Response prediction method of space viscoelastic beams based on absolute nodal coordinate method

Abstract

The present invention discloses a method for predicting the response of a space viscoelastic beam based on an absolute node coordinate method. The method comprises: constructing a space simulation environment that takes into account the damping force of a beam unit; calculating the damping ratio under the influence of different beam unit parameters based on a control variable method; performing curve fitting on the damping ratio under different beam unit parameters to obtain a fitted damping ratio formula for the beam unit; obtaining the damping coefficient of the beam unit by fitting the damping ratio formula based on the expected damping ratio and the beam unit parameters; and selecting a suitable beam unit material based on the damping coefficient of the beam unit to achieve effective suppression of the vibration of the beam unit. By using the present invention, after fitting the damping ratio formula, the corresponding damping coefficient can be quickly solved based on the expected damping ratio and the beam unit parameters, thereby achieving a mitigating effect on the vibration of the beam unit.

Description

Technical Field

[0001] The present invention relates to the field of ultra-large space structure dynamics, and in particular to a space viscoelastic beam response prediction method based on an absolute node coordinate method. Background Art

[0002] Currently, large-scale lightweighting has become a key trend in spacecraft development, embodying various advanced technologies such as satellite communications, astronomical observation, space power generation, and on-orbit servicing. With the advancement of on-orbit construction and assembly technologies, the size of future space structures will reach several kilometers.

[0003] Due to the large size and flexibility of large space structures, their dynamic characteristics are more complex than those of space structures of general size. In complex space environments, large space structures may encounter problems such as large structural vibration, structural instability, and even structural damage. Therefore, it is of great significance to establish a dynamic model of ultra-large space structures and study their on-orbit dynamic behavior. The floating coordinate method is widely used in the dynamic modeling of large space structures considering the dynamic stiffening effect. However, for large deformation cases or complex spacecraft structures, this modeling process can be cumbersome. In contrast, the absolute node coordinate method (ANCF) is more suitable for ultra-large space structures because it takes into account geometric nonlinearity. The absolute node coordinate method has been applied to the dynamic studies of many space structures, such as large solar panels, tethered nets, and solar satellites.

[0004] External damping does not exist in space environments, so internal damping is an indispensable factor influencing the dynamic behavior of spatial structures. However, the ANCF damping unit is limited. The existing proportional damping model is not suitable for purely viscoelastic materials. In addition to affecting structural vibration, it also affects rigid body motion. While existing internal damping models do not affect rigid body motion, their damping ratio must be selected through trial and error, making the selection of the appropriate damping coefficient extremely inconvenient and not conducive to effectively suppressing spatial beam vibration. Summary of the Invention

[0005] In order to solve the above technical problems, the purpose of the present invention is to provide a space viscoelastic beam response prediction method based on the absolute node coordinate method, construct a dynamic equation considering the damping force, and give a solution formula for the damping force and its simplified formula without affecting the rigid body motion. On this basis, the damping ratio formula is obtained, which can realize the convenient solution of the damping coefficient, and then select the appropriate beam unit material to produce a real damping ratio close to the damping ratio formula value, thereby effectively suppressing the vibration of the beam unit.

[0006] The first technical solution adopted by the present invention is: a space viscoelastic beam response prediction method based on the absolute nodal coordinate method, comprising the following steps:

[0007] Construct a space simulation environment that takes damping forces into account;

[0008] Calculate the damping ratio under the influence of different beam element parameters based on the control variable method;

[0009] Perform curve fitting on the damping ratio under different beam element parameters to obtain the fitting damping ratio formula of the beam element;

[0010] The damping coefficient of the beam element is obtained by fitting the damping ratio formula according to the expected damping ratio and beam element parameters;

[0011] Select appropriate beam element material according to the damping coefficient of the beam element to effectively suppress the vibration of the beam element.

[0012] In this technical solution, a simulation environment is established based on the consideration of the damping force. The relationship between the damping ratio and the beam unit parameters is obtained by fitting the calculated damping ratio value, and then the corresponding damping coefficient can be directly calculated, which simplifies the solution of the damping coefficient and facilitates the selection of suitable beam unit materials, effectively suppressing the vibration of the beam unit in spatial motion.

[0013] Furthermore, the space simulation environment is established based on the absolute node coordinate method, wherein the set beam unit parameters include the global position vector r of the beam endpoint, the length l of the beam unit cross section, the cross-sectional area α, the density ρ, the mass m, the Young's modulus E and the moment of inertia I.

[0014] Furthermore, the simulation environment is built based on the dynamic equation of the beam element, which is expressed as:

[0015]

[0016] Where M is the constant mass matrix of the beam element, is the second derivative of the generalized coordinate vector of the beam element, Q ela is the generalized elastic force vector, Q c is the generalized damping force vector, Q ext is the generalized external force vector.

[0017] Furthermore, the generalized damping force vector is specifically expressed as:

[0018] Q c =Q ca +Q ct

[0019] Among them, Q ca is the generalized axial damping force, Q ct is the generalized lateral damping force.

[0020] Furthermore, the simplified damping model of the generalized damping force vector when the rotation speed of the beam element is not large or the time step is small is:

[0021]

[0022] in, i1 is the unit axial vector of the beam element, i2 is the unit transverse vector of the beam element, and the subscript (,e) indicates Represents the first derivative of the generalized coordinate vector of the beam element with respect to time.

[0023] Furthermore, the step of calculating the damping ratio under the influence of different beam unit parameters based on the control variable method specifically includes:

[0024] The logarithmic decrease rate is calculated based on the logarithmic decrease method, and the damping ratio formula is obtained;

[0025] Based on multiple parameters of beam elements, the control variable method is used to calculate the damping ratio under the influence of different beam element parameters.

[0026] Furthermore, the fitting damping ratio formula of the beam element is specifically expressed as:

[0027]

[0028] Among them, Fitting is the fitting damping ratio, θ is the constant in the fitting damping ratio formula, η t is the lateral damping coefficient of the beam element, L is the length of the beam element, E is the Young's modulus of the beam element, I is the moment of inertia of the beam element, ρ is the density of the beam element, and α is the cross-sectional area of ​​the beam element.

[0029] The beneficial effect of the space viscoelastic beam response prediction method based on the absolute node coordinate method provided by the present invention is: the present invention establishes a space simulation environment while considering the damping force, and on this basis studies the influence of the damping ratio on the vibration of the beam unit. The numerical damping ratio under different beam unit parameters is calculated by the control variable method, and the curve fitting is performed to obtain the fitting formula of the damping ratio. It can realize the simple solution of the damping coefficient corresponding to the expected damping ratio, and quickly select the required beam unit material to achieve the effect of suppressing the vibration of the beam unit. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a flowchart of the steps of the space viscoelastic beam response prediction method based on the absolute nodal coordinate method of the present invention;

[0031] Figure 2 It is a beam unit model constructed by solving the beam unit dynamic equation of the present invention;

[0032] Figure 3 It is a beam unit space model constructed based on the absolute node coordinate method of the present invention;

[0033] Figure 4It is the free-rotating beam model set by the present invention;

[0034] Figure 5 is the relative energy error of the damping model in the beam element of the present invention;

[0035] Figure 6 is the relative energy error of the RPDF of the present invention;

[0036] Figure 7 is a comparison diagram of the angular velocities of beams of three damping models of the present invention;

[0037] Figure 8 It is the free-fall beam model set by the present invention;

[0038] Figure 9 It is a comparison diagram of the total energy of the three damping models of the present invention;

[0039] Figure 10 is the absolute displacement error of the three damping models of the present invention at the center of mass;

[0040] Figure 11 This is the vibration diagram of the cantilever beam at point E of the present invention;

[0041] Figure 12 is the fitting damping ratio of the beam element under different parameters of the present invention;

[0042] Figure 13 is the deformation of the beam unit of the present invention on the circular track;

[0043] Figure 14 is the vibration of the beam with different damping ratios and lengths of the present invention;

[0044] Figure 15 is the maximum deformation of beams of different lengths of the present invention;

[0045] Figure 16 is the maximum deformation of the beam at different damping ratios of the present invention;

[0046] Figure 17 is the frequency domain of the beam when the damping ratio of the present invention is 0.01;

[0047] Figure 18 is the frequency domain of the beam when the damping ratio of the present invention is 0.1;

[0048] Figure 19 The invention discloses a space viscoelastic beam response prediction system based on the absolute node coordinate method. DETAILED DESCRIPTION

[0049] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The step numbers in the following embodiments are provided for ease of description only and do not limit the order of the steps. The order of execution of the steps in the embodiments can be adaptively adjusted based on the understanding of those skilled in the art.

[0050] like Figure 1 As shown, the present invention provides a method for predicting the response of a space viscoelastic beam based on an absolute nodal coordinate method, the method comprising the following steps:

[0051] 101. Construct a space simulation environment that takes into account the damping force of beam elements.

[0052] like Figure 2 As shown in the figure, the beam element model is first constructed based on the absolute node coordinate method and the dynamic equations of the beam element are solved. In solving the dynamic equations of the beam element, the Krone integrable theory is used.

[0053] Two matrices P∈R m×n and Q∈R s×t The Krone integrable in any dimension is:

[0054]

[0055] It can also be written as

[0056] The elastic force and damping force are derived using the following theorem:

[0057] Theorem 1:

[0058] Theorem 2:

[0059] Theorem 3: If P∈R m×1 ,but If P∈R 1×m ,but

[0060] Theorem 4: If P is a constant matrix in V, then

[0061] The global position vectors of the two endpoints A and B of the beam element AB in the global coordinate system XOY are denoted as r A 、r B , the length of the cross section is l, the cross-sectional area is α, the density is ρ, the mass is m, the Young's modulus is E, and the moment of inertia is I. The generalized coordinate vector of the beam element is expressed as:

[0062]

[0063] Where x∈[0,l] is the local coordinate of the center line. For the sake of simplicity, the subscript () x express

[0064] The global position vector of any point in the beam element is expressed as:

[0065] r=S(x)e (2)

[0066] Where S(x)∈R 2×8 is the shape function, which can be written using the Kronecker product:

[0067]

[0068] Where I2∈R 2×2 is the identity matrix, Expressed as:

[0069]

[0070] The global velocity vector is expressed as:

[0071]

[0072] Where S represents the shape function, Represents the first derivative of the generalized coordinate vector of the beam element with respect to time.

[0073] Therefore, the kinetic energy of the beam element can be calculated as follows:

[0074]

[0075] Where M = ∫ρS T SdV is the constant mass matrix of the element, V is the volume of the beam, and the mass matrix M can also be expressed as:

[0076]

[0077] According to formula (4), It can also be expressed as:

[0078]

[0079] In the beam element, assuming that the axial direction is from point A to point B, the unit axial vector i1 and the unit transverse vector i2 are specifically expressed as:

[0080]

[0081] i2=[i 21 ,i 22 ] T =[-i 12 ,i11 ] T (10)

[0082] Then the deformation vector of any point is:

[0083] u d =rr A -xi1 (11)

[0084] Therefore, the axial deformation u a and lateral deformation u t It can be expressed as:

[0085]

[0086]

[0087] According to the Euler-Bernoulli beam theory, the axial strain and curvature of the centerline are:

[0088]

[0089]

[0090] The axial stress at the centerline is expressed as:

[0091] σ=Eε (16)

[0092] The virtual work of the axial deformation of the beam element is:

[0093]

[0094] Among them, Q a is the generalized elastic force vector. According to formula (14), the variation of axial strain is:

[0095]

[0096] Among them, the subscript () x express The subscript (,e) indicates Substituting formulas (14) and (18) into formula (17), we obtain:

[0097]

[0098] in,

[0099]

[0100]

[0101]

[0102] Using the definition of Kronecker product, formula (20) becomes:

[0103]

[0104] According to Formula 2 and Theorem 3, Formula (23) is transformed into:

[0105]

[0106] And formula (21) becomes:

[0107]

[0108] Among them, r x1 and r x2 r x Get the first and second rows, S x1 and S x2 S x The first and second rows are obtained. According to the characteristics of the shape function, equation (25) can be further simplified to the following formula:

[0109]

[0110] Where, e1=[e1,e3,e5,e7] T and e2=[e2,e4,e6,e8] T are the coordinates relative to x and y, respectively, e 12 =[e1,e2]∈R 4×2 .

[0111] Define the constant matrix as follows:

[0112]

[0113]

[0114] The generalized axial elastic force vector is:

[0115]

[0116] Similarly, the transverse bending moment is defined as:

[0117] M=EIκ (30)

[0118] The virtual work of bending deformation is:

[0119]

[0120] Then the variation of the centerline curvature is:

[0121]

[0122] Therefore, the generalized bending force vector for the beam element is:

[0123]

[0124] in,

[0125] According to the above formula, the generalized elastic force vector is:

[0126] Q ela =Q a +Q t (35) Using the Lagrange equation, the dynamic equation of the beam element is:

[0127] Among them, Q c is the generalized damping force vector, Q ext is the generalized external force vector.

[0128] At this point, the dynamic equations of the beam element are completed. Next, we solve the generalized damping force vector Q c .

[0129] First, the axial damping stress and bending damping moment are expressed as:

[0130]

[0131]

[0132] where η a and η t are the axial damping coefficient and lateral damping coefficient related to the material, respectively.

[0133] According to formulas (14) and (15), the strain rate and Respectively expressed as:

[0134]

[0135]

[0136] The virtual work of the axial damping force and bending damping moment is:

[0137]

[0138]

[0139] According to formulas (18), (32), (37)-(42), the generalized axial damping force Q can be calculated ca and the generalized lateral damping force Q ct :

[0140]

[0141]

[0142] Using the same steps as formulas (23)-(26), solve for Q ca and Q ct , specifically expressed as:

[0143]

[0144]

[0145] Then, the generalized damping force vector is:

[0146] Q c =Q ca +Q ct (47)

[0147] Since the effect of axial damping on the beam can be neglected in most cases, the effect of lateral deformation is dominant. Therefore, the formula can be simplified to:

[0148] Q c =Q ct (48)

[0149] From formula (40), we can see that It consists of two items. The second term of represents the time derivative of κ in the constant local coordinate system Ai1i2. The first term κ represents the time derivative due to the rotation of the local coordinate system Ai1i2. In general, when the beam rotation speed is not large or the time step is small, the first term in formula (47) can be ignored. Therefore, the damping force can be simplified to:

[0150]

[0151] Among them, formula (48) is called the precise damping model (PDM), and formula (49) is called the simplified damping model (SDM).

[0152] like Figure 3 As shown in the figure, the space beam element model considers the distributed gravity gradient force and viscoelastic damping to study the sun-oriented flexible beam in circular orbit. This model is simplified from the support truss of the solar array of a large solar satellite.

[0153] The origin of the inertial coordinate system OXY is located at the Earth's center of mass. Points D and E are the endpoints of the beam element, and point C is the midpoint of the beam element. A local coordinate system CXY is established to describe the deformation of the beam, with its x-axis tangential to point C. The modeling is performed using viscoelastic ANCF beam elements with a simplified damping model.

[0154] The gravitational force of the ANCF beam element is derived from the gravitational potential energy of the beam element:

[0155]

[0156] Where μ = 3.986 × 10 14 m 3 ·s -2 is the Earth's gravitational constant.

[0157] The generalized universal gravitation of the beam element is:

[0158]

[0159] The approximate expression of formula (51) is obtained by using the second-order Taylor series expansion, which can be applied to various ANCF units. The approximate expression of formula (51) is:

[0160]

[0161] Where r0 is the global position vector of the beam element's center of mass. This gravity model is used because the effects of the gradient torque caused by non-uniform gravity and non-uniform centrifugal force are considered in the formula.

[0162] At this point, the dynamic equations for the ANCF beam element in space have been constructed. Next, the approximate orbit and initial conditions for the beam element are defined.

[0163] Since orbital perturbations are not considered in this paper, the beam will move along a Keplerian orbit. Assume that the initial condition is a circular orbit, where ω0 is the initial orbital angular velocity and r0 is the initial orbital radius. In terms of attitude, an attitude controller must be used to offset the gravity gradient torque and maintain a sun-facing attitude. Assume that the attitude of point C is fully controlled and use geometric constraints in the simulation to replace the implementation of the attitude controller. Let:

[0164] e 2k+3 =0

[0165] where e represents the generalized coordinate vector of the entire beam, and k is the number of elements. The constraint here is that the tangent direction at point C is parallel to the Y axis.

[0166] At this point, the construction of the space simulation environment considering the damping force of the beam unit is completed.

[0167] In this embodiment, the proposed damping force model has the advantage of not affecting the motion of the rigid body, which is verified by combining data below.

[0168] like Figure 3 As shown in the figure, in the simulation model of a freely rotating beam, the results of SDM and PDM are compared with the Rayleigh Proportional Damping Model (RPDM). The calculation formula of the RPDM damping matrix is:

[0169] C=k1M+k2K (53)

[0170] Where k1 and k2 are two constants, K is the linear stiffness matrix of the ANCF unit in the undeformed configuration, and k1 = 0.05 and k2 = 0 are used.

[0171] In a free-rotating beam, the beam rotates around its center (point C) with an initial angular velocity of ω = 10 rad / s, a length of L = 10 m, a Young's modulus of E = 70 GPa, and a cross-sectional area of ​​α = 1.236 × 10 -3 m 2 Density ρ = 2700 kg / m 3 , moment of inertia I = 2.187 × 10 - 6 m 4 , lateral damping coefficient η t = 0.01. The model does not consider external forces, and the beam is initially undeformed. The beam is divided into 4 units, the time step is set to 0.0001s, the simulation time is 1s, and the relative energy error is defined as:

[0172]

[0173] Where H is the total energy of the beam (including kinetic energy and potential energy), and H0 is the initial value of H.

[0174] Simulation comparison chart as follows Figure 4-Figure 6 As shown. Figure 4 and Figure 5 It can be seen that the relative energy errors of the three models increase linearly with time. relative The maximum value is less than 2×10 -8 , ΔH of RPDM relative The maximum value is greater than 8×10 -2 Therefore, the energy dissipation of SDM and PDM is very small compared with that of RPDM. The angular velocity of the beam is Figure 6 As shown in Figure 2, the viscoelastic damping force of the SDM and PDM has a negligible effect on the angular velocity, which is ω = 9.5 rad / s at the end of the simulation. In contrast, the angular velocity of the RPDM decreases linearly with time.

[0175] like Figure 7 As shown in the figure, in the free-fall beam model, the beam is placed horizontally and initially has no deformation. Other beam parameters are the same as those in the free-rotation beam model. The gravity acceleration in the Y direction is set to g = -9.8 m / s. 2 , the absolute displacement error is defined as:

[0176]

[0177] The simulation results are shown in the figure Figure 8-Figure 9 As shown in Figure 3, the total energy of SDM and PDM remains basically unchanged, while the total energy of RPDM decreases rapidly. At the same time, the absolute displacement error of RPDM increases with time.

[0178] Based on the two simulation models above, we can see that the rigid body motion of the beam is not affected by the SDM and PDM, but by the RPDM. Therefore, the proposed SDM and PDM can be applied to the case without environmental damping and have certain advantages.

[0179] 102. Calculate the damping ratio under the influence of different beam unit parameters based on the control variable method.

[0180] Since the damping ratio is affected by boundary conditions and many other parameters such as L, ρ, α, η t , E, I, etc. Therefore, this paper uses the control variable method to study the damping ratio of the beam. In the numerical simulation, the numerical damping ratio is estimated by the logarithmic decrease method. Figure 2 As shown in , it is assumed that the beam element in the model is a cantilever beam, and the vibration of the end point E is used for research. Figure 10 As shown in Figure 2, when the simulation time is long enough, the stable deformation of point E on the Y axis can be obtained. Then, the vibration amplitude A1 of the first cycle and the vibration amplitude A of the nth cycle are calculated. n The logarithmic decreasing rate δ can be calculated as follows:

[0181]

[0182] If the logarithmic decay rate of the amplitude is much less than 2π, the numerical damping ratio of the cantilever beam can be obtained by the following formula:

[0183]

[0184] Table 1

[0185]

[0186]

[0187] Based on the parameters of the beam element in Table 1, the damping ratio of the cantilever beam under the influence of different parameters is calculated using the control variable method. The numerical damping is as follows: Figure 11 shown.

[0188] 103. The damping ratio under different beam parameters is fitted by curve fitting to obtain the fitting damping ratio formula of the beam unit.

[0189] According to the simulation results obtained in the previous step Figure 11 , a simple fitting damping ratio formula is proposed:

[0190]

[0191] In a large number of examples considering parameters of different orders of magnitude, numerical Compare and examine the accuracy of the fitting formula. The results are as follows Figure 11 As shown, it can be seen that in all simulations, ζ Fitting With ζ numerical Highly consistent. Figure 11 In all simulations, ζ Fitting With ζ numerical The maximum relative error is less than 0.7%, indicating the correctness of the formula. Therefore, the lateral damping coefficient of the cantilever beam can be selected according to the damping ratio specified in formula (58).

[0192] In addition, the damping ratio of the simply supported beam under different parameters is also studied. According to the simulation parameters in Table 2 and the numerical damping ratio under the corresponding parameters, a simple expression for the damping ratio of the simply supported beam is obtained:

[0193]

[0194] The damping ratio fitted by the formula is shown in Table 2. Fitting With ζ numerical The relative error is less than 3%, which shows the accuracy of the formula.

[0195] Table 2

[0196] Group L(m) <![CDATA[ρ(kg / m 3 )]]> <![CDATA[a(m 2 )]]> <![CDATA[η t ]]> <![CDATA[I(m 4 )]]> E(GPa) <![CDATA[ζ Simulation ]]> <![CDATA[ζ Fitting ]]> 1 10 4000 0.001 0.02 <![CDATA[1×10 -8 ]]> 70 <![CDATA[1.34×10 -2 ]]> <![CDATA[1.33×10 -2 ]]> 2 12 2700 0.001 0.02 <![CDATA[1×10 -8 ]]> 70 <![CDATA[1.14×10 -2 ]]> <![CDATA[1.12×10 -2 ]]> 3 10 2700 0.001 0.02 <![CDATA[1×10 -8 ]]> 10 <![CDATA[6.14×10 -3 ]]> <![CDATA[6.10×10 -3 ]]> 4 10 2700 0.001 0.02 <![CDATA[1×10 -8 ]]> 70 <![CDATA[1.61×10 -2 ]]> <![CDATA[1.61×10 -2 ]]> 5 10 2700 0.0001 0.02 <![CDATA[1×10 -8 ]]> 70 <![CDATA[5.00×10 -2 ]]> <![CDATA[5.10×10 -2 ]]> 6 10 2700 0.001 0.04 <![CDATA[1×10 -8 ]]> 70 <![CDATA[3.23×10 -2 ]]> <![CDATA[3.23×10 -2 ]]> 7 10 2700 0.001 0.02 <![CDATA[1×10 -7 ]]> 70 <![CDATA[5.03×10 -2 ]]> <![CDATA[5.10×10 -2 ]]>

[0197] Next, we study the influence of the damping ratio on the vibration of the beam unit during its motion in space. Figure 2 As shown in Table 3, the parameters of the beam element based on ANCF are set as follows. The initial damping ratio ζ of the beam element is set to 0, the length L is set to 552m, the beam element in the figure is divided into 10 small elements, and the simulation time is selected to be long enough to show the change of the vibration amplitude of the beam element.

[0198] Table 3

[0199] Parameter Value E 70GPa a <![CDATA[3.86×10 -4 m 2 ]]> ρ <![CDATA[2.7×10 3 kgm 3 ]]> I <![CDATA[7.2468×10 -8 m 4 ]]> <![CDATA[r0]]> 6700km

[0200] The simulation diagram of the beam unit running one circle on the preset track is as follows Figure 12As shown in the figure, it can be seen that the deformation of the beam is significant under the influence of the gravity gradient. Due to the constraint in the formula, the tangent direction of the midpoint remains parallel to the axis, and the deformation of the beam tends to point toward the earth. Since pointing toward the earth is the stable equilibrium posture of the beam, the maximum deformation of the beam occurs at point D or point E.

[0201] Observe the vibration of beam elements of different lengths by adding different damping ratios, such as Figure 13 As shown in the figure, the damping ratio has little effect on the 200-m beam. The high-frequency vibrations of the 200-m beam decay more quickly when ζ = 0.1, while they persist longer when ζ = 0.01. After one orbital cycle (denoted as ), the vibration amplitudes of beams with all damping ratios are around 0.4 m, primarily influenced by the gravity gradient. For the 468-m beam, when ζ = 0.01, the vibration amplitude increases significantly from 43 m in the first cycle to a maximum at 127 m. This behavior resembles a resonance. In contrast, the vibration amplitude is approximately 45 m when ζ = 0.1. The dynamic response of the 552-m beam is similar to that of the 468-m beam. When ζ = 0.01, the vibration amplitude gradually increases over 35 orbital cycles from 114 m to 180 m. However, when ζ = 0.1, the vibration amplitude is less than 100 m after one orbital cycle.

[0202] In addition, if Figure 14 and Figure 15 As shown in the figure, different damping ratios significantly affect the maximum vibration amplitude of beams of different lengths. Specifically, when ζ = 0.01, two resonance points appear near 468m and 552m. When L = 468m, the maximum vibration amplitude of the beam gradually decreases with increasing damping ratio, while the maximum vibration amplitude of the beam at 552m decreases sharply. When the length is less than 450m, the damping ratio has little effect on the vibration amplitude.

[0203] In order to further study the effect of damping ratio, as Figure 16 and Figure 17 As shown in Figure 1, the vibration of the beam under different damping ratios and lengths is transformed into the frequency domain using fast Fourier transform. Symbol A m and f represent the amplitude and frequency respectively. It can be seen that the main vibration frequencies of the beam are 2ω0, 4ω0 and is the lowest natural frequency of the beam with midpoint restraint). The frequencies of 2ω0 and 4ω0 are a result of the gravity gradient. Figure 16 It can be seen that there is a steep peak at the intersection of ω1 and 4ω0. That is, the frequency 4ω0 of the gravity gradient causes the beam to resonate when ζ = 0.01. However, if Figure 17 As shown, the peak is suppressed when ζ = 0.1. In addition, Figure 17 Surface ratio Figure 16 The surface is smoother, Figure 16The surface of the vibration case is smoother than that of the undamped case. The results show that viscoelastic damping has an important influence on the performance of ultra-large space structures and should be taken seriously in structural design.

[0204] 104. The damping coefficient of the beam element is obtained by fitting the damping ratio formula based on the expected damping ratio and beam element parameters.

[0205] Based on the simulation and analysis in the previous step, the damping ratio has a significant effect on the suppression of beam vibration. Therefore, it is crucial to select the corresponding damping coefficient based on the expected damping ratio. According to formulas (58) and (59), the general damping ratio formula for beam elements is summarized as:

[0206]

[0207] First, the numerical damping ratio of the corresponding beam element under different parameters is calculated based on the logarithmic decrease method. Then, the corresponding damping ratio formula is obtained through curve fitting. The expected damping ratio value and the parameters of the corresponding beam element are input to calculate the required damping coefficient.

[0208] 105. Select appropriate beam unit materials according to the damping coefficient of the beam unit to effectively suppress the vibration of the beam unit.

[0209] After calculating the required damping coefficient, the beam element material related to the required damping coefficient is selected. In this way, in the real scene, the beam element will produce a real damping ratio close to the expected damping ratio, thereby reducing the vibration of the beam element and reducing the deformation of the beam element during track operation.

[0210] like Figure 19 As shown, a space viscoelastic beam response prediction system based on the absolute node coordinate method includes:

[0211] Build a simulation environment module and construct a simulation environment for beam elements in space based on the absolute node coordinate method;

[0212] The damping ratio calculation module uses the control variable method and the logarithmic reduction method to calculate the damping ratio under different beam element parameters;

[0213] The damping ratio fitting module performs curve fitting on the damping ratio under different beam element parameters to obtain the fitting damping ratio formula of the beam element;

[0214] The damping coefficient solving module obtains the damping coefficient of the beam element by fitting the damping ratio formula according to the expected damping ratio and beam element parameters;

[0215] The beam unit selection module selects appropriate beam unit materials according to the damping coefficient of the beam unit to effectively suppress the vibration of the beam unit.

[0216] The contents of the above method embodiments are all applicable to the present system embodiments. The functions specifically implemented by the present system embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0217] A control device for a viscoelastic beam element:

[0218] at least one processor;

[0219] at least one memory for storing at least one program;

[0220] When the at least one program is executed by the at least one processor, the at least one processor implements the control method of the viscoelastic beam unit as described above.

[0221] The contents of the above method embodiments are all applicable to the present device embodiments. The functions specifically implemented by the present device embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0222] The above is a specific description of the preferred implementation of the present invention, but the invention is not limited to the embodiments. Those skilled in the art can make various equivalent modifications or substitutions without violating the spirit of the present invention. These equivalent modifications or substitutions are all included in the scope defined by the claims of this application.

Claims

1. A method for predicting the response of a space viscoelastic beam based on the absolute nodal coordinate method, characterized in that: The following steps are involved: Construct a space simulation environment that considers the damping force of beam elements; Calculate the damping ratio under the influence of different beam element parameters based on the control variable method; Perform curve fitting on the damping ratio under different beam element parameters to obtain the fitting damping ratio formula of the beam element; The damping coefficient of the beam element is obtained by fitting the damping ratio formula according to the expected damping ratio and beam element parameters; Select appropriate beam element materials according to the damping coefficient of the beam element to effectively suppress the vibration of the beam element; The simulation environment is built based on the dynamic equation of the beam element, which is expressed as: Where M is the constant mass matrix of the beam element, is the second derivative of the generalized coordinate vector of the beam element with respect to time, Q ela is the generalized elastic force vector, Q c is the generalized damping force vector, Q ext is the generalized external force vector; The generalized damping force vector Q c Specifically expressed as: Q c =Q ca +Q ct ; Among them, Q ca is the generalized axial damping force, Q ct is the generalized lateral damping force.

2. The method for predicting the response of a space viscoelastic beam based on the absolute nodal coordinate method according to claim 1, characterized in that: The space simulation environment is established based on the absolute node coordinate method, in which the set beam unit parameters include the global position vector r of the beam endpoint, the length l of the beam unit cross section, the cross-sectional area α, the density ρ, the mass m, the Young's modulus E and the moment of inertia I.

3. The method for predicting the response of a space viscoelastic beam based on the absolute nodal coordinate method according to claim 1, characterized in that: The step of calculating the damping ratio under the influence of different beam unit parameters based on the control variable method specifically includes: The logarithmic decrease rate is calculated based on the logarithmic decrease method, and the damping ratio formula is obtained; Based on multiple parameters of beam elements, the control variable method is used to calculate the damping ratio under the influence of different beam element parameters.

4. The method for predicting the response of a space viscoelastic beam based on the absolute nodal coordinate method according to claim 1, characterized in that: The fitting damping ratio formula of the beam element is specifically expressed as: Among them, Fitting is the fitting damping ratio, θ is the constant in the fitting damping ratio formula, η t is the lateral damping coefficient of the beam element, L is the length of the beam element, E is the Young's modulus of the beam element, I is the moment of inertia of the beam element, ρ is the density of the beam element, and α is the cross-sectional area of ​​the beam element.

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