CN112182881B - Nonlinear energy well optimal stiffness solving method based on platform phenomenon - Google Patents
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Abstract
The invention discloses a method for solving optimal rigidity of a nonlinear energy trap based on a platform phenomenon, which belongs to the technical field of structural control, combines the actual situation of civil engineering, carries out multi-scale analysis on a double-degree-of-freedom system consisting of a linear main structure and a cubic rigidity nonlinear energy trap under the premise of small initial conditions, observes the platform phenomenon in a slow-change equation, obtains an excitation amplitude interval in which effective target energy transmission can occur to a damper under the condition of fixed system parameters, deduces an analysis solution of optimal design rigidity of the nonlinear damper through formulas of upper and lower limits of the interval, and has outstanding advantages in vibration reduction effect and robustness through the inspection of a numerical method. The optimized rigidity analysis obtained by the method gets rid of the constraint on the initial condition, is applied to the design of the nonlinear damper, has simple actual operation and good optimizing effect, and has great due value and development prospect.
Description
Technical Field
The invention relates to the technical field of structural control, in particular to a nonlinear energy well optimal stiffness solving method based on a platform phenomenon.
Background
Since damage to building structures caused by natural disasters and various artificial damages is not very variable, vibration control of structures is mainly classified into active control, semi-active control and passive control for research hotspots in recent years. Mass tuned dampers (TMDs) are one of the most representative of passive control devices, but have certain limitations because TMDs are linear. First, the effective operation of the TMD relies on precisely tuning, i.e., adjusting the frequency of the TMD to be equal to the primary structure first mode; second, vibration damping is limited because the resonance between the TMD and the host structure continues for a period of time, so that vibration energy is transferred back and forth between the TMD and the host structure, rather than being entirely consumed after unidirectional transfer of the TMD. In view of the two drawbacks of linear dampers, more and more students are beginning to study non-linear dampers, i.e. non-linear energy traps (nes) in general. Since the nonlinear device has no fixed frequency, it has low sensitivity to tuning, can resonate with any frequency, and the resonant mode is destroyed instantaneously, energy is consumed by its damping after being transferred to the NES and is not transferred back to the main structure, so that the NES can realize unidirectional transfer of energy, i.e. target energy transfer (Targeted energy transfer, TET). However, in NES connected systems, the occurrence of TET requires that the NES parameters meet certain conditions. The occurrence of the emphasis response (Strong modulated response, SMR) implies the presence of TET, so in the present invention, the "occurrence of SMR" is taken as the initial measure of the optimization parameters.
Disclosure of Invention
According to the problems existing in the prior art, the invention discloses a nonlinear energy well optimal stiffness solving method based on a platform phenomenon, which comprises the following steps:
s1: a double-freedom-degree system motion equation formed by a linear main structure and a nonlinear energy well is established;
s2: converting real absolute coordinates in a motion equation into complex variable relative coordinates by using a complex variable average method, and eliminating a fast-changing part in the variables;
s3: the complex variable differential equation obtained by the multi-scale method can be respectively converted into a fast-change equation about a fast time scale and a slow-change equation about a slow time scale, the slow-change equation is used for obtaining a slow-change type of a double-degree-of-freedom system and displacement amplitude values at jumping points, and the slow-change equation is used for obtaining steady-state amplitude values of a nonlinear energy trap;
s4, constructing a relation curve by taking the excitation amplitude as an abscissa and the response amplitude of the nonlinear energy trap as an ordinate, defining the relation curve as a horizontal straight line in an emphasized response area, defining the horizontal straight line as a platform, defining the phenomenon that the relation curve in the emphasized response area is a horizontal straight line as a platform phenomenon of the dual-degree-of-freedom system, wherein an abscissa interval corresponding to the platform is an excitation amplitude range of the dual-degree-of-freedom system for emphasized response, and an ordinate corresponding to the platform is an amplitude at a jump point, and the platform phenomenon indicates that the steady-state amplitude of the nonlinear energy trap does not change along with the change of the excitation amplitude when and only when the steady-state response of the dual-degree-of-freedom system is in the emphasized response mode, and the steady-state amplitude is always equal to the amplitude at the jump point of the dual-degree-of-freedom system;
s5, obtaining an analytic expression of an effective excitation amplitude range of the generated emphasis response through a platform phenomenon;
s6, when the damping of the two-degree-of-freedom system is fixed, the effective stiffness range of the two-degree-of-freedom system generating stress damping response is obtained through excitation amplitude range analysis.
Further, the motion equation of the two-degree-of-freedom system is as follows:
wherein: m is m 1 ,m 2 Mass of main structure and NES, c 1 ,c 2 Damping coefficients, k, of the main structure and NES, respectively 1 ,k 2 The stiffness coefficients of the main structure and NES are respectively, F is the excitation amplitude, omega is the excitation frequency, and x 2 ,x 1 Representing the absolute displacement of the main structure and the damper, respectively.
Further, the main structure vibration amplitude ψ 1 And the vibration amplitude ψ of a nonlinear energy trap 2 The relationship of (2) is as follows:
wherein: lambda (lambda) 2 Non-dimensional damping representing an energy trap, i 2 = -1, kn represents the dimensionless stiffness of the energy well.
Further, the excitation amplitude f range of the two-degree-of-freedom system emphasis response is:
f 1 ≤f≤f 2 (70)
further:
wherein: f (f) 1 To lower the excitation amplitude, f 2 To excite the upper limit of the amplitude, N 21 Is the steady-state amplitude of the energy well at the jumping point.
Further, the method comprises the steps of,
wherein: sigma is tuning parameter, lambda 1 Non-dimensional damping, lambda, of the main structure 2 Dimensionless damping for an energy trap:
by adopting the technical scheme, the nonlinear energy trap (NES) optimal rigidity solving method based on the platform phenomenon can be directly used for designing an approximate analytic solution of NES optimal rigidity, so that the system can generate strong braking 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 kn 1 And kn 2 The average of (a) is taken as the optimal stiffness, i.e., the "plateau" midpoint, at which point the system must take place with emphasis on the response. The method has the advantages that the margin is reserved for errors, the system can cope with small-range fluctuation of the excitation amplitude, the robustness is met, and the simulation verification shows that the nonlinear optimized by the method of the inventionThe vibration damping capacity of the energy trap is obviously improved.
Drawings
In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings that are required to be used in the embodiments or the description of the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a simplified diagram of a system model;
FIG. 2 is a graph of invariant flow pattern 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.5)；
FIG. 3 is a constant flow pattern diagram II (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.25)；
FIG. 4 is a graph of displacement time-course (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.25)；
FIG. 5 is a phase diagram I (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.25)；
FIG. 6 is a graph of invariant flow patterns III (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝1)；
FIG. 7 is a displacement time chart II (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝1)；
FIG. 8 is a phase diagram II (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝1)；
FIG. 9 is a graph of invariant flow pattern four (λ) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.5)；
FIG. 10 is a displacement time chart III (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.5)；
FIG. 11 is a phase diagram III (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.5)；
FIG. 12 is a platform diagram I (lambda) 1 ＝0.8,λ 2 ＝0.3)；
FIG. 13 is a displacement time chart IV (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.25)；
FIG. 14 is a displacementTime chart five (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.4)；
FIG. 15 is a graph of displacement time chart six (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.6)；
FIG. 16 is a graph of displacement time course seven (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝5,f＝0.8)；
FIG. 17 is a plot of "excitation amplitude versus slowly varying phase" (λ) 1 ＝0.8,λ 2 ＝0.3,kn＝0.5)；
FIG. 18 is a platform control (lambda) 1 ＝0.8,λ 2 ＝0.3,kn＝0.5)；
FIG. 19 is a diagram of a second stage (lambda) 1 ＝0.8,λ 2 ＝0.3)；
FIG. 20 is a graph of frequency energy versus one (λ) 1 ＝0.8,λ 2 ＝0.3,f＝0.5)；
FIG. 21 is a frequency energy plot II (lambda) 1 ＝0.8,λ 2 ＝0.3,f＝0.5,kn＝kn b ＝6.1692)；
FIG. 22 is a frequency energy plot III (λ) 1 ＝0.8,λ 2 ＝0.3,f＝0.8)；
FIG. 23 is a graph of frequency energy curve four (λ) 1 ＝0.8,λ 2 ＝0.3,f＝0.8,kn＝kn b ＝2.4098)。
Detailed Description
In order to make the technical scheme and advantages of the present invention more clear, the technical scheme in the embodiment of the present invention is clearly and completely described below with reference to the accompanying drawings in the embodiment of the present invention:
a multi-scale analysis is carried out on a double-freedom-degree system excited by resonance as shown in fig. 1 through a complex variable averaging method and a multi-scale method, a platform phenomenon is found when the change rule of steady-state amplitude values along with excitation amplitude values obtained by a slow-change equation is studied, namely, the amplitude values obtained by the slow-change equation are not changed along with the change of excitation in an excitation interval with strong modulation response, but are always stable at a saddle junction bifurcation point of the system, and slow-change phases at the beginning and the end of the platform can be approximately regarded as opposite numbers. By virtue of the above phenomena, through a series of deductions, it is obtained that at a certain system parameter, an excitation amplitude interval of the SMR can occur, and from this formula, an approximate analytical solution of the NES optimal nonlinear stiffness is deduced when the initial condition is near 0. Finally, the outstanding advantage of optimal stiffness in vibration reduction is verified by comparison by using a numerical method.
The research model of the invention is a double-freedom system composed of a linear main structure and a NES, and is excited by simple harmonic, as shown in figure 1. Since in civil engineering, the initial condition is usually 0 or around 0, all subsequent studies of the present invention assume that the initial condition of the system is around 0.
The equation of motion of the system is:
wherein m is 1 ,m 2 Mass of main structure and NES, c 1 ,c 2 Damping coefficients of the main structure and NES, respectively. k (k) 1 ,k 2 The stiffness coefficients of the main structure and NES are respectively, F is the excitation amplitude, omega is the excitation frequency, and x 2 ,x 1 Representing the absolute displacement of the main structure and the damper, respectively.
For the convenience of the following calculation, each parameter is dimensionless:
wherein: ε represents the mass ratio of damper, ω 1 Representing the first natural frequency, lambda, of the main structure 1 λ 2 The non-dimensional damping of the main structure and the energy well, respectively;
substituting the dimensionless parameters of the formula (3) into the original formula, and converting the motion equation into the following dimensionless form:
for the convenience of calculation, the absolute displacement coordinates x of the main structure are respectively calculated 1 And the absolute displacement coordinate x of the energy trap 2 Conversion into a barycentric coordinate u and a relative coordinate v:
u＝x 1 +εx 2 (6)
v＝x 1 -x 2 (7)
substituting the new coordinates into the motion equation yields the motion equations for u and v:
the complex variable averaging method is used for further coordinate transformation, which aims at eliminating the fast variables in the displacements u and v and converting the barycentric coordinates and the relative coordinates into complex variablesAnd->
Wherein:complex variable forms representing the displacement of the main structure and the energy well, respectively;
substituting complex variables into the motion equation and decoupling the differential equation by using the taylor formula to facilitate the following calculation:
through an averaging process, and 0 (epsilon) is eliminated 2 ) The higher order items are sorted:
wherein: sigma represents a tuning parameter;
the method is characterized by converting the complex variable average method into an independent variable index to represent: (16) and (17):
setting new complex variablesThe steady-state amplitude is obtained and substituted into the above formula to obtain:
the equation is divided into two scales for analysis by a multi-scale method, wherein according to the precision requirement of the research, the time is divided into two scales, and the new time parameters are set as follows:
t＝t 0 +εt 1 (18)
wherein: t is a time parameter, t 0 To be a fast-changing time parameter, t 1 Is a slow-change time parameter;
the derivative can therefore also be written correspondingly as:
D＝D 0 +εD 1 (19)
wherein: d represents the derivative of t derivative, D 0 Representation pair t 0 Derivative of derivative D 1 Representation pair t 1 Derivative of derivative;
substituting (19) into complex variable motion equation (16) (17) to obtain:
let epsilon 0 And epsilon 1 The coefficient of the term is 0, and the coefficient equation of each term is:
ε 0 :D 0 ψ 1 ＝0 (22)
obtaining a differential equation represented by a multi-scale time variable by using a multi-scale method; obtaining a constant flow pattern of the system; since the invention is mainly developedThe steady state response of the system is explored, so let ψ in (24) 2 For tau 0 The derivative of (2) is 0, and a constant flow pattern diagram shown in fig. 2, namely (25), can be obtained, wherein parameters are as follows: lambda (lambda) 1 ＝0.8,λ 2 ＝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 portion always operates according to the law of the slow flow pattern except for the jump between the two steady-state branches, thus letting τ follow 0 The scale derivative is 0, and the slow-varying part can be well simulated when only the steady-state response is studied. However, since the jump behavior is a fast-changing behavior and occurs instantaneously, the slow-changing equation of the response path of this part may not be well simulated, and τ is studied later 1 This is demonstrated by the slowly changing equation in scale, which is the theoretical starting point for the derivation of the following equation.
Obtaining the relation between the vibration amplitude of the main structure and the vibration amplitude of the energy neck; writing complex variables in the form of polar coordinates, where N 2 Represents the slowly varying magnitude, delta, of NES 2 Representing the slowly varying phase of NES:
wherein: n (N) 1 ，N 2 Representing the slowly varying magnitudes of the main structure and the energy well, respectively; z is Z 1 ，Z 2 The squares of the slowly varying magnitudes of the main structure and the energy well, respectively;
substituting (26) to (29) into (25), and separating the real and imaginary parts to obtain:
to find jumping points on the invariant stream pattern, the above relation is Z 2 Derivative, and let derivative be 0:
the solution of two local extreme points (jumping points) is: the jumping points of the non-deformation shape are obtained,
wherein: n (N) 21 For steady-state amplitude, N, of the energy trap at the jumping point under small initial conditions 22 Is the steady-state amplitude of the energy well at the jumping point under large initial conditions.
From extreme points of (32) (33), only the dimensionless damping lambda at the energy trap 2 Satisfy the following requirementsTwo extreme points are possible to appear, and when the damping of the energy well does not meet the requirement, no matter how other parameters are changed, the jump phenomenon of the system does not occur. On the basis that the damping satisfies the above conditions, whether NES can produce an emphasized response also depends on the value of the nonlinear stiffness kn, and this value is related to the excitation amplitudef is related.
If the system jumps, N 21 ,N 22 That is, the jumping point, which may also be referred to as a saddle junction bifurcation point, at which 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 is presumed that the jumping point should be N from the constant flow pattern 21 Points, thus later selecting the bifurcation point as N 21 。
Continuing to consider equation (23) (24) from the multiscale method, substituting the invariant stream (25) equation into equation (23) yields a slowly varying time scale with respect to only ψ 2 Is a single variable equation of (2):
the above is written in the following form for ease of observation and calculation:
Ax+Bx * ＝G (35)
A * x * +B * x＝G * (36)
can be solved as follows:
wherein:
a represents psi 2 For tau 1 Coefficient of derivative of B representsFor tau 1 G represents the remainder of the univariate equation other than the derivative term;
psi in the above formula 2 Writing intoSubstituting the above formula and separating real and imaginary parts, and finally finishing the formula into the following formula:
wherein: c, D is the first and second substitution coefficients in the slow-change equation respectively, M is the denominator in the slow-change equation;
writing the formulas (41) (42) into the following form:
wherein:
since the existence of the parent M at the jump points can generate singular points and cannot be normally simulated, the (41) (42) expression is written into the following form under the condition of not changing the mechanical property through verification:
N′ 2 ＝f 1 (50)
δ′ 2 ＝f 2 (51)
the degree and range of accuracy of the above equations (50) and (51) are verified by comparison with the exact equations (8) (9) and complex variable equations (16) (17), and the constant flow pattern and time course curve of the system are made.
FIG. 3 is a constant flow pattern diagram II (lambda) 1 ＝0.8,λ 2 =0.3, kn=5, f=0.25); FIG. 4 is a graph of displacement time-course (lambda) 1 ＝0.8,λ 2 =0.3, kn=5, f=0.25); FIG. 5 is a phase diagram I (lambda) 1 ＝0.8,λ 2 =0.3, kn=5, f=0.25); the parameters of figures 3,4 and 5 are chosen as lambda 1 ＝0.8；λ 2 =0.3; kn=5; f=0.25, it can be seen that the system amplitude has not reached the jump point at this time, no jump has occurred, and the steady state response of the system is stable at this time, always in the low amplitude region of fig. 3.
The parameters of figures 6,7 and 8 are chosen to be lambda 1 ＝0.8；λ 2 =0.3; kn=5; f=1, it can be seen that when the amplitude increases to 1, the steady state response of the system is still stable, but unlike before, at this point 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 braking response occurs, but rather stabilizes in the upper branch. The continuous strong brake response of the two working condition systems does not occur, which belongs to no reasonA desired response state. It can also be seen that at this time, the approximation of the slow change equations (50) (51) is relatively accurate.
The parameters in fig. 9, 10, 11 are selected as: lambda (lambda) 1 ＝0.8；λ 2 =0.3; kn=5; f=0.6, it can be observed from the graph that the steady state response mode is a continuous emphasis response, that is, a jump occurs in the system, however, since the jump is a fast-changing behavior, the (50) (51) equation ignoring the fast-changing part cannot accurately simulate the real response situation. From the invariant flow pattern, (50) (51) the amplitude N 2 After a short increase in experience, finally stabilize at N 21 Where it is located.
In this case, the slow change equation (50) (51) appears to be unavailable for analyzing the emphasis response. It can still be used to judge the occurrence of the emphasis response, taking advantage of the limitations of the slow-varying equations in the face of the fast variables.
Since the nonlinear stiffness kn of the above three sets of graphs is the same, but the excitation amplitude f is varied, when f=0.25; the steady state response is stable when f=0.8, unstable emphasis response when f=0.5, and stable recovery when f=1. It is therefore hypothesized that when the nonlinear stiffness kn is selected, there is a span, and only when f is in this span, the system is likely to occur SMR. The value of this interval may be related to the nonlinear stiffness kn.
To verify the above hypothesis, the excitation amplitude f was varied, kn= 3,5,8,11 was taken separately, and the steady-state amplitude N was determined using (50) (51) 2 As shown in fig. 12.
Fig. 12 verifies that the above-mentioned hypothesis does exist for an interval of f, and that the system can develop an emphasized response if and only if the excitation amplitude is in this interval, this area being referred to herein as the "plateau". The amplitude corresponding to the ordinate of the platform is exactly the saddle bifurcation point N 21 When the excitation amplitude is in the abscissa interval corresponding to the 'platform', the system generates strong excitation response, when the excitation amplitude is out of the range of the platform, the steady state response of the system is restored to be stable, no jump behavior occurs in the monotonically increasing area in front of the platform, and the back partThe divided monotonically increasing regions are attracted by a stable attractor in the high amplitude region after a short number of jumps have begun to occur, and eventually remain stable in the high amplitude region, and no SMR continues to occur, both of which are considered ineffective.
To explain the above plateau phenomenon, the parameter lambda is fixed 1 ＝0.8,λ 2 =0.3, kn=5, taking a certain f value before "plateau", after "plateau" neutralization "plateau", respectively: f=0.2, f=0.4, f=0.6, f=0.8, the above-described phenomenon is explained with the displacement time charts of fig. 13 to 16.
The "plateau" effect is caused by the following reasons: when the excitation amplitude is smaller, the steady-state amplitude cannot reach the jumping point, so that the system presents stable steady-state response in a low-amplitude area; when SMR happens, the jump process cannot be accurately simulated by the (50) (51) equation with the fast-changing part ignored, the amplitude is always stable at the jump point after the short increase of the first few seconds, and the value N is obtained by the jump point (32) 21 And k only n And lambda (lambda) 2 In relation to f, so that the rate of change of amplitude with excitation f is 0 during SMR; when the excitation continues to increase, the system jumps to the upper branch of the constant flow type and does not reciprocate, so that no continuous SMR occurs, and stable steady-state response is restored, and at the moment, the slow-change equation can accurately simulate the amplitude of the system. Therefore, the upper and lower limits of the excitation amplitude of the SMR can occur when the system parameters are fixed and the interval is set as [ f 1 ,f 2 ]In this region, efficient target energy transfer occurs in the system. From FIG. 12 it can also be observed that [ f 1 ,f 2 ]The value of (2) is indeed related to kn.
Make phase delta 2 The change with f, graph 17, found that delta was only in the region where the emphasis response occurred 2 Is decreasing, another magic phenomenon can be observed, the phases at the start and end points of the SMR can be considered approximately opposite to each other, and delta 2 The decreasing range of (2) is fixed and is irrelevant to the value of kn. These phenomena are the solution interval [ f 1 ,f 2 ]The resolution of (a) provides the possibility,let the interval where strong modulation response occurs be [ delta ] 21 ,δ 22 ]From the figure, it can be obtained that: delta 22 ＝-δ 21 。
As can be seen from the time chart obtained by examining the slowly varying equations (50) and (51), N 'is always present in the stable portion where SMR does not occur' 2 ＝δ′ 2 =0, i.e.:
for convenience, the expressions (43) (44) are rewritten here:
from N' 2 ＝δ′ 2 When c=d=0 can be deduced from the equation, it is reasonable to observe 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 not possible to achieve because of the N found by the slow-varying equation (50) (51) 2 Fixed, unchanged cos delta 2 Monotonically increasing, then C, D cannot be fixed all the time as f increases. It can be seen in FIG. 20 that the values of C, D are indeed always 0 when the steady state response is stable, only in the abscissa interval, delta, where both the "plateau" portions where the emphasized response occurs are not 0, and C, D are not 0 2 Monotonically decreasing abscissa interval sum N 2 The abscissa interval of the platform is the superposition of [ f ] 1 ,f 2 ]。
The above observed phenomenon is used to find the excitation amplitude range [ f ] over which SMR can occur 1 ,f 2 ]：
Setting:
wherein:
since D is still equal to 0 at the start point, the simultaneous triangular identity is available from (54) (55):
f 1 2 ＝F 1 2 +F 2 2 (58)
from (58), a lower excitation amplitude limit f is derived at which the emphasis response begins to occur 1 The upper limit f is found as follows 2 ：
Setting the values of C and D at the beginning and the end points as C 1 ,C 2 ,D 1 ,D 2 The following steps are:
C 1 ＝f 1 cosδ 21 -F 1 ＝0 (59)
D 1 ＝f 1 sinδ 21 +F 2 ＝0 (60)
C 2 ＝f 2 cosδ 21 -F 1 (61)
D 2 ＝-f 2 sinδ 21 +F 2 (62)
due to f 2 The starting point phase is known to be obtainable from (59):
from (59) to (62):
/>
and (52) (53) is always 0, and the terminal points of the platform are:
from the triangle identity:
(C 2 +F 1 ) 2 +(D 2 -F 2 ) 2 ＝f 2 2 (67)
substituting the formulas (64) and (66) into (67) to obtain the abscissa f corresponding to the required end point 2 Values, after substitution, are:
simplifying and obtaining:
thus when kn is determined, the excitation amplitude range over which the system can develop a strong modulation response is:
f 1 ≤f≤f 2 (70)
will f 1 And f 2 The values are substituted to obtain:
(71) I.e. the excitation amplitude range formula based on the "plateau" phenomenon, in which an emphasis response can occur, but f is not a design parameter, the purpose of the study is to optimize the nonlinear stiffness kn, and the optimal stiffness value is derived from the formula (71).
As shown in fig. 12, the parameter range [ f ] of the SMR occurs 1 ,f 2 ]Regarding the value of the nonlinear stiffness kn, the smaller the kn value is, the larger the effective excitation range is, which is a good phenomenon, but when the kn value is smaller, the larger the steady-state amplitude of the main structure is, and therefore, the greater the stiffness is, the better or the smaller the stiffness is, and according to the assumption, when the excitation amplitude is, the kn range is certain, and when the stiffness is in the range, the system can generate SMR.
Let the optimal stiffness be kn b Due to f 1 And f 2 Are all about lambda 1 ,λ 2 And kn, thus when damping lambda 1 ,λ 2 And f, determining, by (71) deriving an effective stiffness range, the effective stiffness range in which a strong modulation response can occur is set to [ kn ] 1 ,kn 2 ]. As can be seen in FIG. 12, when the excitation amplitude is determined, the corresponding "plateau" stiffness value (i.e., kn) is found when this amplitude is taken as the "plateau" end abscissa 2 ) And taking this as the optimal stiffness is theoretically most reasonable. However, since the platform range obtained by the expression (71) is only an approximate analytical solution, there is a certain error, and therefore if f is used 2 Derived kn 2 Deviations can occur when the optimal rigidity is regarded as unavoidable, and the robustness is poor.
Embodiment 1: taking the case where the excitation amplitude is f=0.5 as an example, taking the optimum stiffness kn obtained by the above formula b The effect of optimizing stiffness by applying the above method in a two-degree-of-freedom system connecting NES was examined, as compared to the case when other stiffness values were taken.
Taking five nonlinear stiffness values respectively: kn=1<kn 1 ，kn 1 <kn b ＝6.1691<kn 2 ，kn＝kn 2 ＝10.2650，kn＝14,18>kn 2 Since the emphasis is placed on the fact that the response is not of a fixed magnitude, it is appropriate to use the average energy as a measure of the damping effect, and the total transient energy of the system can be written in the form of equation (72):
changing tuning parameter σ, checking to take each stiffness mentioned above, 1: the average energy near resonance within 1, results are shown in fig. 20 and 21.
It can be seen from FIG. 20 that when the stiffness takes the optimal value kn b When the total energy is significantly lower than the other stiffness. Although kn=kn when ω=1 b The total energy at this point is not minimal but is also relatively small and in practice the excitation frequency will not be exactly equal to 1, so robustness in the frequency domain is important, selecting kn=kn b A balance is achieved between damping effect and robustness to frequency. Fig. 21 shows the optimized frequency energy curve alone, with smooth regions on both sides for a steady state response, with emphasis near the middle ω=1, and this response mode is found to be very beneficial for damping.
Taking different excitation amplitudes f=0.8 and five corresponding kn: kn=0.5<kn 1 ，kn 1 <kn b ＝2.4098<kn 2 ，kn＝kn 2 ＝4.0098，kn＝7,9>kn 2 . Re-verifying kn b The results are shown in fig. 22 and 23. The conclusions drawn from fig. 22, 23 are completely identical to the above fig. 20, 21.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (3)
1. A nonlinear energy well optimal stiffness solving method based on a platform phenomenon is characterized by comprising the following steps of: the method comprises the following steps:
s1: a double-freedom-degree system motion equation formed by a linear main structure and a nonlinear energy well is established;
s2: converting real absolute coordinates in a motion equation into complex variable relative coordinates by using a complex variable average method, and eliminating a fast-changing part in the variables;
s3: the complex variable differential equation obtained by the multi-scale method can be respectively converted into a fast-change equation about a fast time scale and a slow-change equation about a slow time scale, the slow-change equation is used for obtaining a slow-change type of a double-degree-of-freedom system and displacement amplitude values at jumping points, and the slow-change equation is used for obtaining steady-state amplitude values of a nonlinear energy trap;
s4, constructing a relation curve by taking the excitation amplitude as an abscissa and the response amplitude of the nonlinear energy trap as an ordinate, defining the relation curve as a horizontal straight line in an emphasized response area, defining the horizontal straight line as a platform, defining the phenomenon that the relation curve in the emphasized response area is a horizontal straight line as a platform phenomenon of the dual-degree-of-freedom system, wherein an abscissa interval corresponding to the platform is an excitation amplitude range of the dual-degree-of-freedom system for emphasized response, and an ordinate corresponding to the platform is an amplitude at a jump point, and the platform phenomenon indicates that the steady-state amplitude of the nonlinear energy trap does not change along with the change of the excitation amplitude when and only when the steady-state response of the dual-degree-of-freedom system is in the emphasized response mode, and the steady-state amplitude is always equal to the amplitude at the jump point of the dual-degree-of-freedom system;
s5, obtaining an analytic expression of an effective excitation amplitude range of the generated emphasis response through a platform phenomenon;
s6, when the damping of the two-degree-of-freedom system is fixed, obtaining the effective stiffness range of the two-degree-of-freedom system generating stress damping response through excitation amplitude range analysis;
the excitation amplitude f range of the two-degree-of-freedom system emphasis response is:
f 1 ≤f≤f 2 (70)
further:
wherein: f (f) 1 To lower the excitation amplitude, f 2 To excite the upper limit of the amplitude, N 21 Steady-state amplitude for the energy traps at the jumping points;
wherein: sigma is tuning parameter, lambda 1 Non-dimensional damping, lambda, of the main structure 2 Is dimensionless damping of the energy trap.
2. The method for solving the optimal stiffness of the nonlinear energy well based on the platform phenomenon according to claim 1, further characterized by: the motion equation of the double-freedom-degree system is as follows:
wherein: m is m 1 ,m 2 Mass of main structure and NES, c 1 ,c 2 Damping coefficients, k, of the main structure and NES, respectively 1 ,k 2 The stiffness coefficients of the main structure and NES are respectively, F is the excitation amplitude, omega is the excitation frequency, and x 2 ,x 1 Representing the absolute displacement of the main structure and the damper, respectively.
3. The method for solving the optimal stiffness of the nonlinear energy well based on the platform phenomenon according to claim 1, further characterized by: the main structure vibration amplitude psi 1 And the vibration amplitude ψ of a nonlinear energy trap 2 The relationship of (2) is as follows:
wherein: lambda (lambda) 2 Non-dimensional damping representing an energy trap, i 2 = -1, kn represents the dimensionless stiffness of the energy well.
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