CN109519499A - The determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency - Google Patents
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Abstract
The invention discloses the determination methods of quasi-zero stiffness vibration isolators vibration isolation initial frequency, belong to nonlinear isolation technical field.The acquisition methods of the vibration isolation initial frequency of existing quasi-zero stiffness vibration isolators there is a problem of time-consuming and laborious.The present invention carries out nondimensionalization processing and approximate processing to kinetics equation, then dimensionless cross-over frequency and dimensionless crest frequency are derived according to the expression formula of transport, compare the two size again to obtain dimensionless vibration isolation initial frequency, finally carries out dimension again.It either explicitly or implicitly include the parameter of vibration isolator in the expression formula of the obtained vibration isolation initial frequency of the present invention, therefore it is readily seen influence of each parameter of vibration isolator to vibration isolation initial frequency, facilitate and suitable parameter is selected to design quasi-zero stiffness vibration isolators, vibration isolation initial frequency is made to meet design requirement.This method can partially substitute emulation and experiment, so as to shorten the design cycle, reduce design cost.
Description
Technical field
The present invention relates to a kind of nonlinear isolation technologies, and in particular to a kind of vibration isolation initial frequency of quasi-zero stiffness vibration isolators
Approximate analysis seek method.
Background technique
Most vibrations are all harmful in engineering.In order to reduce or eliminate the influence of vibration, what is used earliest is passive
Vibration isolation means are linear vibration isolators, but it is but difficult to that low-frequency vibration is isolated.In order to solve the problems, such as low frequency vibration isolation, in recent years
Nonlinear isolation technology flourishes, wherein again most representative with quasi-zero stiffness vibration isolators.Designing quasi- zero stiffness vibration isolation
When device, a highly important performance indicator is vibration isolation initial frequency, because vibration isolation initial frequency is designed as lower than vibration
Some value of source low-limit frequency can just have preferable vibration isolating effect.Obtaining to the vibration isolation initial frequency of quasi-zero stiffness vibration isolators at present
Means main or using emulation or experiment are taken, acquisition process is time-consuming and laborious.It needs to the method for obtaining vibration isolation initial frequency
It evolves, to shorten the design cycle of quasi-zero stiffness vibration isolators.
Summary of the invention
The purpose of the present invention is to solve the presence of the acquisition methods of the vibration isolation initial frequency of existing quasi-zero stiffness vibration isolators
Time-consuming and laborious problem, and propose the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency.
The determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency, the method are realized by following steps:
Step 1: find out the restoring force of vibration isolator and the relational expression of displacement, the restoring force of vibration isolator and displacement relation formula into
The processing of row nondimensionalization, then in equilibrium position Taylor series expansion, the dimensionless for the preceding odd-order for meeting precision is taken to reply
Power expression formula is as the approximation to accurate expression；
Step 2: finding out kinetics equation, and carry out nondimensionalization processing, dimensionless restoring force item therein uses step 1
Obtained approximate form；
Step 3: the steady state solution of dimensionless dynamic response is set as harmonic wave form, frequency is equal to dimensionless driving frequency,
The equation of dimensionless response amplitude is obtained by approximate dimensionless kinetics equation combination steady state solution；
Step 4: finding out the expression formula of transport, and by enabling the null method of transport, solve dimensionless response width
It is worth the expression formula about dimensionless driving frequency；
Step 5: expression formula obtained in step 4 being substituted into dimensionless response amplitude equation obtained in step 3, is obtained only
About the equation of dimensionless driving frequency, and thus obtain dimensionless cross-over frequency；
Step 6: dimensionless response amplitude equation obtained in step 3 is rewritten as dimensionless response amplitude about dimensionless
The implicit function of frequency, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then obtained nothing
The corresponding dimensionless frequency of the maximum of dimension response amplitude is dimensionless crest frequency；
Step 7: comparing the size of dimensionless crest frequency Yu dimensionless cross-over frequency, using the greater as dimensionless vibration isolation
Initial frequency；
Step 8: dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency.
Preferably, the excitation includes the two ways of power excitation and basic excitation, and the transport includes that power swashs
Encourage the transport generated with basic excitation mode；Wherein,
Pass through expression formulaThe transmissibility under power excitation is obtained,
In formula, f0It is dimensionless power excitation amplitude, Ω is dimensionless driving frequency, and A is dimensionless response amplitude,It indicates
Phase；
Pass through expression formulaThe displacement transport under basic excitation is obtained,
In formula, b0It is dimensionless basic excitation amplitude, H is the dimensionless response of relative motion between vibration isolation object and basis
Amplitude,Indicate phase；For the dimensionless response amplitude H comprising relative motion between vibration isolation object and basis and immeasurable
The function of guiding principle driving frequency Ω.
Preferably, in the step 5, expression formula obtained in step 4 is substituted into dimensionless obtained in step 3 and is responded
Amplitude equation obtains the equation only with respect to dimensionless driving frequency, when solving this equation and obtaining dimensionless cross-over frequency, if excitation
The case where motivating based on mode, if the dimensionless cross-over frequency acquired is imaginary number, enabling the value of dimensionless cross-over frequency is zero.
Preferably, in the step 6, dimensionless response amplitude equation obtained in step 3 is rewritten as dimensionless and is rung
Implicit function of the amplitude about dimensionless frequency is answered, the maximum and its corresponding dimensionless frequency of dimensionless response amplitude are thus found out
Rate, then the corresponding dimensionless frequency of maximum of obtained dimensionless response amplitude is the process of dimensionless crest frequency,
Specifically: dimensionless response amplitude equation obtained in step 3 is made implicit function derivation to dimensionless frequency, and enables the derivative value be
Zero, obtain a new algebraic equation, then with dimensionless response amplitude equations simultaneousness, while solving the pole of dimensionless response amplitude
Big value and dimensionless crest frequency.If the dimensionless response amplitude equation that step 3 obtains can be write as about Ω2Quadratic power
Journey then directly finds out the maximum of dimensionless response amplitude using quadratic function discriminate, then generation goes back to dimensionless response amplitude side
Journey solves dimensionless crest frequency, then the maximum of dimensionless response amplitude, then generation are directly found out using quadratic function discriminate
Dimensionless response amplitude equation is returned, dimensionless crest frequency is solved.
The invention has the benefit that
It is symmetrical that the vibration isolation initial frequency approximate analysis of quasi-zero stiffness vibration isolators of the invention asks method to be suitable for any rigidity
The vibration isolation initial frequency of quasi-zero stiffness vibration isolators solves.
In the expression formula of obtained vibration isolation initial frequency either explicitly or implicitly include the parameter of vibration isolator, therefore is easy to see
Influence of each parameter of vibration isolator to vibration isolation initial frequency out facilitates and suitable parameter is selected to design quasi-zero stiffness vibration isolators, letter
The calculating process for having changed vibration isolation initial frequency makes vibration isolation initial frequency meet design requirement.Approximate analysis provided by the invention is asked
Method can partially substitute emulation and experiment, to greatly shorten the design cycle, reduce design cost.
Detailed description of the invention
Fig. 1 is that the vibration isolation initial frequency approximate analysis of a kind of quasi-zero stiffness vibration isolators seeks the flow chart of method；
Fig. 2 is the schematic diagram as the quasi-zero stiffness vibration isolators of embodiment one；
Fig. 3 is the schematic diagram as the quasi-zero stiffness vibration isolators of embodiment two；
Specific embodiment
Specific embodiment 1:
The determination method of the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment, the method includes following steps
It is rapid:
Step 1: find out the restoring force of vibration isolator and the relational expression of displacement, the restoring force of vibration isolator and displacement relation formula into
The processing of row nondimensionalization, then in equilibrium position Taylor series expansion, the dimensionless for the preceding odd-order for meeting precision is taken to reply
Power expression formula is as the approximation to accurate expression；
Step 2: finding out kinetics equation, and carry out nondimensionalization processing, dimensionless restoring force item therein uses step 1
Obtained approximate form；
Step 3: the steady state solution of dimensionless dynamic response is set as harmonic wave form, frequency is equal to dimensionless driving frequency,
The equation of dimensionless response amplitude is obtained by approximate dimensionless kinetics equation combination steady state solution；
Steady state solution is substituted into the approximate dimensionless kinetics equation that step 2 obtains, omits higher order term, then enable equation both ends
Corresponding first harmonic term coefficient is equal, obtains two equations about dimensionless amplitude and phase, and cancellation phase obtains immeasurable
The equation of guiding principle response amplitude；
Step 4: finding out the expression formula of transport, and by enabling the null method of transport, solve dimensionless response width
It is worth the expression formula about dimensionless driving frequency；
Step 5: expression formula obtained in step 4 being substituted into dimensionless response amplitude equation obtained in step 3, is obtained only
About the equation of dimensionless driving frequency, and solves this equation and thus obtain dimensionless cross-over frequency；
Step 6: dimensionless response amplitude equation obtained in step 3 is rewritten as dimensionless response amplitude about dimensionless
The implicit function of frequency, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then obtained nothing
The corresponding dimensionless frequency of the maximum of dimension response amplitude is dimensionless crest frequency；
Step 7: comparing the size of dimensionless crest frequency Yu dimensionless cross-over frequency, using the greater as dimensionless vibration isolation
Initial frequency；
Step 8: dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency.
Specific embodiment 2:
Unlike specific embodiment one, the determination of the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment
Method in the step 4, finds out the expression formula of transport, enables transport be equal to zero, solves dimensionless response amplitude about nothing
The process of the expression formula of dimension driving frequency, specifically:
The excitation includes the two ways of power excitation and basic excitation, and the transport includes transmissibility and position
Move transport；Wherein,
Pass through expression formulaThe transmissibility under power excitation is obtained,
In formula, f0It is dimensionless power excitation amplitude, Ω is dimensionless driving frequency, and A is dimensionless response amplitude,It indicates
Phase；
Pass through expression formulaThe displacement transport under basic excitation is obtained,
In formula, b0It is dimensionless basic excitation amplitude, H is the dimensionless response of relative motion between vibration isolation object and basis
Amplitude,Indicate phase；For the dimensionless response amplitude H comprising relative motion between vibration isolation object and basis and immeasurable
The function of guiding principle driving frequency Ω.
Specific embodiment 3:
Unlike specific embodiment two, the determination of the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment
Expression formula obtained in step 4 in the step 5, is substituted into dimensionless response amplitude equation obtained in step 3 by method,
The equation only with respect to dimensionless driving frequency is obtained, when solving this equation and obtaining dimensionless cross-over frequency, if energisation mode is base
The case where plinth motivates, if the dimensionless cross-over frequency acquired is imaginary number, enables immeasurable when dimensionless basic excitation amplitude is too small
The value of guiding principle cross-over frequency is zero.
Specific embodiment 4:
Unlike specific embodiment three, the determination of the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment
Method, in the step 6, by dimensionless response amplitude equation obtained in step 3 be rewritten as dimensionless response amplitude about
The implicit function of dimensionless frequency, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then it is required
The corresponding dimensionless frequency of maximum of the dimensionless response amplitude obtained is the process of dimensionless crest frequency, specifically:
Dimensionless response amplitude equation obtained in step 3 is made implicit function derivation to dimensionless frequency, and enables derivative value
Be zero, obtain a new algebraic equation, then with dimensionless response amplitude equations simultaneousness, while solving dimensionless response amplitude
Maximum and dimensionless crest frequency.
Specific embodiment 5:
Unlike specific embodiment four, the determination of the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment
Method, the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency according to claim 4, which is characterized in that if step
Rapid 3 obtained dimensionless response amplitude equations can be write as about Ω2Quadratic equation, then using quadratic function discriminate it is direct
The maximum of dimensionless response amplitude is found out, then generation returns dimensionless response amplitude equation, solves dimensionless crest frequency, then utilizes
Quadratic function discriminate directly finds out the maximum of dimensionless response amplitude, then generation returns dimensionless response amplitude equation, can solve
Dimensionless crest frequency out.
Specific embodiment 6:
Unlike specific embodiment four or five, the quasi-zero stiffness vibration isolators vibration isolation initial frequency of present embodiment
Determine method, in the step 7, when dimensionless crest frequency is greater than dimensionless cross-over frequency, the frequency response curve of vibration isolator
Certainly exist chattering with transport curve, i.e., there are more solutions in some frequency range, wherein one unstable, at this time without
Dimension crest frequency is approximately equal to frequency hopping rate under dimensionless, so by comparing dimensionless crest frequency and dimensionless cross-over frequency
Size, using the greater as dimensionless vibration isolation initial frequency.
Embodiment 1:
The determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency, specifically includes the following steps:
Step 1, the restoring force vibration isolator and displacement relation formula carry out nondimensionalization processing, then in equilibrium position Thailand
Series expansion is strangled, takes the dimensionless restoring force expression formula for the preceding odd-order that can satisfy precision as the approximation to accurate expression
Form:
As shown in Fig. 2, the quasi-zero stiffness vibration isolators being made of horizontal spring, connecting rod and uprighting spring, vibration isolation object by
The excitation of simple harmonic quantity power, length of connecting rod L, the rigidity of horizontal spring are kh, the rigidity of uprighting spring is kv, m is the object being activated
Quality.At vibration isolator equipoise, when vibration isolation object has vertical displacement X relative to equipoise, the pressure of horizontal spring
Power are as follows:
The total restoring force expression formula of vibration isolator is as follows:
Wherein, FresIndicate the total restoring force of vibration isolator；It is vertical that X indicates that vibration isolation object is generated relative to equipoise
Displacement；kvIndicate the rigidity of uprighting spring；khIndicate horizontal spring rigidity be；δhWhen indicating the decrement maximum of horizontal spring,
The decrement of horizontal spring, δvWhen indicating the decrement maximum of horizontal spring, the decrement of uprighting spring；FsprIndicate horizontal bullet
The pressure of spring, L indicate length of connecting rod；The angle of θ expression connecting rod and horizontal direction；Introduce fres=Fres/(kvL), x=X/L, λ
=kh/kv, e=δh/ L, the total restoring force expression formula of vibration isolator are changed to dimensionless restoring force expression formula shown in formula (3):
In formula (3), dimensionless rigidity is obtained to nondimensional displacement derivation are as follows:
Realize quasi- zero stiffness when k (0)=0, thus quasi- zero stiffness condition are as follows:
Taylor series expansion is carried out to dimensionless restoring force expression formula and takes first three rank, then the quasi- zero stiffness item of wushu (5)
Part substitutes into wherein, obtains:
In formula, β=(1-e)/(2e).
Step 2 finds out the lower kinetics equation of power excitation, and progress nondimensionalization processing, using the approximate form of step 1
Dimensionless restoring force obtain final dimensionless kinetics equation:
1) expression formula of power excitation, is set are as follows:
F (t)=F0cos(ωt) (7)
ω indicates driving frequency, and t indicates time, F0Indicate excitation amplitude；
Determine the kinetics equation in the case where power motivates are as follows:
In formula, c is damped coefficient, and g is acceleration of gravity,Indicate that displacement X asks second order to lead time t,Indicate displacement X
Single order is asked to lead time t, m indicates the mass of object being activated；
2), at equipoise, the gravity of object is undertaken by uprighting spring completely, therefore the k of equation left endvL and equation
The mg of right end is offseted；It introducesf0=F0/(kvL)、τ=ω0T, Ω=ω/ω0, right
The kinetics equation of formula (8) carries out nondimensionalization, then the dimensionless restoring force of the approximate form using step 1, obtains approximate shape
The dimensionless kinetics equation of formula:
x″+2ζx′+βx3=f0cos(Ωτ) (9)
In formula, x " indicates that nondimensional displacement x asks second order to lead nondimensional time τ, and x ' expression nondimensional displacement x is to dimensionless
Time, τ asked single order to lead, and ζ indicates damping ratio, f0Indicate dimensionless excitation amplitude, ω0Indicate that the undamped of equivalent linear vibration isolator is solid
There is frequency, Ω indicates dimensionless driving frequency, and τ indicates nondimensional time.
Step 3 sets the steady state solution of dimensionless dynamic response as harmonic wave form, and frequency is equal to dimensionless driving frequency；
Steady state solution is substituted into the dimensionless kinetics equation that step 2 obtains, obtains the expression formula about dimensionless response amplitude:
1) steady state solution, is set are as follows:
In formula, A indicates dimensionless response amplitude,Indicate phase；
Later, it substitutes into the dimensionless kinetics equation equation of formula (9) and ignores triple-frequency harmonics, enable first harmonic humorous accordingly
Wave system number is equal, obtains two equations about dimensionless amplitude and phase:
2) it, will obtain being added cancellation phase again with two equation both sides square of phase about dimensionless amplitude?
To the equation of the polynomial form about dimensionless response amplitude:
Step 4, the expression formula for finding out transmissibility enable transport be equal to zero, solve the nothing about dimensionless driving frequency
The expression formula of dimension response amplitude:
Transmissibility expression formula is acquired by derivation are as follows:
Lower dimensionless cross-over frequency is motivated to be denoted as Ω powerfc；
According to the definition of cross-over frequency, when dimensionless driving frequency is ΩfcWhen, there is Tf=1, response amplitude at this time is known as
Amplitude is passed through, A is denoted asc, the transmissibility expression formula of formula (13) is substituted into, the expression formula for passing through amplitude is obtained, it is as follows；
Step 5 will substitute into step about the expression formula of the dimensionless response amplitude of dimensionless driving frequency obtained in step 4
The expression formula of dimensionless response amplitude obtained in rapid 3, obtains the equation only with respect to dimensionless driving frequency, solves this equation and obtains
To dimensionless cross-over frequency:
The expression formula for passing through amplitude of wushu (14) substitutes into dimensionless response amplitude equation shown in step 3 formula (12) and obtains
Only with respect to the equation of dimensionless driving frequency, following equation is obtained:
Dimensionless cross-over frequency Ω is solved by formula (15)fc。
The expression of dimensionless response amplitude obtained in step 3 is rewritten as dimensionless response amplitude about immeasurable by step 6
The implicit function of guiding principle frequency, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then it is obtained
The corresponding dimensionless frequency of the maximum of dimensionless response amplitude is dimensionless crest frequency:
Dimensionless response amplitude equation is rewritten as about Ω2Quadratic equation form, be shown below:
To solve dimensionless response amplitude maximum and its corresponding dimensionless frequency, and by the dimensionless of solution ring
Answer dimensionless frequency corresponding to the maximum of amplitude as dimensionless crest frequency；
If the maximum value A of amplitude on frequency response curvepIt indicates.At the peak value of frequency response curve, shown in formula (16) about
Ω2Quadratic equation there are two double root, so the discriminate of root is equal to zero, i.e.,
Dimensionless amplitude maximum A can be solvedpExpression formula be
ApExpression formula (18) substituted back into equation (16), the dimensionless crest frequency Ω under capable excitation can be solvedfpFor
Step 7, the size for comparing dimensionless crest frequency Yu dimensionless cross-over frequency, using the greater as dimensionless vibration isolation
Initial frequency:
Dimensionless vibration isolation initial frequency to be asked, first compares ΩfcWith ΩfpSize.Comparative approach is as follows:
By the Ω in the equation of formula (15)fcReplace with Ωfp, obtain following expression:
It enablesEquation (20) is converted into the equation about μ, as follows:
μ3-82μ2- 660 μ+2600=0 (21)
Three roots that the equation of solution formula (21) obtains μ are respectively 89.081,2.919 and -10；Obviously, negative root -10 is to give up
It goes, and 2.9187 will also reject, reason are as follows: if μ value is 2.9187, according to formula (19) counted ΩfpIt is imaginary number, then
Dimensionless frequency response curve does not occur peak value actually.Therefore, the root for taking equation (21) is μ=89.081, then by equation
(20) f solved0Are as follows:
Therefore, work as f0It is less thanWhen, Ωfc< Ωfp；Work as f0It is greater thanWhen, Ωfc>
Ωfp；So, dimensionless vibration isolation initial frequency ΩfsExpression formula it is as follows:
Wherein, ΩfcThe equation as shown in formula (15) solves.
Dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency by step 8:
The practical vibration isolation that the dimensionless vibration isolation initial frequency of formula (23) is converted into circular frequency form is originated by formula (24)
Frequencies omegafs:
Wherein, practical cross-over frequency ωfcIt is solved by formula (25):
It can also be by the vibration isolation initial frequency ω of the circular frequency form of formula (24)fsDivided by 2 π, being transformed into unit is hertz
Vibration isolation initial frequency.
Embodiment 2:
The determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency, specifically includes the following steps:
Step 1, the restoring force vibration isolator and displacement relation formula carry out nondimensionalization processing, then in equilibrium position Thailand
Series expansion is strangled, takes the dimensionless restoring force expression formula for the preceding odd-order that can satisfy precision as the approximation to accurate expression
Form
As shown in figure 3, the quasi-zero stiffness vibration isolators being made of horizontal spring, sliding block, cam, uprighting spring, by simple harmonic quantity
Basic excitation, the restoring force and displacement relation formula of vibration isolator are as follows:
Wherein, U indicates the relative displacement between vibration isolation object and basis, k2Indicate horizontal spring rigidity, k1Indicate vertical bullet
Spring rigidity, R indicate cam radius, and m indicates the quality of vibration isolation object；The decrement of the horizontal spring of equipoise is δv, quiet
The uprighting spring decrement of equilbrium position is δv, introduce fres=Fres/(2k1R), u=U/ (2R), λ=k2/k1, ε=δh/ (2R),
Restoring force expression formula is changed to following Dimensionless Form:
Formula (27) obtains dimensionless rigidity to nondimensional displacement u differential are as follows:
Wherein, quasi- zero stiffness is realized when meeting formula (29), it may be assumed that
To fres(u) it Taylor series expansion is carried out, and takes first three rank, then wushu (29) substitutes into obtain quasi-zero stiffness vibration isolators
Approximate dimensionless power and displacement relation formula are as follows:
In formula, β=λ -1/2.
Step 2 finds out the lower kinetics equation of power excitation, and progress nondimensionalization processing, using the approximate form of step 1
Dimensionless restoring force obtain final dimensionless kinetics equation:
1) when, setting by basic excitation, the basic equation of motion are as follows:
B (t)=B0cos(ωt) (31)
In formula, ω indicates driving frequency, and t indicates time, B0Indicate excitation amplitude；
The relative displacement between vibration isolation object and basis is indicated with U (t), then has X (t)=B (t)+U (t).The power of system
Learning equation is
In formula, c is damped coefficient, and g is acceleration of gravity.
2), the 2k of equation left end1R is offseted with the mg of equation right end.It introducesb0=B0/(2R)、τ=ω0T, Ω=ω/ω0, kinetics equation (32) can be carried out with nondimensionalization, then use approximate form
Dimensionless restoring force, finally obtained dimensionless kinetics equation is
u″+2ζu′+βu3=b0Ω2cos(Ωτ) (33)
In formula, u " indicates that dimensionless relative displacement asks second order to lead nondimensional time τ, u ' expression dimensionless relative displacement pair
Nondimensional time τ asks single order to lead, and ζ indicates damping ratio, and Ω indicates dimensionless driving frequency, and τ indicates nondimensional time.
Step 3 sets the steady state solution of dimensionless dynamic response as harmonic wave form, and frequency is equal to dimensionless driving frequency；
Steady state solution is substituted into the dimensionless kinetics equation that step 2 obtains, obtains the expression formula about dimensionless response amplitude:
1), set steady state solution as
In formula, H indicates dimensionless response amplitude,Indicate phase；
Substitute into equation (33) and simultaneously ignore triple-frequency harmonics, enable the corresponding harmonic constant of first harmonic equal, obtain about amplitude and
Two equations of phase:
2), upper two formula both sides square are added cancellation phase againIt can obtain
Step 4, the expression formula for finding out transmissibility enable transport be equal to zero, solve the nothing about dimensionless driving frequency
The expression formula of dimension response amplitude:
Absolute displacement transport is by deriving are as follows:
Dimensionless cross-over frequency under basic excitation is denoted as Ωbc。
According to the definition of " cross-over frequency ", when dimensionless driving frequency is equal to ΩbcWhen, there is Tb=1, response amplitude at this time
It is denoted as Hc, substitute into expression formula (37) Shi Ke get of transport
Step 5 will substitute into step about the expression formula of the dimensionless response amplitude of dimensionless driving frequency obtained in step 4
The expression formula of dimensionless response amplitude obtained in rapid 3, obtains the equation only with respect to dimensionless driving frequency, solves this equation and obtains
To dimensionless cross-over frequency:
Wushu (38) substitution dimensionless response amplitude equation (36) can solve dimensionless cross-over frequency and be
As can be seen that working asWhen, ΩbcFor imaginary number, Ω is taken at this timebc=0.
The expression of dimensionless response amplitude obtained in step 3 is rewritten as dimensionless response amplitude about immeasurable by step 6
The implicit function of guiding principle frequency, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then it is obtained
The corresponding dimensionless frequency of the maximum of dimensionless response amplitude is dimensionless crest frequency:
Dimensionless response amplitude equation is rewritten as about dimensionless frequency Ω2Quadratic equation, be shown below:
To solve dimensionless response amplitude maximum and its corresponding dimensionless frequency, and by the dimensionless of solution ring
Answer dimensionless frequency corresponding to the maximum of amplitude as dimensionless crest frequency:
If the maximum value H of amplitude on frequency response curvepIt indicates.At the peak value of frequency response curve, about Ω2Quadratic equation
(40) there are two double roots, so the discriminate of root is equal to zero, i.e.,
Dimensionless amplitude maximum H can therefrom be solvedpExpression formula be
HpExpression formula (42) substitute into equation (40), can solve dimensionless crest frequency is
Step 7, the size for comparing dimensionless crest frequency Yu dimensionless cross-over frequency, using the greater as dimensionless vibration isolation
Initial frequency:
Compare ΩbcWith ΩbpSize.It enables formula (39) equal with formula (43), can solve
Therefore, whenWhen, Ωbc≤Ωbp；WhenWhen, Ωbc>
Ωbp。
In conclusion dimensionless vibration isolation initial frequency Ω under basic excitationbsExpression formula are as follows:
Dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency by step 8:
Dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency ωbs, indicate are as follows:
It can also be by the vibration isolation initial frequency ω for the circular frequency form that formula (46) providebcDivided by 2 π, it is conspicuous for being transformed into unit
Vibration isolation initial frequency hereby.
The present invention can also have other various embodiments, without deviating from the spirit and substance of the present invention, this field
Technical staff makes various corresponding changes and modifications in accordance with the present invention, but these corresponding changes and modifications all should belong to
The protection scope of the appended claims of the present invention.
Claims (5)
1. the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency, which comprises the following steps:
Step 1: finding out the restoring force of vibration isolator and the relational expression of displacement, the restoring force and displacement relation formula of vibration isolator are carried out nothing
Dimensionization processing, then in equilibrium position Taylor series expansion, take the dimensionless restoring force table for the preceding odd-order for meeting precision
Up to formula as the approximation to accurate expression；
Step 2: finding out kinetics equation, and carry out nondimensionalization processing, dimensionless restoring force item therein is obtained using step 1
Approximate form；
Step 3: setting the steady state solution of dimensionless dynamic response as harmonic wave form, frequency is equal to dimensionless driving frequency, passes through
Approximate dimensionless kinetics equation combination steady state solution obtains the equation of dimensionless response amplitude；
Step 4: finding out the expression formula of transport, and by enabling the null method of transport, solve dimensionless response amplitude pass
In the expression formula of dimensionless driving frequency；
Step 5: by expression formula obtained in step 4 substitute into step 3 obtained in dimensionless response amplitude equation, obtain only with respect to
The equation of dimensionless driving frequency, and thus obtain dimensionless cross-over frequency；
Step 6: dimensionless response amplitude equation obtained in step 3 is rewritten as dimensionless response amplitude about dimensionless frequency
Implicit function, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then obtained dimensionless
The corresponding dimensionless frequency of the maximum of response amplitude is dimensionless crest frequency；
Step 7: comparing the size of dimensionless crest frequency Yu dimensionless cross-over frequency, originated the greater as dimensionless vibration isolation
Frequency；
Step 8: dimensionless vibration isolation initial frequency is transformed into practical vibration isolation initial frequency.
2. the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency according to claim 1, which is characterized in that described
Step 4 in, find out the expression formula of transport, enable transport be equal to zero, solve dimensionless response amplitude about dimensionless motivate
The process of the expression formula of frequency, specifically:
The excitation includes the two ways of power excitation and basic excitation, and the transport includes that transmissibility and displacement pass
Pass rate；Wherein,
Pass through expression formulaThe transmissibility under power excitation is obtained,
In formula, f0It is dimensionless power excitation amplitude, Ω is dimensionless driving frequency, and A is dimensionless response amplitude,Indicate phase
Position；
Pass through expression formulaThe displacement transport under basic excitation is obtained,
In formula, b0It is dimensionless basic excitation amplitude, H is the dimensionless response amplitude of relative motion between vibration isolation object and basis,Indicate phase.
3. the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency according to claim 2, which is characterized in that described
Step 5 in, by expression formula obtained in step 4 substitute into step 3 obtained in dimensionless response amplitude equation, obtain only with respect to
The equation of dimensionless driving frequency, when solving this equation and obtaining dimensionless cross-over frequency, if the feelings motivated based on energisation mode
Condition, if the dimensionless cross-over frequency acquired is imaginary number, enabling the value of dimensionless cross-over frequency is zero.
4. the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency according to claim 3, which is characterized in that described
Step 6 in, dimensionless response amplitude equation obtained in step 3 is rewritten as dimensionless response amplitude about dimensionless frequency
Implicit function, thus find out dimensionless response amplitude maximum and its corresponding dimensionless frequency, then obtained dimensionless
The corresponding dimensionless frequency of the maximum of response amplitude is the process of dimensionless crest frequency, specifically:
Dimensionless response amplitude equation obtained in step 3 is made implicit function derivation to dimensionless frequency, and enabling derivative value is zero,
Obtain a new algebraic equation, then with dimensionless response amplitude equations simultaneousness, while solving the very big of dimensionless response amplitude
Value and dimensionless crest frequency.
5. the determination method of quasi-zero stiffness vibration isolators vibration isolation initial frequency according to claim 4, which is characterized in that if step
Rapid 3 obtained dimensionless response amplitude equations can be write as about Ω2Quadratic equation, then using quadratic function discriminate it is direct
The maximum of dimensionless response amplitude is found out, then generation returns dimensionless response amplitude equation, solves dimensionless crest frequency.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
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| CN201811625094.6A CN109519499B (en) | 2018-12-28 | 2018-12-28 | Method for determining vibration isolation initial frequency of quasi-zero stiffness vibration isolator |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113648638A (en) * | 2021-08-12 | 2021-11-16 | 北京理工大学 | Traction device for air posture training |
| CN114934972A (en) * | 2022-06-10 | 2022-08-23 | 上海交通大学 | Damping-containing nonlinear spring limiter for steel frame structure of power plant boiler |
Citations (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN1997999A (en) * | 2004-03-29 | 2007-07-11 | 彼德·T·杰尔曼 | Systems and methods to determine elastic properties of materials |
| CN203362993U (en) * | 2013-05-02 | 2013-12-25 | 中国人民解放军军事医学科学院卫生装备研究所 | Single degree-of-freedom vertical vibration isolation system having quasi zero stiffness |
| CN203641365U (en) * | 2014-01-07 | 2014-06-11 | 湖南大学 | Quasi zero stiffness vibration isolator |
| CN105046008A (en) * | 2015-07-30 | 2015-11-11 | 南京航空航天大学 | Piecewise nonlinear vibration isolator and design method therefor |
| CN105183025A (en) * | 2015-07-16 | 2015-12-23 | 南京航空航天大学 | Quasi-zero stiffness vibration isolation system and nonlinear feedback control method thereof |
| RU2669914C2 (en) * | 2017-02-07 | 2018-10-16 | Акционерное общество "МЕРА" | Method and device for vibration protection |
| CN108916283A (en) * | 2018-07-17 | 2018-11-30 | 大连大学 | Quasi- zero stiffness metal-rubber composite vibration isolator |
-
2018
- 2018-12-28 CN CN201811625094.6A patent/CN109519499B/en not_active Expired - Fee Related
Patent Citations (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN1997999A (en) * | 2004-03-29 | 2007-07-11 | 彼德·T·杰尔曼 | Systems and methods to determine elastic properties of materials |
| CN203362993U (en) * | 2013-05-02 | 2013-12-25 | 中国人民解放军军事医学科学院卫生装备研究所 | Single degree-of-freedom vertical vibration isolation system having quasi zero stiffness |
| CN203641365U (en) * | 2014-01-07 | 2014-06-11 | 湖南大学 | Quasi zero stiffness vibration isolator |
| CN105183025A (en) * | 2015-07-16 | 2015-12-23 | 南京航空航天大学 | Quasi-zero stiffness vibration isolation system and nonlinear feedback control method thereof |
| CN105046008A (en) * | 2015-07-30 | 2015-11-11 | 南京航空航天大学 | Piecewise nonlinear vibration isolator and design method therefor |
| RU2669914C2 (en) * | 2017-02-07 | 2018-10-16 | Акционерное общество "МЕРА" | Method and device for vibration protection |
| CN108916283A (en) * | 2018-07-17 | 2018-11-30 | 大连大学 | Quasi- zero stiffness metal-rubber composite vibration isolator |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113648638A (en) * | 2021-08-12 | 2021-11-16 | 北京理工大学 | Traction device for air posture training |
| CN114934972A (en) * | 2022-06-10 | 2022-08-23 | 上海交通大学 | Damping-containing nonlinear spring limiter for steel frame structure of power plant boiler |
| CN114934972B (en) * | 2022-06-10 | 2023-03-14 | 上海交通大学 | Damping-containing nonlinear spring limiter for power plant boiler steel frame structure |
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