using the natural frequencies of the first and the second bending modes in the two normal directions, the first torsional and the first axial modes of the fine FE model of the shell structure (totally six modes). The flow chart of premodel updating is shown in Fig. 2, which includes five steps:  [Friswell and Mottershead (1995)] between the mode shapes of the beam model and those of the stiffened shell are calculated. The MAC value between the th mode shape of the beam model and the th targeted mode is calculated from The closer the MAC value is to 1, the higher correlation of the two mode shapes is in. For each targeted mode shape, one mode shape of the beam model is identified that brings the MAC value of the two closest to 1. Then such two modes are paired. This process is called mode matching. b) The ratio of the natural frequencies of the first two bending modes, , is improved.
where b1 and b2 are the natural frequencies of the first and second bending modes. The area moments of inertia, 1 , and, 2 , are selected as the design parameters to improve . In addition, for the case in which the frequencies of the first bending modes in two perpendicular out-of-plane directions are different, the same design parameters are taken to improve the ratio of the two frequencies, .
where the superscripts (1) and (2) represent two bending directions. c) The first bending modes are improved. The Young's modulus, , is selected as the design parameter. d) Now the first torsional mode is improved. The polar moment of area, p , is selected as the design parameter. e) Afterwards, the first axial mode is improved. The stiffness of the spring elements, , is selected as the design parameter, and the initial value come from where, 0 is the initial Young's modulus value of the beam element, � is the Young's modulus value after step c), and are the cross-sectional area and the length of the beam element. The above steps are repeated several times until the frequencies of the afore-mentioned six modes of the equivalent beam model match the corresponding frequency of the fine FE model to the required accuracy. The values of design parameters after pre-model updating are quite different from those of the initial beam model, but the bending rigidity, torsional rigidity and tensile stiffness of the beam model before and after pre-model updating have not much, as shown in Section 6.1, which is expected and indirectly reflects that the finial equivalent beam model shows similar global static properties as the original model. And, it should be noted that, because of the beam approximation method presented in this paper is focused on the global dynamic performances of the original model, the equivalent beam model can reflect only the global static properties of the original model. On this basis, the proper updating can be carried out to consider the higher frequencies and other quantities of interest. The ranges of the design parameters can be given if needed. For a simple stiffened cylindrical shell, its equivalent beam model can be obtained in good accuracy just by pre-model updating and this is presented in the next section. The modal analyses of the finite element models in the paper are made in ANSYS using block Lanczos method.

Stiffened cylindrical shell of circular cross-section
The stiffened cylindrical shell of circular cross-section is 50 m long and 5 m in diameter. Its skin thickness is 0.01 m. 8 circular longitudinal ribs with a diameter of 0.15 m and 11 circular circumferential ribs with a diameter of 0.6 m are distributed uniformly. There are totally 3200 SHELL181 elements, 1152 BEAM188 elements and 3232 nodes in the fine FEM model. A normal mode analysis of the fine FEM model is taken and frequencies and modes below 100 Hz are extracted. The beam model is established to reflect the first two bending, torsional and axial modes of the stiffened cylindrical shell, totally 6 modes, and the errors of the natural frequencies should be less than 2%, and the MAC values should be greater than 0.95. There are totally 10 BEAM188 elements, 10 COMBIN14 elements and 11 nodes in the beam model. The parameter values of the normal mode analysis of the equivalent beam model are the same as those of the fine FEM model of the shell. The values of design parameters before and after model updating are shown in Tab. 1. The calculation results of the bending rigidity, torsional rigidity and tensile stiffness are given in Tab. 2. The shear modulus, , is calculated by where is Poisson's ratio. The tensile stiffness, , of the beam model is calculated by The torsional rigidity between brackets in Tab. 2 is calculated from the modified polar moment of area calculated by Eq. (13). In Tab. 1, the values of design parameters after pre-model updating are quite different from those of the initial beam model. And in Tab. 2, it is shown that the bending rigidity, torsional rigidity and tensile stiffness of the beam model before and after pre-model updating are not changed a lot. This indirectly reflects that the finial equivalent beam model shows similar static properties as the original model.  The results of the modal analysis are shown in Tabs   As it is shown in Tab. 3, after pre-model updating, the first two bending, torsional and axial natural frequencies of the stiffened cylindrical shell predicted by the equivalent beam model are already in good agreement with the original structure, and thus the subsequent proper model updating is not necessary except for accurate prediction of higher frequencies if needed. And in Tab. 4, the relative errors of the natural frequencies of the first four bending, axial modes and the first three torsional modes are below 5%. It is shown that, for a simple stiffened cylindrical shell, its equivalent beam model can be obtained in good accuracy just by pre-model updating.

Composite structure of stiffened cylindrical shells
As shown in Fig. 3    In this example, only the first several modes of each component have been modified, and as shown in Tab. 5, the largest relative error among the first 16 natural frequencies of the whole structure is 2% (for the 6th natural frequency). And in Fig. 4, the mode shapes of the equivalent beam model are similar to those of the fine FEM model. For a composite structure of stiffened cylindrical shells, like a launch vehicle, the lower modes are usually most concerned and they are made up from the lower modes of individual components. The construction of the equivalent beam model for each component that captures the lower modes of the original structure can provide an accurate reduced model for the whole composite structure.

Conclusions
An equivalent beam model for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells is successfully established. Model updating method is applied to improve the beam model of the stiffened cylindrical shells. Three aspects of this work lead to its success. In establishing the initial equivalent beam model, some improvements are made to the cross-sectional area equivalence principle. The mass distribution of the initial equivalent beam model is the same as that of the stiffened cylindrical shell along the longitudinal direction. In comparing the mode shapes, by introducing the weighted nodal displacement coupling relationship, the relation between the mode shapes of the detailed FE model of the stiffened cylindrical shell and its equivalent beam model is constructed. In model updating, the whole process is split into two procedures, which helps prevent the design parameters from converging to incorrect values. The method proposed in this paper is an improvement of the method proposed by Pan et al. [Pan, Wang, Ma et al. (2014)]. There are three main advantages of the improved method compared with the common methods. a) The physical meaning of the equivalent beam model is clear. It is easy to build an intuitive FE model of the equivalent beam using conventional beam elements in commercial software. The improved method and the method proposed by Pan et al. both have clear physical meanings. The whole model reduction process does not depend on the experience of the analyst, and the high-accuracy equivalent beam models of the stiffened cylindrical shells can be established by follow some simple steps. b) Compared with the method proposed by Pan et al., the improved method can take into account the influences of the circumferential ribs of the stiffened cylindrical shells, so the mass distributions of the beam models are much closer to those of the shells along the longitudinal direction. And the model updating against the modal analysis results of the shells is given to the beam models, so they can have a higher accuracy. c) The method proposed by Pan et al. has a limitation of applicability. A good accuracy can be obtained only if the cross-sectional moment of area of the equivalent beam is close to that of the original structure. It means that the method proposed by Pan et al. is more suitable for establishing the equivalent beam models of the stiffened cylindrical shells with large diameter of the shell skins, small cross-section size and densely distributed of the longitudinal ribs, and few circumferential ribs. The improved method is not subjected to these limitations. It should be pointed out that although the proposed method improves the accuracy of the equivalent beam model through model updating, it needs to establish a fine FEM model of the original structure, which undoubtedly increases some extra work of the model reduction process compared with the method proposed by Pan et al. A real launch vehicle consists of several kinds of stiffened cylindrical shells which need to be classified at first. The stiffened cylindrical shells which constitute the main structures of the launch vehicle need to be reduced separately. If the length of the stiffened cylindrical shell is short, it should be conceptually grouped with a few other stiffened cylindrical shells so that all together the grouped shells are long enough to be modelled as a beam. Generally, when the slenderness ratio of a stiffened cylindrical shell is more than 10, its global dynamic behaviour becomes similar to that of a beam. For the other stiffened cylindrical shells which are very short and mainly used for connection. The equivalent beam models of them need to be updated together with the equivalent beam models of the stiffened cylindrical shells connected by them. After all the stiffened cylindrical shells are reduced properly, they are assembled together to determine the connections between the different stages and boosters. For applying this method to the general situation, the basic ideas are the same. Firstly, the mass distribution of the equivalent beam model should be the same as that of the stiffened cylindrical shell along the longitudinal direction, which is relatively easy to achieve mathematically. The next step is the adjustment of the stiffness distribution, which is the difficult part in beam model equivalence. If the stiffened cylindrical shell is very complex along the longitudinal direction, it needs to be conceptually divided into several simple segments, and many design parameters have to be selected to get the equivalent model, which will bring many problems. This method is more suitable for establishing the equivalent beam model of the structure composed of several simple slender stiffened cylindrical shells. For further development of this method, one action is introducing additional degrees of freedom to enrich the representativeness of the equivalent beam model, to improve the accuracy and to expand the application, so that it can be used to describe accurately such structures as stiffened cylindrical shells with large openings on their side walls. The other is the application of model updating method combined with weighted nodal displacement coupling relationship. Such an improved method will have the ability to establish the equivalent models of other kinds of large and complex structures.