A mechanical modeling method for clustered tensegrity structures

Abstract

This invention discloses a mechanical modeling method for clustered tensioned monolithic structures. This invention can be used for quasi-static and dynamic simulations of this type of structure. The provided model has simple motion boundary conditions, is easy to implement, and has high computational efficiency. The method of this invention includes: a mechanical modeling method for clustered tensioned monolithic structures based on arbitrary Lagrange-Euler descriptions.

Description

Technical Field

[0001] This invention relates to the field of flexible multibody dynamics technology, and in particular to a mechanical modeling method for aggregated tension integral structures. Background Technology

[0002] Aggregate tensioned monolithic structures are a new structural form, improved from classic tensioned monolithic structures. This type of structure consists of compression members (compression rods) and tension members (ropes). It can undergo a large range of rigid body displacements, can be stored in confined spaces for easy transport and storage, and automatically deploys to its working state during use. Due to its excellent properties such as light weight and high flexibility, it stands out among many deployable spatial structures and has been widely used in civil engineering, aerospace, and robotics design. Because this type of structure contains sliding ropes, its mechanical properties exhibit high nonlinearity. To analyze its complex mechanical behavior, an accurate and efficient computational model is needed. Currently, most modeling methods are limited to quasi-static deployment analysis of aggregate tensioned monolithic structures, and quasi-static deployment is achieved by changing the effective length of the ropes without mentioning the specific driving method. For the dynamic analysis of aggregate tensioned monolithic structures, some methods based on the Lagrangian finite element method treat the sliding ropes as a super-element, establishing a dynamic model of the structure. However, the number of nodes in this unit depends on the number of pulleys the sliding rope passes through, making unit assembly cumbersome, and the kinematic constraint equations describing rope-driven motion are very complex. Therefore, establishing a more efficient dynamic model is of great significance for the numerical simulation of clustered tensioned monolithic structures in engineering. Summary of the Invention

[0003] To overcome the problems of existing technologies, this invention proposes a mechanical modeling method for clustered tensioned monolithic structures based on arbitrary Lagrange-Euler descriptions. This model enables quasi-static and dynamic numerical simulations of such structures. It features simpler motion boundary conditions, namely, kinematic constraint equations describing cable-driven motion. This method is easy to implement, computationally efficient, and the calculation results agree well with existing models.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for mechanical modeling of aggregated tensioned integral structures includes the following steps:

[0006] Step 1: Establish the geometric model of the clustered tension integral structure, that is, define the position coordinates of the structural nodes and the connection relationship of each node;

[0007] Step 2: Divide the sliding rope in the aggregated tensioned integral structure into several arbitrary Lagrange-Euler time-varying length cable elements. The compression members and ordinary ropes in the structure are modeled using nonlinear bar elements. Among them, the nonlinear bar elements are only related to the position coordinates of the nodes. The arbitrary Lagrange-Euler time-varying length cable element is, in addition to the position coordinates of the nodes, an arc length variable is introduced to represent the material coordinates of any point within the sliding rope. Each element has the same position coordinates at the same nodes.

[0008] Step 3: Establish the overall dynamic equations of the clustered tensioned monolithic structure. Assemble the dynamic equations of the clustered tensioned monolithic structure based on the dynamic equations of each unit. Select the constraint equations according to the required driving strategy and process the constraint equations using the Lagrange multiplier method.

[0009] Step 4: Solve the dynamic equations using the generalized α algorithm to obtain the position coordinates of the aggregated tensioned integral structure.

[0010] Furthermore, the time-varying mass matrix, generalized force matrix, and corresponding Jacobian matrix of the time-varying length cable element are defined as follows:

[0011] Generalized elastic force:

[0012]

[0013] Where g = (r2-r1) / ||r2-r1|| is the direction vector of the current configuration of the time-varying length cable unit; λ = ||r2-r1|| / (s2-s1) is the elongation of the time-varying length cable unit; r i Let be the global position coordinates of the node, i = 1, 2, s i Let be the material coordinates of the node; the natural length of the cable element is l0 = s2 - s1, and ε is the axial Green strain of the cable element, expressed as:

[0014] Jacobian matrix of generalized elastic force:

[0015]

[0016] in, It is a 3×3 identity matrix; symbol Represents the Kronecker product of matrices;

[0017] Time-varying mass matrix:

[0018]

[0019] In the above formula, Where ρ and A are the density and cross-sectional area of ​​the rope, respectively;

[0020] Additional inertial force:

[0021]

[0022] in

[0023]

[0024] The specific expressions for A1, A2, B1-B3, C1-C3, D1-D6 are as follows:

[0025]

[0026] In the integral of the above equation Let ξ be the interpolation function of the element, and let ξ = (2s - s1 - s2) / (s2 - s1) be the natural coordinates of the element.

[0027] The Jacobian matrix of the additional inertial force with respect to q is:

[0028]

[0029] in

[0030]

[0031] Additional inertial force about The Jacobian matrix is:

[0032]

[0033] in

[0034]

[0035] Furthermore, the overall dynamic equation of the aggregated tensioned monolithic structure is:

[0036]

[0037] M is the time-varying mass matrix of the system; Φ(q) D ,t) represents the kinematic constraints. It is the Jacobian matrix of the kinematic constraints; λ represents the Lagrange multiplier; Q is the generalized force of the system.

[0038] Right now In the above formula, Q f Q a Q e These are the system's generalized external force, additional inertial force, and generalized elastic force, respectively.

[0039] Furthermore, the constraint equations comprise three types:

[0040] The first type of constraint equation is used to eliminate rigid body displacements of the structure, and its specific expression is as follows:

[0041] r k -r k0 =0(15)

[0042] Where, r k0 These are the initial values ​​of the position coordinates at the fixed point;

[0043] The second type of constraint fixes the material coordinates at the end of the sliding rope, degenerating the node into an Eulerian node, the specific expression of which is as follows:

[0044] s k -s k0 =0(16)

[0045] Among them, s k0 These are the initial values ​​for the material coordinates of the constraint nodes;

[0046] The third type of constraint is the constraint on the sliding rope driving point, and its specific expression is as follows:

[0047]

[0048] in It depends on the driving strategy.

[0049] Compared with existing technologies, the advantages of adopting the above technical solution are as follows:

[0050] This invention provides a mechanical modeling method for clustered tensile monolithic structures based on arbitrary Lagrange-Euler descriptions, avoiding the shortcomings of traditional Lagrange-based modeling. This method features simpler motion boundary conditions and higher computational efficiency. Attached Figure Description

[0051] Figure 1 This is a flowchart illustrating the calculation method for the system dynamics equations.

[0052] Figure 2 This is a schematic diagram of a clustered tensioned monolithic structure;

[0053] Figure 3 The figure shows the quasi-static simulation results obtained using this invention.

[0054] Figure 4 The figure shows the dynamic simulation results obtained using the present invention. Detailed Implementation

[0055] This invention provides a mechanical modeling method for clustered tensioned monolithic structures. The method is used for quasi-static and dynamic unfolding simulations of such structures. The provided model has simple motion boundary conditions, is easy to implement, and has high computational efficiency. Using this invention, numerical simulations of clustered tensioned monolithic structures can be performed, providing theoretical guidance for the design of such structures and possessing engineering significance.

[0056] This invention provides a method for modeling clustered tensioned monolithic structures in engineering, enabling numerical simulation of the quasi-static and dynamic development processes of such structures. To achieve the above objectives, this invention provides a mechanical modeling method for clustered tensioned monolithic structures, comprising the following steps:

[0057] Step 1: Establish the geometric model of the structure. The clustered tension monolithic structure contains ordinary ropes, sliding ropes, and compression members, all of which are one-dimensional components. Therefore, the geometric model of the structure only needs to be described by the coordinates of the nodes and the connections between them. Establishing the geometric model of the structure involves defining the initial values ​​of all node coordinates and the connections between each node;

[0058] Step 2: Divide the sliding rope in the aggregated tensioned integral structure into several arbitrary Lagrange-Euler time-varying length cable elements. The compression members and ordinary ropes in the structure are modeled using nonlinear bar elements. Among them, the nonlinear bar elements are only related to the position coordinates of the nodes. The arbitrary Lagrange-Euler time-varying length cable element is, in addition to the position coordinates of the nodes, an arc length variable is introduced to represent the material coordinates of any point within the sliding rope. Each element has the same position coordinates at the same nodes.

[0059] Step 3: Establish the overall dynamic equations of the clustered tensioned monolithic structure. Assemble the dynamic equations of the clustered tensioned monolithic structure based on the dynamic equations of each unit. Select the constraint equations according to the required driving strategy and process the constraint equations using the Lagrange multiplier method.

[0060] Step 4: Solve the dynamic equations using the generalized α algorithm to obtain the position coordinates of the aggregated tensioned integral structure.

[0061] Furthermore, the time-varying mass matrix, generalized force matrix, and corresponding Jacobian matrix of the time-varying length cable element are defined as follows:

[0062] Generalized elastic force:

[0063]

[0064] Where g = (r2-r1) / ||r2-r1|| is the direction vector of the current configuration of the time-varying length cable unit; λ = ||r2-r1|| / (s2-s1) is the elongation of the time-varying length cable unit; r i Let be the global position coordinates of the node, i = 1, 2, s i Let be the material coordinates of the node; the natural length of the cable element is l0 = s2 - s1, and ε is the axial Green strain of the cable element, expressed as:

[0065] Jacobian matrix of generalized elastic force:

[0066]

[0067] in, It is a 3×3 identity matrix; symbol Represents the Kronecker product of matrices;

[0068] Time-varying mass matrix:

[0069]

[0070] In the above formula, Where ρ and A are the density and cross-sectional area of ​​the rope, respectively;

[0071] Additional inertial force:

[0072]

[0073] in

[0074]

[0075] The specific expressions for A1, A2, B1-B3, C1-C3, D1-D6 are as follows:

[0076]

[0077] In the integral of the above equation Let ξ be the interpolation function of the element, and let ξ = (2s - s1 - s2) / (s2 - s1) be the natural coordinates of the element.

[0078] The Jacobian matrix of the additional inertial force with respect to q is:

[0079]

[0080] in

[0081]

[0082] Additional inertial force about The Jacobian matrix is:

[0083]

[0084] in

[0085]

[0086] Furthermore, the overall dynamic equation of the aggregated tensioned monolithic structure is:

[0087]

[0088] M is the time-varying mass matrix of the system; Φ(q) D ,t) represents the kinematic constraints. It is the Jacobian matrix of the kinematic constraints; λ represents the Lagrange multiplier; Q is the generalized force of the system.

[0089] Right now In the above formula, Q f Q a Q e These are the system's generalized external force, additional inertial force, and generalized elastic force, respectively.

[0090] Furthermore, the constraint equations comprise three types:

[0091] The first type of constraint equation is used to eliminate rigid body displacements of the structure, and its specific expression is as follows:

[0092] r k -r k0 =0(15)

[0093] Where, r k0 These are the initial values ​​of the position coordinates at the fixed point;

[0094] The second type of constraint fixes the material coordinates at the end of the sliding rope, degenerating the node into an Eulerian node, the specific expression of which is as follows:

[0095] s k -s k0 =0(16)

[0096] Among them, s k0 These are the initial values ​​for the material coordinates of the constraint nodes;

[0097] The third type of constraint is the constraint on the sliding rope driving point, and its specific expression is as follows:

[0098]

[0099] in It depends on the driving strategy.

[0100] This invention is applicable to quasi-static and dynamic simulation of aggregated tension integral structures. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.

Claims

1. A method for modeling the mechanical properties of a clustered tensioned integral structure, characterized in that, It includes the following steps: Step 1: Establish the geometric model of the clustered tension integral structure, that is, define the position coordinates of the structural nodes and the connection relationship of each node; Step 2: Divide the sliding rope in the aggregated tensioned integral structure into several arbitrary Lagrange-Euler time-varying length cable elements. The compression members and ordinary ropes in the structure are modeled using nonlinear bar elements. Among them, the nonlinear bar elements are only related to the position coordinates of the nodes. The arbitrary Lagrange-Euler time-varying length cable element is, in addition to the position coordinates of the nodes, an arc length variable is introduced to represent the material coordinates of any point within the sliding rope. Each element has the same position coordinates at the same nodes. The time-varying mass matrix, generalized force matrix, and corresponding Jacobian matrix of the time-varying length cable element are expressed as follows: Generalized elastic force: Where g = (r2-r1) / ||r2-r1|| is the direction vector of the current configuration of the time-varying length cable unit; λ = ||r2-r1|| / (s2-s1) is the elongation of the time-varying length cable unit; r i Let be the global position coordinates of the node, i = 1, 2, s i Let be the material coordinates of the nodes; the natural length of the cable element is l0 = s2 - s1, and ε is the axial Green strain of the cable element, expressed as: Jacobian matrix of generalized elastic force: in, It is a 3×3 identity matrix; symbol Represents the Kronecker product of matrices; Time-varying mass matrix: In the above formula, Where ρ and A are the density and cross-sectional area of ​​the rope, respectively; Additional inertial force: in The specific expressions for A1, A2, B1-B3, C1-C3, D1-D6 are as follows: In the integral of the above equation Let ξ be the interpolation function of the element, and let ξ = (2s - s1 - s2) / (s2 - s1) be the natural coordinates of the element. The Jacobian matrix of the additional inertial force with respect to q is: in Additional inertial force about The Jacobian matrix is: in Step 3: Establish the overall dynamic equations of the clustered tensioned monolithic structure; assemble the dynamic equations of the clustered tensioned monolithic structure based on the dynamic equations of each unit; select the constraint equations according to the required driving strategy, and process the constraint equations using the Lagrange multiplier method. Step 4: Solve the dynamic equations using the generalized α algorithm to obtain the position coordinates of the aggregated tensioned integral structure.

2. The method for mechanical modeling of a clustered tensioned integral structure according to claim 1, characterized in that, The overall dynamic equation of the aggregated tensioned monolithic structure is: M is the time-varying mass matrix of the system; Φ(q) D ,t) represents the kinematic constraints. It is the Jacobian matrix of the kinematic constraints; λ represents the Lagrange multiplier; Q is the generalized force of the system. Right now In the above formula, Q f Q a Q e These are the system's generalized external force, additional inertial force, and generalized elastic force, respectively.

3. The method for mechanical modeling of a clustered tensioned integral structure according to claim 1, characterized in that, The constraint equations include three types: The first type of constraint equation is used to eliminate rigid body displacements of the structure, and its specific expression is as follows: r k -r k0 ＝0(15) Where, r k0 The initial values ​​are the coordinates of the fixed point. The second type of constraint fixes the material coordinates at the end of the sliding rope, degenerating the node into an Eulerian node, the specific expression of which is as follows: s k -s k0 =0(16)。

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