CN112182881A

CN112182881A - Nonlinear energy trap optimal stiffness solving method based on platform phenomenon

Info

Publication number: CN112182881A
Application number: CN202011035770.1A
Authority: CN
Inventors: 李芦钰; 张天娇
Original assignee: Dalian University of Technology
Current assignee: Dalian University of Technology
Priority date: 2020-09-27
Filing date: 2020-09-27
Publication date: 2021-01-05
Anticipated expiration: 2040-09-27
Also published as: CN112182881B

Abstract

The invention discloses a nonlinear energy trap optimal stiffness solving method based on a platform phenomenon, which belongs to the technical field of structure control, and is characterized in that the method is combined with the actual situation of civil engineering, under the premise of small initial conditions, a double-freedom-degree system consisting of a linear main structure and a cubic stiffness nonlinear energy trap is subjected to multi-scale analysis, the platform phenomenon in a slow varying equation is observed, so that an excitation amplitude interval in which effective target energy transmission can occur to a damper under the condition that system parameters are fixed is obtained, an analytic solution of nonlinear damper optimal design stiffness is derived through an interval upper and lower limit formula, and the design value has outstanding advantages in the aspects of vibration reduction effect and robustness after the examination of a numerical method. The optimized rigidity analytic solution obtained by the invention gets rid of the constraint on the initial condition, is applied to the design of the nonlinear damper, is simple in actual operation and good in optimization effect, and has great due value and development prospect.

Description

Nonlinear energy trap optimal stiffness solving method based on platform phenomenon

Technical Field

The invention relates to the technical field of structural control, in particular to a nonlinear energy trap optimal stiffness solving method based on a platform phenomenon.

Background

Since the damage to the building structure caused by natural disasters and various man-made damages is not a little worth, the structural vibration control is a research focus in recent years, and the vibration control is mainly divided into active control, semi-active control and passive control. Mass tuned dampers (TMDs) are the most representative of passive control devices, but have certain limitations because TMDs are linear. Firstly, efficient operation of the TMD relies on precise tuning, i.e. the frequency of the TMD is tuned to be equal to the primary mode shape of the main structure; second, because the resonance between the TMD and the main structure persists for some time, vibration energy is transmitted back and forth between the TMD and the main structure rather than being dissipated entirely after unidirectional transmission of the conductive TMD, and thus vibration damping is of limited efficiency. In response to the two-sided disadvantages of linear dampers, more and more researchers are beginning to study nonlinear dampers, namely the so-called nonlinear energy traps (NESS). Because the nonlinear device has no fixed frequency, the nonlinear device has low sensitivity to tuning, can resonate with any frequency, and a resonant mode can be destroyed instantly, energy can be consumed by damping after being transferred to the NES and can not be transferred back to a main structure, so that the NES can realize unidirectional energy transfer, namely Target Energy Transfer (TET). However, in a system connecting NES, the occurrence of TET requires that the parameters of NES meet certain conditions. The presence of TET is implied by the occurrence of a Strong Modulated Response (SMR), and therefore in the present invention "occurring SMR" is used as the initial measure of the optimization parameter.

Disclosure of Invention

According to the problems existing in the prior art, the invention discloses a nonlinear energy trap optimal stiffness solving method based on a platform phenomenon, which comprises the following steps:

s1: establishing a two-degree-of-freedom system motion equation consisting of a linear main structure and a nonlinear energy trap;

s2: converting real absolute coordinates in a motion equation into complex variable relative coordinates by using a complex variable averaging method, and eliminating a fast-changing part in a variable;

s3: the complex variable differential equation obtained by the multi-scale method can be respectively converted into a fast variable equation related to a fast time scale and a slow variable equation related to a slow time scale, the fast variable equation obtains a slow variable flow pattern of the two-degree-of-freedom system and a displacement amplitude at a jumping point, and the slow variable equation obtains a steady-state amplitude of the nonlinear energy trap;

s4, constructing a relation curve by taking the amplitude of the excitation as an abscissa and the response amplitude of the nonlinear energy trap as an ordinate, wherein the relation curve is a horizontal straight line in an emphasis control response region, the horizontal straight line is defined as a platform, and the phenomenon that the relation curve in the emphasis control region is the horizontal straight line is defined as a platform phenomenon of a two-degree-of-freedom system, the abscissa interval corresponding to the platform is the excitation amplitude range of the two-degree-of-freedom system for emphasizing control response, the ordinate corresponding to the platform is the amplitude at a jump point, and the platform phenomenon shows that when and only when the steady-state response of the two-degree-of-freedom system is in an emphasis control response mode, the steady-state amplitude of the nonlinear energy trap does not change along with the change of the excitation amplitude, and the steady-state amplitude is always equal to the amplitude at the jump point of;

s5, obtaining an analytic expression of the effective excitation amplitude range with strong modulation response through a platform phenomenon;

and S6, when the damping of the two-degree-of-freedom system is fixed, obtaining the effective rigidity range of the two-degree-of-freedom system with strong modulated response through the excitation amplitude range analytic expression.

Further, the two-degree-of-freedom system motion equation is as follows:

wherein: m is₁,m₂Mass of main structure and NES, respectively, c₁,c₂Damping coefficient, k, of the main structure and NES, respectively₁,k₂The stiffness coefficients of the main structure and the NES are respectively, F is the excitation amplitude, omega is the excitation frequency, and x₂,x₁Indicating the absolute displacement of the main structure and the damper, respectively.

Further, the main structure vibration amplitude psi₁And the amplitude psi of the vibration of the nonlinear energy trap₂The relationship of (a) to (b) is as follows:

wherein: lambda [ alpha ]₂Representing dimensionless damping of energy traps, i²Kn represents the dimensionless stiffness of the energy trap at-1.

Further, the excitation amplitude/range of the two-degree-of-freedom system emphasized the modulation response is:

f₁≤f≤f₂ (70)

further:

wherein: f. of₁To excite the lower limit of amplitude, f₂Upper limit of excitation amplitude, N₂₁Is the steady state amplitude of the energy well at the jump point.

Further, the air conditioner is provided with a fan,

wherein: σ is a tuning parameter, λ₁As dimensionless damping of the main structure, λ₂Dimensionless damping for energy traps:

due to the adoption of the technical scheme, the optimal stiffness solving method of the nonlinear energy trap (NES) based on the platform phenomenon, provided by the invention, obtains an approximate analytical solution of the NES optimal stiffness which can be directly used for design, so that the system can generate strong control response, and the total energy of the system is as low as possible. And the approximate solution is required to have certain robustness, the invention takes kn₁And kn₂The average value of (c) is taken as the optimum stiffness, i.e. the "plateau" midpoint, at which time it must happen that the system emphasizes the control response. Thus, a margin is set for errors, the system can be guaranteed to cope with small-range fluctuation of the excitation amplitude, robustness is met, and the nonlinear energy trap vibration reduction capability optimized by the method is obviously improved as proved by simulation.

Drawings

In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings can be obtained by those skilled in the art without creative efforts.

FIG. 1 is a system model diagram;

FIG. 2 is a constant current pattern diagram I (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.5)；

FIG. 3 is a constant current pattern II (. lamda.)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.25)；

FIG. 4 is a graph of displacement time course one (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.25)；

FIG. 5 is a phase diagram of phase diagram one (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.25)；

FIG. 6 is a constant flow pattern diagram III (. lamda.)₁＝0.8,λ₂＝0.3,kn＝5,f＝1)；

FIG. 7 is a graph of displacement time course two (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝1)；

FIG. 8 is a phase diagramTwo (lambda)₁＝0.8,λ₂＝0.3,kn＝5,f＝1)；

FIG. 9 is a constant current pattern diagram IV (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.5)；

FIG. 10 is a graph of displacement time course three (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.5)；

FIG. 11 is a phase diagram of phase III (. lamda.)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.5)；

FIG. 12 is a plateau view one (λ)₁＝0.8,λ₂＝0.3)；

FIG. 13 is a displacement time chart of four (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.25)；

FIG. 14 is a graph of displacement versus time (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.4)；

FIG. 15 is a graph of displacement versus time for a sixth time period (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.6)；

FIG. 16 is a graph of displacement versus time (λ)₁＝0.8,λ₂＝0.3,kn＝5,f＝0.8)；

FIG. 17 is a graph of "excitation amplitude-slowly changing phase" (λ)₁＝0.8,λ₂＝0.3,kn＝0.5)；

FIG. 18 is a table map (. lamda.)₁＝0.8,λ₂＝0.3,kn＝0.5)；

FIG. 19 is a table diagram of two (. lamda.)₁＝0.8,λ₂＝0.3)；

FIG. 20 is a plot of frequency energy plot one (λ)₁＝0.8,λ₂＝0.3,f＝0.5)；

FIG. 21 is a graph of frequency energy plot two (λ)₁＝0.8,λ₂＝0.3,f＝0.5,kn＝kn_b＝6.1692)；

FIG. 22 is a frequency energy plot of three (λ)₁＝0.8,λ₂＝0.3,f＝0.8)；

FIG. 23 is a frequency energy plot of four (λ)₁＝0.8,λ₂＝0.3,f＝0.8,kn＝kn_b＝2.4098)。

Detailed Description

In order to make the technical solutions and advantages of the present invention clearer, the following describes the technical solutions in the embodiments of the present invention clearly and completely with reference to the drawings in the embodiments of the present invention:

the multi-scale analysis is carried out on a two-degree-of-freedom system excited by resonance as shown in fig. 1 by a complex variable averaging method and a multi-scale method, and when the change rule of a steady-state amplitude value obtained by a slow-varying equation along with the change of an excitation amplitude value is researched, a 'platform phenomenon' is found, namely in an excitation interval with strong modulation response, the amplitude value obtained by the slow-varying equation does not change along with the change of excitation, but is always stabilized at a saddle node bifurcation point of the system, and the slow-varying phases at the beginning and the end of a platform can be approximately regarded as opposite numbers and the like. By means of the phenomenon, a series of deductions shows that the excitation amplitude interval of the SMR can occur when the system parameters are fixed, and the approximate analytic solution of the optimal nonlinear rigidity of the NES under the initial condition of being near 0 is deduced by the formula. Finally, the outstanding advantage of the optimal rigidity in terms of vibration damping is obtained through comparison verification by using a numerical method.

The research model of the invention is a two-degree-of-freedom system consisting of a linear main structure and an NES, and is excited by simple harmonic, which is shown in figure 1. Since the initial condition is usually 0 or around 0 in civil engineering, all studies of the present invention to follow assume that the initial condition of the system is around 0.

The system equation of motion is:

wherein m is₁,m₂Mass of main structure and NES, respectively, c₁,c₂Damping coefficients of the main structure and NES, respectively. k is a radical of₁,k₂The stiffness coefficients of the main structure and the NES are respectively, F is the excitation amplitude, omega is the excitation frequency, and x₂,x₁Indicating absolute position of main structure and damper respectivelyAnd (6) moving.

For the following calculation convenience, each parameter is subjected to non-dimensionalization:

wherein: representing the mass ratio, ω, of the damper₁Representing the first natural frequency, λ, of the main structure₁λ₂Represents the dimensionless damping of the main structure and the energy sink, respectively;

substituting the dimensionless parameters of the formula (3) into the original formula, and converting the motion equation into the following dimensionless form:

respectively calculating the absolute displacement coordinate x of the main structure₁And absolute displacement coordinates x of the energy well₂And (3) converting into a barycentric coordinate u and a relative coordinate v:

u＝x₁+x₂ (6)

v＝x₁-x₂ (7)

substituting the new coordinates into the equation of motion yields equations of motion for u and v:

further coordinate transformation is performed using complex variable averaging to remove fast variations in the displacements u and v, and to align the barycentric coordinates and the phaseConversion of coordinates into complex variables

And

wherein:

complex variable forms representing displacement of the main structure and the energy trap, respectively;

substituting the complex variable into the motion equation, and decoupling the differential equation by using a Taylor formula to obtain the following calculation convenience:

after an averaging process, 0 is eliminated²) The above high-order terms are arranged:

wherein: σ represents a tuning parameter;

converting the parameters into independent variable indexes by a complex variable averaging method for representation: (16) and (17):

setting new complex variables

A steady state amplitude is obtained, which is substituted into the equation:

in the following, an equation is divided into two scales of speed and speed by using a multi-scale method for analysis, wherein only time is divided into two scales according to the precision requirement of research, and new time parameters are set as follows:

t＝t₀+t₁ (18)

wherein: t is a time parameter, t₀For fast-changing time parameters, t₁Is a slow-varying time parameter;

the derivative can therefore also be written as:

D＝D₀+D₁ (19)

wherein: d represents the derivative of the derivative on t, D₀Represents a pair t₀Derivative of derivation, D₁Represents a pair t₁A derivative of the derivative;

substituting (19) into complex variable equation of motion (16) (17) yields:

order to⁰And¹the coefficient of the term is 0, then the coefficient equation for each term is:

⁰:D₀ψ₁＝0 (22)

obtaining a differential equation expressed by a multi-scale time variable by using a multi-scale method; obtaining the constant flow pattern of the system; since the present invention mainly studies the steady state response of the system, let ψ in (24)₂For tau₀Is 0, an invariant flow pattern diagram as shown in fig. 2 can be obtained, i.e., (25), where the parameters: lambda [ alpha ]₁＝0.8,λ₂＝0.3,kn＝5,f＝0.5。

In fig. 2, the thick line is the constant flow pattern of the system, the thin line is the actual response of the original equation of motion of the system, and it can be seen that the steady state part of the system is always operated according to the law of the slow flow pattern except the jump between the two steady state branches, so that τ is set later₀The scale derivative is 0 and the slow-varying part can be modeled well when only the steady-state response is studied. However, since the jump behavior belongs to the fast-changing behavior and occurs in a moment, the slow-changing course of the response path in this part may not be well simulated, and τ is studied later₁This is verified by the slow-varying equation in scale, which is a theoretical starting point for the derivation of the following formula.

Obtaining the relation between the vibration amplitude of the main structure and the vibration amplitude of the energy neck; writing complex variables in polar form, where N₂A slowly varying magnitude of NES is represented,₂slow-varying phase representing NES:

wherein: n is a radical of₁，N₂The slow-varying amplitudes of the main structure and the energy well are respectively represented; z₁，Z₂The squares of the slow varying amplitudes of the main structure and the energy wells, respectively;

substituting (26) to (29) into (25), and separating the real and imaginary parts gives:

for finding jump points on the constant flow pattern, the above formula is applied to Z₂Derivative, and let the derivative be 0:

two local extreme points (jump points) are solved as follows: a jumping point of a non-changing manifold is obtained,

wherein: n is a radical of₂₁Is the steady state amplitude of the energy trap at the jump point under small initial conditions, N₂₂Is the steady state amplitude of the energy well at the jump point for large initial conditions.

From the extreme points of (32) and (33), only dimensionless damping λ in the energy trap₂Satisfy the requirement of

Two extreme points are possible to appear, and when the damping of the energy trap does not meet the requirement, no matter how other parameters are changed, the system cannot jump. On the basis that the damping satisfies the above condition, whether the NES can generate strong modulation response depends on the value of the nonlinear stiffness kn, and the value is related to the excitation amplitude f.

If the system jumps, N₂₁,N₂₂Which is the jump point, also called saddle-bifurcation point, and at which point the jump actually occurs depends on the initial conditions of the system. From the practical point of view of civil engineering, the initial conditions are generally around 0, in which case it can be speculated from the non-changing flow pattern that the jump point should be N₂₁Point, hence the following branch point is chosen as N₂₁。

By continuing to consider equations (23) and (24) from the multiscale method, and substituting equation (25) for equation (23), we can derive a slowly varying time scale for phi alone₂The univariate equation of (c):

writing the above equation to the following form is convenient for observation and calculation:

Ax+Bx^*＝G (35)

A^*x^*+B^*x＝G^* (36)

can be solved to obtain:

wherein:

a denotes psi₂For tau₁B represents the coefficient of the derivative of

For tau₁G represents the remainder of the univariate equation other than the derivative term;

will be psi in the above formula₂Is written into

Substituting the formula into the formula, performing real-imaginary part separation, and finally finishing the formula into the following forms:

wherein: c and D are respectively a first substitution coefficient and a second substitution coefficient in the slow varying equation, and M is a denominator in the slow varying equation;

the expressions (41) and (42) are written as follows:

wherein:

since the existence of the denominator M at the jumping point generates singular points and cannot be simulated normally, it is verified that the following forms are written in the formulas (41) and (42) without changing the mechanical properties:

N′₂＝f₁ (50)

′₂＝f₂ (51)

the degree of accuracy and the range of accuracy of the above equations (50) and (51) are verified by comparison with the accurate equations (8) (9) and the complex variable equations (16) (17), and the constant flow pattern and time course curve of the system are made.

FIG. 3 is a constant current pattern II (. lamda.)₁＝0.8,λ₂0.3, kn 5, f 0.25); FIG. 4 is a graph of displacement time course one (λ)₁＝0.8,λ₂0.3, kn 5, f 0.25); FIG. 5 is a phase diagram of phase diagram one (λ)₁＝0.8,λ₂0.3, kn 5, f 0.25); the parameters of FIGS. 3, 4, and 5 are chosen to be λ₁＝0.8；λ₂0.3; kn is 5; f is 0.25, and it can be seen that the system amplitude has not yet reached this timeBy the jump point, no jump occurs, and the steady state response of the system is stable, always in the low amplitude region of fig. 3.

The parameters of FIGS. 6, 7, and 8 are chosen to be λ₁＝0.8；λ₂0.3; kn is 5; when the amplitude increases to 1, it can be seen that the steady state response of the system is still stable, but unlike before, the system eventually jumps to the upper region in the phase diagram of fig. 7 due to the increase in excitation, and no longer returns to the lower region where a sustained strong modulated response occurs, but stabilizes at the upper branch. The above two working condition systems do not generate continuous strong control response, and belong to an undesirable response state. It can also be seen that the approximation of the slow varying equations (50) (51) is relatively accurate at this time.

The parameters in fig. 9, 10, 11 are selected as: lambda [ alpha ]₁＝0.8；λ₂0.3; kn is 5; when f is 0.6, it can be observed that the steady-state response mode is a continuous strong modulated response, that is, a jump occurs in the system, however, since the jump is a fast-changing behavior, equation (50) (51) which ignores the fast-changing part cannot accurately simulate the real response situation. From the invariant flow chart, the amplitude N is obtained (50) (51)₂After a short increase in experience, eventually stabilized at N₂₁To (3).

In this case, the slowly varying equations (50) (51) appear to be unavailable to analyze the strong modulation response. It can still be used to judge the occurrence of strong modulation responses, and it is the limitations of slow-varying equations in the face of fast-varying variables that are utilized.

Since the nonlinear stiffness kn of the above three sets of graphs is the same, but the excitation amplitude f is changed, when f is 0.25; when f is 0.8, the steady state response is stable, when f is 0.5, the steady state response is an unstable strong modulation response, and when f is 1, the steady state response is again stable. It is therefore hypothesized that when the nonlinear stiffness kn is selected, there is an interval where the system is likely to generate SMR only when f is in this interval. The value of this interval may be related to the nonlinear stiffness kn.

To verify the above guess, the excitation amplitude f is changed by taking kn as 3,5,8,11, respectively, and using (50)) Formula (51) steady state amplitude N₂As shown in fig. 12.

FIG. 12 verifies the above hypothesis that there does exist an interval of f where the system can generate a strong modulation response if and only if the excitation amplitude is in this interval, this region being referred to herein as the "plateau". The amplitude value corresponding to the longitudinal coordinate of the platform is just the saddle node bifurcation point N₂₁When the excitation amplitude is in the abscissa interval corresponding to the platform, the system generates a strong modulation response, when the excitation amplitude exceeds the range of the platform, the steady-state response of the system recovers to be stable again, no jumping behavior occurs in the monotone increasing area in front of the platform, the monotone increasing area of the rear part is attracted by a certain stable attractor of the high-amplitude area after the monotone increasing area starts to generate jumping behaviors for a few times, and finally, the system is still stable in the high-amplitude area and no SMR continuously occurs, and the two cases are regarded as invalid.

To account for the above plateau phenomenon, the parameter λ is fixed₁＝0.8,λ₂When kn is 5, a certain f value in front of the platform and in the middle of the platform and after the platform is respectively taken as: f is 0.2, f is 0.4, f is 0.6, and f is 0.8, and the displacement time charts of fig. 13 to 16 are used to explain the above-described phenomenon.

The reason for the "plateau" effect is as previously described: when the excitation amplitude is small, the steady-state amplitude does not reach the jump point, so that the system presents stable steady-state response in a low-amplitude region; when SMR occurs, the formula (50) (51) neglecting the fast-changing part can not accurately simulate the jumping process, the amplitude is stabilized at the jumping point after the short-term increase of the first few seconds, and the jumping point value N is obtained by the formula (32)₂₁Only with k_nAnd λ₂Is related, independent of f, so the rate of change of amplitude with excitation f is 0 during SMR; when the excitation is increased continuously, the system jumps to the constant flow pattern and branches off, and does not reciprocate, and no continuous SMR occurs, so that the stable steady-state response is recovered, and at the moment, the slowly-varying equation can accurately simulate the amplitude of the system. Therefore, the abscissa of the starting point and the ending point of the 'platform' corresponds to the upper limit and the lower limit of the excitation amplitude of the SMR when the system parameters are fixed, and the interval is set as f₁,f₂]In this region, efficient target energy transfer occurs in the system. Also from FIG. 12, [ f ] can be observed₁,f₂]The value of (c) is indeed related to kn.

Make phase place₂As a function of f, the graph 17, finds that only in the region where the strong modulation response occurs,₂is decreasing, another surprising phenomenon can also be observed, the phases at the SMR start and end points can be viewed approximately as being opposite numbers to each other, and₂the decreasing range of (c) is fixed and is independent of the value of kn. These phenomena are the solution interval [ f ]₁,f₂]The analysis of (2) is possible, and the section where the strong modulation response occurs is defined as [ 2 ]₂₁,₂₂]From the figure, it can be derived:₂₂＝-₂₁。

from the time charts obtained by observing the slow varying equations (50) and (51), it is found that N 'is always present in the stable portion where SMR does not occur'₂＝′₂0, namely:

for convenience, here the rewrite equation (43) (44) is given by:

from N'₂＝′₂It can be deduced that C-D-0 is 0, and it is reasonable to observe the equations that C and D are 0 when the steady state response is stable, but when the parameters are in the plateau region in fig. 12, C-D-0 is impossible to achieve because N is found by the slow varying equations (50) (51)₂Constant, cos₂Monotonically increasing, then C, D may not be constant at all times as f increases. It can be seen in fig. 20 that while the steady state response is stable, the values of C, D are indeed always 0, both are not 0 only in the "plateau" portion where the emphasized response occurs, and that the abscissa interval where C, D are not 0,₂monotonically decreasing abscissa interval and N₂The horizontal coordinate interval of the platform is coincident with the three, and all the three are [ f₁,f₂]。

The following uses the above-observed phenomenon to find the range of excitation amplitudes [ f ] in which SMR can occur₁,f₂]：

Setting:

wherein:

since at the starting point C, D is still equal to 0, it can be derived from the simultaneous trigonometric identities of (54) (55):

f₁ ²＝F₁ ²+F₂ ² (58)

the lower excitation amplitude limit f at which the onset of the strong modulation response occurs is given by (58)₁The upper limit f is determined as follows₂：

C and D values at the beginning and the end are respectively C₁,C₂,D₁,D₂Then, there are:

C₁＝f₁ cos₂₁-F₁＝0 (59)

D₁＝f₁ sin₂₁+F₂＝0 (60)

C₂＝f₂ cos₂₁-F₁ (61)

D₂＝-f₂ sin₂₁+F₂ (62)

due to f₂As known, the starting point phase obtained from (59) is:

from (59) to (62) can be obtained:

and the tail end points of the platform are that (52) and (53) are always 0:

from the trigonometric identity:

(C₂+F₁)²+(D₂-F₂)²＝f₂ ² (67)

substituting the expressions (64) and (66) into the expression (67) can solve the abscissa f corresponding to the required end point₂The value, substituted, gives:

simplifying to obtain:

thus when kn is determined, the range of excitation amplitudes over which the system can generate a strong modulation response is:

f₁≤f≤f₂ (70)

will f is₁And f₂Substituting the value to obtain:

(71) i.e. an excitation amplitude range formula based on the "plateau" phenomenon, in which a strong modulation response can occur, but f is not a design parameter, the objective of the study is to optimize the nonlinear stiffness kn, and the optimal stiffness value is derived from the formula (71) below.

As shown in FIG. 12, the parameter range [ f ] for which SMR occurs₁,f₂]The method is related to the value of nonlinear stiffness kn, the smaller the value of kn is, the larger the range of effective excitation is, which is a good phenomenon, but when the value of kn is smaller, the larger the steady-state amplitude of a main structure is, a trade-off relation exists between the range of effective excitation and the steady-state amplitude, so that the larger the stiffness is, the better the stiffness is, or the smaller the stiffness is, the better the stiffness is, therefore, the range of kn is supposed to exist when the excitation amplitude is taken, and when the stiffness is in the range, the system can generate SMR.

Let the optimum stiffness be kn_bDue to f₁And f₂All with respect to lambda only₁,λ₂And kn, so when damping λ₁,λ₂And f, a range of effective stiffness can be derived by equation (71), and the range of effective stiffness in which strong modulation response can occur is set to [ kn [ k ] n ]₁,kn₂]. As can be seen from FIG. 12, when the excitation amplitude is determined, the amplitude is found as the corresponding stiffness value of "plateau" (i.e., kn) when the abscissa of the end point of a certain "plateau" is determined₂) And it is theoretically most reasonable to take this as the optimum stiffness. However, since the plateau range obtained by equation (71) is only an approximate analytical solution and has a certain error, if f is used, f is used₂Derived kn₂And deviation occurs when the optimal rigidity is taken as inevitable, and the robustness is poor.

Embodiment 1: taking the case of the excitation amplitude f being 0.5 as an example, the optimal stiffness kn obtained by the above formula is taken_bThe effect of applying the above method to optimize stiffness in a two degree of freedom system connected NES was examined in comparison to the situation when other stiffness values were taken.

Five nonlinear rigidity values are respectively taken: kn is 1<kn₁，kn₁<kn_b＝6.1691<kn₂，kn＝kn₂＝10.2650，kn＝14,18>kn₂Since there is no fixed amplitude when emphasizing the damping response, it is appropriate to use the average energy as the test criterion for the damping effect, and the transient total energy of the system can be written in the form of (72):

changing the tuning parameter σ, and checking each stiffness described above, 1: the average energy around resonance within 1, the results are shown in fig. 20 and 21.

It can be seen from FIG. 20 that when stiffness takes the optimum value kn_bThe total energy is significantly lower than the other stiffnesses. Although when ω is 1, kn is kn_bThe total energy of (a) is not the minimum but is also relatively small, and in practical cases, the excitation frequency is not exactly equal to 1, so the robustness in the frequency domain is important, and kn is chosen to be kn_bA balance is achieved between damping effect and robustness to frequency. Fig. 21 separately shows the optimized frequency energy curve, the smooth regions on both sides are stable steady-state responses, and the strong modulation response occurs near the middle ω ═ 1, so that the response mode is very favorable for vibration damping.

Taking the different excitation amplitudes f as 0.8 and five corresponding kn: kn is 0.5<kn₁，kn₁<kn_b＝2.4098<kn₂，kn＝kn₂＝4.0098，kn＝7,9>kn₂. Revalidation of kn_bThe results are shown in fig. 22 and 23. The conclusions drawn from fig. 22, 23 are in full agreement with fig. 20, 21 above.

The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims

1. A nonlinear energy trap optimal stiffness solving method based on a platform phenomenon is characterized by comprising the following steps: the method comprises the following steps:

2. The method for solving the optimal rigidity of the nonlinear energy trap based on the platform phenomenon according to claim 1, further characterized by comprising the following steps: the motion equation of the two-degree-of-freedom system is as follows:

3. The method for solving the optimal rigidity of the nonlinear energy trap based on the platform phenomenon according to claim 1, further characterized by comprising the following steps: the main structure vibration amplitude psi₁And the amplitude psi of the vibration of the nonlinear energy trap₂The relationship of (a) to (b) is as follows:

4. The method for solving the optimal rigidity of the nonlinear energy trap based on the platform phenomenon according to claim 1, further characterized by comprising the following steps: the excitation amplitude/range of the two-degree-of-freedom system strong modulation response is:

f₁≤f≤f₂ (70)

further:

5. The method for solving the optimal rigidity of the nonlinear energy trap based on the platform phenomenon according to claim 4, further characterized by comprising the following steps:

wherein: σ is a tuning parameter, λ₁As dimensionless damping of the main structure, λ₂Is the dimensionless damping of the energy sink.