CN115238415A - Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system - Google Patents

Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system Download PDF

Info

Publication number
CN115238415A
CN115238415A CN202210866695.6A CN202210866695A CN115238415A CN 115238415 A CN115238415 A CN 115238415A CN 202210866695 A CN202210866695 A CN 202210866695A CN 115238415 A CN115238415 A CN 115238415A
Authority
CN
China
Prior art keywords
stiffness
rigidity
layer
linear
lower layer
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202210866695.6A
Other languages
Chinese (zh)
Inventor
利云云
陈洋
汤杰
盛鹏
王者蓝
赵凡
吴欣凯
吴佳伟
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Shanghai Space Precision Machinery Research Institute
Original Assignee
Shanghai Space Precision Machinery Research Institute
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Shanghai Space Precision Machinery Research Institute filed Critical Shanghai Space Precision Machinery Research Institute
Priority to CN202210866695.6A priority Critical patent/CN115238415A/en
Publication of CN115238415A publication Critical patent/CN115238415A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16F—SPRINGS; SHOCK-ABSORBERS; MEANS FOR DAMPING VIBRATION
    • F16F15/00—Suppression of vibrations in systems; Means or arrangements for avoiding or reducing out-of-balance forces, e.g. due to motion
    • F16F15/02—Suppression of vibrations of non-rotating, e.g. reciprocating systems; Suppression of vibrations of rotating systems by use of members not moving with the rotating systems
    • F16F15/04—Suppression of vibrations of non-rotating, e.g. reciprocating systems; Suppression of vibrations of rotating systems by use of members not moving with the rotating systems using elastic means
    • F16F15/06—Suppression of vibrations of non-rotating, e.g. reciprocating systems; Suppression of vibrations of rotating systems by use of members not moving with the rotating systems using elastic means with metal springs
    • F16F15/067—Suppression of vibrations of non-rotating, e.g. reciprocating systems; Suppression of vibrations of rotating systems by use of members not moving with the rotating systems using elastic means with metal springs using only wound springs
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/04—Constraint-based CAD
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Geometry (AREA)
  • General Engineering & Computer Science (AREA)
  • Theoretical Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Computer Hardware Design (AREA)
  • Evolutionary Computation (AREA)
  • Acoustics & Sound (AREA)
  • Aviation & Aerospace Engineering (AREA)
  • Mechanical Engineering (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Vibration Prevention Devices (AREA)

Abstract

The invention provides a vibration reduction design and optimization method and system of a double-layer high-static low-dynamic stiffness vibration isolation system, which relate to the technical field of vibration reduction and comprise the following steps: step S1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system; step S2: carrying out principle analysis on high static stiffness and low dynamic stiffness, and giving a relational expression between external force and displacement borne by the whole system; and step S3: linear term k giving the total stiffness of the upper and lower layers of the structure 11 And kappa 12 Obtaining an effective value area through the constraint relation between the two groups; and step S4: establishment systemA dynamic equation of the system is solved; step S5: changing the total linear rigidity k of the upper layer and the lower layer according to the effective value area obtained in the step S2 11 And kappa 12 And obtaining the optimal double-layer high-static-low dynamic stiffness system. The invention can play excellent vibration damping performance in a pipeline test of a carrier rocket type, and reduces fatigue damage of a large amount of vibration to a vibration table and supporting hardware facilities.

Description

Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system
Technical Field
The invention relates to the technical field of vibration damping, in particular to a vibration damping design and optimization method and system of a double-layer high-static low-dynamic stiffness vibration isolation system.
Background
Both natural science and engineering applications place ever-increasing demands on the vibration environment. In recent years, high static and low dynamic stiffness vibration isolation systems have attracted wide attention of domestic and foreign scholars because of providing large static load supporting capacity and having extremely low natural frequency. The basic principle of the high-static low-dynamic stiffness vibration isolation system is that positive and negative stiffness mechanisms are connected in parallel, and when the positive and negative stiffness are just offset, the system can reach quasi-zero stiffness.
The invention mainly analyzes a double-layer high-static low-dynamic stiffness vibration isolation system with a negative stiffness adjusting structure sharing one foundation, and the total stiffness of the system can be regarded as the combination of the total stiffness of a lower layer system (the linear stiffness and the nonlinear stiffness of the lower layer system are connected in parallel) and the linear stiffness in series, and then the combination of the linear stiffness and the nonlinear stiffness in parallel. The rigidity of the upper layer and the lower layer of the system adopting the combination mode is not independent, so that certain constraint relation exists between the rigidity of the upper layer and the rigidity of the lower layer, and the condition that the integral rigidity cannot be less than 0 needs to be met. The effective value range of the rigidity is obtained according to the constraint condition, the influence of linear rigidity and nonlinear rigidity on the vibration isolation performance of the system is researched aiming at the boundary and the internal area of the system, and an optimal vibration isolation scheme is provided aiming at the double-layer high-static-low dynamic rigidity vibration isolation system.
The invention patent with the publication number of CN109737178A discloses a semi-active control quasi-zero rigidity vibration isolation system, which comprises a quasi-zero rigidity structure module, a magnetorheological damper and a semi-active control system; the semi-active control system is used for controlling the damping force of the magnetorheological damper according to vibration information acquired by the sensor on the quasi-zero stiffness structure module, so that the ratio of the excitation frequency to the natural frequency of the system is always in the range of the vibration isolation area.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a vibration reduction design and optimization method and system of a double-layer high-static low-dynamic stiffness vibration isolation system.
According to the vibration reduction design and optimization method and system of the double-layer high-static-low dynamic stiffness vibration isolation system provided by the invention, the scheme is as follows:
in a first aspect, a vibration damping design and optimization method for a double-layer high static stiffness and low dynamic stiffness vibration isolation system is provided, and the method includes:
step S1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system;
step S2: performing principle analysis on high static stiffness and low dynamic stiffness, and giving a relational expression between external force borne by the whole system and displacement;
and step S3: linear term k giving the total stiffness of the upper and lower layers of the structure 11 And kappa 12 Obtaining an effective value area through the constraint relation between the two groups;
and step S4: establishing a dynamic equation of the system and solving;
step S5: changing the total linear rigidity k of the upper layer and the lower layer according to the effective value area obtained in the step S2 11 And kappa 12 And obtaining the optimal double-layer high-static-low dynamic stiffness system.
Preferably, the step S2 includes:
the principle analysis is carried out on high static stiffness and low dynamic stiffness, the compressed Euler beam only generates deformation in the horizontal direction, and when the vibration-isolated object moves up and down, the vertical restoring force provided by the compressed Euler beam is as follows:
Figure BDA0003759503000000021
wherein L is the length of the euler beam when undeformed; pe = EI (pi/L) 2 The method is characterized in that hinged supports at two ends are used as boundary constraint conditions, and EI is the bending rigidity of the Euler beam; k is a radical of 1 And k 3 Is related to the initial tilt angle theta and the initial deflection omega 0 Coefficient of correlation, initial deflection ω 0 A euler beam critical buckling load of zero; x is the displacement from the initial position downwards;
after the negative stiffness mechanism is connected with the linear spring in parallel, a single-layer vibration isolator with high static stiffness and low dynamic stiffness is obtained; the rigidity of the existing upper layer linear spring is k v1 Then the overall system's resilience expression is written as:
Figure BDA0003759503000000022
wherein the first term for displacement represents the linear stiffness of the system and the third term for displacement represents the nonlinear stiffness of the system; when the linear rigidity of the system is zero, the system is in a quasi-zero rigidity state in high static rigidity and low dynamic rigidity.
Preferably, the step S3 includes:
is set to act on m 1 Has an excitation force of F m1 Are respectively aligned with m 1 、m 2 And (3) carrying out stress analysis and non-dimensionalization treatment to obtain the following two equations:
Figure BDA0003759503000000023
Figure BDA0003759503000000024
the following parameters were introduced:
Figure BDA0003759503000000031
wherein,
Figure BDA0003759503000000032
representing a dimensionless force; epsilon represents the stiffness ratio of the linear springs at the lower layer and the upper layer of the structure;
Figure BDA0003759503000000033
representing a dimensionless displacement; λ represents the stiffness ratio of the euler beam to the spring; k is a radical of formula 1 And k 3 Is a constant term related to a system parameter, P e The lower corner marks 1 and 2 represent the upper layer and the lower layer of the structure for the critical instability load of the Euler beam with hinged supports at two ends and zero initial deflection in the middle; l is the length of the Euler beam when undeformed, k v1 Is the stiffness of the upper linear spring; k is a radical of v2 Is the stiffness of the lower linear spring;
respectively solving first derivatives of the upper layer and the lower layer of the recovery force to obtain a dimensionless rigidity expression:
Figure BDA0003759503000000034
Figure BDA0003759503000000035
wherein, κ 11 ＝(1-λ 1 k 11 ) And kappa 12 ＝(ε-λ 2 k 12 ) Linear terms, κ, representing the overall stiffness of the upper and lower layers of the structure, respectively 31 ＝λ 1 k 31 And kappa 32 ＝λ 2 k 32 Respectively representing nonlinear terms of the total rigidity of the upper layer and the lower layer of the structure;
for the double-layer high-static-low dynamic stiffness vibration isolation system, the total stiffness K of the system at the balance position total_n Non-negative requirements need to be met; from this, k can be obtained 11 And kappa 12 The constraint relation of (c) is set to e =1:
Figure BDA0003759503000000036
binding kappa 11 ，κ 12 ∈[0,1]To obtain kappa 11 ，κ 12 An effective value area graph; the upper boundary and the right boundary of the structure respectively correspond to the upper layer rigidity linearity kappa of the structure 11 ＝1，κ 31 =0 and lower layer stiffness linearity κ 12 ＝1，κ 32 Case of = 0;
within the effective value range, when k is 12 And when the stiffness is not less than 0, the lower layer of the structure realizes quasi-zero stiffness, and the stiffness of the upper layer of the structure can only be linear k 11 =1, quasi-zero stiffness cannot be achieved simultaneously.
Preferably, the step S4 includes:
establishing a kinetic equation of the system: according to Newton's second law, a dynamic equation set of the active vibration isolation model is obtained:
Figure BDA0003759503000000037
Figure BDA0003759503000000038
the following parameters were introduced:
Figure BDA0003759503000000039
Figure BDA00037595030000000310
the equation becomes:
Figure BDA0003759503000000041
Figure BDA0003759503000000042
wherein μ represents the mass ratio of the lower layer and the upper layer of the structure, ε represents the linear spring rate ratio of the lower layer structure to the upper layer structure, and ζ represents the linear spring rate ratio 1 、ζ 2 The damping ratios of the upper and lower layer vibration isolation systems are respectively; and solving the equation by using a harmonic balancing method.
Preferably, the step S5 includes:
step S5.1: taking values along the upper boundary of the effective value area, and analyzing the influence of the lower layer rigidity on the force transfer rate of the system when the upper layer rigidity is linear;
step S5.2: taking a value along the right boundary of the effective value area, and analyzing the influence of changing the rigidity of the upper layer on the force transfer rate of the system when the rigidity of the lower layer is linear;
step S5.3: taking a value along the lower boundary of the effective value area, analyzing the influence of simultaneously changing the rigidity of the upper layer and the lower layer on the force transfer rate of the system, wherein the corresponding situation is that the rigidity of the upper layer and the lower layer is nonlinear at the same time; finally, an optimal double-layer high-static-low dynamic stiffness system is found.
In a second aspect, a vibration damping design and optimization system for a double-layer high static stiffness and low dynamic stiffness vibration isolation system is provided, the system comprising:
a module M1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system;
a module M2: performing principle analysis on high static stiffness and low dynamic stiffness, and giving a relational expression between external force borne by the whole system and displacement;
a module M3: linear term k giving the total stiffness of the upper and lower layers of the structure 11 And kappa 12 Obtaining an effective value area through the constraint relation between the two groups;
a module M4: establishing a dynamic equation of the system and solving;
a module M5: according to the effective value-taking area obtained in the module M2, the total linear rigidity k of the upper layer and the lower layer is changed 11 And kappa 12 And obtaining the optimal double-layer high-static-low dynamic stiffness system.
Preferably, said module M2 comprises:
the principle analysis is carried out on high static stiffness and low dynamic stiffness, the compressed Euler beam only generates deformation in the horizontal direction, and when the vibration-isolated object moves up and down, the vertical restoring force provided by the compressed Euler beam is as follows:
Figure BDA0003759503000000043
wherein L is the length of the euler beam when undeformed; pe = EI (Pi/L) 2 The method is characterized in that hinged supports at two ends are used as boundary constraint conditions, and EI is the bending rigidity of the Euler beam; k is a radical of 1 And k 3 Is related to the initial tilt angle theta and the initial deflection omega 0 Coefficient of correlation, initial deflection ω 0 A critical buckling load of eulerian beams of zero; x is the displacement from the initial position downwards;
after the negative stiffness mechanism is connected with the linear spring in parallel, a single-layer high-static-low dynamic stiffness vibration isolator is obtained; the rigidity of the existing upper layer linear spring is k v1 Then the overall system's resilience expression is written as:
Figure BDA0003759503000000051
wherein the first term for displacement represents the linear stiffness of the system and the third term for displacement represents the nonlinear stiffness of the system; when the linear rigidity of the system is zero, the system is in a quasi-zero rigidity state in high static rigidity and low dynamic rigidity.
Preferably, said module M3 comprises:
is set to act on m 1 Has an excitation force of F m1 Each to m 1 、m 2 And (3) carrying out stress analysis and non-dimensionalization to obtain the following two equations:
Figure BDA0003759503000000052
Figure BDA0003759503000000053
the following parameters were introduced:
Figure BDA0003759503000000054
wherein,
Figure BDA0003759503000000055
representing a dimensionless force; epsilon represents the stiffness ratio of the linear springs at the lower layer and the upper layer of the structure;
Figure BDA0003759503000000056
representing a dimensionless displacement; λ represents the stiffness ratio of the euler beam to the spring; k is a radical of 1 And k 3 Is a constant term related to a system parameter, P e The lower corner marks 1 and 2 represent the upper layer and the lower layer of the structure for the critical instability load of the Euler beam with hinged supports at two ends and zero initial deflection in the middle; l is the length of the Euler beam when undeformed, k v1 Is the stiffness of the upper linear spring; k is a radical of formula v2 Is the stiffness of the lower linear spring;
respectively solving first derivatives of the upper layer and the lower layer of the recovery force to obtain a dimensionless rigidity expression:
Figure BDA0003759503000000057
Figure BDA0003759503000000058
wherein, κ 11 ＝(1-λ 1 k 11 ) And kappa 12 ＝(ε-λ 2 k 12 ) Linear terms, κ, representing the overall stiffness of the upper and lower layers of the structure, respectively 31 ＝λ 1 k 31 And kappa 32 ＝λ 2 k 32 Respectively representing nonlinear terms of the total rigidity of the upper layer and the lower layer of the structure;
for double layer heightStatic low dynamic stiffness vibration isolation system, total stiffness K of system at balance position total_n Non-negative requirements need to be met; from this, k can be obtained 11 And kappa 12 The constraint relation of (c) is set to e =1:
Figure BDA0003759503000000059
binding kappa 11 ，κ 12 ∈[0,1]To obtain kappa 11 ，κ 12 An effective value area graph; the upper boundary and the right boundary of the structure respectively correspond to the upper layer rigidity linearity kappa of the structure 11 ＝1，κ 31 =0 and lower layer stiffness linearity κ 12 ＝1，κ 32 Case of = 0;
within the valid value range, when κ 12 When the stiffness is not less than 0, the lower layer of the structure realizes quasi-zero stiffness, and the stiffness of the upper layer of the structure can only be linear k 11 =1, quasi-zero stiffness cannot be achieved simultaneously.
Preferably, said module M4 comprises:
establishing a kinetic equation of the system: according to Newton's second law, a dynamic equation set of the active vibration isolation model is obtained:
Figure BDA0003759503000000061
Figure BDA0003759503000000062
the following parameters were introduced:
Figure BDA0003759503000000063
Figure BDA0003759503000000064
the equation becomes:
Figure BDA0003759503000000065
Figure BDA0003759503000000066
wherein μ represents the mass ratio of the lower layer and the upper layer, ε represents the linear spring rate ratio of the lower layer structure and the upper layer structure, and ζ represents the linear spring rate ratio of the lower layer structure and the upper layer structure 1 、ζ 2 The damping ratios of the upper and lower layer vibration isolation systems are respectively; and solving the equation by using a harmonic balance method.
Preferably, said module M5 comprises:
block M5.1: values are taken along the upper boundary of the effective value area, and the influence of the lower layer rigidity on the force transfer rate of the system is analyzed when the upper layer rigidity is linear;
block M5.2: taking values along the right boundary of the effective value area, and analyzing the influence of the rigidity of the upper layer on the force transmission rate of the system when the rigidity of the lower layer is linear;
block M5.3: taking a value along the lower boundary of the effective value area, and analyzing the influence of simultaneously changing the rigidity of the upper layer and the lower layer on the force transfer rate of the system, wherein the situation that the rigidity of the upper layer and the lower layer is nonlinear at the same time corresponds to the situation; finally, an optimal double-layer high-static-low dynamic stiffness system is found.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention provides a new vibration isolation idea and a vibration isolation structure;
2. the invention greatly increases the vibration isolation efficiency and ensures the vibration isolation effect at low frequency band and high frequency band;
3. the system has the excellent effect of realizing full-band vibration isolation after the parameters are optimized.
Drawings
Other features, objects and advantages of the invention will become more apparent upon reading of the detailed description of non-limiting embodiments with reference to the following drawings:
FIG. 1 is a schematic structural view of a double-layer high static stiffness and low dynamic stiffness vibration isolation system according to an embodiment of the invention;
FIG. 2 is a schematic view of a negative stiffness mechanism according to an embodiment of the present invention;
FIG. 3 shows an embodiment κ of the present invention 11 ，κ 12 And (4) taking a value area map.
Detailed Description
The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that it would be obvious to those skilled in the art that various changes and modifications can be made without departing from the spirit of the invention. All falling within the scope of the present invention.
The embodiment of the invention provides a vibration reduction design and optimization method of a double-layer high-static-low dynamic stiffness vibration isolation system, which is shown by referring to fig. 1 and fig. 2 and specifically comprises the following steps:
step S1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system; m, k v C respectively represents the vibration-isolated mass, the linear spring stiffness and the viscous damping coefficient; the lower corner marks 1, 2 represent the upper and lower layers of the structure.
Step S2: and (4) carrying out principle analysis on high static stiffness and low dynamic stiffness, and giving out a relational expression of external force and displacement of the whole system.
Wherein, step S2 specifically includes:
principle analysis is carried out on high static stiffness and low dynamic stiffness, and if the compressed Euler beam only generates deformation in the horizontal direction, when the vibration-isolated object moves up and down, the vertical restoring force provided by the compressed Euler beam can be approximated as follows:
Figure BDA0003759503000000071
wherein L is the length of the euler beam when undeformed; pe = EI (pi/L) 2 The method is characterized in that hinged supports at two ends are used as boundary constraint conditions, and EI is the bending rigidity of the Euler beam; k is a radical of formula 1 And k 3 Is at an initial angle of inclinationTheta and initial deflection omega 0 Coefficient of correlation, initial deflection ω 0 A euler beam critical buckling load of zero; x is the displacement from the initial position downwards.
After the negative stiffness mechanism is connected with the linear spring in parallel, a single-layer high-static-low dynamic stiffness vibration isolator is obtained; now assume that the stiffness of the upper linear spring is k v1 Then the restoring force expression for the entire system can be written as:
Figure BDA0003759503000000072
wherein the first term for displacement represents the linear stiffness of the system and the third term for displacement represents the nonlinear stiffness of the system; when the linear rigidity of the system is zero, the system is in an optimal state of high static rigidity and low dynamic rigidity, namely quasi-zero rigidity.
And step S3: linear term k giving the total stiffness of the upper and lower layers of the structure 11 And kappa 12 The effective value area is obtained through the constraint relation between the two.
Specifically, step S3 includes:
acting on m 1 Has an excitation force of F m1 Are respectively aligned with m 1 、m 2 And (3) carrying out stress analysis and non-dimensionalization to obtain the following two equations:
Figure BDA0003759503000000081
Figure BDA0003759503000000082
the following parameters were introduced:
Figure BDA0003759503000000083
wherein,
Figure BDA0003759503000000084
representing a dimensionless force; epsilon represents the stiffness ratio of the linear springs at the lower layer and the upper layer of the structure;
Figure BDA0003759503000000085
representing a dimensionless displacement; λ represents the stiffness ratio of the euler beam to the spring; k is a radical of 1 And k 3 Is a constant term related to a system parameter, P e For the critical instability load of the Euler beam with zero initial deflection at the middle of hinged supports at two ends, lower corner marks 1 and 2 represent an upper layer and a lower layer of the structure; l is the length of the Euler beam when undeformed, k v1 Is the stiffness of the upper linear spring; k is a radical of formula v2 Is the stiffness of the lower linear spring;
respectively solving first derivative of upper and lower layer reversion force to obtain dimensionless rigidity expression:
Figure BDA0003759503000000086
Figure BDA0003759503000000087
wherein, κ 11 ＝(1-λ 1 k 11 ) And kappa 12 ＝(ε-λ 2 k 12 ) Linear terms, κ, representing the overall stiffness of the upper and lower layers of the structure, respectively 31 ＝λ 1 k 31 And kappa 32 ＝λ 2 k 32 Respectively representing nonlinear terms of the total rigidity of the upper layer and the lower layer of the structure;
for the double-layer high-static-low dynamic stiffness vibration isolation system, the total stiffness K of the system at the balance position total_n Non-negative requirements need to be met; from this, k can be obtained 11 And kappa 12 (assuming ε = 1):
Figure BDA0003759503000000088
binding kappa 11 ，κ 12 ∈[0,1]Obtaining the value range ofTo kappa 11 ，κ 12 An effective value area graph; the dark regions as in FIG. 3 are κ 11 And kappa 12 The upper boundary and the right boundary of the effective value area of (c) respectively correspond to the structural upper layer stiffness linearity (k) 11 ＝1，κ 31 = 0) and lower layer stiffness linearity (κ) 12 ＝1，κ 32 Case of = 0);
it is noted that within the valid range of values, when κ is present 12 When the stiffness is not less than 0, the lower layer of the structure realizes quasi-zero stiffness, and the stiffness of the upper layer of the structure can only be linear (kappa) 11 = 1), quasi-zero stiffness cannot be achieved simultaneously.
And step S4: and establishing a kinetic equation of the system and solving.
Specifically, step S4 includes:
establishing a kinetic equation of the system: according to Newton's second law, a dynamic equation set of the active vibration isolation model is obtained:
Figure BDA0003759503000000091
Figure BDA0003759503000000092
the following parameters were introduced:
Figure BDA0003759503000000093
Figure BDA0003759503000000094
the equation becomes:
Figure BDA0003759503000000095
Figure BDA0003759503000000096
wherein μ represents the mass ratio of the lower layer and the upper layer of the structure, ε represents the linear spring rate ratio of the lower layer structure to the upper layer structure, and ζ represents the linear spring rate ratio 1 、ζ 2 The damping ratios of the upper and lower layer vibration isolation systems are respectively; and solving the equation by using a harmonic balancing method.
Step S5: changing the total linear rigidity k of the upper layer and the lower layer according to the effective value area obtained in the step S2 11 And kappa 12 And obtaining an optimal double-layer high-static-stiffness and low-dynamic-stiffness system.
Wherein, step S5 specifically includes, according to the effective value area obtained in step 2:
step S5.1: values are taken along the upper boundary of the effective value area, and the influence of the lower layer rigidity on the force transfer rate of the system is analyzed when the upper layer rigidity is linear;
step S5.2: taking values along the right boundary of the effective value area, and analyzing the influence of the rigidity of the upper layer on the force transmission rate of the system when the rigidity of the lower layer is linear;
step S5.3: taking a value along the lower boundary of the effective value area, analyzing the influence of simultaneously changing the rigidity of the upper layer and the lower layer on the force transfer rate of the system, wherein the corresponding situation is that the rigidity of the upper layer and the lower layer is nonlinear at the same time; finally, an optimal double-layer high-static-low dynamic stiffness system is found.
The vibration reduction design and optimization method and system of the double-layer high-static-low dynamic stiffness vibration isolation system provide a new vibration isolation thought and a vibration isolation structure, greatly increase the vibration isolation efficiency, ensure the vibration isolation effect in a low frequency band and a high frequency band, and have the excellent effect of realizing full-frequency-band vibration isolation after the parameters of the related system are optimized. The vibration reduction design and optimization method for the double-layer high-static-low dynamic stiffness vibration isolation system can play an excellent vibration reduction role in a pipeline test of a carrier rocket type, and reduces fatigue damage of a large amount of vibration to a vibration table and supporting hardware facilities.
It is well within the knowledge of a person skilled in the art to implement the system and its various devices, modules, units provided by the present invention in a purely computer readable program code means that the same functionality can be implemented by logically programming method steps in the form of logic gates, switches, application specific integrated circuits, programmable logic controllers, embedded microcontrollers and the like. Therefore, the system and various devices, modules and units thereof provided by the invention can be regarded as a hardware component, and the devices, modules and units included in the system for realizing various functions can also be regarded as structures in the hardware component; means, modules, units for realizing various functions can also be regarded as structures in both software modules and hardware components for realizing the methods.
The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes or modifications may be made by one skilled in the art within the scope of the appended claims without departing from the spirit of the invention. The embodiments and features of the embodiments of the present application may be combined with each other arbitrarily without conflict.

Claims (10)

1. A vibration reduction design and optimization method for a double-layer high-static-stiffness and low-dynamic-stiffness vibration isolation system is characterized by comprising the following steps:
step S1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system;
step S2: carrying out principle analysis on high static stiffness and low dynamic stiffness, and giving a relational expression between external force and displacement borne by the whole system;
and step S3: linear term k giving the overall stiffness of the upper and lower layers of the structure 11 And kappa 12 Obtaining an effective value area through the constraint relation between the two groups;
and step S4: establishing a dynamic equation of the system and solving;
step S5: changing the total linear rigidity kappa of the upper layer and the lower layer according to the effective value-taking area obtained in the step S2 11 And kappa 12 And obtaining the optimal double-layer high-static-low dynamic stiffness system.
2. The vibration damping design and optimization method for the double-layer high static and low dynamic stiffness vibration isolation system according to claim 1, wherein the step S2 comprises:
the principle analysis is carried out on high static stiffness and low dynamic stiffness, the compressed Euler beam only generates deformation in the horizontal direction, and when the vibration-isolated object moves up and down, the vertical restoring force provided by the compressed Euler beam is as follows:
Figure FDA0003759502990000011
wherein L is the length of the euler beam when undeformed; pe = EI (pi/L) 2 The method is characterized in that hinged supports at two ends are used as boundary constraint conditions, and EI is the bending rigidity of the Euler beam; k is a radical of 1 And k 3 Is related to the initial tilt angle theta and the initial deflection omega 0 Coefficient of correlation, initial deflection ω 0 A critical buckling load of eulerian beams of zero; x is the displacement from the initial position downwards;
after the negative stiffness mechanism is connected with the linear spring in parallel, a single-layer vibration isolator with high static stiffness and low dynamic stiffness is obtained; the rigidity of the existing upper layer linear spring is k v1 Then the restoring force expression for the entire system is written as:
Figure FDA0003759502990000012
wherein the first term for displacement represents the linear stiffness of the system and the third term for displacement represents the nonlinear stiffness of the system; when the linear rigidity of the system is zero, the system is in a quasi-zero rigidity state in high static rigidity and low dynamic rigidity.
3. The vibration damping design and optimization method for the double-layer vibration isolation system with high static stiffness and low dynamic stiffness according to claim 1, wherein the step S3 comprises the following steps:
is set to act on m 1 Has an excitation force of F m1 Are respectively aligned with m 1 、m 2 The stress analysis and the dimensionless treatment are carried out to obtain the following twoThe equation:
Figure FDA0003759502990000021
Figure FDA0003759502990000022
the following parameters were introduced:
Figure FDA0003759502990000023
wherein,
Figure FDA0003759502990000024
representing a dimensionless force; epsilon represents the stiffness ratio of the linear springs at the lower layer and the upper layer of the structure;
Figure FDA0003759502990000025
representing a dimensionless displacement; λ represents the stiffness ratio of the euler beam to the spring; k is a radical of formula 1 And k 3 Is a constant term, P, related to a system parameter e For the critical instability load of the Euler beam with zero initial deflection at the middle of hinged supports at two ends, lower corner marks 1 and 2 represent an upper layer and a lower layer of the structure; l is the length of the Euler beam when undeformed, k v1 Is the stiffness of the upper linear spring; k is a radical of formula v2 Is the stiffness of the lower linear spring;
respectively solving first derivatives of the upper layer and the lower layer in response to obtain a dimensionless rigidity expression:
Figure FDA0003759502990000026
Figure FDA0003759502990000027
wherein, κ 11 ＝(1-λ 1 k 11 ) And kappa 12 ＝(ε-λ 2 k 12 ) Linear terms, κ, representing the overall stiffness of the upper and lower layers of the structure, respectively 31 ＝λ 1 k 31 And kappa 32 ＝λ 2 k 32 Respectively representing nonlinear terms of the total rigidity of the upper layer and the lower layer of the structure;
for the double-layer high static and low dynamic stiffness vibration isolation system, the total stiffness K of the system at the balance position total_n Non-negative requirements need to be met; from this, k can be obtained 11 And kappa 12 With respect to the constraint relation of (c), let ε =1:
Figure FDA0003759502990000028
binding kappa 11 ，κ 12 ∈[0,1]To obtain kappa 11 ，κ 12 An effective value area graph; the upper boundary and the right boundary of the structure respectively correspond to the upper layer rigidity linearity kappa of the structure 11 ＝1，κ 31 =0 and lower layer stiffness linearity κ 12 ＝1，κ 32 Case of = 0;
within the valid value range, when κ 12 When the stiffness is not less than 0, the lower layer of the structure realizes quasi-zero stiffness, and the stiffness of the upper layer of the structure can only be linear k 11 =1, quasi-zero stiffness cannot be achieved simultaneously.
4. The vibration damping design and optimization method for the double-layer high static and low dynamic stiffness vibration isolation system according to claim 1, wherein the step S4 comprises:
establishing a kinetic equation of the system: according to Newton's second law, a dynamic equation set of the active vibration isolation model is obtained:
Figure FDA0003759502990000029
Figure FDA00037595029900000210
the following parameters were introduced:
Figure FDA00037595029900000211
τ＝ω n t，
Figure FDA00037595029900000212
Figure FDA0003759502990000031
the equation becomes:
Figure FDA0003759502990000032
Figure FDA0003759502990000033
wherein μ represents the mass ratio of the lower layer and the upper layer, ε represents the linear spring rate ratio of the lower layer structure and the upper layer structure, and ζ represents the linear spring rate ratio of the lower layer structure and the upper layer structure 1 、ζ 2 The damping ratios of the upper and lower layer vibration isolation systems are respectively; and solving the equation by using a harmonic balance method.
5. The vibration damping design and optimization method for the double-layer high static and low dynamic stiffness vibration isolation system according to claim 1, wherein the step S5 comprises:
step S5.1: taking values along the upper boundary of the effective value area, and analyzing the influence of the lower layer rigidity on the force transfer rate of the system when the upper layer rigidity is linear;
step S5.2: taking values along the right boundary of the effective value area, and analyzing the influence of the rigidity of the upper layer on the force transmission rate of the system when the rigidity of the lower layer is linear;
step S5.3: taking a value along the lower boundary of the effective value area, and analyzing the influence of simultaneously changing the rigidity of the upper layer and the lower layer on the force transfer rate of the system, wherein the situation that the rigidity of the upper layer and the lower layer is nonlinear at the same time corresponds to the situation; finally, an optimal double-layer high-static-low dynamic stiffness system is found.
6. The utility model provides a damping design and optimizing system of double-deck high quiet low dynamic stiffness vibration isolation system which characterized in that includes:
a module M1: building a system: a pair of Euler bending beam negative rigidity regulators are respectively connected in parallel on the upper layer and the lower layer of the double-layer linear vibration isolation system;
a module M2: carrying out principle analysis on high static stiffness and low dynamic stiffness, and giving a relational expression between external force and displacement borne by the whole system;
a module M3: linear term k giving the total stiffness of the upper and lower layers of the structure 11 And kappa 12 Obtaining an effective value area through the constraint relation between the two groups;
a module M4: establishing a dynamic equation of the system and solving;
a module M5: according to the effective value-taking area obtained in the module M2, the total linear rigidity k of the upper layer and the lower layer is changed 11 And kappa 12 And obtaining the optimal double-layer high-static-low dynamic stiffness system.
7. The vibration damping design and optimization system for the vibration isolation system with double layers of high static stiffness and low dynamic stiffness as claimed in claim 6, wherein the module M2 comprises:
the principle analysis is carried out on high static stiffness and low dynamic stiffness, the compressed Euler beam only generates deformation in the horizontal direction, and when the vibration-isolated object moves up and down, the vertical restoring force provided by the compressed Euler beam is as follows:
Figure FDA0003759502990000034
wherein L is the length of the euler beam when undeformed; pe = EI (pi/L) 2 The method is characterized in that hinged supports at two ends are used as boundary constraint conditions, and EI is the bending rigidity of the Euler beam; k is a radical of 1 And k 3 Is related to the initial tilt angle theta and the initial deflection omega 0 Related toCoefficient, initial deflection ω 0 A critical buckling load of eulerian beams of zero; x is the displacement from the initial position downwards;
after the negative stiffness mechanism is connected with the linear spring in parallel, a single-layer high-static-low dynamic stiffness vibration isolator is obtained; the rigidity of the existing upper layer linear spring is k v1 Then the restoring force expression for the entire system is written as:
Figure FDA0003759502990000041
wherein the first term for displacement represents the linear stiffness of the system and the third term for displacement represents the nonlinear stiffness of the system; when the linear rigidity of the system is zero, the system is in a quasi-zero rigidity state in high static rigidity and low dynamic rigidity.
8. The vibration damping design and optimization system for a double-deck high static and low dynamic stiffness vibration isolation system according to claim 6, wherein the module M3 comprises:
is set to act on m 1 Has an excitation force of F m1 Are respectively aligned with m 1 、m 2 And (3) carrying out stress analysis and non-dimensionalization treatment to obtain the following two equations:
Figure FDA0003759502990000042
Figure FDA0003759502990000043
the following parameters were introduced:
Figure FDA0003759502990000044
wherein,
Figure FDA0003759502990000045
representing a dimensionless force; epsilon represents the stiffness ratio of the linear springs at the lower layer and the upper layer of the structure;
Figure FDA0003759502990000046
representing a dimensionless displacement; λ represents the stiffness ratio of the euler beam to the spring; k is a radical of 1 And k 3 Is a constant term related to a system parameter, P e The lower corner marks 1 and 2 represent the upper layer and the lower layer of the structure for the critical instability load of the Euler beam with hinged supports at two ends and zero initial deflection in the middle; l is the length of the Euler beam when undeformed, k v1 Is the stiffness of the upper linear spring; k is a radical of v2 Is the stiffness of the lower linear spring;
respectively solving first derivatives of the upper layer and the lower layer in response to obtain a dimensionless rigidity expression:
Figure FDA0003759502990000047
Figure FDA0003759502990000048
wherein, κ 11 ＝(1-λ 1 k 11 ) And kappa 12 ＝(ε-λ 2 k 12 ) Linear terms, κ, representing the overall stiffness of the upper and lower layers of the structure, respectively 31 ＝λ 1 k 31 And kappa 32 ＝λ 2 k 32 Respectively representing nonlinear terms of the total rigidity of the upper layer and the lower layer of the structure;
for the double-layer high static and low dynamic stiffness vibration isolation system, the total stiffness K of the system at the balance position total_n Non-negative requirements need to be met; from this, k can be obtained 11 And kappa 12 With respect to the constraint relation of (c), let ε =1:
Figure FDA0003759502990000049
binding kappa 11 ，κ 12 ∈[0,1]To obtain kappa 11 ，κ 12 An effective value area graph; the upper boundary and the right boundary of the structure respectively correspond to the upper layer rigidity linearity kappa of the structure 11 ＝1，κ 31 =0 and lower layer stiffness linearity κ 12 ＝1，κ 32 Case of = 0;
within the effective value range, when k is 12 When the stiffness is not less than 0, the lower layer of the structure realizes quasi-zero stiffness, and the stiffness of the upper layer of the structure can only be linear k 11 =1, quasi-zero stiffness cannot be achieved simultaneously.
9. The vibration damping design and optimization system for the vibration isolation system with two layers of high static stiffness and low dynamic stiffness as claimed in claim 6, wherein the module M4 comprises:
establishing a kinetic equation of the system: according to Newton's second law, a dynamic equation set of the active vibration isolation model is obtained:
Figure FDA0003759502990000051
Figure FDA0003759502990000052
the following parameters were introduced:
Figure FDA0003759502990000053
τ＝ω n t，
Figure FDA0003759502990000054
Figure FDA0003759502990000055
the equation becomes:
Figure FDA0003759502990000056
Figure FDA0003759502990000057
wherein μ represents the mass ratio of the lower layer and the upper layer of the structure, ε represents the linear spring rate ratio of the lower layer structure to the upper layer structure, and ζ represents the linear spring rate ratio 1 、ζ 2 The damping ratios of the upper and lower layer vibration isolation systems are respectively; and solving the equation by using a harmonic balance method.
10. The vibration damping design and optimization system for a double-deck high static and low dynamic stiffness vibration isolation system according to claim 6, wherein the module M5 comprises:
block M5.1: values are taken along the upper boundary of the effective value area, and the influence of the lower layer rigidity on the force transfer rate of the system is analyzed when the upper layer rigidity is linear;
block M5.2: taking a value along the right boundary of the effective value area, and analyzing the influence of changing the rigidity of the upper layer on the force transfer rate of the system when the rigidity of the lower layer is linear;
block M5.3: taking a value along the lower boundary of the effective value area, analyzing the influence of simultaneously changing the rigidity of the upper layer and the lower layer on the force transfer rate of the system, wherein the corresponding situation is that the rigidity of the upper layer and the lower layer is nonlinear at the same time; finally, an optimal double-layer high-static-low dynamic stiffness system is found.
CN202210866695.6A 2022-07-22 2022-07-22 Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system Pending CN115238415A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202210866695.6A CN115238415A (en) 2022-07-22 2022-07-22 Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202210866695.6A CN115238415A (en) 2022-07-22 2022-07-22 Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system

Publications (1)

Publication Number Publication Date
CN115238415A true CN115238415A (en) 2022-10-25

Family

ID=83676132

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202210866695.6A Pending CN115238415A (en) 2022-07-22 2022-07-22 Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system

Country Status (1)

Country Link
CN (1) CN115238415A (en)

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115773612A (en) * 2022-11-28 2023-03-10 中国科学院西安光学精密机械研究所 Combined vibration isolation system of multiple refrigerators of space optical equipment and design method

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115773612A (en) * 2022-11-28 2023-03-10 中国科学院西安光学精密机械研究所 Combined vibration isolation system of multiple refrigerators of space optical equipment and design method

Similar Documents

Publication Publication Date Title
Jiang et al. A novel bio-inspired multi-joint anti-vibration structure and its nonlinear HSLDS properties
Chen et al. Influence of inerter on natural frequencies of vibration systems
Narasimhan et al. Smart base‐isolated benchmark building. Part I: problem definition
Jung et al. Application of some semi‐active control algorithms to a smart base‐isolated building employing MR dampers
Wang Time‐delayed dynamic output feedback ℋ︁∞ controller design for civil structures: A decentralized approach through homotopic transformation
Tan et al. Benchmark structural control problem for a seismically excited highway bridge—part II: phase I sample control designs
Cha et al. Direct performance-based design with 200 kN MR dampers using multi-objective cost effective optimization for steel MRFs
Domaneschi Simulation of controlled hysteresis by the semi-active Bouc-Wen model
Lavan et al. Full resources utilization seismic design of irregular structures using multiple tuned mass dampers
CN113312713A (en) Optimal parameter design method for tuned inertial mass damper
Azadi et al. Performance analysis of a semi-active mount made by a new variable stiffness spring
Cimellaro et al. Design of passive systems for control of inelastic structures
CN115238415A (en) Vibration reduction design and optimization method and system for double-layer high-static-low-dynamic-stiffness vibration isolation system
Cao et al. New control strategies for active tuned mass damper systems
Cimellaro et al. Algorithm for design of controlled motion of adjacent structures
Ou et al. Seismic response analysis of structures with velocity-dependent dampers
Wang et al. Curved surface-based vibration isolation mechanism with designable stiffness: Modeling, simulation, and applications
CN105160100A (en) TMD (Tuned Mass Damper) optimal design method of installation spring-quality system
Kim et al. Optimal design of smart mid-story isolated control system for a high-rise building
Mathew et al. Seismic response control of a building by negative stiffness devices
Lee et al. Experimental and analytical study of sliding mode control for isolated bridges with MR dampers
Huang et al. Optimal arrangement of viscoelastic dampers for seismic control of adjacent shear-type structures
Cha et al. Seismic retrofit of MRF buildings using decentralized semi‐active control for multi‐target performances
Fakhry et al. Optimized seismic response control of coupled FEM-BEM high-rise structural models using magnetorheological dampers
Elias et al. Seismic response control of steel benchmark building with a tuned mass damper

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination