Design method of ultra-low frequency vibration isolation structure based on linear positive and negative stiffness compensation mechanism

Abstract

The application discloses a kind of based on linear positive and negative stiffness compensation mechanism's ultra-low frequency vibration isolation structure design method, the method includes the following steps: step 1, based on linear positive and negative stiffness compensation mechanism's ultra-low frequency vibration isolation structure design;Step 2, design the symmetric polygon linear negative stiffness unit containing inclined spring;Step 3, determine the linear positive and negative stiffness compensation mechanism of ultra-low frequency vibration isolation structure;Step 4, construct the nonlinear dynamics model and motion equation of ultra-low frequency vibration isolation structure containing vibration isolation object;Step 5, ultra-low frequency vibration isolation structure vibration isolation performance evaluation.The ultra-low frequency vibration isolation structure based on positive and negative stiffness compensation mechanism designed in the application not only has large bearing capacity, but also has ultra-wide quasi-zero stiffness area, which can reach 88% of the total stroke of the structure;And have wider vibration isolation band, lower vibration frequency and resonance amplitude, and effective vibration isolation frequency can be reduced to 0.15Hz.

Description

Technical Field

[0001] This invention relates to a design method for an ultra-low frequency vibration isolation structure, specifically a design method for an ultra-low frequency quasi-zero stiffness vibration isolation structure composed of a symmetrical polygonal linkage structure and a linear vertical spring. Background Technology

[0002] Vibration problems are prevalent and difficult to eliminate in engineering applications. Harmful vibrations can affect the function of precision equipment, shorten the lifespan of engineered components, reduce the precision of mechanical manufacturing, and even lead to the damage of engineered components. Therefore, in practical engineering, it is necessary to control harmful vibrations within acceptable limits to avoid irreparable damage.

[0003] To meet diverse vibration isolation and reduction requirements, existing research has developed various vibration control strategies, such as active control, semi-active control, and passive control methods. While semi-active and active control strategies can easily achieve large control forces and desired vibration control performance, they require piezoelectric materials with significant energy input, electromagnetic components, and electromechanical equipment with high maintenance costs, leading to system complexity and poor stability. Passive vibration control methods, with their good stability and cost-effectiveness, overcome the shortcomings of semi-active and active control methods, and offer wide isolation bandwidth and excellent vibration reduction performance even at high frequencies.

[0004] For linear passive vibration isolation systems, only excitation frequencies higher than [the specified frequency] can be suppressed. Vibrations at frequencies exceeding the resonant frequency. While reducing the stiffness of a linear system can effectively lower the resonant frequency and widen the isolation zone, it also leads to a significant deviation from the equilibrium point, reducing system stability. Therefore, nonlinear isolation systems with high static and low dynamic stiffness have attracted widespread attention from researchers and are applied to low-frequency vibration isolation. However, most nonlinear isolation systems have nonlinear stiffness characteristics in their negative stiffness elements, and methods to achieve quasi-zero stiffness are limited to compensating for the nonlinear negative stiffness of these elements using linear or electromagnetic springs. This significantly limits the realization of a wider quasi-zero stiffness region and ultra-low-frequency vibration isolation performance. Summary of the Invention

[0005] To address the low-frequency vibration isolation and reduction requirements faced in existing engineering applications, this invention provides a design method for ultra-low frequency vibration isolation structures based on a linear positive and negative stiffness compensation mechanism. This method utilizes the positive stiffness of a vertical linear spring to compensate for the linear negative stiffness generated by the negative stiffness unit, thereby increasing the quasi-zero stiffness region of the structure and improving its ultra-low frequency vibration isolation characteristics.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] An ultra-low frequency vibration isolation structure based on a linear positive and negative stiffness compensation mechanism includes a polygonal linkage structure and a linear vertical spring, wherein:

[0008] The polygonal linkage structure is composed of symmetrical polygonal linear negative stiffness units.

[0009] The symmetrical polygonal linear negative stiffness element is composed of a first polygonal element and a second polygonal element, which are symmetrically distributed about the linear vertical spring as the axis of symmetry.

[0010] Both the first polygonal unit and the second polygonal unit are composed of an upper quadrilateral structure and a lower quadrilateral structure;

[0011] The top of the upper quadrilateral structure is connected to the upper support platform, the bottom of the upper quadrilateral structure is connected to the top of the lower quadrilateral structure, and the bottom of the lower quadrilateral structure is connected to the lower support platform.

[0012] A stiffness of k is installed on one diagonal of the upper quadrilateral structure and the lower quadrilateral structure. s The two tilting springs share a common rotation node.

[0013] A rigidity k is installed between the upper and lower load-bearing platforms. v A linear vertical spring.

[0014] A design method for ultra-low frequency vibration isolation structures based on linear positive and negative stiffness compensation mechanism includes the following steps:

[0015] Step 1: Design of Ultra-Low Frequency Vibration Isolation Structure Based on Linear Positive and Negative Stiffness Compensation Mechanism

[0016] Step 1.1: Define the ultra-low frequency vibration isolation structure.

[0017] The ultra-low frequency vibration isolation structure includes a polygonal linkage structure and a linear vertical spring, wherein:

[0018] The polygonal linkage structure is composed of symmetrical polygonal linear negative stiffness units.

[0019] The symmetrical polygonal linear negative stiffness element is composed of a first polygonal element and a second polygonal element, which are symmetrically distributed about the linear vertical spring as the axis of symmetry.

[0020] Both the first polygonal unit and the second polygonal unit are composed of an upper quadrilateral structure and a lower quadrilateral structure;

[0021] The top of the upper quadrilateral structure is connected to the supporting platform, the bottom of the upper quadrilateral structure is connected to the top of the lower quadrilateral structure, and the bottom of the lower quadrilateral structure is connected to the foundation.

[0022] A stiffness of k is installed on one diagonal of the upper quadrilateral structure and the lower quadrilateral structure. sThe two tilting springs share a common rotation node.

[0023] The lengths of the inclined rod and the horizontal rod in the upper quadrilateral structure and the lower quadrilateral structure are L1 and L2, respectively, and the initial angle between the inclined rod and the horizontal plane is α.

[0024] A rigidity k is installed between the load-bearing platform and the foundation. v A linear vertical spring, by adjusting the distance d between the bearing platform and the linear vertical spring. h Enhance the tunability of ultra-low frequency vibration isolation structures;

[0025] Step 1.2: Design a quasi-zero stiffness passive vibration isolation platform

[0026] The quasi-zero stiffness vibration isolation platform consists of an ultra-low frequency vibration isolation structure, a load-bearing platform, and a foundation.

[0027] The horizontal displacement Δh and rotation angle β of the ultra-low frequency vibration isolation structure are expressed as follows:

[0028]

[0029]

[0030] The initial length L of the tilting spring s1 and the length L after deformation s2 They are represented as follows:

[0031]

[0032]

[0033] The deformation length ΔL of the inclined spring s Represented as:

[0034]

[0035] Where z1 is the vertical displacement of the bearing platform, Δh is the horizontal displacement of the horizontal bar, and β is the rotation angle of the inclined bar;

[0036] Step 2: Design a symmetrical polygonal linear negative stiffness element containing a tilting spring.

[0037] Step 2.1: For the load-bearing platform, its force balance equation is:

[0038]

[0039] Where f1 and f2 are the axial forces generated by connecting rods A1B1 and A2B2, respectively, and their magnitudes are the same as the axial forces generated by connecting rods A4B4 and A3B3. pThe vertical load f borne by a symmetric polygonal linear negative stiffness element s =k s ΔL s The tension of the inclined spring. For the spring force f s The angle between the upper quadrilateral and the horizontal axis; nodes A1, A2, A3, and A4 are the connection nodes between the top of the upper quadrilateral structure and the supporting platform, respectively; nodes B1, B2, B3, and B4 are the connection nodes between the bottom of the upper quadrilateral structure and the top of the lower quadrilateral structure, respectively; nodes C1, C2, C3, and C4 are the connection nodes between the bottom of the lower quadrilateral structure and the foundation, respectively.

[0040] Step 2.2: For nodes B1 and B2, the force balance equations are expressed as follows:

[0041]

[0042]

[0043] f2cos(α-β)+f5cos(α-β)-f3=0 (9)

[0044] f2sin(α-β)-f5sin(α-β)=0 (10)

[0045] Among them, f3, f4 and f5 are the axial tensile forces generated by connecting rods B1B2, B1C1 and B2C2;

[0046] Step 2.3: Based on the equilibrium equations of the forces at the bearing platform, nodes B1 and B2, the vertical bearing capacity F of the symmetrical polygonal linear negative stiffness element is calculated. p Represented as:

[0047]

[0048] in:

[0049]

[0050] Step 3: Determine the linear positive and negative stiffness compensation mechanism of ultra-low frequency vibration isolation structures.

[0051] Step 3.1: When the displacement of the bearing platform is 0 ≤ z1 ≤ d h When the load-bearing platform is not in contact with the linear vertical spring, the total restoring force F of the ultra-low frequency vibration isolation structure and the vertical bearing force F of the symmetrical polygonal linear negative stiffness element are... p same;

[0052] Step 3.2, when the displacement of the bearing platform z1 ≥ d h When the symmetrical polygonal linear negative stiffness element and the linear vertical spring are connected in parallel, the total restoring force F is:

[0053]

[0054] Step 4: Construct the nonlinear dynamic model and equations of motion for the ultra-low frequency vibration isolation structure containing vibration isolation objects.

[0055] Step 4.1: The vertical displacements of the bearing platform and foundation of the ultra-low frequency vibration isolation structure are respectively represented by z. p and z b The total mass of the vibration isolation object and the load-bearing platform is represented by M. t The relative displacement between the basic excitation and the supporting platform is expressed as z1 = z p -z b The kinetic energy T of the ultra-low frequency vibration isolation structure s and potential energy U s Represented as:

[0056]

[0057] In the formula, ΔL v Let g be the deformation length of the linear vertical spring, and g be the acceleration due to gravity.

[0058] Step 4.2: Considering air damping and nodal rotational friction damping effects, the generalized force G in the vertical direction of the ultra-low frequency vibration isolation structure... z Represented as:

[0059]

[0060] In the formula, e1 is the air damping coefficient, and e2 is the nodal rotational friction damping coefficient;

[0061] Step 4.3: Apply Lagrange's principle to obtain the nonlinear equation of motion for the ultra-low frequency vibration isolation structure:

[0062]

[0063] Step 4.4: By solving equations (14)-(16), the nonlinear motion equations of the ultra-low frequency vibration isolation structure are obtained as follows:

[0064]

[0065] Where the nonlinear function P c (z1) and P f (z1) is represented as:

[0066]

[0067] Step 4.5: When the vibration-isolated object acts on the load-bearing platform, the ultra-low frequency vibration isolation structure will reach a new static equilibrium position. When the vibration isolation structure is subjected to foundation excitation, the vibration-isolated object will vibrate up and down near the static equilibrium position. The static equilibrium position is selected as the vibration origin, and the vibration displacement z at the static equilibrium position is... s Represented as:

[0068] z s =z1-z st (19)

[0069] Among them, z st The static displacement of the ultra-low frequency vibration isolation structure under the vibration isolation mass;

[0070] Step 4.6: According to equations (17)-(19), the nonlinear motion equations of the ultra-low frequency vibration isolation structure are further expressed as:

[0071]

[0072] Step 5: Evaluation of the vibration isolation performance of ultra-low frequency vibration isolation structures

[0073] Step 5.1: For the nonlinear damping and restoring force function P... c (z s ) and P f (z s Polynomial fitting is performed to transform the fractional terms in the nonlinear function into polynomial form:

[0074]

[0075]

[0076] Step 5.2: Substitute equations (21) and (22) into equation (20) to obtain:

[0077]

[0078] Step 5.3: Define the following parameters:

[0079]

[0080]

[0081] The nonlinear motion equations of ultra-low frequency vibration isolation structures are uniformly written as follows:

[0082]

[0083] Step 5.4, Assume the basic stimulus z b=z0cos(2πft), where f is the external excitation frequency and z0 is the basic excitation amplitude. The steady-state response of the nonlinear motion equation is obtained by solving equation (26) using the multi-scale method.

[0084] Step 5.5, Vibration displacement z at static equilibrium position s The amplitude-frequency response curve is expressed as:

[0085]

[0086] Where A1 is the vibration amplitude, s is the ratio of the external excitation frequency to the system's natural frequency, and ω0 is the system's natural frequency;

[0087] Vibration displacement z of ultra-low frequency vibration isolation structure s Represented as:

[0088] z s =A1cos(2πft+θ) (28)

[0089] Where θ is the phase angle;

[0090] Vibration transmissivity T of ultra-low frequency vibration isolation structure R Represented as:

[0091]

[0092] Step 5.6: Perform ultra-low frequency vibration isolation characteristic analysis on ultra-low frequency vibration isolation structure according to equations (27) and (29).

[0093] Compared with the prior art, the present invention has the following advantages:

[0094] (1) This invention designs and develops a nonlinear ultra-low frequency vibration isolation structure based on the linear positive and negative stiffness compensation mechanism, clarifies the design method of the symmetrical polygonal linkage vibration isolation structure with linear spring, and reveals the linear positive and negative stiffness compensation mechanism to achieve an ultra-wide quasi-zero stiffness region, providing a new design and development idea for the design of nonlinear vibration isolators in practical engineering applications.

[0095] (2) The ultra-low frequency vibration isolation structure based on positive and negative stiffness compensation mechanism designed in this invention not only has a large load-bearing capacity, but also has an ultra-wide quasi-zero stiffness region, which accounts for up to 88% of the total stroke of the structure.

[0096] (3) Compared with traditional quasi-zero stiffness vibration isolation systems, the ultra-low frequency vibration isolation structure designed in this invention has an ultra-wide quasi-zero stiffness range while ensuring load-bearing capacity and stability; and has a wider vibration isolation band, a lower vibration isolation frequency and resonance amplitude, and the effective vibration isolation frequency can be reduced to 0.15Hz.

[0097] (4) This invention can easily achieve low-frequency and ultra-low-frequency vibration isolation performance by simply adjusting the positive and negative stiffness spring parameters of the structure. It has good versatility and flexibility for low-frequency vibration reduction and isolation problems in mechanical engineering structures. Attached Figure Description

[0098] Figure 1 This is an ultra-low frequency vibration isolation structure based on the linear positive and negative stiffness compensation mechanism;

[0099] Figure 2 The curves show the positive and negative stiffness compensation principle of ultra-low frequency vibration isolation structures based on the linear positive and negative stiffness compensation mechanism.

[0100] Figure 3 This is a quasi-zero stiffness passive vibration isolation platform designed based on an ultra-low frequency vibration isolation structure.

[0101] Figure 4 Deformation analysis of ultra-low frequency vibration isolation structures;

[0102] Figure 5 For vertical load F p Deformation of a symmetrical polygonal linear negative stiffness element under action;

[0103] Figure 6 Stress analysis for the supporting platform and nodes B1 and B2;

[0104] Figure 7 For the vertical displacement of the vibration-isolated object and the foundation excitation;

[0105] Figure 8 Nonlinear force-displacement curves of a symmetrical polygonal linear negative stiffness element under different initial angles α;

[0106] Figure 9 For different initial angles α and spring stiffness k v Results of positive and negative stiffness compensation for ultra-low frequency vibration isolation structures;

[0107] Figure 10 This is a simulation model of an ultra-low frequency vibration isolation structure in ADAMS.

[0108] Figure 11 The force-displacement curves are obtained based on simulation and the method of this invention;

[0109] Figure 12 The amplitude-frequency response curves of the structure under numerical integration and multi-scale calculation methods;

[0110] Figure 13 The amplitude-frequency response curves of the relative vibration displacement of the vibration isolation structure under different rod lengths L2 are shown.

[0111] Figure 14 The vibration isolation mass varies with the equilibrium position;

[0112] Figure 15 The vibration isolation characteristics of the structure under different static equilibrium positions. Detailed Implementation

[0113] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0114] Quasi-zero stiffness vibration isolation systems composed of negative stiffness elements and linear springs exhibit high static and low dynamic stiffness characteristics. However, the negative stiffness elements of traditional quasi-zero stiffness vibration isolation systems exhibit nonlinear negative stiffness characteristics. Most methods for achieving quasi-zero stiffness compensate for the nonlinear negative stiffness of the elements using linear or electromagnetic springs, thus limiting the quasi-zero stiffness region of the isolation structure. Furthermore, traditional quasi-zero stiffness vibration isolation systems are prone to instability phenomena such as frequency jumping and bifurcation under large excitation, restricting the realization of ultra-low frequency vibration isolation. Therefore, this invention provides a design method for ultra-low frequency vibration isolation structures based on a linear positive and negative stiffness compensation mechanism. The main technical problems to be solved by this method include:

[0115] (1) Design a novel ultra-low frequency quasi-zero stiffness vibration isolation structure consisting of a symmetrical polygonal linkage structure and a linear vertical spring, and explore its specific advantages in ultra-low frequency vibration isolation.

[0116] (2) Clarify the design method of symmetrical polygonal linkage vibration isolation structure with linear spring, and reveal the linear positive and negative stiffness compensation mechanism of vibration isolation structure to achieve large load capacity and ultra-wide quasi-zero stiffness region;

[0117] (3) Construct the nonlinear dynamic model and its motion differential equation of the quasi-zero stiffness vibration isolation system under load, and solve its amplitude-frequency characteristics and vibration transmissibility using the multi-scale method and numerical method.

[0118] (4) Clarify the key structural parameter adjustment rules for achieving ultra-wide quasi-zero stiffness region and ultra-low frequency vibration isolation performance of vibration isolation system;

[0119] (5) The feasibility and effectiveness of the linear positive and negative stiffness compensation mechanism provided by the present invention have been verified, and the vibration isolation system has both large load-bearing capacity and ultra-large stroke quasi-zero stiffness region. The vibration isolation structure has achieved excellent ultra-low frequency vibration isolation performance while ensuring good stability.

[0120] The specific implementation steps are as follows:

[0121] Step 1: Design of Ultra-Low Frequency Vibration Isolation Structure Based on Linear Positive and Negative Stiffness Compensation Mechanism

[0122] Figure 1An ultra-low frequency vibration isolation structure is presented, where the black circles in the structure represent the rotation nodes of the connecting rods. Figure 2 A schematic diagram of the linear positive and negative stiffness compensation mechanism of an ultra-low frequency vibration isolation structure is given. The ultra-low frequency vibration isolation structure includes a polygonal linkage structure and a linear vertical spring, wherein: the polygonal linkage structure is composed of symmetrical polygonal linear negative stiffness units; the symmetrical polygonal linear negative stiffness units are composed of first polygonal units and second polygonal units, which are symmetrically distributed about the linear vertical spring as an axis of symmetry; both the first and second polygonal units consist of an upper quadrilateral structure and a lower quadrilateral structure; the top of the upper quadrilateral structure is connected to the bearing platform, the bottom of the upper quadrilateral structure is connected to the top of the lower quadrilateral structure, and the bottom of the lower quadrilateral structure is connected to the foundation; the lengths of the inclined rod and the horizontal rod in the upper and lower quadrilateral structures are L1 and L2, respectively, and the initial angle between the inclined rod and the horizontal plane is α. Since the mass of the vibration isolation object is generally much larger than the mass of the linkages (inclined rod and horizontal rod), the mass of the structural rotation nodes and linkages can be ignored. A stiffness k is installed on one diagonal of the upper and lower quadrilateral structures. s The inclined springs form a negative stiffness unit, and the two inclined springs share a common rotation node; a stiffness unit of k is installed between the bearing platform and the foundation. v A linear vertical spring provides linear positive stiffness. Since the elastic deformation of the link is very small compared to the spring deformation, the link can be considered a lightweight rigid body. Furthermore, the distance d between the bearing platform and the linear vertical spring can be adjusted. h Enhance the tunability of ultra-low frequency vibration isolation structures.

[0123] During structural compression under load, the symmetrical polygonal elements initially exhibit positive stiffness characteristics. When the load on the structure exceeds its bearing capacity, the symmetrical polygonal linear negative stiffness elements exhibit linear negative stiffness characteristics, while the vertical spring can provide linear positive stiffness to compensate for the linear negative stiffness of the structure, as shown in Figure 2. By adjusting the parameters of the symmetrical polygonal structure, the absolute value of the linear negative stiffness of the negative stiffness elements can be made equal to the positive stiffness of the vertical spring, thereby obtaining a large quasi-zero stiffness range. In summary, as shown in Figure 2... Figure 2 As shown, the compression process of the vibration isolation structure under load can be divided into two stages: I (positive stiffness characteristics) and II (quasi-zero stiffness characteristics). Since the structure exhibits excellent high static and low dynamic stiffness characteristics in stage II, we hope that this vibration isolation structure will always work in compression stage II.

[0124] Based on the ultra-low frequency vibration isolation structure proposed in this invention, it is possible to design, as follows: Figure 3 The quasi-zero stiffness passive vibration isolation platform shown is... Figure 3It can be seen that the quasi-zero stiffness vibration isolation platform consists of three linkage-type ultra-low frequency vibration isolation structures, a linear vertical spring, an upper load-bearing platform, and a foundation. The deformation analysis of the linkage-type ultra-low frequency vibration isolation structure is as follows: Figure 4 As shown, z1 is the vertical displacement of the bearing platform, Δh is the horizontal displacement of the horizontal rod, and β is the rotation angle of the inclined rod. According to... Figure 4 The geometric relationships shown, the horizontal displacement Δh and the rotation angle β can be expressed by z1 as:

[0125]

[0126]

[0127] The initial length L of the tilting spring s1 And the length L of the deformed spring s2 They can be represented as:

[0128]

[0129]

[0130] Therefore, the tilting spring deformation length ΔL s It can be represented as:

[0131]

[0132] Step 2: Design a symmetrical polygonal linear negative stiffness element containing a tilting spring.

[0133] Figure 5 For vertical load F p The symmetrical polygonal linear negative stiffness element deforms under the action, where nodes A1, A2, A3, and A4 are the connection nodes between the top of the upper quadrilateral structure and the bearing platform, respectively; B1, B2, B3, and B4 are the connection nodes between the bottom of the upper quadrilateral structure and the top of the lower quadrilateral structure, respectively; and C1, C2, C3, and C4 are the connection nodes between the bottom of the lower quadrilateral structure and the foundation, respectively. Figure 6 This is a schematic diagram illustrating the force analysis of the supporting platform and nodes B1 and B2. For the supporting platform, the force equilibrium equation is:

[0134]

[0135] In equation (6), f1 and f2 are the axial forces generated by connecting rods A1B1 and A2B2, respectively, and their magnitudes are the same as the axial forces generated by connecting rods A4B4 and A3B3. p The vertical load f borne by a symmetric polygonal linear negative stiffness element s =k s ΔL s The tension of the inclined spring. For the spring force f s The angle between the horizontal axis and the horizontal axis.

[0136] For nodes B1 and B2, the force balance equations can be expressed as:

[0137]

[0138]

[0139] f2cos(α-β)+f5cos(α-β)-f3=0 (9)

[0140] f2sin(α-β)-f5sin(α-β)=0 (10)

[0141] Where f3, f4, and f5 are the axial tensile forces generated by connecting rods B1B2, B1C1, and B2C2. According to formulas (5)-(10), the vertical bearing capacity F of the symmetrical polygonal linear negative stiffness element is... p It can be represented as:

[0142]

[0143] in:

[0144]

[0145] Step 3: Linear positive and negative stiffness compensation mechanism of ultra-low frequency vibration isolation structure

[0146] Based on the linear positive and negative stiffness compensation mechanism, symmetrical polygonal linear negative stiffness elements with large stroke and beneficial linear negative stiffness can be combined with traditional positive stiffness elastic elements to form a novel ultra-low frequency vibration isolation structure with a wider quasi-zero stiffness region. Figure 1 and Figure 2 It can be seen that a gap d is designed between the quasi-zero stiffness bearing platform and the linear vertical spring. h This enhances the tunability of the ultra-low frequency vibration isolation structure proposed in this invention, enabling the structure to achieve a wider quasi-zero stiffness region and greater load-bearing capacity.

[0147] When the displacement of the bearing platform is 0≤z1≤d h When the load-bearing platform is not in contact with the linear vertical spring, the total restoring force F of the ultra-low frequency vibration isolation structure is the same as the vertical bearing capacity of the symmetrical polygonal linear negative stiffness element (F = F). p If the displacement of the bearing platform z1 ≥ d h When the symmetrical polygonal linear negative stiffness element and the linear vertical spring are connected in parallel, the total restoring force F is:

[0148]

[0149] Step 4: Construct the nonlinear dynamic model and equations of motion for the ultra-low frequency vibration isolation structure containing vibration isolation objects.

[0150] like Figure 7 As shown, the vertical displacements of the bearing platform and foundation of the ultra-low frequency vibration isolation structure are represented by z. p and z b The total mass of the vibration isolation object and the load-bearing platform is represented by M. t The relative displacement between the basic excitation and the supporting platform can be expressed as z1 = z p -z b The kinetic energy T of the ultra-low frequency vibration isolation structure s and potential energy U s It can be represented as:

[0151]

[0152] In the formula ΔL v Let g be the deformation length of the linear vertical spring, g = 9.8 m / s 2 This is the acceleration due to gravity.

[0153] Considering air damping and nodal rotational friction damping effects, the generalized force G in the vertical direction of the ultra-low frequency vibration isolation structure z It can be represented as:

[0154]

[0155] In the formula, e1 is the air damping coefficient and e2 is the nodal rotational friction damping coefficient.

[0156] The nonlinear equations of motion for ultra-low frequency vibration isolation structures can be obtained by applying Lagrange's principle:

[0157]

[0158] By solving equations (14)-(16), the nonlinear motion equations of the ultra-low frequency vibration isolation structure can be obtained as follows:

[0159]

[0160] Where the nonlinear function P c (z1) and P f (z1) can be represented as:

[0161]

[0162] When the vibration-isolated object acts on the supporting platform, the ultra-low frequency vibration isolation structure will reach a new static equilibrium position. When the vibration isolation structure is subjected to foundation excitation, the vibration-isolated object will vibrate up and down near the static equilibrium position. Furthermore, the stiffness of the ultra-low frequency vibration isolation structure is independent of its static equilibrium position. However, for the ultra-low frequency vibration isolation structure designed in this invention, different static equilibrium positions result in different stiffness characteristics. Therefore, the static equilibrium position is selected as the vibration origin, and the vibration displacement z at the static equilibrium position... s It can be represented as:

[0163] z s =z1-z st (19)

[0164] Where z st This refers to the static displacement of the ultra-low frequency vibration isolation structure under the vibration isolation mass.

[0165] Therefore, according to equations (17)-(19), the nonlinear motion equation of the ultra-low frequency vibration isolation structure designed in this invention can be further expressed as:

[0166]

[0167] Step 5: Evaluation of the vibration isolation performance of ultra-low frequency vibration isolation structures

[0168] For nonlinear damping and restoring force function P c (z s ) and P f (z s Polynomial fitting is performed to transform the fractional terms in the nonlinear function into polynomial form:

[0169]

[0170]

[0171] Substituting equations (21) and (22) into equation (20), we get:

[0172]

[0173] By defining the following parameters:

[0174]

[0175]

[0176] The nonlinear equations of motion for ultra-low frequency vibration isolation structures can be uniformly written as:

[0177]

[0178] Assume the basic stimulus zb =z0cos(2πft), where f is the external excitation frequency and z0 is the fundamental excitation amplitude. Solving equation (26) using the multi-scale method yields the steady-state response of the nonlinear motion equation. The static equilibrium position vibration displacement z s The amplitude-frequency response curve can be expressed as:

[0179]

[0180] Where A1 is the vibration amplitude, s is the ratio of the external excitation frequency to the system's natural frequency, and ω0 is the system's natural frequency. The vibration displacement z of the ultra-low frequency vibration isolation structure... s It can be represented as:

[0181] z s =A1cos(2πft+θ) (28)

[0182] Where θ is the phase angle. The vibration transmissibility T of the ultra-low frequency vibration isolation structure. R It can be represented as:

[0183]

[0184] The ultra-low frequency vibration isolation characteristics of the ultra-low frequency vibration isolation structure provided by the present invention can be analyzed according to equations (27) and (29).

[0185] Example:

[0186] 1. Calculation Example

[0187] Choose such Figure 7 The ultra-low frequency vibration isolation structure containing the vibration-isolated object shown is used as an example for calculation and analysis of the present invention. The relevant parameters are set as follows: L1 = L2 = 0.1m, α = 80°, k s =2000N / m, k v =500N / m, d h =0m,z st =0.1m, e1=3N·s / m, e2=0.01N·m·s / rad, z0=0.02m.

[0188] 2. Calculation process

[0189] (1) Step 1 of the technical solution of the present invention clarifies the design of ultra-low frequency vibration isolation structure based on linear positive and negative stiffness compensation mechanism;

[0190] (2) A symmetrical polygonal linear negative stiffness unit with inclined springs was designed according to step 2 of the technical solution of the present invention.

[0191] (3) Step 3 of the present invention clarifies the linear positive and negative stiffness compensation mechanism of the ultra-low frequency vibration isolation structure and matches the linear positive and negative stiffness of the ultra-low frequency vibration isolation structure.

[0192] (4) According to step 4 of the technical solution of the present invention, a nonlinear dynamic model and motion equation of an ultra-low frequency vibration isolation structure containing vibration isolation objects are constructed.

[0193] (5) The multi-scale method provided in step 5 of the present invention is used to evaluate the vibration transmissibility and ultra-low frequency vibration isolation performance of the ultra-low frequency vibration isolation structure.

[0194] 3. Example verification:

[0195] (1) Considering the structural parameters as: L1=L2=0.1m, k s =2000N / m, Figure 8 Nonlinear force-displacement curves for a symmetric polygonal linear negative stiffness element with different initial angles α are presented. Figure 8 It can be observed that when the initial angle α = 50°, 60°, and 70°, the symmetrical polygonal linear negative stiffness element exhibits nonlinear negative stiffness characteristics after reaching the maximum load-bearing capacity. However, when α = 80°, the symmetrical polygonal linear negative stiffness element exhibits linear negative stiffness characteristics within the displacement range of 0.022m–0.197m, which helps to significantly improve the quasi-zero stiffness range and load-bearing capacity of the ultra-low frequency vibration isolation structure.

[0196] (2) Considering the structural parameters as: L1 = L2 = 0.1m, k s =2000N / m, Figure 9 Different initial angles α and spring stiffness k are given. v Curves showing the positive and negative stiffness compensation results of the ultra-low frequency vibration isolation structure. Figure 9 It can be observed that when α = 60°, k v When α = 267 N / m, the symmetrical polygonal negative stiffness element exhibits nonlinear negative stiffness characteristics. The vibration isolation structure composed of the symmetrical polygonal negative stiffness element and the linear vertical spring has a relatively small quasi-zero stiffness region and load-bearing capacity. However, when α = 80°, k... v When the stiffness is 500 N / m, the symmetrical polygonal negative stiffness element exhibits linear negative stiffness characteristics within the displacement range of 0.022 m–0.197 m. The linear negative stiffness can be just offset by the positive stiffness of the linear vertical spring, thus forming a wider quasi-zero stiffness region and a greater load-bearing capacity, which is extremely beneficial for the realization of ultra-low frequency vibration isolation.

[0197] (3) Calculate the force-displacement curve of the ultra-low frequency vibration isolation structure according to steps 1 to 3, then perform numerical simulation in ADAMS simulation software, and compare the result with the force-displacement curve obtained by the method provided in this invention. Figure 10and Figure 11 As shown. From Figure 10 and Figure 11 It can be seen that the force-displacement curve obtained by the method provided by the present invention matches the simulation results well (error less than 0.5%), which proves the correctness and effectiveness of the static stiffness theoretical analysis and calculation program of the present invention.

[0198] (4) Calculate the amplitude-frequency response characteristics of the ultra-low frequency vibration isolation structure of this invention according to steps 1 to 5. The structural parameters are considered as follows: L1 = L2 = 0.1m, α = 80°, k s =2000N / m, k v =500N / m, d h =0m,z st =0.1m, e1=3N·s / m, e2=0.01N·m·s / rad, z0=0.02m. Figure 12 The structural amplitude-frequency response curves obtained based on the direct numerical integration method and the multi-scale method provided in this invention are presented. Figure 12 As can be seen, the ultra-low frequency vibration isolation structure exhibits hard nonlinear characteristics, and the results obtained by numerical integration and multi-scale methods are in good agreement, which proves the effectiveness of the multi-scale method provided by this invention in analyzing this nonlinear dynamic system.

[0199] 4. Calculate the revenue

[0200] (1) The method of improving the quasi-zero stiffness region of the ultra-low frequency vibration isolation structure based on the linear positive and negative stiffness compensation mechanism of the present invention has been verified. The quasi-zero stiffness region accounts for 88% of the total stroke of the structure. Furthermore, the ultra-low frequency vibration isolation structure has a wider vibration isolation band, a lower vibration isolation frequency and resonance amplitude, and the effective vibration isolation frequency can be reduced to 0.15Hz.

[0201] (2) Figure 13 The influence of rod length L2 on the vibration isolation characteristics of the ultra-low frequency (ULF) vibration isolation structure is demonstrated. When the rod length L2 is 0.1m and 0.098m, the ULF vibration isolation structure exhibits hard nonlinear characteristics; when the rod length L2 increases to 0.102m, the ULF vibration isolation structure exhibits excellent soft nonlinear characteristics. By adjusting the rod length L2, the ULF vibration isolation structure provided by this invention possesses soft nonlinear characteristics, which can reduce the structural resonance region, widen the vibration isolation zone, and improve structural stability and ULF vibration isolation performance.

[0202] (3) In order to further verify the superior vibration isolation characteristics of the ultra-low frequency vibration isolation structure based on the positive and negative stiffness compensation mechanism, Figure 14 and Figure 15 It was clarified that in Q1(z) st =0.10m), Q2(z) st =0.13m), Q3(z) st=0.10m) and Q4(z st Vibration transmissibility curves of the structure at different equilibrium positions (0.13m). Figure 14 and Figure 15 As can be seen, the ultra-low frequency vibration isolation structure with linear positive and negative stiffness compensation mechanism has the smallest resonance frequency and resonance amplitude at the equilibrium position Q1, and its vibration isolation frequency can be reduced to about 0.15Hz, which proves that the ultra-low frequency vibration isolation structure provided by the present invention has ultra-low frequency vibration isolation characteristics.

[0203] (4) Compared with the traditional quasi-zero stiffness vibration isolation system, the ultra-low frequency vibration isolation structure provided by the present invention can improve its quasi-zero stiffness region by about 70% while ensuring the load-bearing capacity, and has a wider quasi-zero stiffness range; in addition, when the external excitation frequency is 1Hz, the vibration isolation efficiency of the ultra-low frequency vibration isolation structure provided by the present invention is 53% higher than that of the traditional nonlinear vibration isolator.

Claims

1. An ultra-low frequency vibration isolation structure based on a linear positive and negative stiffness compensation mechanism, characterized in that... The ultra-low frequency vibration isolation structure includes a polygonal linkage structure and a linear vertical spring, wherein: The polygonal linkage structure is composed of symmetrical polygonal linear negative stiffness units. The symmetrical polygonal linear negative stiffness element is composed of a first polygonal element and a second polygonal element, which are symmetrically distributed about the linear vertical spring as the axis of symmetry. Both the first polygonal unit and the second polygonal unit are composed of an upper quadrilateral structure and a lower quadrilateral structure; The top of the upper quadrilateral structure is connected to the supporting platform, the bottom of the upper quadrilateral structure is connected to the top of the lower quadrilateral structure, and the bottom of the lower quadrilateral structure is connected to the foundation. An inclined spring is installed on one diagonal of the upper quadrilateral structure and the lower quadrilateral structure, and the two inclined springs share a common rotation node. A linear vertical spring is installed between the support platform and the foundation.

2. A design method for ultra-low frequency vibration isolation structures based on linear positive and negative stiffness compensation mechanism, characterized in that... The method includes the following steps: Step 1: Design of Ultra-Low Frequency Vibration Isolation Structure Based on Linear Positive and Negative Stiffness Compensation Mechanism Step 1.1: Define the ultra-low frequency vibration isolation structure. The ultra-low frequency vibration isolation structure includes a polygonal linkage structure and a linear vertical spring, wherein: The polygonal linkage structure is composed of symmetrical polygonal linear negative stiffness units. The symmetrical polygonal linear negative stiffness element is composed of a first polygonal element and a second polygonal element, which are symmetrically distributed about the linear vertical spring as the axis of symmetry. Both the first polygonal unit and the second polygonal unit are composed of an upper quadrilateral structure and a lower quadrilateral structure; The top of the upper quadrilateral structure is connected to the supporting platform, the bottom of the upper quadrilateral structure is connected to the top of the lower quadrilateral structure, and the bottom of the lower quadrilateral structure is connected to the foundation. A stiffness of k is installed on one diagonal of the upper quadrilateral structure and the lower quadrilateral structure. s The two tilting springs share a common rotation node. The lengths of the inclined rod and the horizontal rod in the upper quadrilateral structure and the lower quadrilateral structure are L1 and L2, respectively, and the initial angle between the inclined rod and the horizontal plane is α. A rigidity k is installed between the load-bearing platform and the foundation. v A linear vertical spring; Step 1.2: Design a quasi-zero stiffness passive vibration isolation platform The quasi-zero stiffness vibration isolation platform consists of an ultra-low frequency vibration isolation structure, a load-bearing platform, and a foundation. The horizontal displacement Δh and rotation angle β of the ultra-low frequency vibration isolation structure are expressed as follows: （1） , （2） The initial length L of the tilting spring s1 and the length L after deformation s2 They are represented as follows: （3） （4） The deformation length ΔL of the inclined spring s Represented as: （5） Where z1 is the vertical displacement of the bearing platform, Δh is the horizontal displacement of the horizontal bar, and β is the rotation angle of the inclined bar; Step 2: Design a symmetrical polygonal linear negative stiffness element containing a tilting spring. Step 2.1: For the load-bearing platform, its force balance equation is: （6） Where f1 and f2 are the axial forces generated by connecting rods A1B1 and A2B2, respectively, and their magnitudes are the same as the axial forces generated by connecting rods A4B4 and A3B3. p The vertical load f borne by a symmetric polygonal linear negative stiffness element s = k s ΔL s The tension of the inclined spring. For the spring force f s The angle between the upper quadrilateral and the horizontal axis; nodes A1, A2, A3, and A4 are the connection nodes between the top of the upper quadrilateral structure and the supporting platform, respectively; nodes B1, B2, B3, and B4 are the connection nodes between the bottom of the upper quadrilateral structure and the top of the lower quadrilateral structure, respectively; nodes C1, C2, C3, and C4 are the connection nodes between the bottom of the lower quadrilateral structure and the foundation, respectively. Step 2.2: For nodes B1 and B2, the force balance equations are expressed as follows: （7） （8） （9） （10） Among them, f3, f4 and f5 are the axial tensile forces generated by connecting rods B1B2, B1C1 and B2C2; Step 2.3: Based on the equilibrium equations of the forces at the bearing platform, nodes B1 and B2, the vertical bearing capacity F of the symmetrical polygonal linear negative stiffness element is calculated. p Represented as: （11） in: （12） Step 3: Determine the linear positive and negative stiffness compensation mechanism of ultra-low frequency vibration isolation structures. Step 3.1: When the displacement of the bearing platform is 0 ≤ z1 ≤ d h Inside, d h The distance between the bearing platform and the linear vertical spring is such that the bearing platform and the linear vertical spring are not in contact. At this time, the total restoring force F of the ultra-low frequency vibration isolation structure and the vertical bearing force F of the symmetrical polygonal linear negative stiffness element are considered. p same; Step 3.2, when the displacement of the bearing platform z1 ≥ d h When the symmetrical polygonal linear negative stiffness element and the linear vertical spring are connected in parallel, the total restoring force F is: （13） Step 4: Construct the nonlinear dynamic model and equations of motion for the ultra-low frequency vibration isolation structure containing vibration isolation objects. Step 4.1: The vertical displacements of the bearing platform and foundation of the ultra-low frequency vibration isolation structure are respectively represented by z. p and z b The total mass of the vibration isolation object and the load-bearing platform is represented by M. t The relative displacement between the basic excitation and the supporting platform is expressed as z1 = z p -z b The kinetic energy T of the ultra-low frequency vibration isolation structure s and potential energy U s Represented as: , （14） In the formula, ΔL v Let g be the deformation length of the linear vertical spring, and g be the acceleration due to gravity. Step 4.2: Considering air damping and nodal rotational friction damping effects, the generalized force G in the vertical direction of the ultra-low frequency vibration isolation structure... z Represented as: （15） In the formula, e1 is the air damping coefficient, and e2 is the nodal rotational friction damping coefficient; Step 4.3: Apply Lagrange's principle to obtain the nonlinear equation of motion for the ultra-low frequency vibration isolation structure: （16） Step 4.4: By solving equations (14)-(16), the nonlinear motion equations of the ultra-low frequency vibration isolation structure are obtained as follows: （17） Where the nonlinear function P c (z1) and P f (z1) is represented as: , （18） Step 4.5: When the vibration-isolated object acts on the load-bearing platform, the ultra-low frequency vibration isolation structure will reach a new static equilibrium position. When the vibration isolation structure is subjected to foundation excitation, the vibration-isolated object will vibrate up and down near the static equilibrium position. The static equilibrium position is selected as the vibration origin, and the vibration displacement z at the static equilibrium position is... s Represented as: （19） Among them, z st The static displacement of the ultra-low frequency vibration isolation structure under the vibration isolation mass; Step 4.6: According to equations (17)-(19), the nonlinear motion equations of the ultra-low frequency vibration isolation structure are further expressed as: （20） Step 5: Evaluation of the vibration isolation performance of ultra-low frequency vibration isolation structures Step 5.1: For the nonlinear damping and restoring force function P... c (z s ) and P f (z s Polynomial fitting is performed to transform the fractional terms in the nonlinear function into polynomial form: （21） （22） Step 5.2: Substitute equations (21) and (22) into equation (20) to obtain: （23） Step 5.3: Define the following parameters: , , , , （24） , , , , （25） The nonlinear motion equations of ultra-low frequency vibration isolation structures are uniformly written as follows: （26） Step 5.4, Assume the basic stimulus z b =z0cos(2πft), where f is the external excitation frequency and z0 is the basic excitation amplitude. The steady-state response of the nonlinear motion equation is obtained by solving equation (26) using the multi-scale method. Step 5.5, Vibration displacement z at static equilibrium position s The amplitude-frequency response curve is expressed as: （27） Where A1 is the vibration amplitude, s is the ratio of the external excitation frequency to the system's natural frequency, and ω0 is the system's natural frequency; Vibration displacement z of ultra-low frequency vibration isolation structure s Represented as: （28） in, Phase angle; Vibration transmissivity T of ultra-low frequency vibration isolation structure R Represented as: （29） Step 5.6: Perform ultra-low frequency vibration isolation characteristic analysis on ultra-low frequency vibration isolation structure according to equations (27) and (29).

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