CN212272921U - Local resonance beam structure for low-frequency vibration reduction of precision instrument - Google Patents
Local resonance beam structure for low-frequency vibration reduction of precision instrument Download PDFInfo
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- CN212272921U CN212272921U CN202021714378.5U CN202021714378U CN212272921U CN 212272921 U CN212272921 U CN 212272921U CN 202021714378 U CN202021714378 U CN 202021714378U CN 212272921 U CN212272921 U CN 212272921U
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Abstract
The utility model discloses a local resonance beam structure for low-frequency vibration damping of a precision instrument, which comprises a simply supported beam and a precision instrument arranged on the simply supported beam, wherein the precision instrument is arranged on the upper surface of the simply supported beam in a vibration isolation way; m local resonance cells are also periodically arranged on the upper surface of the simply supported beam along the axial direction, and each local resonance cell comprises N vibration absorbers; MN vibration absorbers are arranged at equal intervals, and a precision instrument is arranged between any two adjacent vibration absorbers; the utility model provides a local resonance beam structure can effectively restrain the low frequency vibration of precision instruments, and local resonance beam structure comprises local resonance cell and precision instruments on the roof beam by setting up, contains the bump leveller of a plurality of differences among the local resonance cell, and this effective damping scope and the damping efficiency that can increase the bump leveller by a wide margin reaches good low frequency vibration isolation effect. The utility model discloses simple structure, be convenient for processing, installation and later maintenance have reduced the application cost.
Description
Technical Field
The utility model relates to a low frequency vibration control technical field, especially a local resonance beam structure for precision instruments low frequency damping.
Background
The vibration is one of important environmental factors influencing the measurement accuracy of a precision instrument, and the low-frequency vibration is difficult to eliminate and influences deeply. Therefore, the method has very important significance for effectively inhibiting the low-frequency vibration of the precision instrument. In the currently common vibration reduction mode, the vibration absorber can achieve vibration reduction performance of a low-frequency broadband and is widely researched and applied. The vibration absorber has the basic working principle that a resonance subsystem containing damping, rigidity and mass, namely the vibration absorber, is arranged at a specific position on a main vibration system, and when the natural frequency of the vibration absorber is close to the natural frequency of the main vibration system, the resonance energy of the main vibration system is transferred into the vibration absorber, so that the vibration amplitude of the main vibration system is effectively reduced. The proposal of the local resonance concept provides a new solution for the problem of low-frequency vibration control, the local resonance type structure is an artificially constructed periodic structure, the vibration damping performance of the local resonance type structure mainly depends on the local resonance of the microstructures in the constituent units, and the effective range of the vibration damping frequency can be effectively widened due to the diversification of the vibrator parameters in each local resonance unit.
In the practical engineering application process, the complex and changeable working environment puts higher requirements on the effective vibration reduction range of the vibration absorber, the vibration reduction frequency band of the single type of vibration absorber is narrow, and the working frequency facing the change of a main vibration system cannot achieve the ideal vibration reduction effect.
SUMMERY OF THE UTILITY MODEL
An object of the utility model is to overcome prior art's not enough, provide a local resonance beam structure for precision instruments low frequency damping to solve the problem that proposes in the above-mentioned technical background.
In order to achieve the above object, the present invention provides a method for manufacturing a semiconductor device, comprising:
a local resonance beam structure for low-frequency vibration reduction of a precision instrument comprises a simply supported beam and the precision instrument mounted on the simply supported beam, wherein the precision instrument is mounted on the upper surface of the simply supported beam in a vibration isolation manner; m local resonance cells are also periodically arranged on the upper surface of the simply supported beam along the axial direction, and each local resonance cell comprises N vibration absorbers; the MN (namely M multiplied by N) vibration absorbers are arranged at equal intervals, and the precision instrument is arranged between any two adjacent vibration absorbers;
wherein the local resonant beam structure satisfies the vibration equation of formula (1):
wherein: ρ is the density of the simply supported beam, A is the cross-sectional area of the simply supported beam, E is the elastic modulus of the simply supported beam, I is the moment of inertia of the beam section about the neutral axis, w (x, t) is the lateral displacement of the beam section neutral axis at the coordinate x at the time t, f (t) is the lateral external force applied to the beam at the time t, f (t)0(t) is the reaction force of the precision instrument on the beam, fij(xj+ ia, t) is the reaction force of the jth absorber in the ith cell from left to right on the beam, xj+ ia is the position of the jth vibration absorber on the beam in the ith cell, a is the length of each local resonance cell, xj(x-x) distance of the jth vibration absorber relative to the first vibration absorber in each cell0) And (x-x)ij) Is a unit impulse function and is used for describing the reaction force of a precision instrument or a vibration absorber on a beamPoint of action.
Further, the lateral external force applied to the simply supported beam may be represented by the following formula (2):
f(t)＝F0sinωt (2)
wherein F (t) is the transverse external force applied to the simply supported beam at the moment t, F0Is the amplitude of the lateral external force, and ω is the excitation frequency of the lateral external force.
Further, when the simply supported beam is acted by the transverse external force, the reaction force of the precision instrument on the simply supported beam is obtained by the following formula (3):
wherein k is0The rigidity corresponding to vibration isolation of the precision instrument; c. C0Damping corresponding to vibration isolation of a precision instrument; w (x)0T) is the coordinate x0The transverse displacement of the neutral axis of the beam section at the moment t; x is the number of0The coordinates of the installation position of the precision instrument on the beam; u. of0(t) is the lateral displacement of the precision instrument at time t;
is the coordinate x0The transverse velocity of the beam section neutral axis at the time t is w (x)0The first derivative of t);
is the transverse velocity of the precision instrument at time t, u0(t) first derivative; the damping force of the precision instrument is the product of the damping and the speed difference at the two ends of the damping; the spring force of a precision instrument is equal to the stiffness multiplied by the difference in displacement across the spring.
Further, when the simply supported beam is subjected to a lateral external force, the reaction force of the vibration absorber on the simply supported beam is obtained by the following formula (4):
wherein：kijAnd cijRespectively the rigidity and the damping of the jth vibration absorber in the ith cell from left to right on the beam; w (x)j+ ia, t) being the coordinate xjThe lateral displacement of the beam section neutral axis at + ia at time t; x is the number ofj+ ia is the position coordinate of the jth vibration absorber in the ith cell on the beam; u. ofij(xj+ ia, t) is the lateral displacement of the jth vibration absorber in the ith cell at the time t;
is the coordinate xjThe transverse velocity of the beam section neutral axis at + ia at time t is w (x)jThe first derivative of + ia, t);
is the lateral velocity of the jth vibration absorber in the ith cell at the time t, and is uij(xjThe first derivative of + ia, t); the spring force of the vibration absorber is equal to the product of the rigidity multiplied by the displacement difference at the two ends of the spring, and the damping force of the vibration absorber is the product of the damping and the speed difference at the two ends of the damping.
Further, according to the simple support boundary condition of the simple support beam, the transverse inherent vibration displacement of the simple support beam can be obtained by a modal superposition method, and can be expressed as:
wherein w (x, t) is the lateral displacement of the beam section neutral axis at the coordinate x at the time t; l is the length of the simply supported beam; n is the number of vibration modes of the beam; omega is the excitation frequency of the transverse external force; wsn，WcnThe sine and cosine components of the simply supported beam amplitude, respectively.
Further, the vibration of the precision instrument and the vibration absorber is expressed as:
ui(t)＝Usi sinωt+Uci cosωt i＝0,1,2,…,MN (6)
wherein, when i is 0, u0(t) is the lateral displacement of the precision instrument at time t; when i takes a value within the range of 1-MN, ui(t) representsThe ith vibration absorber from left to right on the beam or arranged at x0The transverse displacement of the precision instrument at the time t; u shapesi，UciThe sine and cosine components of the vibration amplitude of the vibration absorber on the beam and the precision instrument are respectively, and MN is the total number of the vibration absorbers.
In the above technical scheme, the total number MN of the vibration absorbers on the simply supported beam takes a value of 6, and each local resonance cell is a single-vibrator cell, a double-vibrator cell or a triple-vibrator cell, that is, the number N of the vibration absorbers in each local resonance cell takes a value of 1, 2 or 3.
In the technical scheme, in each local resonance unit cell, the mass ratio of each vibration absorber is 0.16, the natural frequency ratio is 0.5-1.5, and the damping ratio is 0.05-0.25. The mass ratio of the vibration absorber is the ratio of the mass of the vibration absorber to the mass of the precision instrument; the natural frequency ratio of the vibration absorber means the ratio between the natural frequency of the vibration absorber and the natural frequency of the precision instrument, the damping ratio of the vibration absorber means the ratio between the damping of the vibration absorber and the damping of the precision instrument, and the mass ratio, the natural frequency ratio and the damping ratio of the vibration absorber referred to herein are all meant.
In the above technical solution, when each local resonance unit cell is a single-vibrator unit cell, that is, M is 6 and N is 1, the mass ratio of the vibration absorber in each local resonance unit cell is μ is 0.16, the natural frequency ratio is γ 0.7221, and the damping ratio is ζ 0.1321;
when each local resonance unit cell is a double-vibrator unit cell, namely M is 3 and N is 2, the mass ratio of one vibration absorber in each local resonance unit cell is respectively mu-0.16, the natural frequency ratio is respectively gamma-0.8273 and the damping ratio is respectively zeta-0.0959; the mass ratio of the other vibration absorber is 0.16, the natural frequency ratio is 0.7338, and the damping ratio is 0.0969;
when each local resonance unit cell is a three-vibrator unit cell, namely M is 2 and N is 3, the mass ratio of one vibration absorber in each local resonance unit cell is 0.16, the natural frequency ratio is 0.7534, and the damping ratio is ζ 0.1186; the mass ratio of the other vibration absorber is mu-0.16, the natural frequency ratio is gamma-0.7281, and the damping ratio is zeta-0.0653; the mass ratio of the remaining vibration absorbers is 0.16, the natural frequency ratio is 0.8553, and the damping ratio is 0.0859.
Preferably, each local resonance unit cell is a two-vibrator unit cell.
Compared with the prior art, the beneficial effects of the utility model are that:
1. the utility model discloses a quantity, quality, damping and the rigidity to the bump leveller of installation on the simply supported beam to the research of damping performance influence, obtained the influence law of each parameter of bump leveller to system's damping performance, this selection optimization that is favorable to the design of local resonance roof beam structure and bump leveller parameter range can provide theoretical guidance for the research of low frequency vibration control problem.
2. The utility model discloses based on the research result discovery to local resonance beam structure low frequency vibration isolation performance, local resonance beam structure can effectively restrain the low frequency vibration of precision instruments, and local resonance structure comprises local resonance cellula and main vibration system (like precision instruments) on the roof beam by setting up, contains the bump leveller of a plurality of differences in the local resonance cellula, and this effective damping scope and the damping efficiency that can increase the bump leveller by a wide margin reaches good low frequency vibration isolation effect.
3. The utility model discloses simple structure, be convenient for processing, installation and later maintenance have reduced the application cost.
Drawings
Fig. 1 is a schematic structural view of the present invention;
FIG. 2 is a schematic diagram of the dynamic structure of the present invention;
FIG. 3 is a schematic diagram of the dynamic structure of a localized resonance unit cell;
fig. 4 is a calculation result of the vibration isolation performance of the local resonance beam structure by using the analytic method and the finite element method of the present invention, in which the solid line represents the calculation result of the analytic method of the present invention and the dotted line represents the calculation result of the finite element method;
FIG. 5 is an amplitude image of a precision instrument with different numbers of vibration absorbers, wherein the abscissa is the excitation frequency and the ordinate is the amplitude;
FIG. 6 is an image of the amplitude of the device at different mass ratios of the vibration absorber to the precision instrument, with the abscissa representing the excitation frequency and the ordinate representing the amplitude;
FIG. 7 is an image of the amplitude of the device at different ratios of the natural frequency of the vibration absorber to the natural frequency of the precision instrument, with the excitation frequency on the abscissa and the amplitude on the ordinate;
FIG. 8 is an image of the amplitude of the device under different damping ratios of the vibration absorber to the precision instrument, with the excitation frequency on the abscissa and the amplitude on the ordinate;
FIG. 9 is a diagram of the device vibration image contrast before and after optimization of the cell parameters of the single vibrator, with the abscissa being the excitation frequency and the ordinate being the amplitude;
FIG. 10 is a vibration image of the device after the double-vibrator cell optimization, wherein the abscissa is the excitation frequency and the ordinate is the amplitude;
FIG. 11 is a vibration image of the device after the optimization of the three-vibrator unit cell, wherein the abscissa is the excitation frequency and the ordinate is the amplitude;
FIG. 12 is a graph of damping efficiency for different localized resonating cells;
in fig. 4 to 11, the ordinate apparatus amplitude is the amplitude of the precision instrument, i.e., the amplitude in the frequency band of 5-50 Hz;
in the figure, 1, a simply supported beam; 2. a precision instrument; 3. a local resonance cell; 3.1, vibration absorber.
Detailed Description
The following description of the embodiments of the present invention is provided for illustrative purposes, and other advantages and effects of the present invention will be readily apparent to those skilled in the art from the disclosure herein. The present invention can also be implemented or applied through other different specific embodiments, and various details in the present specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It is to be noted that the features in the following embodiments and examples may be combined with each other without conflict.
It should be noted that the drawings provided in the following embodiments are only for illustrating the basic concept of the present invention, and the components related to the present invention are only shown in the drawings rather than drawn according to the number, shape and size of the components in actual implementation, and the form, amount and ratio of the components in actual implementation may be changed at will, and the layout of the components may be more complicated.
Referring to fig. 1 and 2, the utility model provides a local resonance beam structure for low frequency vibration damping of a precision instrument, the local resonance beam structure comprises a simply supported beam 1 and a precision instrument 2 installed on the simply supported beam 1, the precision instrument 2 is installed on the upper surface of the simply supported beam 1 in a vibration isolation manner; m local resonance cells 3 are also periodically arranged on the upper surface of the simply supported beam 1 along the axial direction, and each local resonance cell 3 comprises N vibration absorbers 3.1; the MN (namely M multiplied by N) vibration absorbers 3.1 are arranged at equal intervals, and the precision instrument 2 is arranged between any two adjacent vibration absorbers 3.1; the structure of each vibration absorber comprises a mass block, a damping element and a spring, wherein the spring and the damping element are connected in parallel between the mass block and the simply supported beam;
referring to fig. 2, fig. 2 is a single periodic cell of the local resonance beam structure, that is, a local resonance cell, in the drawing, N vibration absorbers are disposed in the single periodic cell, and each vibration absorber includes a damping element, which not only can effectively widen a vibration attenuation band, but also can consume energy on the main system through damping to a certain extent. Suppose the location of the ith vibration absorber is xjThe mass, spring rate and damping of the ith vibration absorber are respectively mi、kiAnd ci. The length of the cell is a, the N vibration absorbers in the cell divide the whole cell into N +1 sections, and the length of each section of beam is marked as aiThen, the vibration absorber position relationship can be expressed as:
for the low-frequency vibration of the simply supported beam, the influence of the shearing deformation of the beam and the rotational inertia of the cross section around the neutral axis can be ignored. This beam model is called a Bernoulli-Euler beam. According to its vibration principle and combine the utility model provides a local resonance beam structure's structural feature, the vibration equation that can obtain local resonance beam structure is:
wherein: rho is the density of the simply supported beam, A is the cross-sectional area of the simply supported beam, E is the elastic modulus of the simply supported beam, I is the moment of inertia of the beam section about the neutral axis, w (x, t) is the transverse displacement of the neutral axis of the beam section at the coordinate x at the moment t, f (t) is the transverse external force applied to the beam, f (t)0(t) is the reaction force of the precision instrument on the beam, fij(xj+ ia, t) is the reaction force of the j vibration absorber in the i cell from left to right on the beam, xj+ ia is the position of the jth vibration absorber in the ith cell, a is the length of the local resonance cell, xjThe distance between the jth vibration absorber in each unit cell and the 1 st vibration absorber in the unit cell is calculated; (x-x)0) And (x-x)ij) Is a unit pulse function, when the inside of the bracket is 0 (i.e. x ═ x)0Or x ═ xij)，(x-x0) 1 or (x-x)ij) 1, the description is given for the position of the reaction force of the precision instrument or the vibration absorber acting on the simply supported beam.
Further, the simply supported beam is subjected to a transverse external force:
f(t)＝F0sinωt (2)
wherein F (t) is the transverse external force applied to the simply supported beam at the moment t, F0The amplitude of the lateral external force is shown, and omega is the excitation frequency of the lateral external force.
When the simply supported beam is subjected to transverse external force, the stress of the precision instrument is as follows:
wherein k is0The rigidity corresponding to vibration isolation of the precision instrument; c. C0Damping corresponding to vibration isolation of a precision instrument; w (x)0T) is the coordinate x0The transverse displacement of the neutral axis of the beam section at the moment t; x is the number of0The coordinates of the installation position of the precision instrument on the beam; u. of0(t) is the lateral displacement of the precision instrument at time t;
is the coordinate x0The transverse velocity of the beam section neutral axis at the time t is w (x)0The first derivative of t);
is the transverse velocity of the precision instrument at time t, u0(t) first derivative; the spring force of the precision instrument is equal to the rigidity multiplied by the displacement difference of two ends of the spring; the damping force of the precision instrument is the product of the damping and the speed difference between the two ends of the damping.
When the simply supported beam is subjected to transverse external force, the stress of the precision instrument is as follows:
wherein: k is a radical ofijAnd cijThe rigidity and the damping of the jth vibration absorber of the ith cellular from left to right on the beam are respectively set; w (x)j+ ia, t) being the coordinate xjThe lateral displacement of the beam section neutral axis at + ia at time t; x is the number ofj+ ia the position coordinate of the jth vibration absorber in the ith cell; u. ofij(xj+ ia, t) is the lateral displacement of the jth vibration absorber in the ith cell at the time t;
is the coordinate xjThe transverse velocity of the beam section neutral axis at + ia at time t is w (x)jThe first derivative of + ia, t);
is the lateral velocity of the jth vibration absorber in the ith cell at the time t, and is uij(xjThe first derivative of + ia, t); the spring force of the vibration absorber is equal to the rigidity multiplied by the displacement difference of two ends of the spring; the damping force of the vibration absorber is the product of the damping and the speed difference between the two ends of the damping.
Research on transverse external force F (t) ═ F applied to simply supported beam0The vibration of the precision instrument in sin ω t is overlapped by mode according to the simple supporting boundary condition of the beamThe transverse natural vibration displacement of the simply supported beam obtained by addition can be expressed as:
w (x, t) is the transverse displacement of a beam section neutral axis at a coordinate x at a moment t, l is the length of the simply supported beam, n is the number of beam vibration modes, and omega is the excitation frequency of a transverse external force; wsn，WcnThe sine and cosine components of the simply supported beam amplitude, respectively.
Taking the Q-order mode of the beam to participate in calculation, the vibration of the precision instrument and the vibration absorber can be expressed as follows:
ui(t)＝Usi sinωt+Uci cosωt i＝0,1,2,…,MN (6)
wherein, when i is 0, u0(t) is the lateral displacement of the precision instrument at time t; when i takes a value within the range of 1-MN, ui(t) denotes the ith vibration absorber or set at x0The transverse displacement of the precision instrument at the time t; u shapesi，UciSine and cosine components of the vibration amplitude of the vibrator (comprising a plurality of vibration absorbers and a precision instrument) on the beam are respectively, and MN (M multiplied by N) is the total number of the vibration absorbers;
substituting the equations (2), (3), (4), (5) and (6) into the vibration equation (1) of the system, and multiplying the left end and the right end of the equation (1) by the same value
And integrating x from 0 to l, taking the Q-order mode of the beam to participate in calculation, and integrating the vibration equation into a matrix form by utilizing the orthogonality of the vibration modes of the beam:
wherein: q ═ Ws,Wc,Us,Uc]TAn unknown vector to be solved of 2(Q + MN +1) x 1 (2(Q + MN +1) x 1 is the dimension of the unknown vector, wherein Q is the mode number, MN is the total number of the vibration absorbers, and +1 is the precisionMiss, multiply 2, is two because the cosine and sine components are considered separately), Ws，WcSine and cosine components, U, of the beam amplitude, respectivelys，UcSine and cosine components of the amplitude of the on-beam vibrator (including the vibration absorber and the precision instrument) respectively; q11 is a Q diagonal matrix, and Q11 can be decomposed into two parts: q11 ═ Λb+ C1. Wherein ΛbRepresentative are the inherent properties of the beam itself, of which the elements are derived:
c1 represents the effect of the stiffness of the vibrator (including the device) on the beam on the vibration of the beam, wherein the element is
p and q correspond to the number of rows and columns of the matrix, respectively; q12The Q × Q matrix represents the effect of the damping of the vibrator (including the device) on the beam on the vibration of the beam, and the elements in the matrix are derived as follows:
Q13and Q14The matrix of Q × (MN +1), representing the effect of stiffness and damping of the vibrator (without equipment) on the beam vibration, respectively, is derived from the following elements:
Q33is a diagonal matrix of (MN +1) × (MN +1), is mainly determined by the inertia and rigidity of a vibrator (including equipment) on the beam, and is calculated by Q33The element in (A) is Q33pq＝-ω2mq-1+kq-1；Q34Is a diagonal matrix of (MN +1) × (MN +1), is mainly determined by the damping of the vibrator (including equipment) on the beam, and the element in the calculated matrix is
The rest of the component matrixes in the left coefficient matrix with the same number in the formula (7) have similar expression forms and the same meanings with the known matrixes, and can be obtained by transforming the matrixes, and specifically, the expression forms are as follows: q21＝-Q12，Q22＝Q11，Q23＝-Q14，Q24＝Q13。
Q31＝Q13 T，Q32＝Q14 T，Q41＝-Q32，Q42＝Q31，Q43＝-Q34，Q44＝Q33. Equation (7) the equal-sign right vector F is a force vector of 2(Q + MN) × 1, including lateral external forces acting on the beam, which can be expressed as: f ═ F1,01×(Q+MN+1)]TWherein:
in conclusion, the vibration amplitude of the precision instrument under different excitation frequencies can be obtained by solving the matrix equation (7).
1. The utility model discloses a vibration isolation performance research
In order to reduce the low frequency vibration of precision instruments, the utility model discloses a this kind of damping mode of local resonance type bump leveller, precision instruments installs on the base member, just assume here that precision instruments installs and carry out the analysis on the most common mechanical structure roof beam, solve to obtain the vibration law of precision instruments along with excitation frequency through the equation, analysis contrast installation bump leveller front and back, and the vibration damping effect contrast when installing different kinds of local resonance cells, the finite element method utilizes emulation software comsol to simulate, verify the exactness of analytic solution, in order to obtain the best solution of the different cellular damping of this model, each bump leveller parameter of having carried out the analysis in the cell and obtained the vibration damping law, optimize the parameter of bump leveller with the particle swarm optimization algorithm at last, the target function and the optimization parameter scope have been confirmed according to the vibration damping law that obtains above.
(1) Verify and pass the utility model provides an analytic validity of vibration equation
For verifying the utility model discloses gained vibration equation carries out the validity of analytic precision instruments vibration amplitude under different excitation frequencies, adopts here respectively the utility model discloses gained vibration equation (formula 7) and finite element method calculate the contrast to the vibration amplitude of precision instruments under different excitation frequencies in the local resonance beam structure.
Installing a precision instrument at the midpoint position of the simply supported beam, and taking the initial calculation parameters of the beam and the precision instrument as follows: the length l of the beam is 0.7m, the width B is 0.05m, the height H is 0.008m, and the modulus of elasticity E is 2.1 × 1011Pa, density rho 7.8 × 103kg/m3The first order natural frequency of the available beam is 38.4 Hz; mass m of precision instrument00.5kg, stiffness k0＝9×103N/m, damping c0The natural frequency of the available device was 19.7Hz at 46N · s/m. Substituting the above parameters into the utility model discloses the resulting coupling equation (equation 7) and finite element software respectively obtain the pair ratio of analytic result and finite element result for as shown in fig. 4.
As can be seen from fig. 4, the first formant (located between 10 to 30 Hz) reflects the coupling between the precision instrument and the simply supported beam, the corresponding frequency is the natural frequency of the precision instrument, the second formant (located between 30 to 50 Hz) reflects the resonance characteristic of the simply supported beam itself, and the corresponding frequency is the first-order natural frequency of the simply supported beam. As can be seen from the calculation and analysis, the maximum error between the analysis result and the finite element result is 1.01 × 10 in the target frequency band-6m, average error of 7.23X 10-7And m is selected. Therefore, the utility model discloses a frequency that the formant of analytic method calculated result and finite element result corresponds is unanimous basically, and the image has higher goodness of fit, has verified the validity of analytic method.
(2) Single factor method for analyzing low frequency vibration isolation characteristic
When a plurality of vibration absorbers are periodically arranged on the simply supported beam, the influence of the quantity, the rigidity, the quality and the damping of the vibration absorbers on the vibration reduction performance is required to be researched to analyze the vibration reduction effect of the vibration absorbers on a precision instrument. The low-frequency vibration isolation characteristic of the single-vibrator unit cell is analyzed, and the vibration attenuation rule of vibrators in the unit cell under different parameters is researched. Assuming the initial calculated parameters of the vibration absorber are: the mass ratio μ of the vibration absorber to the precision instrument is 0.16, the natural frequency ratio γ is 0.8, and the damping ratio ζ is 0.1.
2.1 Effect of additional vibrators on the Natural frequency of the Beam
In order to study the coupling effect between the additional vibrator (a plurality of vibration absorbers and precision instruments) on the beam and the beam, the change of the natural frequency of the front beam and the rear beam of the additional vibrator on the beam is analyzed. Taking the example of mounting a precision instrument on the beam, the natural frequencies of the front beam and the rear beam coupled with the equipment and the beam are respectively calculated. Front n-order natural frequency formula capable of utilizing simply supported beam
The first 4 order natural frequencies of the beam are calculated and the finite element software is used to calculate the natural frequencies of the various orders of the beam when the device is coupled to the beam. The control ratios are shown in table 1.
TABLE 1 Natural frequency of the Beam
As can be seen from table 1, the natural frequency of each order of the coupled back beam is changed. Due to the coupling effect of the precision instrument and the beam, when the excitation frequency is the same as the natural frequency of the precision instrument, the resonance of the precision instrument can also cause the beam to vibrate greatly, so that the first-order natural frequency of the beam is the same as the natural frequency of the precision instrument. The natural frequencies of the rest orders of the beam are shifted, the natural frequencies of the beam are changed into a complex form due to the existence of damping in the equipment, and the influence of the coupling action on the natural frequency of the high-frequency-band beam is smaller than that of the low-frequency band.
2.2 Effect of number of vibration absorbers on vibration isolation Performance
Assuming that the mass ratio μ of the vibration absorbers is 0.16, the natural frequency ratio γ is 0.8, and the damping ratio ζ is 0.1, in the case of a single-vibrator cell (i.e., N is 1), the influence of the number M × N of vibration absorbers arranged at equal intervals on the beam on the vibration isolation performance is first analyzed.
Respectively recording the maximum amplitude (namely the maximum amplitude within the frequency range of 5-50 Hz) of front and rear precision instruments with vibration absorbers arranged on the beam as Wm0And WmThe damping efficiency of the vibration absorber can be expressed as:
as can be seen from fig. 5, when the number of vibration absorbers (M is 5 and N is 1) is 5, the vibration damping performance is optimal, the vibration damping efficiency is 11.34%, and the amplitude of the precision equipment gradually decreases and then increases as the number of vibration absorbers increases (the amplitude of the precision equipment refers to the variation of the maximum amplitude value within a certain frequency band, such as 10 to 30Hz, the same applies hereinafter), and the vibration damping efficiency increases from 1.21% to 11.34% and then decreases to 8.69%. Due to the coupling effect of the vibration absorber and the beam, an original one formant is changed into two formants within the frequency range of 10-30 Hz, and the frequency corresponding to the newly added formant is the inherent frequency of the vibration absorber. The number of vibration absorbers affects the damping efficiency given the initial parameters of the vibration absorbers, but not the greater the number of vibration absorbers, the better the damping efficiency, but the number of vibration absorbers should be selected in combination with the actual requirements.
2.3 influence of mass ratio on vibration isolation Properties
Assuming that the natural frequency ratio γ of the vibration absorbers is 0.8 and the damping ratio ζ is 0.1, in the case of a single-vibrator cell, the influence law of the vibration-damping performance by the mass ratio μ of the vibration absorbers was examined, taking the number N of the vibration absorbers as 6.
As can be seen from fig. 6, as the mass ratio increases, the mass of the vibration absorber gradually increases, and its natural frequency gradually decreases and is away from the natural frequency of the precision instrument. A new resonance peak appears in the frequency range of 10-20 Hz, which is caused by the natural frequency of the vibration absorber. The maximum value of the amplitude (the amplitude of a precision instrument, namely the maximum value of the amplitude in a frequency band range of 5-30 Hz, the same applies below) of the equipment without arranging the vibration absorber on the beam is known to be 2.14 multiplied by 10-5m, when the mass ratio mu is 0.08, the maximum value of the device amplitude is 1.91 multiplied by 10-5m, the vibration reduction efficiency is 10.46%; when the mass ratio mu is 0.16, the maximum value of the amplitude of the device is 1.95X 10-5m, the vibration reduction efficiency is 8.69%; when the mass ratio mu is 0.24, the maximum value of the amplitude of the device is 2.09X 10-5m, damping efficiency of 2.18%. The vibration reduction performance of the vibration absorber is improved along with the increase of the mass ratio in the frequency range of 30-50 Hz, and the vibration of the beam is restrainedAnd gradually reduces the amplitude of the precision instrument.
2.4 influence of the natural frequency ratio on the vibration isolation Properties
Assuming that the damping ratio ζ of the vibration absorbers is 0.1, in the case of a single-vibrator cell, the influence law of the natural frequency ratio γ of the vibration absorbers on the vibration isolation performance was studied, taking the number N of the vibration absorbers as 6 and the mass ratio μ of the vibration absorbers as 0.16.
As can be seen from fig. 7, as the natural frequency ratio of the vibration absorber gradually changes from 0.4 to 0.83, the resonance peak corresponding to the natural frequency of the vibration absorber gradually shifts to the right and gradually increases in the frequency range of 5 to 20Hz due to the coupling effect between the vibration absorber and the beam. Meanwhile, the resonance peak value corresponding to the natural frequency of the precision instrument is gradually reduced. When the natural frequency ratio is 0.4 to 0.6, the resonance peak value corresponding to the natural frequency of the vibration absorber is far smaller than that corresponding to the natural frequency of the equipment, and the vibration reduction efficiency is improved from 2.71% to 8.69% along with the increase of the natural frequency ratio. When the natural frequency ratio is 0.8 to 0.83, the resonance peak value corresponding to the natural frequency of the vibration absorber exceeds the resonance peak value corresponding to the natural frequency of the equipment, so that the vibration reduction efficiency is mainly influenced, the maximum amplitude value of the precision instrument in the target frequency band is increased, and the vibration reduction efficiency is reduced to 2.58 percent.
2.5 influence of damping ratio on vibration isolation Properties
The influence rule of the damping ratio zeta of the vibration absorbers on the vibration isolation performance is researched, wherein the number N of the vibration absorbers is 6, the mass ratio mu of the vibration absorbers to a precision instrument is 0.16, and the natural frequency ratio gamma is 0.8.
Fig. 8 is a comparison of the variation curve of the amplitude of the equipment with frequency under different damping ratios, and it can be seen from fig. 8 that two formants appear when the damping ratio of the vibration absorber is 0.08 and 0.1 within the frequency range of 10 to 30Hz, which respectively correspond to the natural frequency of the vibration absorber and the natural frequency of the equipment, and the vibration reduction efficiency is 4.43% and 8.69%. When the damping ratio of the vibration absorber is 0.18 to 0.2, the original two formants become a formant with a smaller peak value and a wider bandwidth, which is caused by the superposition of the two formants due to the gradual increase of the damping component in the vibration absorber, and the vibration reduction efficiency is improved to 12.20 percent at the moment due to the reduction of the maximum value of the amplitude of the equipment in the target frequency band. Within the frequency range of 30-50 Hz, the amplitude of the precision instrument is gradually reduced and the vibration tends to be smooth along with the increase of the damping ratio.
(3) Parameter optimization and analysis of local resonance type vibration absorber
3.1 parameter optimization
After the quantity of the vibration absorbers and the influence rule of each parameter on the vibration reduction performance are obtained, the function of the parameters of the vibration absorbers in the vibration reduction process can be preliminarily known. The mass ratio should be as large as possible, considering that the mass ratio of the vibration absorber may affect the vibration damping performance of the vibration absorber when the mass ratio is small in the multi-vibrator cell. The total number N of vibration absorbers is 6 and the mass ratio μ is 0.16, and when the total number of vibration absorbers is constant, the vibration absorbers are divided into single-oscillator, double-oscillator, and three-oscillator cells, respectively, and the natural frequency ratio γ and the damping ratio ζ of the vibration absorbers are optimally designed by the particle swarm optimization.
Recording the maximum amplitude value of the precision instrument within the frequency range of 5-50Hz as WmAverage value is denoted as Wp. Considering that the optimization aims to better reduce the resonance peak value of the low-frequency precision instrument under each excitation frequency, and the optimization effect is required to be stable, the optimization objective function can be selected as follows: 0.7 XWm+0.3×Wp(since the main factors affecting the operation performance of the precision instrument are the maximum value of the vibration amplitude and the fluctuation amplitude of the precision instrument in the target frequency band, and the requirement for the maximum value is higher than the requirement for the fluctuation amplitude, the sum of 70% of the maximum value and 30% of the average value is taken as the optimized objective function). When the natural frequency of the vibration absorber is near the natural frequency of the controlled object, the vibration amplitude of the controlled object is effectively reduced, and the optimization range of the natural frequency ratio of the vibration absorber is 0.5-1.5; the vibration reduction performance is better when the damping ratio of the vibration absorber in the single-vibrator unit cell is taken to be near 0.18, and the optimized range of the damping ratio of the vibration absorber is 0.05-0.25.
(1) A single oscillator unit cell (M is 6, N is 1). The 6 vibration absorbers uniformly arranged on the beam are divided into 6 local resonance cells, and each cell comprises 1 vibrator. In order to verify the effectiveness of the optimization result, the amplitudes of the precision instrument before and after the oscillator unit cell optimization are compared, and oscillator initial calculation parameters are taken as shown in table 2.
TABLE 2 original parameters of the single-vibrator cell
The optimized parameters obtained by optimally designing the natural frequency ratio and the damping ratio of the oscillator in the cellular are as follows: γ is 0.7221 and ζ is 0.1321. And substituting the optimized parameters into a vibration equation (equation 7) to obtain the vibration amplitude value pairs of the front and rear devices before and after optimization, such as shown in fig. 8.
As can be seen from fig. 9, in the frequency range of 10 to 30Hz, two resonance peaks exist respectively corresponding to the natural frequency of the vibration absorber and the natural frequency of the equipment. Before optimization, the resonance peak value corresponding to the natural frequency of the vibration absorber is larger than the resonance peak value corresponding to the natural frequency of the equipment, and the main influence is generated on the vibration reduction efficiency. After optimization, the resonance peak value corresponding to the natural frequency of the vibration absorber is obviously reduced and is lower than the resonance peak value corresponding to the natural frequency of the equipment. The data is further processed, and the maximum value of the amplitude of the precision instrument in the target frequency band is 1.95 multiplied by 10 before optimization-5m is reduced to 1.87 multiplied by 10 after optimization-5m, the damping efficiency is also improved from 2.45% to 8.69%.
Example 2
(2) Double-vibrator unit cell (M is 3, N is 2). The 6 vibration absorbers uniformly arranged on the beam are divided into three local resonance cells, each cell comprises two vibrators, and the mass ratio of the vibrators in the cell is assumed to be mu1、μ2The natural frequency ratios of the oscillators are respectively gamma1、γ2Damping ratio of the vibrator is ζ1、ζ2In which μ1＝μ20.16. The natural frequency ratio and the damping ratio of the oscillator are optimally designed by utilizing a particle swarm optimization, and the parameters of the optimized oscillator are shown in table 3.
TABLE 3 optimized parameters of the two-vibrator cell
The optimized parameters are substituted into the vibration equation (equation 7) to obtain the variation curve of the vibration amplitude of the precision instrument along with the frequency, which is shown in fig. 9.
As can be seen from fig. 10, the two-vibrator unit cell has more excellent vibration damping performance than the single-vibrator unit cell, and the maximum value of the amplitude of the precision instrument corresponding to the optimized two-vibrator unit cell is 1.75 × 10 within the target frequency band-5m, 1.87 × 10 optimized for the contrast single-vibrator cell-5m is obviously improved. The optimized damping efficiency of the double-vibrator unit cell is 18%, and compared with the optimized damping efficiency of the single-vibrator unit cell, the damping efficiency is improved by more than two times.
(3) Three-vibrator unit cell (M is 2, N is 3). The 6 vibration absorbers uniformly arranged on the beam are divided into two cells, each cell comprises three vibrators, and the mass ratio of the vibrators in each cell is assumed to be mu1、μ2、μ3The natural frequency ratios of the oscillators are respectively gamma1、γ2、γ3Damping ratio of the vibrator is ζ1、ζ2、ζ3In which μ1＝μ2＝μ3The natural frequency ratio and the damping ratio of the oscillator were optimized by the particle swarm optimization, and the optimized parameters are shown in table 4.
TABLE 4 parameters after optimization of three-vibrator cell
The optimized parameters are substituted into the vibration equation (equation 7) to obtain a vibration image of the equipment (referred to as a precision instrument) as shown in fig. 11.
The vibration damping performance of the three-vibrator unit cell is close to that of the double-vibrator unit cell and slightly superior to that of the double-vibrator unit cell, three resonance peaks appear in the frequency range of 10-30 Hz, and the maximum value of the amplitude of the precision instrument corresponding to the optimized three-vibrator unit cell is 1.747 multiplied by 10 due to the coupling result of vibrators with different natural frequencies in the equipment and the unit cell and the beam-5m, the damping efficiency is 18.3%.
The biggest problem of a single oscillator unit cell is that the damping frequency band is narrow, and the problem can be effectively improved by a plurality of oscillator unit cells. Next, the experimental data are further processed, and the specific values of the vibration reduction efficiency in the range of 5 to 50Hz are calculated based on the vibration reduction data before and after the optimization of the cells including different vibrators, so that the vibration reduction efficiency of different cells can be obtained as shown in fig. 11.
Analysis shows that the vibration reduction efficiency of the three-vibrator unit cell is relatively close to that of the two-vibrator unit cell in the whole frequency band where the vibration absorber plays a role. The damping efficiency of the three-vibrator and double-vibrator unit cells in the frequency band of 10-38 Hz is higher than that of the single-vibrator unit cells, and the damping efficiency of the three-vibrator, double-vibrator and single-vibrator unit cells is respectively as follows: 18.3%, 18% and 12.45%. In order to simplify the design, a two-vibrator cell design may be selected in the local resonance type structure.
The above-mentioned embodiments only express the specific embodiments of the present invention, and the description thereof is specific and detailed, but not construed as limiting the scope of the present invention. It should be noted that, for those skilled in the art, without departing from the spirit of the present invention, several variations and modifications can be made, which are within the scope of the present invention.
Claims (5)
1. A local resonance beam structure for low-frequency vibration reduction of a precision instrument is characterized by comprising a simply supported beam (1) and a precision instrument (2) arranged on the simply supported beam (1), wherein the precision instrument (2) is arranged on the upper surface of the simply supported beam (1) in a vibration isolation manner; m local resonance cells (3) are also periodically arranged on the upper surface of the simply supported beam (1) along the axial direction, and each local resonance cell (3) comprises N vibration absorbers (3.1); the M multiplied by N vibration absorbers (3.1) are arranged at equal intervals, and the precision instrument (2) is arranged between any two adjacent vibration absorbers (3.1).
2. A local resonance beam structure for low frequency damping of precision instruments according to claim 1, characterized in that each vibration absorber (3.1) comprises a mass, a damping element and a spring, said spring and damping element being arranged in parallel between the mass and the simply supported beam.
3. The local resonance beam structure for low frequency vibration damping of a precision instrument according to claim 1, characterized in that the total number M x N of vibration absorbers on the simply supported beam (1) is 6, and each local resonance cell (3) is a single-vibrator cell, a double-vibrator cell or a triple-vibrator cell, i.e. the number N of vibration absorbers in each local resonance cell is 1, 2 or 3.
4. The local resonance beam structure for low-frequency vibration reduction of a precision instrument as claimed in claim 3, wherein in each local resonance unit cell (3), the mass ratio of each vibration absorber is 0.16 μ, the natural frequency ratio γ is 0.5-1.5, and the damping ratio ζ is 0.05-0.25.
5. The local resonance beam structure for low frequency vibration damping of precision instruments according to claim 4, characterized in that when each local resonance cell (3) is a single-vibrator cell, i.e. M6, N1, the mass ratio of the vibration absorber (3.1) in each local resonance cell (3) is μ 0.16, the natural frequency ratio is γ 0.7221, the damping ratio is ζ 0.1321;
when each local resonance unit cell (3) is a double-vibrator unit cell, namely M is 3 and N is 2, the mass ratio of each vibration absorber (3.1) in each local resonance unit cell (3) is mu is 0.16, and the natural frequency ratio is gamma respectively1＝0.8273、γ20.7338, the damping ratios are ζ1＝0.0959、ζ2＝0.0969；
When each local resonance unit cell (3) is a three-vibrator unit cell, namely M is 2 and N is 3, the mass ratio of each vibration absorber (3.1) in each local resonance unit cell (3) is mu is 0.16, and the natural frequency ratio is gamma respectively1＝0.7534、γ2＝0.7281、γ30.8553, the damping ratios are ζ1＝0.1186、ζ2＝0.0653、ζ3＝0.0859。
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