CN112580210A - Vibration isolation frequency band regulation and control design method for one-dimensional periodic cushion layer vibration reduction ballast bed - Google Patents

Vibration isolation frequency band regulation and control design method for one-dimensional periodic cushion layer vibration reduction ballast bed Download PDF

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CN112580210A
CN112580210A CN202011533103.6A CN202011533103A CN112580210A CN 112580210 A CN112580210 A CN 112580210A CN 202011533103 A CN202011533103 A CN 202011533103A CN 112580210 A CN112580210 A CN 112580210A
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姜博龙
刘冀钊
胡文林
王少林
何宾
齐春雨
杨宝峰
杨其振
刘道通
谭大正
程保青
徐彩彩
韩朝霞
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China Railway Design Corp
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • E—FIXED CONSTRUCTIONS
    • E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
    • E01B—PERMANENT WAY; PERMANENT-WAY TOOLS; MACHINES FOR MAKING RAILWAYS OF ALL KINDS
    • E01B1/00—Ballastway; Other means for supporting the sleepers or the track; Drainage of the ballastway
    • E01B1/002—Ballastless track, e.g. concrete slab trackway, or with asphalt layers
    • E—FIXED CONSTRUCTIONS
    • E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
    • E01B—PERMANENT WAY; PERMANENT-WAY TOOLS; MACHINES FOR MAKING RAILWAYS OF ALL KINDS
    • E01B19/00—Protection of permanent way against development of dust or against the effect of wind, sun, frost, or corrosion; Means to reduce development of noise
    • E01B19/003—Means for reducing the development or propagation of noise
    • E—FIXED CONSTRUCTIONS
    • E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
    • E01B—PERMANENT WAY; PERMANENT-WAY TOOLS; MACHINES FOR MAKING RAILWAYS OF ALL KINDS
    • E01B2204/00—Characteristics of the track and its foundations
    • E01B2204/09—Ballastless systems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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Abstract

The invention discloses a simplified calculation and vibration isolation frequency band design method for a one-dimensional periodic cushion damping track bed energy band structure. The rule that the forbidden band and the vibration isolation frequency band change along with geometric and material parameters can be obtained through analysis of influence factors, and active regulation and control of the forbidden band and the vibration isolation frequency band are achieved. The method has the advantages that the method for calculating the energy band structure and designing the vibration isolation frequency band of the one-dimensional periodic cushion damping ballast bed is provided, and theoretical basis is provided for the design and engineering practice of the periodic damping ballast bed. The method has the advantages of small calculated amount, clear theoretical basis, convenient use, wide applicability and wide calculation result, and the calculation result can completely meet the requirements of engineering design.

Description

Vibration isolation frequency band regulation and control design method for one-dimensional periodic cushion layer vibration reduction ballast bed
Technical Field
The invention relates to the field of vibration reduction and noise reduction of rail transit, in particular to a method for regulating and controlling a vibration isolation frequency section of a one-dimensional periodic cushion layer vibration reduction ballast bed.
Background
At present, the track vibration reduction measures widely adopt a method for reducing the support rigidity or increasing the vibration participating quality to control the environmental vibration, and the method can cause abnormal abrasion of the wheel track under the conditions of excessive vibration reduction or improper vibration reduction, so that the rapid abnormal damage of equipment and parts is caused, the maintenance workload of the wheel track system is increased, and even the stability and the safety of the driving are influenced. In addition, the conventional track vibration reduction design lacks scientific frequency planning and comprehensive vibration reduction and isolation concepts, so that the effect of achieving half the effort is achieved.
Solid physics studies have found that periodic media arranged in a manner have elastic wave band gap characteristics, i.e., incident waves cannot propagate in the periodic medium when the frequency of the incident wave falls within the band gap range. This brings a completely new angle to the rail damping. The rail transit field has begun to utilize the forbidden band characteristic of the periodic structure to carry out periodic innovative design on the structure involved in rail transit, so as to realize active regulation and control of vibration. At present, a periodic vibration reduction ballast bed based on a periodic structure forbidden band theory has no engineering embodiment, so that the periodic vibration reduction ballast bed has no engineering experience for reference and has no theoretical calculation of a scientific system for support. Therefore, a suitable method for calculating the energy band structure and designing the vibration isolation frequency band of the periodic vibration reduction ballast bed is urgently needed to be provided, and a theoretical basis is provided for the design and engineering practice of the periodic vibration reduction ballast bed.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a vibration isolation frequency band regulation and control design method for a one-dimensional periodic cushion damping ballast bed.
In order to achieve the purpose, the invention adopts the following technical scheme:
a vibration isolation frequency band regulation and control design method for a one-dimensional periodic cushion damping ballast bed comprises the following steps:
s1: determining a simplified form, a periodic characteristic, a basic assumption and a boundary condition of a one-dimensional periodic cushion damping track bed energy band structure calculation model;
s2: establishing a wave equation of longitudinal waves transmitted by the one-dimensional periodic cushion damping ballast bed along the periodic direction:
Figure BDA0002850055840000011
in the formula: u (x, t) is a displacement vector;
Figure BDA0002850055840000012
is the velocity of longitudinal wave; lambda and mu are Lame constants; rho is density; x is a position vector; t is a time variationThe amount of the compound (A) is,
considering that both sides of the equation have a time-varying simple harmonic term eiωtAnd ω is the angular frequency, equation (1) is reduced to the form:
Figure BDA0002850055840000021
solving the equation to obtain an energy band structure along the periodic direction;
s3: giving a general solution to the state quantity of an elastic wave in a single homogeneous medium:
for the first medium (0) in the basic unit<x<a1) The general solution of the state quantity of the compressional wave in a single homogeneous medium is:
Figure BDA0002850055840000022
in the formula: a. the1 +、A1 -Amplitude in the positive and negative directions, q, respectively, of propagation in the x-direction in the first medium1Is the wave vector:
q1＝ω/cp,1 (4)
in the formula: c. Cp,1Is the compressional wave velocity of the first medium in the basic unit;
s4: converting the fluctuation problem into a characteristic value problem containing an elastic wave propagation dispersion relation by using a Bloch theorem:
due to the periodicity of the one-dimensional periodic cushion damping track bed in the x direction, the solution of u (x) is written into a form of Bloch waves according to the Bloch-Floquet theorem:
u(x)＝eikxuk(x) (5)
in the formula:
uk(x)＝uk(x+a) (6)
wherein a ═ a1+a2Is a period constant;
obtained by the formulae (3) and (5):
Figure BDA0002850055840000023
in consideration of the periodic boundary condition (6), equation (7) is changed to the following form:
Figure BDA0002850055840000024
wherein: na<x<na+a1，
Bringing formula (5) into formula (8) gives:
when na<x<na+a1When the temperature of the water is higher than the set temperature,
Figure BDA0002850055840000025
when na + a1<x<na+a1+a2When the temperature of the water is higher than the set temperature,
Figure BDA0002850055840000026
in the formula: q. q.s2＝ω/cp,2Is the wave vector;
s5: determining undetermined coefficient by applying continuous boundary conditions of displacement and stress
Figure BDA0002850055840000031
Obtaining a dispersion equation:
there are two material boundaries within a cycle:
Figure BDA0002850055840000032
Figure BDA0002850055840000033
at the adjacent cycle boundaries there are:
Figure BDA0002850055840000034
Figure BDA0002850055840000035
obtaining a relation curve of an energy band structure, namely a relation between frequency and wave vector, by using a sufficient condition that the equation system has a non-zero solution:
cos(ka)＝cosh(iq1a1)cosh(iq2a2)+(F+1/F)sinh(iq1a1)sinh(iq2a2)/2 (15)
wherein,
Figure BDA0002850055840000036
at this point, the wave equation is converted into a frequency dispersion equation;
for the transverse wave propagating in the structure, the wave velocity c of the longitudinal wave in the equation is measuredpUsing the wave velocity c of the transverse wavesInstead of calculating the band structure relationship curve of the transverse wave, wherein
Figure BDA0002850055840000037
S6: sweeping the wave vector through a first irreducible Brillouin zone to obtain the relationship between the frequency and the wave vector:
the first irreducible Brillouin zone of the one-dimensional periodic cushion damping bed is a line segment, the end points are respectively 0 and 2 pi/a, m of the first irreducible Brillouin zone can be equally divided when the first irreducible Brillouin zone is swept by wave vectors, and then the first irreducible Brillouin zone is swept to obtain the relation between the frequency and the wave vectors;
s7: drawing a frequency-wave vector relation curve and capturing a forbidden band;
s8: carrying out parametric analysis to obtain the influence rule of each factor on the forbidden band initial frequency and the band width;
s9: according to the target vibration isolation frequency band, the geometric dimension of the one-dimensional periodic cushion damping ballast bed and the value of material parameters are adjusted and designed, so that the purposes of regulating and controlling forbidden bands and vibration isolation frequency bands are achieved.
In step S1, the simplified form, the periodic characteristics, the basic assumption and the boundary conditions of the one-dimensional periodic cushion damping track bed energy band structure calculation model are as follows: (1) the periodic vibration reduction ballast bed is characterized in that a unidirectional infinite periodic structure is formed by infinitely and repeatedly arranging certain basic units along a single direction, has periodicity only in the single direction, and can be simplified into a one-dimensional periodic structure; (2) the one-dimensional periodic cushion layer damping ballast bed extends infinitely along the line direction; (3) the one-dimensional periodic cushion damping ballast bed meets the continuous condition of displacement and stress at a certain medium boundary, and meets the Bloch-Floquet periodic boundary condition on the boundary of a basic unit besides the continuous boundary condition of displacement and stress.
In step S8, when the road bed slab is of a concrete structure and the periodic cushion layer is made of an elastic material, the initial frequency of the first-order complete forbidden band is effectively reduced by increasing the thicknesses of the slab and the periodic cushion layer, reducing the elastic moduli of the slab and the periodic cushion layer, and increasing the impedance ratio of the slab and the periodic cushion layer; the bandwidth of the first-order complete forbidden band is increased by increasing the thickness of the concrete slab, reducing the thickness of the periodic cushion layer, increasing the elastic modulus of the concrete slab and the periodic cushion layer and increasing the impedance ratio of the concrete slab and the periodic cushion layer.
Under the assumption that the one-dimensional periodic cushion damping track bed is of a one-dimensional infinite periodic structure and extends infinitely along the aperiodic direction, a single-period transmission matrix is obtained by starting from a fluctuation equation of continuous state parameters and combining with a section continuous condition, an analytic solution of an energy band structure curve can be obtained by introducing a period boundary condition, and a vibration isolation frequency band is determined according to a forbidden band distribution range. The rule that forbidden bands and vibration isolation frequency bands change along with geometric and material parameters is obtained through analysis of influence factors, and active regulation and control of the forbidden bands and the vibration isolation frequency bands are achieved.
Compared with the prior art, the one-dimensional periodic cushion damping ballast bed energy band structure calculation and vibration isolation frequency band design method provides theoretical basis for design and engineering practice of the periodic damping ballast bed, the method is small in calculation amount, clear in theoretical basis and convenient to use, the calculation result can completely meet the requirement of engineering design, and the method has wide applicability.
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FIG. 1 is a schematic structural view of a one-dimensional periodic cushion damping track bed according to the present invention;
FIG. 2 is a simplified diagram of a forbidden band calculation model of a one-dimensional periodic structure consisting of a concrete slab A and a periodic cushion layer B in the invention;
FIG. 3 is an irreducible Brillouin zone of a one-dimensional periodic cushion damping track bed structure according to the present invention;
fig. 4 is a distribution diagram of forbidden bands in an embodiment of the present invention.
Wherein:
1. concrete base and limiting structure 2, rail bearing platform 3, sleeve 4, elastic cushion layer 5, grouting hole 6, protrusion A, concrete slab B and periodic cushion layer
Detailed Description
To further illustrate the technical means and effects of the present invention adopted to achieve the intended purpose, the following detailed description of the embodiments, calculation methods and steps according to the present invention is provided with the accompanying drawings and the preferred embodiments. In the description that follows, certain features or characteristics of one or more embodiments may be combined in any suitable manner.
One embodiment of a one-dimensional periodic cushion layer damping track bed is shown in fig. 1, the damping track bed is a one-dimensional scattering periodic composite track bed plate structure formed by alternately and repeatedly arranging concrete plates A and periodic cushion layers B along the vertical direction, a concrete base and limiting structure 1 is arranged below the periodic composite track bed plate structure, a track bearing table 2 and grouting holes 5 are arranged on the top surface of the periodic composite track bed plate structure, and a sleeve 3 is arranged on the track bearing table 2.
The horizontal cross-sectional shapes and the sizes of the concrete slabs A and the periodic cushion layers B which are alternately and repeatedly arranged along the vertical direction are consistent, the concrete slabs A and the periodic cushion layers B are rectangular and are aligned up and down, and the concrete slabs A and the periodic cushion layers B are fastened by mortar or are solidified by glue sealing.
Through the mode, the effect that the periodic cushion layer B is fully paved under the concrete slab A is achieved, vibration energy transmitted by the concrete slab A is evenly spread and weakened, and deformation can be coordinated.
As shown, the basic unit consisting of the concrete slabs a and the periodic underlayment B repeats two cycles in the vertical direction, wherein the thickness of the single concrete slab a is 0.26m, the thickness of the single-layered periodic underlayment B is 0.02m, both have the same width of 2.2m, both have the same length of 6 m. Thus, in the simplified diagram of the computational model of FIG. 2, a10.26m and 0.28 m. The concrete slab material parameters are as follows: elastic modulus 30000MPa, Poisson's ratio 0.2, density 2500kg/m3(ii) a The periodic cushion layer B is made of rubber materials, and the material parameters are as follows: elastic modulus 0.12MPa, Poisson's ratio 0.46, density 1300kg/m3。
The vibration isolation frequency band regulation and control design method of the one-dimensional periodic cushion damping ballast bed comprises the following steps of:
s1, determining a simplified form, a periodic characteristic, a basic assumption and a boundary condition of the one-dimensional periodic cushion damping track bed energy band structure calculation model:
the simplified form, the periodic characteristics, the basic assumption and the boundary conditions of the one-dimensional periodic cushion damping track bed energy band structure calculation model are as follows: (1) the periodic vibration reduction ballast bed is a periodic structure formed by repeatedly arranging basic units consisting of concrete plates A and periodic cushion layers B along the vertical direction, has periodicity in a single direction only and can be simplified into a one-dimensional periodic structure; (2) the one-dimensional periodic cushion damping ballast bed extends infinitely along the line direction; (3) the one-dimensional periodic cushion damping ballast bed meets the continuous condition of displacement and stress at a certain medium boundary, and meets the Bloch-Floquet periodic boundary condition on the boundary of a basic unit besides the continuous boundary condition of displacement and stress.
S2, establishing a wave equation of longitudinal waves transmitted along the periodic direction of the one-dimensional periodic cushion damping ballast bed:
Figure BDA0002850055840000051
in the formula: u (x, t) is a displacement vector;
Figure BDA0002850055840000052
is the velocity of longitudinal wave; lambda and mu are Lame constants; rho is density; x is a position vector; t is a time variable.
Considering that both sides of the equation have a time-varying simple harmonic term eiωtAnd ω is the angular frequency, equation (1) can be expressed as follows:
Figure BDA0002850055840000053
the energy band structure along the periodic direction can be obtained by solving the longitudinal wave equation, so that the vibration isolation frequency band in the vertical direction is designed. In the same way, the wave equation of the shear wave can be established to solve the energy band structure in the horizontal direction, and the vibration isolation frequency band design in the horizontal direction can be carried out.
S3, giving a general solution of the state quantity of the elastic wave in a single homogeneous medium:
for the first medium (0) in the basic unit<x<a1) The general solution of the state quantity of the compressional wave in a single homogeneous medium is:
Figure BDA0002850055840000054
in the formula: a. the1 +、A1 -Amplitude in the positive and negative directions, q, respectively, of propagation in the x-direction in the first medium1Is the wave vector:
q1＝ω/cp,1 (4)
in the formula: c. Cp,1Is the compressional wave velocity of the first medium in the basic cell.
S4, converting the fluctuation problem into a characteristic value problem containing an elastic wave propagation dispersion relation according to the Bloch-Floquet theorem due to the periodicity of the one-dimensional periodic cushion damping track bed in the x direction:
the solution of u (x) can be written in the form of a Bloch wave:
u(x)＝eikxuk(x) (5)
in the formula:
uk(x)＝uk(x+a) (6)
wherein a ═ a1+a2Is a period constant.
From the formulae (3) and (5)
Figure BDA0002850055840000061
In view of the periodic boundary condition (6), equation (7) may be changed into the following form:
Figure BDA0002850055840000062
wherein: na<x<na+a1
The compound represented by formula (5) is obtained by bringing formula (8):
when na<x<na+a1Time of flight
Figure BDA0002850055840000063
When na + a1<x<na+a1+a2Time of flight
Figure BDA0002850055840000064
In the formula: q. q.s2＝ω/cp,2Is the wave vector.
Up to now, the problem of fluctuation has been converted into a problem of eigenvalue including a dispersion relation of propagation of elastic waves.
S5, introducing continuous conditions of stress and displacement at continuous boundary, and determining undetermined coefficient
Figure BDA0002850055840000065
Obtaining a frequency dispersion equation:
there are two material boundaries within a cycle:
Figure BDA0002850055840000066
Figure BDA0002850055840000067
at the adjacent cycle boundaries there are:
Figure BDA0002850055840000068
Figure BDA0002850055840000069
the relation curve of the energy band structure, namely the relation system of the frequency and the wave vector can be obtained through the sufficient condition that the equation system has non-zero solution:
cos(ka)＝cosh(iq1a1)cosh(iq2a2)+(F+1/F)sinh(iq1a1)sinh(iq2a2)/2 (15)
wherein,
Figure BDA0002850055840000071
to this end, the wave equation has been converted to a dispersion equation. For the transverse wave propagating in the structure, the wave speed c of the longitudinal wave in the equation is calculatedpUsing the wave velocity c of the transverse wavesInstead, a band structure curve of the transverse wave can likewise be calculated, wherein
Figure BDA0002850055840000072
S6, scanning the wave vector in the first irreducible Brillouin area to obtain the relationship between the frequency and the wave vector:
the first irreducible Brillouin zone of the one-dimensional periodic cushion damping bed is a line segment, the end points are respectively 0 and 2 pi/a, when the first irreducible Brillouin zone is swept by a wave vector, the first irreducible Brillouin zone can be divided into 10 equal parts, and then the sweep is carried out to obtain the relation between the frequency and the wave vector.
S7, drawing a frequency-wave vector relation curve, capturing a forbidden band:
and (5) drawing a frequency-wavevector relation curve, namely a dispersion curve. Each curve in the dispersion curve represents a vibration mode corresponding to the dispersion curve, and if any curve does not exist under certain frequencies, a blank band is formed, and the frequency band is a forbidden band of the periodic structure. Generally, the vibration within the forbidden band of the infinite periodic structure cannot propagate through the infinite periodic structure, which is represented by the large attenuation of the frequency band vibration in the finite periodic structure. In general, the lowest band is referred to as the first-order band.
Fig. 4 shows a one-dimensional periodic cushion damping track bed frequency dispersion curve obtained by calculation according to the method, wherein the first-order complete forbidden band of the one-dimensional periodic cushion damping track bed frequency dispersion curve appears in a range of 58-140 Hz, the range is within a range of 0-200 Hz of interest of rail transit environment vibration, and the common significant frequency 63Hz is covered, so that a good damping effect can be generated in the frequency band.
S8, carrying out parametric analysis to obtain the influence rule of each factor on the forbidden band initial frequency and the band width:
when the road bed plate is of a concrete structure and the periodic cushion layer is made of elastic materials, the thicknesses of the concrete slab and the periodic cushion layer are increased, the elastic moduli of the concrete slab and the periodic cushion layer are reduced, and the impedance ratio of the concrete slab and the periodic cushion layer is increased, so that the initial frequency of a first-order complete forbidden band can be effectively reduced, and a better low-frequency vibration isolation effect is generated; the thickness of the concrete slab is increased, the thickness of the periodic cushion layer is reduced, the elastic modulus of the concrete slab and the periodic cushion layer is increased, and the impedance ratio of the concrete slab and the periodic cushion layer is increased, so that the bandwidth of a first-order complete forbidden band can be increased, and the vibration isolation effect of a wider frequency band is generated.
S9, designing the geometric dimension and material parameter values of the one-dimensional periodic cushion damping ballast bed according to the target vibration isolation frequency band:
according to the parametric analysis result and the obtained influence rule of each factor on the forbidden band initial frequency and the bandwidth, the geometric dimension of the damping track bed of the one-dimensional periodic cushion layer and the value of the material parameter are adjusted and designed according to the target vibration isolation frequency band, so that the purpose of regulating and controlling the forbidden band and the vibration isolation frequency band is achieved.
In the embodiment shown in fig. 1, a concrete foundation and spacing structure 1 is formed with protrusions 6, a periodic composite track bed plate structure formed by alternately and repeatedly arranging concrete slabs a and periodic bed layers B in the vertical direction is formed with corresponding through holes, the protrusions 6 are inserted into the through holes, and an elastic bed layer 4 is arranged between the protrusions and the through holes. The position of the periodic composite roadbed slab structure and the position of the concrete basement and limiting structure are relatively fixed through the structure.
Of course, the fixing manner of the position of the periodic composite track bed plate structure and the concrete foundation and limiting structure is not limited to the structure in this embodiment, and different limiting structures have no influence on the design method of the invention.
The one-dimensional periodic cushion damping ballast bed in the embodiment is a periodic cushion full-paved working condition, but the ballast bed to which the energy band structure vibration isolation frequency band design method of the invention is applicable is not limited thereto, and the method is also applicable to other paving conditions, such as one-dimensional periodic strip pavement and the like, that is, the design method of the invention is applicable to any one-dimensional periodic cushion damping ballast bed.
The present embodiment is only for explaining the present invention, and the dimension and material setting of the one-dimensional periodic cushion damping ballast bed are various and not limited to the present embodiment, and those skilled in the art can make modifications without creative contribution to the present embodiment as required after reading the present specification, as long as they are protected by the patent laws within the scope of the claims of the present invention.

Claims (3)

1. A vibration isolation frequency band regulation and control design method for a one-dimensional periodic cushion damping ballast bed is characterized by comprising the following steps:
s1: determining a simplified form, a periodic characteristic, a basic assumption and a boundary condition of a one-dimensional periodic cushion damping track bed energy band structure calculation model;
s2: establishing a wave equation of longitudinal waves transmitted by the one-dimensional periodic cushion damping ballast bed along the periodic direction:
Figure FDA0002850055830000011
in the formula: u (x, t) is a displacement vector;
Figure FDA0002850055830000012
is the velocity of longitudinal wave; lambda and mu are Lame constants; rho is density; x is a position vector; t is a variable of time and t is,
considering that both sides of the equation have a time-varying simple harmonic term eiωtAnd ω is the angular frequency, equation (1) is reduced to the form:
Figure FDA0002850055830000013
solving the equation to obtain an energy band structure along the periodic direction;
s3: giving a general solution to the state quantity of an elastic wave in a single homogeneous medium:
for the first medium (0) in the basic unit<x<a1) The general solution of the state quantity of the compressional wave in a single homogeneous medium is:
Figure FDA0002850055830000014
in the formula: a. the1 +、A1 -Amplitude in the positive and negative directions, q, respectively, of propagation in the x-direction in the first medium1Is the wave vector:
q1＝ω/cp,1 (4)
in the formula: c. Cp,1Is the compressional wave velocity of the first medium in the basic unit;
s4: converting the fluctuation problem into a characteristic value problem containing an elastic wave propagation dispersion relation by using a Bloch theorem:
due to the periodicity of the one-dimensional periodic cushion damping track bed in the x direction, the solution of u (x) is written into a form of Bloch waves according to the Bloch-Floquet theorem:
u(x)＝eikxuk(x) (5)
in the formula:
uk(x)＝uk(x+a) (6)
wherein a ═ a1+a2Is a period constant;
obtained by the formulae (3) and (5):
Figure FDA0002850055830000015
in consideration of the periodic boundary condition (6), equation (7) is changed to the following form:
Figure FDA0002850055830000016
wherein: na<x<na+a1，
Bringing formula (5) into formula (8) gives:
when na<x<na+a1When the temperature of the water is higher than the set temperature,
Figure FDA0002850055830000021
when na + a1<x<na+a1+a2When the temperature of the water is higher than the set temperature,
Figure FDA0002850055830000022
in the formula: q. q.s2＝ω/cp,2Is the wave vector;
s5: determining undetermined coefficient by applying continuous boundary conditions of displacement and stress
Figure FDA0002850055830000023
Obtaining a dispersion equation:
there are two material boundaries within a cycle:
Figure FDA0002850055830000024
Figure FDA0002850055830000025
at the adjacent cycle boundaries there are:
Figure FDA0002850055830000026
Figure FDA0002850055830000027
obtaining a relation curve of an energy band structure, namely a relation between frequency and wave vector, by using a sufficient condition that the equation system has a non-zero solution:
cos(ka)＝cosh(iq1a1)cosh(iq2a2)+(F+1/F)sinh(iq1a1)sinh(iq2a2)/2 (15)
wherein,
Figure FDA0002850055830000028
at this point, the wave equation is converted into a frequency dispersion equation;
for the transverse wave propagating in the structure, the wave velocity c of the longitudinal wave in the equation is measuredpUsing the wave velocity c of the transverse wavesInstead of calculating the band structure relationship curve of the transverse wave, wherein
Figure FDA0002850055830000029
S6: sweeping the wave vector through a first irreducible Brillouin zone to obtain the relationship between the frequency and the wave vector:
the first irreducible Brillouin zone of the one-dimensional periodic cushion damping bed is a line segment, the end points are respectively 0 and 2 pi/a, m of the first irreducible Brillouin zone can be equally divided when the first irreducible Brillouin zone is swept by wave vectors, and then the first irreducible Brillouin zone is swept to obtain the relation between the frequency and the wave vectors;
s7: drawing a frequency-wave vector relation curve and capturing a forbidden band;
s8: carrying out parametric analysis to obtain the influence rule of each factor on the forbidden band initial frequency and the band width;
s9: according to the target vibration isolation frequency band, the geometric dimension of the one-dimensional periodic cushion damping ballast bed and the value of material parameters are adjusted and designed, so that the purposes of regulating and controlling forbidden bands and vibration isolation frequency bands are achieved.
2. The vibration isolation frequency band regulation and control design method of the one-dimensional periodic cushion damping ballast bed according to claim 1, which is characterized in that:
in step S1, the simplified form, the periodic characteristics, the basic assumption and the boundary conditions of the one-dimensional periodic cushion damping track bed energy band structure calculation model are as follows: (1) the periodic vibration reduction ballast bed is characterized in that a unidirectional infinite periodic structure is formed by infinitely and repeatedly arranging certain basic units along a single direction, has periodicity only in the single direction, and can be simplified into a one-dimensional periodic structure; (2) the one-dimensional periodic cushion damping ballast bed extends infinitely along the line direction; (3) the one-dimensional periodic cushion damping ballast bed meets the continuous condition of displacement and stress at a certain medium boundary, and meets the Bloch-Floquet periodic boundary condition on the boundary of a basic unit besides the continuous boundary condition of displacement and stress.
3. The vibration isolation frequency band regulation and control design method of the one-dimensional periodic cushion damping ballast bed according to claim 1, which is characterized in that: in step S8, when the road bed slab is of a concrete structure and the periodic cushion layer is made of an elastic material, the initial frequency of the first-order complete forbidden band is effectively reduced by increasing the thicknesses of the slab and the periodic cushion layer, reducing the elastic moduli of the slab and the periodic cushion layer, and increasing the impedance ratio of the slab and the periodic cushion layer; the bandwidth of the first-order complete forbidden band is increased by increasing the thickness of the concrete slab, reducing the thickness of the periodic cushion layer, increasing the elastic modulus of the concrete slab and the periodic cushion layer and increasing the impedance ratio of the concrete slab and the periodic cushion layer.
CN202011533103.6A 2020-12-22 2020-12-22 Vibration isolation frequency band regulation and control design method for one-dimensional periodic cushion layer vibration reduction ballast bed Pending CN112580210A (en)

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