Modeling method study for transfer matrix of space rigid bodies supported by multiple elastic hinges

When rigid bodies as well as elastic hinges connected are modeled by using a transfer matrix method, they need to be defined as multi-input and multi-output space rigid bodies, resulting in higher dimensions of the whole transfer matrix and complexity of modeling. To reduce the dimensions of the whole matrix and improve computational efficiency and accuracy, the elastic hinges and the rigid bodies are integrated into a unified model in this paper. Then a single 12×12 matrix is obtained through theoretical derivation. The resulting transfer matrix of space rigid bodies supported by multiple elastic hinges not only reduces the dimensions of the whole matrix but also simplifies the configuration of the transfer matrix model by which some tree-like systems can be simplified into chain-like systems. Subsequently, the effectiveness of the matrix is validated through computational comparative analysis of typical systems. The modeling method is an important optimization to the calculation method of the dynamics characteristics of the transfer matrix.


Introduction
Multi-body dynamics [1][2][3] is widely used for calculating and analyzing the dynamics characteristics of typical multi-body systems such as artillery [4], multiple-launch rocket systems [5] and warship pipelines [6].In order to enhance the efficiency of the transfer matrix method, Wan [7] proposed an improved transfer matrix method to improve computational efficiency.Xu [8] applied this method to the identification of noise sources in rectangular plate structures and Guo [9] applied it to analyze the vibration and stability of intelligent beams under axial force [10], significantly improving computational efficiency.In specific applications, unit transfer matrices with a specific standard form are written for each of the components constituting a system and then assembled into a whole transfer matrix for calculation based on the topological structure of the system.Multi-body systems are topologically classified into chain-like structures, tree-like structures and network structures [11][12].The chain-like structures have the lowest whole matrix dimensions, faster computation speed, and higher accuracy.
For a typical system shown in Figure 1, when using a transfer matrix method to calculate dynamics characteristics, U1, U2, U3 (including multiple elastic hinges) and U4 are often modeled separately and assembled into a whole transfer matrix.In the modeling process, U3 should be defined as a spacerigid body with multiple interfaces.This system is a typical tree-like system with a 72×12 whole transfer matrix.
In order to reduce the whole transfer matrix dimensions of the system, He [13] deduces a generalized elastic hinge rigidity matrix.Multiple elastic hinges in U3 are modeled by a generalized elastic hinge to reduce the dimensions of the whole transfer matrix to 24×12, thereby significantly downscaling the whole transfer matrix.
To further simplify the system modeling approach, U3 and U4 may be modeled as a whole due to the subsidiary relationship to reduce the dimensions of the whole transfer matrix to 12x12, thus simplifying the modeling configuration by which the system topology is simplified from the tree-like configuration to the chain-like configuration.

Theoretical inference of transfer matrix of space rigid bodies supported by multiple elastic hinges
A space-vibration rigid body with a mass of m is shown in Figure 2. In a connected system with an input point I as the origin of coordinates, the inertia matrix of the rigid body relative to point I is denoted as JI.The coordinates of an output point O are b1, b2 and b3.The coordinates of the center of mass C are cc1, cc2 and cc3.Assuming that the rigid body is supported by n vibration isolators, the coordinates of the i-th (i = 1, 2, 3, …, n) support point Si are Si1, Si2 and Si3.In the case of tiny linear vibrations of a rigid body, it can be assumed that the input point I, the output point O, the center of mass C and various support points Si in the space have the same rotation angles, which is equal to those of the rigid body around the point I. Considering the tiny angle and its vector property, the displacement of the output point O can be expressed as the displacement of the input point I from it and the angle by which the input point I rotates around it.
In the equation, IO ∃ l is the displacement cross-product matrix.Similarly, the displacements of the center of mass C and the support point Si can also be given in the above form.The cross-product matrix can be expressed as: First of all, only one support point, Si, is considered as the output point.According to the definition of the vector direction in the coordinate system as well as considering supporting reaction at the point Si, a translational equation is obtained from the theorem of motion of the center of mass.
Since 00 00 00 By converting Equation ( 2) into the formula of the center of mass and substituting Equation ( 6) into Equation ( 5): Equation ( 9) can be rearranged as:

Characteristics analysis of transfer matrix of space rigid bodies supported by multiple elastic hinges
According to the analysis of the derived transfer matrix Si U of space rigid bodies supported by multiple elastic hinges in Section 2.1, the matrix has the following characteristics.
1) The boundary condition for integrating rigid bodies with elastic hinges should be fixed.That is, there are no interconnected transfer components below the elastic hinges.
2) The transfer matrix of space rigid bodies with multiple elastic hinges simplifies the transfer branch of elastic hinges.That is, a tree-like model with two inputs and one output is reducible to a single-input single-output chain model.

Example validation
To verify the accuracy of the space rigid bodies supported by multiple elastic hinges, a specific actual ship system was chosen for modeling with the transfer matrix method and the finite element method, comparative calculation and analysis.

Introduction to the calculation model of a transfer matrix
The system includes equipment and accessories such as pumps, valves, filters, pipe supports and expansion joints, to which pipes are connected.There are diverse vibration excitation sources with complex coupling and transmission.The specific layout of the system is shown in Figure 2. According to the requirements of the calculation method, the system is divided into pipe segment units, single input-single output rigid body units, pipe elastic support units, and concentrated mass units.
1) The pipes are divided into straight pipes or steering straight pipe beam units; 2) Filters and valves are used as single-input and single-output rigid body units without elastic hinges, while pumps are used as space rigid body units supported by multiple elastic hinges; 3) Pipe Vibration isolators (PVI) supports are used as elastic support units; 4) Expansion joints (EJ) are treated as flexible connection pipes with rigidity, length, and mass properties; By using the above discrete modeling method, the pipes are divided into 30 calculation nodes.The model is shown in Figure 3.By using the modeling method of space rigid bodies supported by multiple elastic hinges described in this paper, the system now becomes a typical chain-like system model.In contrast, when space rigid bodies without elastic hinge supports are used for modeling, vibration isolators supporting the pumps need to be separately modeled.The whole system then topologically becomes a tree-like system with two inputs and a single output.Therefore, the modeling approach is significantly optimized.

Introduction to finite element calculation model
According to the system layout and vibration isolator parameters, ANSYS is used for modeling and calculation (as shown in Figure 4).
a) The pumps, filters, and valves are modeled by using concentrated mass units.The vibration isolators, pipe supports and expansion joints are modeled by using spring units.The pipes are modeled by using beam units with circular cross-sections.Like the transfer matrix model, hinge-supported boundary conditions are adopted at the starting and tail ends of the system pipes.The rigidities of remaining vibration isolators, pipe supports, and expansion joints are input from standard database data to calculate inherent frequency and static deformation.

Comparison of calculated results for inherent frequency
The comparison of calculated results for inherent frequency in the system is shown in Table 1.
Table 1 From Table 1, it can be seen that the calculated results of the inherent frequency obtained by modeling space rigid bodies supported by multiple elastic hinges for pumps and calculating the dynamics characteristics of a specific system with the transfer matrix method almost agree with those from the finite element method, except for a slight difference observed beyond the 17th order, indicating that modeling with space rigid bodies supported by multiple elastic hinges not only reduces the dimensions of the calculated model but also ensures calculation accuracy.

Comparison of calculated static deformation results
Similarly, to validate the accuracy of the static response calculation for modeling space rigid bodies with multiple elastic hinge supports, the static displacement response is calculated.The calculated results are presented in Table 2. From Table 2, it can be observed that the calculated deformation results for elastic components in the system including vibration isolators (i.e., elastic hinges in the rigid bodies), pipe supports, and expansion joints are almost consistent with the results obtained from finite element analysis.By comparing the static response calculations, it is demonstrated that the calculation accuracy in response analysis of space rigid bodies modeling with multiple elastic hinge supports can also be guaranteed.

Conclusions
In this paper, the rigid bodies and elastic hinges connected thereto are modeled together to form a 12x12 matrix for the space rigid bodies supported by multiple elastic hinges based on the transfer matrix method, which not only reduces the dimensions of the whole matrix but also simplifies the topological configuration of the transfer matrix model, allowing for the simplification of some treelike systems into chain-like systems.Subsequently, the effectiveness of the matrix is validated through computational comparative analysis of a typical system.The modeling method is an important optimization to the calculation method of the dynamics characteristics of the transfer matrix.

Figure 1 .
Figure 1.Schematic diagram of a typical system model with rigid bodies and elastic hinges.

Figure 2 .
Figure 2. Schematic diagram of a space rigid body with elastic hinges.In the case of tiny linear vibrations of a rigid body, it can be assumed that the input point I, the output point O, the center of mass C and various support points Si in the space have the same rotation angles, which is equal to those of the rigid body around the point I.

Figure 2 .
Figure 2. Schematic layout diagram of the system piping.According to the requirements of the calculation method, the system is divided into pipe segment units, single input-single output rigid body units, pipe elastic support units, and concentrated mass units.1)The pipes are divided into straight pipes or steering straight pipe beam units;2) Filters and valves are used as single-input and single-output rigid body units without elastic hinges, while pumps are used as space rigid body units supported by multiple elastic hinges;3) Pipe Vibration isolators (PVI) supports are used as elastic support units; 4) Expansion joints (EJ) are treated as flexible connection pipes with rigidity, length, and mass properties; By using the above discrete modeling method, the pipes are divided into 30 calculation nodes.The model is shown in Figure3.

Figure 3 .
Figure 3. Schematic diagram of transfer matrix modeling of system pipes and nodes.The starting and tail ends of the model are subject to hinge-supported boundary conditions.The rigidities of remaining vibration isolators, pipe supports and expansion joints are input from standard database data to calculate inherent frequency and static deformation.By using the modeling method of space rigid bodies supported by multiple elastic hinges described in this paper, the system now becomes a typical chain-like system model.In contrast, when space rigid bodies without elastic hinge supports are used for modeling, vibration isolators supporting the pumps need to be separately modeled.The whole system then topologically becomes a tree-like system with two inputs and a single output.Therefore, the modeling approach is significantly optimized.
b) Remote device ports of inlet and outlet pipes, vibration isolators and locations where pipe supports are fixed are subject to simply supported constraints.c)The weight of the fluid inside the pipes is taken into account.d)The masses of the expansion joints, as well as the weight of fluid therein, are implemented by applying concentrated mass units at both ends of the expansion joints.

Figure 4 .
Figure 4. Finite element model of system pipes.Like the transfer matrix model, hinge-supported boundary conditions are adopted at the starting and tail ends of the system pipes.The rigidities of remaining vibration isolators, pipe supports, and expansion joints are input from standard database data to calculate inherent frequency and static deformation.
are torsional moments.From the angular momentum theorem of the center of the moment, the rotation equation around point I is obtained by omitting the high-order tiny terms according to the small-angle vibration, which is as follows:

.
Comparison of calculated results of inherent frequency in the piping system (Hz).

Table 2 .
Comparison of calculated static response results in the piping system (mm).