Method and device for analyzing energy dissipation characteristics of weakly damped nonlinear energy sink system

Abstract

The embodiment of the application discloses a weak damping nonlinear energy sink system energy dissipation characteristic analysis method and device, the analysis method comprises the following steps: establishing a damped nonlinear energy sink system model; based on the damped nonlinear energy sink system model, a weak damping nonlinear energy sink system model is established, and the energy dissipation characteristics of the weak damping nonlinear energy sink system are analyzed; a segmented stiffness is introduced to the weak damping nonlinear energy sink system to establish a weak damping segmented stiffness nonlinear energy sink system, correspondingly, a topological characteristic model of the weak damping segmented stiffness nonlinear energy sink system is established, and the energy dissipation characteristics of the weak damping segmented stiffness nonlinear energy sink system are analyzed. The energy dissipation characteristics of the weak damping segmented stiffness nonlinear energy sink system are analyzed through the research on the dynamic characteristics and vibration suppression effect of the segmented stiffness nonlinear energy sink system coupled with a single degree of freedom main system.

Description

Technical Field

[0001] This invention relates to the field of nonlinear energy traps, and in particular to a method and apparatus for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system. Background Technology

[0002] Current research on piecewise stiffness models mainly focuses on numerical simulation and experimental verification. It compares the frequency response characteristics before and after the introduction of piecewise stiffness using frequency response characteristics and frequency sweep tests, and analyzes the impact response of the system before and after the introduction of piecewise stiffness using external impact excitation. However, there is relatively little analysis on the approximate analytical solution of piecewise stiffness systems.

[0003] In studies that analyze piecewise stiffness analytical solutions, few studies have investigated the energy dissipation path during the damped process of the system by analyzing the topology of the undamped free vibration process of the system. Summary of the Invention

[0004] To address the aforementioned technical problems, embodiments of the present invention aim to provide a method and apparatus for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system. The technical solution of the present invention is implemented as follows:

[0005] In a first aspect, the present invention provides a method for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system, the method comprising the following steps:

[0006] S101. Establish a model of a damped nonlinear energy trap system;

[0007] S102. Based on the damped nonlinear energy trap system model, establish a weakly damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system.

[0008] S103. Introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system. Accordingly, establish a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system and analyze the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system.

[0009] Secondly, the present invention also provides an apparatus for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system. The apparatus includes a first establishment section, a first establishment and analysis section, and a second establishment and analysis section. The first establishment section is configured to establish a damped nonlinear energy trap system model. The first establishment and analysis section is configured to establish a weakly damped nonlinear energy trap system model based on the damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system. The second establishment and analysis section is configured to introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system, and correspondingly establish a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system, and analyze the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system.

[0010] This invention provides a method for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system. Addressing the insufficient research on combined stiffness nonlinear energy trap (NES) systems combining cubic stiffness and piecewise stiffness, this invention analyzes the energy dissipation characteristics of a weakly damped piecewise stiffness nonlinear energy trap system by studying the dynamic characteristics and vibration suppression effect of a piecewise stiffness nonlinear energy trap system coupled with a single-degree-of-freedom master system. Attached Figure Description

[0011] Figure 1 This is a schematic diagram of the energy dissipation characteristic analysis method of a weakly damped nonlinear energy trap system according to an embodiment of the present invention;

[0012] Figure 2 This is a schematic diagram of a weakly damped piecewise stiffness nonlinear energy trap system model in an embodiment of the present invention;

[0013] Figure 3 This is a schematic diagram of the energy dissipation ratio of the weakly damped piecewise stiffness nonlinear energy trap system in an embodiment of the present invention.

[0014] Figure 4 This is the definition of the three-dimensional energy column space coordinate system in the embodiments of the present invention;

[0015] Figure 5 This is the energy dissipation trajectory of the low-energy-level weakly damped nonlinear energy trap system in this embodiment of the invention;

[0016] Figure 6 The energy dissipation trajectory of the weakly damped nonlinear energy trap system at the middle energy level in this embodiment of the invention;

[0017] Figure 7 This is the energy dissipation trajectory of a higher-level weakly damped nonlinear energy trap system in an embodiment of the present invention;

[0018] Figure 8The energy dissipation trajectory of the high-energy-level weakly damped nonlinear energy trap system in this embodiment of the invention;

[0019] Figure 9 This is the energy dissipation trajectory of a weakly damped nonlinear energy trap system without piecewise stiffness in an embodiment of the present invention.

[0020] Figure 10 This invention provides the energy dissipation trajectory of a weakly damped nonlinear energy trap system with piecewise stiffness in an embodiment of the invention.

[0021] Figure 11 The energy dissipation trajectory of a weakly damped nonlinear energy trap system under the condition of introducing additional linear stiffness in an embodiment of the present invention;

[0022] Figure 12 The changes in the orbital branching characteristics of the weakly damped nonlinear energy trap system before and after the introduction of segmented stiffness in this embodiment of the invention;

[0023] Figure 13 This is the energy level decomposition and corresponding feature map of the dissipation trajectory under the first operating condition in this embodiment of the invention;

[0024] Figure 14 The input energy-decrease time characteristics of the system after introducing piecewise stiffness in the second working condition in this embodiment of the invention;

[0025] Figure 15 This is a schematic diagram of an energy dissipation characteristic analysis device for a weakly damped nonlinear energy trap system according to an embodiment of the present invention. Detailed Implementation

[0026] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0027] See appendix Figure 1 The diagram illustrates a flowchart of a method for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system, which includes the following steps:

[0028] S101. Establish a model of a damped nonlinear energy trap system;

[0029] S102. Based on the damped nonlinear energy trap system model, establish a weakly damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system.

[0030] S103. Introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system. Accordingly, establish a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system and analyze the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system.

[0031] See appendix Figure 2 It shows a schematic diagram of a damped nonlinear energy trap system, for example, as shown in the attached diagram. Figure 2 The system shown has a damped single-degree-of-freedom linear oscillator, coupled through a damper, a segmented stiffness spring, and an ungrounded attachment (NES) to obtain a two-degree-of-freedom system. Based on the attachment... Figure 2 For the system shown, establish the equation of motion (1) for the damped nonlinear energy trap system, which is expressed as:

[0032]

[0033]

[0034] Where m1 and m2 are the masses of the linear principal system and the NES oscillator, respectively; c1 and c2 are the damping of the linear principal system and the damping of the NES oscillator, respectively; k1 and k2 are the stiffness of the linear principal system and the cubic stiffness term of the connection between the NES oscillator and the linear principal system, respectively; x is the displacement of the linear principal system; and v is the displacement of the NES oscillator. By canceling the mass term, the equation of motion (1) is simplified to obtain the equation of motion (2):

[0035]

[0036]

[0037] in,

[0038] The proportion of energy dissipated on a nonlinear energy trap (NES) oscillator is defined as:

[0039]

[0040] See appendix Figure 3 , attached Figure 3 A schematic diagram of the energy dissipation ratio of a segmented stiffness NES is shown. By inputting different initial energy levels, the changes in the energy dissipation characteristics on the NES attachment are observed. Figure 3 It is evident that the energy dissipation characteristics of the linear master system and the NES accessory differ depending on the input energy level. Within a certain timeframe, the energy on the NES can even constitute the majority of energy dissipation. However, when the energy level of the linear master system increases to a certain threshold, the energy dissipation proportion of the NES system increases significantly. Frequency characteristics also have a substantial impact on the energy dissipation efficiency of the NES, and a certain relationship exists between frequency and energy, jointly influencing the vibration suppression characteristics of the system.

[0041] Under the assumption that the motion trajectory under weak damping is a family of undamped trajectories spliced ​​together, a model of a weakly damped nonlinear energy trap system is established based on the damped nonlinear energy trap system model shown in the motion equation (1), and the energy dissipation characteristics of the weakly damped nonlinear energy trap system are analyzed.

[0042] Referring to equation (2), considering the damped nonlinear energy trap system as a weakly damped nonlinear energy trap system, and assuming λ1 = 0 and λ2 is a weakly damped term, the equation (3) of motion for the weakly damped nonlinear energy trap system model is obtained:

[0043]

[0044]

[0045] Considering the dynamic characteristics of the S11 branch, and assuming that the motion trajectory under weak damping is a family of undamped trajectories spliced ​​together, where S11 refers to the trajectory corresponding to the motion characteristics in phase space, S11+ refers to the trajectory corresponding to the same-phase motion characteristics in phase space, and S11- refers to the trajectory corresponding to the opposite-phase motion characteristics in phase space:

[0046]

[0047] Since a damping term is introduced, the total energy conservation of the weakly damped nonlinear energy trap system no longer holds; therefore, an energy evaluation term is introduced:

[0048] Where r is a parameter that measures the total energy.

[0049] The formula used to describe the relationship between the amplitude of the linear master system and the nonlinear energy trap oscillator. And the energy evaluation term (7) is substituted into the kinematic equation (3) of the weakly damped nonlinear energy trap system shown in formula (5), where a1 is the complex amplitude A of the principal system and a2 is the complex amplitude B of the nonlinear energy trap. The model of the weakly damped nonlinear energy trap system is simplified to a model in spherical coordinates (R + ×S 1 ×S 1 For a three-dimensional dynamic system on a surface, the diffuse dynamic model can be obtained as shown in expression (8):

[0050]

[0051]

[0052]

[0053] As can be seen, compared with the undamped system model (8-2) in the prior art, the energy consumption rate of the damped system is... Two damping-related quantities were added to represent the contributions of the two different dampers to the system's energy dissipation. When the damping is zero, the system degenerates into the energy-conserving state of an undamped system. The variables ψ, expressing the energy ratio between the main system and the NES, and Φ, the phase difference between the two oscillators, also have a certain impact after the damping is added.

[0054]

[0055]

[0056] To more clearly describe the energy transfer process of the system and obtain the energy dissipation trajectory of the damped nonlinear energy trap system, a topological transformation is performed on the cylindrical space to which the flow dynamics model shown in expression (8) is converted to obtain the topological characteristic model of the weakly damped nonlinear energy trap system. Specifically, the original model mapped to the spherical coordinate system (R) is transformed. + ×S 1 ×S 1 The system model was changed to a new cylindrical coordinate system, and the original angular quantity ψ(S) was changed to a new cylindrical coordinate system. 1 The polar moment of the transformation is π / 2 - ψ(R) + The definition of the cylindrical space coordinate system is as follows: Figure 4 As shown, the cylindrical space coordinate system adds a height coordinate axis representing energy r to the two-dimensional unit disk. When the trajectory is closer to the center (ψ→π / 2), the energy is more concentrated in the main system; conversely, when the trajectory is closer to the edge of the disk (ψ→-π / 2), the energy is more concentrated on the NES attachment.

[0057] Next, by changing the energy level of the input energy of the weakly damped nonlinear energy trap system, the energy dissipation characteristics of the system are analyzed. See Appendix for details. Figure 5 To be continued Figure 8 The diagrams show the energy dissipation trajectories of a damped system at low, medium, and high energy levels, respectively.

[0058] When the input energy is at the lower energy level r = 0.2, see Appendix. Figure 5 It can be seen that in the extremely low energy level state, the system motion process is basically a small-range motion around the Z-axis (i.e. the main system), and there is almost no energy transfer between the main system and the NES attachment, that is, it does not have the conditions for energy transfer in the nonlinear energy trap.

[0059] When the input energy is at the intermediate level r = 0.5, see Appendix. Figure 6It can be seen that in the middle energy level, the system energy is gradually transferred from the main system (i.e., near the Z-axis) to the NES (external), proving that compared with the low energy level, energy begins to be transferred from the main system to the NES. However, it can be seen that the energy returns to the main system earlier, indicating that it is in a transitional state in which energy exchange can occur.

[0060] When the input energy is at a higher energy level r=1, see Appendix. Figure 7 As can be seen, under higher energy input conditions, when the total energy of the system is still relatively large, the energy begins to be rapidly transferred from the main system to the NES attachment. This is manifested as the initial point of the trajectory being at the center of the disk (i.e., the energy is in the main system). During the motion, when the vertical coordinate is higher, it moves closer to the edge of the disk, representing the energy distribution gradually moving closer to the NES. Finally, when the total energy of the entire system approaches 0, the energy proportion of the main system gradually recovers to a higher proportion, which is manifested as a spiral trajectory on the r=0 plane returning to the center. This indicates that under higher energy conditions, there is a characteristic of directional energy transfer. Although the directional transfer is manifested as a spiral towards the NES attachment in the trajectory, it also reflects the situation where the beat frequency will occur during the energy transfer process under damped dissipation.

[0061] When the input energy is at a higher energy level r=3, see Appendix. Figure 8 As can be seen, when the input is at a high energy level, the directional transfer of energy is more pronounced when the total energy of the system is higher. At the same time, the oscillation state generated by the trajectory in this process is relatively smaller, which more intuitively reflects the characteristic of energy transfer to the NES after the system input energy reaches the threshold.

[0062] After introducing a weakly damped term, the original slow-flow dynamics model is transformed into cylindrical space by performing a topological transformation on the slow-flow dynamics of the system. This allows for a qualitative analysis of the influence of the system's input energy characteristics on its dynamic features. It is easy to see that the dynamic characteristics in the intermediate energy level are quite complex, and the system trajectory can be distributed in different energy proportion regions and undergo transformations.

[0063] Based on the established weakly damped nonlinear energy trap system and the analysis of its energy dissipation characteristics, the present invention provides a method for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system, which further includes: introducing piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system; correspondingly, establishing a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system based on the piecewise stiffness; and analyzing the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system.

[0064] First, when establishing a weakly damped piecewise stiffness nonlinear energy trap system, a piecewise stiffness kn is introduced into the weakly damped piecewise stiffness nonlinear energy trap system to obtain the dynamic model of the weakly damped piecewise stiffness nonlinear energy trap system. The equation of motion (9) of the weakly damped piecewise stiffness nonlinear energy trap system is as follows:

[0065]

[0066]

[0067] Using the transformation shown in equation (9-1), the motion equation (9) is converted into the equation of the displacement of the system's center of mass and the relative displacement of the NES oscillator, resulting in equation (10) as shown in expression (10):

[0068]

[0069] Among them, the new variable w1 represents the motion characteristics of the center of mass during the system's motion, and w2 represents the relative displacement between the main system and the NES oscillator.

[0070]

[0071]

[0072] in,

[0073] k0 — the linear stiffness between the linear master system and the nonlinear energy trap oscillator;

[0074] λ — the damping of the nonlinear energy trap oscillator;

[0075] k3 — the cubic stiffness term of the connection between the nonlinear energy trap oscillator and the linear master system;

[0076] k n —The piecewise stiffness term of the connection between the nonlinear energy trap oscillator and the linear master system;

[0077] L—the gap between the nonlinear energy trap oscillator and the linear main system;

[0078] ε—the mass ratio of the nonlinear energy trap oscillator to the linear master system;

[0079] y1 — the centroid displacement of the linear principal system;

[0080] y2 — the displacement of the nonlinear energy trap oscillator relative to the linear master system.

[0081] Transformation is performed by introducing complex variables as shown in expression (11):

[0082]

[0083] Among them, for complex variables φ1 and φ2, the complex amplitudes A and B and the phase variables α and β are used to represent their complex amplitude characteristics, and the expression (12) for complex variables φ1 and φ2 can be obtained:

[0084] φ1=Ae jα ,φ2=Be jβ (12)

[0085] Based on the above equations, the slow-flow dynamics model of the system is obtained (13):

[0086]

[0087]

[0088]

[0089]

[0090] The first two terms of the slow-flow dynamics model (13) still correspond to the total energy characteristics of the system. Due to the introduction of the damping term, the system energy is no longer conserved. Consider using the variable substitution shown in expression (14):

[0091] r 2 =A 2 +εB 2

[0092] φ=α-β

[0093]

[0094] In the variable expression (14), the first term corresponds to the total energy characteristic of the weakly damped piecewise stiffness nonlinear energy trap system, the second term corresponds to the phase difference characteristic of the weakly damped piecewise stiffness nonlinear energy trap system, and the third term corresponds to the energy characteristics occupied by the linear main system and the nonlinear energy trap oscillator in the weakly damped piecewise stiffness nonlinear energy trap system. Accordingly, the closer ψ is to -π / 2, the more concentrated the energy is in the NES; conversely, the closer ψ is to π / 2, the more concentrated the energy is in the main system. Substituting the variable expression (14) into the slow-flow dynamics equation of the transformed system represented by the slow-flow dynamics model (13), we can obtain the system's diffuse-flow dynamics equation (15):

[0095]

[0096]

[0097]

[0098]

[0099] The energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap were analyzed using the cylindrical space coordinate system definition and topological transformation as described above. The influence of introducing piecewise stiffness on the characteristics of the weakly damped piecewise stiffness nonlinear energy trap system was studied by setting different operating conditions for comparison. (See Appendix) Figure 9 To be continued Figure 11 The diagrams illustrate the energy dissipation trajectories of a weakly damped nonlinear energy trap system without piecewise stiffness, with piecewise stiffness, and with additional linear stiffness, respectively. The linear stiffness is equal to the piecewise stiffness.

[0100] In the first working condition, the cubic stiffness k3 = 0.1, and the piecewise stiffness k n =0, mass ratio ε = 0.05, linear stiffness k0 = 0.5, input system energy characteristic r0 = 1, gap value L = 0.2, damping ratio λ = 0.1. The system energy is initially mainly concentrated on the main system. At this time, piecewise stiffness has not yet been introduced. Observe the topological characteristics of the system trajectory. See Appendix. Figure 9 Without a piecewise stiffness term, the energy dissipation trajectory of the system first shifts outward from the main system, then gradually descends along a spiral trajectory to the zero energy surface at r=0, and converges to a point relatively close to the middle of the plane. The convergence point is still inside ψ=0, meaning that a large part of the energy is still concentrated in the main system. This shows that, without piecewise stiffness, although the energy dissipates to the NES and dissipates some energy, the energy of the main system still accounts for a high proportion in the final stage of dissipation.

[0101] In the second working condition, the cubic stiffness k3 = 0.1, and the piecewise stiffness k n =0.5, mass ratio ε =0.05, linear stiffness k0 =0.5, input system energy characteristic r0 =1, gap value L =0.2, damping ratio λ =0.1. The system energy is initially mainly concentrated on the main system. At this time, piecewise stiffness is introduced into the system. Observe the topological characteristics of the system trajectory. See Appendix. Figure 10After introducing piecewise stiffness, the energy dissipation trajectory of the system gradually shifts significantly outward from the main system and gradually crosses the boundary of ψ=0. The energy is mainly concentrated on the NES (Neural Energy Element) attachments. Subsequently, it gradually descends along a spiral trajectory to the zero energy surface of r=0 and hovers around a point relatively close to the boundary of the plane (the energy is still mainly concentrated on the NES). Finally, it returns to a point near the main system when it is extremely close to the zero energy surface. This shows that after introducing piecewise stiffness, compared with the energy dissipation trajectory of a system containing only cubic stiffness, a larger portion of the energy is dissipated on the NES attachments, the energy proportion on the NES is higher, and the response of the main system is better suppressed.

[0102] In the third working condition, the cubic stiffness k3 = 0.1, and the piecewise stiffness k n =0, mass ratio ε = 0.05, linear stiffness k0 = 1, input system energy characteristic r0 = 1, gap value L = 0.2, damping ratio λ = 0.1. The system energy is initially mainly concentrated on the main system. At this time, no piecewise stiffness is introduced into the system, but a linear stiffness with the same value as the piecewise stiffness is introduced to verify that the better effect of NES is due to the piecewise characteristics of the piecewise stiffness, rather than the effect of introducing an additional linear stiffness value. See Appendix. Figure 11 When introducing a piecewise stiffness term k n With the same linear stiffness, the energy transfer characteristics of the system are greatly changed, causing the energy dissipation trajectory to enter the convergence point near the main system earlier, and the trajectory characteristics no longer change significantly. This shows that simply introducing such a linear stiffness is not conducive to the energy transfer and dissipation of NES. This also shows that the energy transfer characteristics generated in the second working condition to NES are due to the introduction of non-smooth piecewise stiffness, and are unrelated to the change in the linear stiffness value.

[0103] Observe the changes in energy dissipation trajectories under the first, second, and third operating conditions, and project the energy dissipation trajectories under the first and second operating conditions onto the unit disk, as shown in the attached figure. Figure 12 The diagram shows the trajectory bifurcation characteristics of the system before and after the introduction of the piecewise stiffness term. It can be observed that without the introduction of the piecewise stiffness term (see Appendix...), the system exhibits... Figure 12 a, which shows the energy dissipation trajectory projection for the first operating condition, indicates that the system's energy dissipation trajectory mainly converges to the limit point where the phase difference is close to 0. In this operating condition, the out-of-phase trajectory is mainly dissipated by the main system. Although there is an initial tendency to transfer to the in-phase branch, the transfer fails due to insufficient input energy levels. However, after introducing piecewise stiffness (see Appendix...), Figure 12b, which shows the energy dissipation trajectory projection of the second operating condition, under the same initial energy level excitation, the system trajectory successfully entered the in-phase orbit (S11+ branch) with a phase difference φ close to π, and realized a large amount of energy transfer and dissipation to the NES. After dissipating a certain amount of energy near the NES, the system finally started to return to the anti-phase orbit branch when the total energy of the system was near the 0 energy level.

[0104] Energy dissipation trajectories and their projection diagrams for the first and second operating conditions (see appendix) Figure 9 Appendix Figure 10 and appendix Figure 12 The analysis was conducted by inputting the phase φ and energy percentage parameter ψ corresponding to different energy level stages, and observing the correspondence between the corresponding topological feature diagrams and the characteristics in weakly damped dissipation. The results of the first working condition are shown in the attached figure. Figure 13 As shown, it illustrates the energy level decomposition of the dissipation trajectory under the first operating condition and its corresponding characteristic diagram, with appended... Figure 13 Figure a shows the energy level decomposition diagram of the dissipation trajectory under the first operating condition, with appendix. Figure 13 b illustrates the characteristics of the initial high energy level in the first operating condition; Appendix Figure 13 c shows the characteristics of the transition energy level segment under the first operating condition. Figure 13 d shows the characteristics corresponding to the low-energy segment of the first operating condition.

[0105] The energy dissipation process of the first operating condition is decomposed according to the energy level segments. The total energy level segment is r∈(0,1), the initial high energy level segment is r∈(0.9,1), the transition energy level segment is r∈(0.4,0.9), and the low energy segment is r∈(0,0.4). It can be seen that in the first operating condition, a certain amount of energy transfer occurs in the initial high energy level segment, followed by a significant amount of energy transfer to the NES in the transition energy level segment, and finally, the energy is captured by the reverse orbit S11- in the low energy segment. In fact, although a considerable amount of energy is occupied in the main system in the low energy state, due to the low total energy, there will be no significant response, and the energy will eventually be exhausted.

[0106] Similarly, the energy dissipation process of the second working condition is decomposed according to energy level segments. The total energy level segment is r∈(0,1), the initial high energy level segment is r∈(0.9,1), the transition energy level segment is r∈(0.1,0.9), and the low energy segment is r∈(0,0.1). It can be seen that in the second working condition, a certain amount of energy transfer occurs in the initial high energy level segment. At this time, the energy transfer situation is slightly lower than that without the introduction of segmented stiffness. Subsequently, a wider transition energy level segment generates more energy transfer characteristics to NES and stays near a state with a high proportion of NES energy for a long time. At this time, it is attracted by the in-phase orbit S11+. The energy proportion state of the transition segment after the introduction of segmented stiffness is better than that before the introduction of segmented stiffness. Finally, it is captured by the reverse orbit S11- in the low energy segment. In the low energy level state, the energy is finally exhausted.

[0107] In view of this, in order to obtain the interaction relationship between the input energy level, linear stiffness and piecewise stiffness, this invention considers the energy decay factor as the natural base e, studies the relationship between the energy input characteristics of the system and the decay time, and defines the decay time to satisfy the relationship shown in equation (16):

[0108] r(T)=r(0)e -1 (16)

[0109] The input energy-decrease time characteristics of the system shown in the second operating condition are analyzed. See Appendix. Figure 14 It shows the input energy-decrease time characteristics of the system after introducing piecewise stiffness, where the attached... Figure 14 a shows the energy decay trajectory of point a, r(0) = 0.088; Appendix Figure 14 b shows the energy decay trajectory at point b, r(0) = 0.1456; Appendix Figure 14 c shows the energy decay trajectory at point c, r(0) = 0.288; Appendix Figure 14 Figure d shows the relationship between the system input energy and the system energy decay time.

[0110] See appendix Figure 14 As can be seen from point d, there is a period between point a and point b where the decay time T decreases significantly, indicating a substantial improvement in vibration reduction efficiency. (Analysis appendix) Figure 14 The planar phase space characteristic diagrams of three special points (a, b, c) in d show that, from point a to point b, the projection of the system's dynamic trajectory in phase space exhibits a sudden shift (see appendix). Figure 14 a and appendix Figure 14 b. The system's dynamic trajectory suddenly shows a tendency to shift outward towards S11+, and the system's motion characteristics transition from being limited to the motion of the main system to having a large relative motion between the main system and the NES oscillator, and then reach strong NES oscillator motion (see Appendix). Figure 14 The trajectory in c is at the edge of the unit disk, indicating that the NES has a large energy proportion. This sudden shift is the reason why the vibration suppression effect of the system is improved after introducing piecewise stiffness.

[0111] Based on the same concept as the aforementioned technical solutions, see Appendix Figure 15 This invention illustrates an energy dissipation characteristic analysis device 150 for a weakly damped nonlinear energy trap system, characterized in that the device includes a first establishment section 1501, a first establishment and analysis section 1502, and a second establishment and analysis section 1503; wherein,

[0112] The first establishment part 1501 is configured to establish a damped nonlinear energy trap system model;

[0113] The first analysis section 1502 is configured to establish a weakly damped nonlinear energy trap system model based on the damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system.

[0114] The second analysis section 1503 is configured to introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system. Accordingly, it establishes a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system and analyzes the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system.

[0115] Understandably, in this embodiment, "part" can be a part of a circuit, a part of a processor, a part of a program or software, etc., or it can be a unit, a module, or a non-modular one.

[0116] Furthermore, in this embodiment, the components can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional module.

[0117] If the integrated unit is implemented as a software functional module and not sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this embodiment, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) or processor to execute all or part of the steps of the method described in this embodiment. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0118] It should be noted that the technical solutions described in the embodiments of the present invention can be combined arbitrarily without conflict.

[0119] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system, characterized in that, The analytical method includes the following steps: S101. Establish a model of a damped nonlinear energy trap system; S102. Based on the damped nonlinear energy trap system model, establish a weakly damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system. S103. Introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system. Correspondingly, establish a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system and analyze the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system. The analysis of the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system specifically includes the following steps: S601. When the weakly damped piecewise stiffness nonlinear energy trap system has no piecewise stiffness, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system. S602. When piecewise stiffness is introduced into the weakly damped piecewise stiffness nonlinear energy trap system, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system. S603. When an additional linear stiffness is introduced into the weakly damped piecewise stiffness nonlinear energy trap system, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system, wherein the value of the linear stiffness is equal to the value of the piecewise stiffness.

2. The analytical method according to claim 1, characterized in that, The equations of motion for the damped nonlinear energy trap system model are shown in expression (1): Where m1 and m2 are the masses of the linear master system and the nonlinear energy trap oscillator, respectively; c1 and c2 are the damping of the linear master system and the damping of the nonlinear energy trap oscillator, respectively; k1 and k2 are the stiffness of the linear master system and the cubic stiffness term of the connection between the nonlinear energy trap oscillator and the linear master system, respectively; x is the displacement of the linear master system; and v is the displacement of the nonlinear energy trap oscillator.

3. The analytical method according to claim 2, characterized in that, The establishment of a weakly damped nonlinear energy trap system model based on the damped nonlinear energy trap system model specifically includes the following steps: S301. By canceling the mass term, the equation of motion (1) is simplified to obtain the equation of motion (2) as shown in expression (2): in, S302. Considering the weakly damped model, it is assumed that... , Assuming a weakly damped term, the equations of motion (3) for the weakly damped nonlinear energy trap system model are obtained:

4. The analytical method according to claim 3, characterized in that, The analysis of the energy dissipation characteristics of the weakly damped nonlinear energy trap system specifically includes the following steps: S401. Substitute expressions (4) and (5) into the equation of motion (3), where expression (4) represents the relationship between the amplitude of the linear master system and the amplitude of the nonlinear energy trap oscillator, and expression (5) is the energy evaluation term. in, A parameter for measuring total energy, Let A be the complex amplitude of the linear principal system. The complex amplitude B of the nonlinear energy trap; S402. Simplify the equation of motion (5) into a three-dimensional dynamic system in circular space to obtain the slow flow dynamic model (6) as shown in expression (6): in, This refers to the rate at which the system consumes energy. This refers to the energy ratio between the linear master system and the nonlinear energy trap. This refers to the phase difference between the two oscillators; S403. Transform the slow-flow dynamics model (6) into a cylindrical space for topological transformation to obtain the energy dissipation trajectory of the damped nonlinear energy trap system. S404. Change the energy level of the input energy of the weakly damped nonlinear energy trap system, and analyze the energy dissipation trajectory of the weakly damped nonlinear energy trap system under different energy levels of the input energy.

5. The analytical method according to claim 4, characterized in that, The process of introducing piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system, and correspondingly establishing a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system, specifically includes the following steps: S501. After introducing piecewise stiffness into the weakly damped nonlinear energy trap system, the equation of motion (7) is obtained: S502. Converting the equation of motion (7) into the relative displacement between the center of mass and the nonlinear energy trap oscillator, we can obtain the expression (8): in, This is the linear stiffness term between the linear master system and the nonlinear energy trap oscillator; The damping of the nonlinear energy trap oscillator; The cubic stiffness term is the connection between the nonlinear energy trap oscillator and the linear master system. This is the piecewise stiffness term for the connection between the nonlinear energy trap oscillator and the linear master system; The gap between the nonlinear energy trap oscillator and the linear main system; The ratio of the mass of the nonlinear energy trap oscillator to the mass of the linear master system. This refers to the centroid displacement of the linear principal system. The displacement of the nonlinear energy trap oscillator relative to the linear master system; S503. Using the complex variables φ1 and φ2 as shown in equation (9), perform the transformation. Wherein, for the complex variable and Using complex amplitudes A and B and phase variables , To represent its complex amplitude characteristics, the expression for the complex variable is obtained (10): ， (10)； S504. Based on the above expressions (8) and (10), the slow-flow dynamic model (11) of the weakly damped piecewise stiffness nonlinear energy trap system is obtained: S505. Due to the introduction of a damping term, the energy of the weakly damped piecewise stiffness nonlinear energy trap system is no longer conserved. Therefore, the variable shown in expression (12) is used to replace the slow-flow dynamics model (11): The first term of expression (12) corresponds to the total energy characteristic of the weakly damped piecewise stiffness nonlinear energy trap system, the second term corresponds to the phase difference characteristic of the weakly damped piecewise stiffness nonlinear energy trap system, and the third term corresponds to the energy characteristics of the linear master system and the nonlinear energy trap oscillator in the weakly damped piecewise stiffness nonlinear energy trap system. After the substitution, the slow flow dynamics equation (13) is obtained. S506. Transform the slow-flow dynamics equation (13) into the cylindrical space and perform a topological transformation to obtain the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system.

6. A device for analyzing the energy dissipation characteristics of a weakly damped nonlinear energy trap system, characterized in that, The device includes a first establishment section, a first establishment analysis section, and a second establishment analysis section; wherein... The first establishment part is configured to establish a damped nonlinear energy trap system model; The first analysis section is configured to establish a weakly damped nonlinear energy trap system model based on the damped nonlinear energy trap system model, and analyze the energy dissipation characteristics of the weakly damped nonlinear energy trap system. The second analysis section is configured to introduce piecewise stiffness into the weakly damped nonlinear energy trap system to establish a weakly damped piecewise stiffness nonlinear energy trap system. Correspondingly, a topological characteristic model of the weakly damped piecewise stiffness nonlinear energy trap system is established, and the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system are analyzed. Specifically, the analysis of the energy dissipation characteristics of the weakly damped piecewise stiffness nonlinear energy trap system includes: S601. When the weakly damped piecewise stiffness nonlinear energy trap system has no piecewise stiffness, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system. S602. When piecewise stiffness is introduced into the weakly damped piecewise stiffness nonlinear energy trap system, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system. S603. When an additional linear stiffness is introduced into the weakly damped piecewise stiffness nonlinear energy trap system, observe the energy dissipation trajectory of the weakly damped piecewise stiffness nonlinear energy trap system, wherein the value of the linear stiffness is equal to the value of the piecewise stiffness.

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