CN115130347B - Acoustic-vibration response calculation method of constrained damping structure considering frequency-dependent characteristics - Google Patents
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Abstract
The invention discloses a method for calculating sound-vibration response of a constrained damping structure in consideration of frequency-dependent characteristics, which comprises the following steps of: s1: acquiring the initial natural vibration frequency and the natural vibration mode of the structure without laying the constrained damping layer; s2: establishing a finite element model of a constrained damping structure, and performing iterative modal analysis; s3: taking the self-vibration mode of the constrained damping structure in the analysis frequency band as a space substrate, and calculating the vibration response of the constrained damping structure based on a modal superposition method; s4: and calculating the acoustic response of the constraint damping structure by taking the vibration response of the constraint damping structure as an acoustic boundary condition. The invention can provide technical support for dynamic response analysis of the constrained damping structure and parameter optimization design of the constrained damping structure in various fields.
Description
Technical Field
The invention belongs to the technical field of vibration reduction and noise reduction of rail transit bridges, and particularly relates to a sound-vibration response calculation method of a constrained damping structure considering frequency-variable characteristics.
Background
The constrained damping layer is widely applied to the fields of aerospace, ships and submarines, rail transit and the like with excellent wide-spectrum vibration suppression characteristics. However, most studies on constrained damping structures consider damping layer materials as constants whose material parameters do not vary with the loading frequency. In fact, the material of the damping core has visco-elastic properties, the loss factor eta of which v And shear modulus G d The shear modulus and loss factor of the damping layer material will in turn affect the vibration response of the constrained damping structure, depending on the frequency of the applied load. Therefore, when analyzing the acoustic-vibration response of the constrained damping structure, how to consider the parameter frequency-dependent characteristics of the viscoelastic material of the damping layer is a problem to be solved by those skilled in the art.
Disclosure of Invention
In order to solve the problems, the invention provides a method for calculating the sound-vibration response of a constrained damping structure considering frequency-dependent characteristics.
The technical scheme of the invention is as follows: a sound-vibration response calculation method of a constrained damping structure considering frequency-dependent characteristics comprises the following steps:
s1: acquiring the initial natural vibration frequency and the natural vibration mode of the structure without laying the constrained damping layer;
s2: establishing a finite element model of a constraint damping structure, taking material parameters corresponding to the initial natural vibration frequency of the structure as iteration initial conditions, performing iteration modal analysis, and entering step S3 after the iteration modal analysis is completed;
s3: the self-vibration mode of the constrained damping structure in the analysis frequency band obtained after iterative modal analysis is used as a space substrate, and the vibration response of the constrained damping structure is calculated based on a modal superposition method;
s4: and calculating the acoustic response of the constraint damping structure by taking the vibration response of the constraint damping structure as an acoustic boundary condition.
Further, in step S1, a specific method for obtaining the initial natural vibration frequency and the natural vibration mode of the structure on which the constrained damping layer is not laid is as follows: establishing a structural finite element model without laying a constrained damping layer, and carrying out modal analysis on the structural finite element model to obtain an initial natural vibration frequency omega m,0 And self-vibration mode [ phi ] m,0 And m is the order of the mode shape.
Further, in step S2, a finite element model of the constrained damping structure is established, and the initial natural vibration frequency ω of the structure without the constrained damping layer is set m,0 Corresponding damping layer shear modulus G d (ω m,0 ) And material loss factor eta v (ω m,0 ) Performing iterative modal analysis as an iterative initial condition until the relative error | G of the shear modulus of the viscoelastic damping material obtained by two adjacent modal analyses d (ω m,n+1 )-G d (ω m,n )|/G d (ω m,n ) Or relative error | η of loss factor v (ω m,n+1 )-η v (ω m,n )|/η v (ω m,n ) And if the amplitude is smaller than the set value, performing next-order iterative modal analysis until all order self-vibration modes of the constrained damping structure in the analysis frequency band are obtained, and performing step S3.
Further, in step S2, the eigenvalue equation obtained by performing iterative modal analysis is:
wherein [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]Being the imaginary part of the stiffness matrix of the damping layer,
for constraining the self-vibration mode of the damping structure, m is the order of the mode, n is the iteration number, and omega is m,n Calculating the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the control conditions of the mth order iterative modal analysis are as follows:
wherein epsilon G Is the relative error in shear modulus of the viscoelastic damping material,
is the relative error in the dissipation factor of the viscoelastic damping material,
is an iterative control condition corresponding to the shear modulus of the viscoelastic damping material,
for iterative control conditions corresponding to the dissipation factor of the viscoelastic damping material, G d (ω m,n ) Calculated value of shear modulus, G, of viscoelastic damping material after nth iteration d (ω m,n+1 ) Is the calculated value of the shear modulus, eta, of the viscoelastic damping material after the n +1 iteration v (ω m,n ) Is a calculated value of the loss factor, eta, of the viscoelastic damping material after the nth iteration v (ω m,n+1 ) And calculating the loss factor of the viscoelastic damping material after the (n + 1) th iteration.
Further, in step S3, a specific method for calculating the vibration response of the constrained damping structure is as follows: and establishing a vibration differential equation of the constraint damping structure, solving generalized coordinates corresponding to all orders of self-vibration modes according to the vibration differential equation of the constraint damping structure, and calculating the vibration response of the constraint damping structure according to the generalized coordinates corresponding to all orders of self-vibration modes.
Further, the expression of the vibration differential equation of the constrained damping structure is:
where i is the imaginary unit, { F } is the extrinsic payload vector, [ mu ] m]For the mass matrix of the constrained damping structure, N is the number of the vibration modes intercepted in the analysis frequency band,
mth order vibration mode, x of constrained damping structure m Is a generalized coordinate, [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]As the imaginary part, ω, of the stiffness matrix of the damping layer m,n Calculating a value of the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the formula for the vibration response { u } of the constrained damping structure is:
further, in step S4, a specific method for calculating the acoustic response of the constrained damping structure is as follows: and extracting the boundary grid of the constraint damping structure by using acoustic calculation software, introducing the vibration response { u } of the constraint damping structure as an acoustic boundary condition, and calculating the acoustic response of the constraint damping structure by using the acoustic calculation software.
The invention has the beneficial effects that: the invention provides a sound-vibration response calculation method of a constrained damping structure considering frequency-dependent characteristics, which solves the problem that parameters of a viscoelastic damping material change along with frequency in dynamic analysis of the constrained damping structure through an iterative algorithm. The invention can provide technical support for dynamic response analysis of the constrained damping structure and parameter optimization design of the constrained damping structure in various fields.
Drawings
FIG. 1 is a schematic flow chart of an acousto-vibration response calculation method of a constrained damping structure considering frequency-dependent characteristics according to the present invention;
FIG. 2 is a schematic diagram of the dimension of an I-beam and the layout of a constrained damping layer according to an embodiment of the present invention;
FIG. 3 is a graph of the shear modulus and loss factor of a viscoelastic damping material of the present invention as a function of frequency.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings.
Before describing specific embodiments of the present invention, in order to make the solution of the present invention more clear and complete, the definitions of abbreviations and key terms appearing in the present invention will be explained:
finite element model: the model established when the finite element analysis method is applied is a group of unit combinations which are only connected at nodes, only pass force by the nodes and are only restrained at the nodes.
And (3) modal analysis: a method for researching the dynamic characteristics of a structure is generally applied to the field of engineering vibration. The modes refer to the natural vibration characteristics of the mechanical structure, and each mode has a specific natural frequency, a specific damping ratio and a specific mode shape. The process of analyzing these modal parameters is called modal analysis.
A modal superposition method: a method of calculating the dynamic response of a structure to transient or steady state harmonic excitation using the natural frequencies and mode shapes in a modal analysis.
As shown in fig. 1, the present invention provides a method for calculating an acousto-acoustic response of a constrained damping structure considering frequency-dependent characteristics, comprising the following steps:
s1: acquiring the initial natural vibration frequency and the natural vibration mode of the structure without laying the constrained damping layer;
s2: establishing a finite element model of a constraint damping structure, taking material parameters corresponding to the initial natural vibration frequency of the structure as iteration initial conditions, performing iteration modal analysis, and entering step S3 after the iteration modal analysis is completed;
s3: calculating the vibration response of the constrained damping structure based on a modal superposition method by taking the self-vibration mode of the constrained damping structure in the analysis frequency band obtained after the iterative modal analysis as a space substrate;
s4: and calculating the acoustic response of the constraint damping structure by taking the vibration response of the constraint damping structure as an acoustic boundary condition.
In the embodiment of the present invention, in step S1, a specific method for obtaining the initial natural vibration frequency and the natural vibration mode of the structure on which the constrained damping layer is not laid is as follows: establishing a structural finite element model without laying a constrained damping layer, and carrying out modal analysis on the structural finite element model to obtain an initial natural vibration frequency omega m,0 And self-vibration mode [ phi ] m,0 And m is the order of the mode shape.
In the embodiment of the invention, in the step S2, a finite element model of the constraint damping structure is established, and the initial natural vibration frequency omega of the structure without the constraint damping layer is set m,0 Corresponding damping layer shear modulus G d (ω m,0 ) And material loss factor eta v (ω m,0 ) Performing iterative modal analysis as an iterative initial condition until the relative error | G of the shear modulus of the viscoelastic damping material obtained by two adjacent modal analyses d (ω m,n+1 )-G d (ω m,n )|/G d (ω m,n ) Or relative error | η of loss factor v (ω m,n+1 )-η v (ω m,n )|/η v (ω m,n ) And if the amplitude is smaller than the set value, performing next-order iterative modal analysis until all order self-vibration modes of the constrained damping structure in the analysis frequency band are obtained, and performing step S3.
In the embodiment of the present invention, in step S2, an eigenvalue equation obtained by performing iterative modal analysis is:
wherein i is an imaginary unit, [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]Being the imaginary part of the stiffness matrix of the damping layer,
for constraining the self-vibration mode of the damping structure, m is the order of the mode, n is the iteration number, and omega is m,n Calculating the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the control conditions of the mth order iterative modal analysis are as follows:
wherein epsilon G Is the relative error in shear modulus of the viscoelastic damping material,
is the relative error in the dissipation factor of the viscoelastic damping material,
is an iterative control condition corresponding to the shear modulus of the viscoelastic damping material,
for iterative control conditions corresponding to the dissipation factor of the viscoelastic damping material, G d (ω m,n ) Is the calculated value of the shear modulus of the viscoelastic damping material after the nth iteration, G d (ω m,n+1 ) Is the calculated value of the shear modulus of the viscoelastic damping material after the (n + 1) th iteration, eta v (ω m,n ) Is a calculated value of the loss factor, eta, of the viscoelastic damping material after the nth iteration v (ω m,n+1 ) The calculated value of the loss factor of the viscoelastic damping material after the (n + 1) th iteration is obtained.
In the embodiment of the present invention, in step S3, a specific method for calculating the vibration response of the constrained damping structure is as follows: and establishing a vibration differential equation of the constraint damping structure, solving generalized coordinates corresponding to all orders of self-vibration modes according to the vibration differential equation of the constraint damping structure, and calculating the vibration response of the constraint damping structure according to the generalized coordinates corresponding to all orders of self-vibration modes.
In the embodiment of the invention, the expression of the vibration differential equation of the constraint damping structure is as follows:
where i is the imaginary unit, { F } is the extrinsic payload vector, [ mu ] m]For the mass matrix of the constrained damping structure, N is the number of the vibration modes intercepted in the analysis frequency band,
mth order vibration mode, x of constrained damping structure m As a generalized coordinate, [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]As the imaginary part, ω, of the stiffness matrix of the damping layer m,n Calculating a value of the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the transposition of the m-th order mode vector left-times at both ends of the above equation can result in:
according to the orthogonality of each order vibration mode of the constraint damping structure with respect to the mass matrix and the rigidity matrix:
the vibration differential equation of the constrained damping structure can be simplified into a plurality of single-degree-of-freedom vibration differential equations:
defining the mth-order generalized mass, the generalized stiffness coefficient and the generalized load as follows:
then the above formula can be written as:
solving the above formula can obtain the generalized coordinate x corresponding to the mth order vibration mode m . Similarly, generalized coordinates x corresponding to other order modes l (l is more than or equal to 1 and less than or equal to N) can be solved according to the method. After the solution of the generalized coordinates is completed, the vibration response of the constrained damping structure can be expressed as:
in the embodiment of the present invention, in step S4, a specific method for calculating the acoustic response of the constrained damping structure is as follows: and extracting the boundary grid of the constraint damping structure by using acoustic calculation software, introducing the vibration response { u } of the constraint damping structure as an acoustic boundary condition, and calculating the acoustic response of the constraint damping structure by using the acoustic calculation software. The acoustic computation software may use LMS Virtual lab (LMS VL) software.
The present invention will be described with reference to specific examples.
In the present embodiment, as shown in fig. 2, a Q355B i-beam having a height of 0.5m and a length of 2.5m is taken as an example, and the i-beam in the present embodiment has a length of 2.5m and a height of 0.5m, a wing plate thickness of 12mm, and a web plate thickness of 8mm, and is fixed to the support base by 8 bolts. The I-beam material is Q355B steel, the elastic modulus is 206GPa, the Poisson ratio is 0.3, and the density is 7850kg/m 3 . The material of the constraint layer is aluminum, the elastic modulus is 70GPa, the Poisson ratio is 0.3, and the density is 2700kg/m 3 . The viscoelastic material has a density of 1500kg/m 3 Poisson's ratio of 0.495, shear modulus and lossSee figure 3 for factors.
(1) With G s 、G c And G d Respectively representing the shear modulus of the structural layer, the constraint layer and the damping layer, and the analysis frequency band of the acoustic vibration response is 20-2000 Hz. And setting the I-beam without the constraint damping layer as an initial structure, establishing a finite element model of the I-beam, and carrying out modal analysis on the I-beam based on the bolt fixing boundary condition to obtain all-order natural vibration frequency and the natural vibration mode of the I-beam in the analysis frequency band.
(2) The iterative process is explained by taking the solution of the mth order natural vibration frequency of the constrained damping I-beam as an example. Setting the mth order initial natural vibration frequency of the I-beam to be omega m,0 Determining the shear modulus G of the viscoelastic material according to the relation curve of the parameters of the viscoelastic material and the frequency shown in FIG. 3 d (ω m,0 ) And loss factor eta v (ω m,0 ). And then, establishing an I-beam finite element model for laying the constrained damping layer, and endowing each layer with corresponding material parameters. Carrying out modal analysis on the constrained damping I-beam to obtain the mth order natural vibration frequency omega m,1 And calculating the corresponding viscoelastic material G of the frequency d (ω m,1 ) And loss factor eta v (ω m,1 )。
(3) If the calculation result in the step (2) does not meet the control condition of iteration: | G d (ω m,1 )-G d (ω m,0 )|/G d (ω m,0 )<10 -3 Or | η v (ω m,1 )-η v (ω m,0 )|/η v (ω i,0 )<10 -3 Updating the viscoelastic material parameter G d (ω m,2 ) And eta v (ω m,2 ) And carrying out modal analysis again until the control condition is met: | G d (ω m,n+1 )-G d (ω m,n )|/G d (ω m,n )<10 -3 Or | η v (ω m,n+1 )-η v (ω m,n )|/η v (ω m,n )<10 -3 。
The above process can be realized by circularly calling a finite element program to solve the kernel through an external program, and can also be realized by directly programming a circular command stream in finite element software. The other self-vibration frequencies of all orders of the constraint damping I-beam can be solved according to the same method.
(4) The m-th order natural vibration frequency omega of the I-beam is subjected to constrained damping m And self-vibration mode
After the solution, a mode superposition method is adopted to develop the self-vibration mode only considering the m-th order
And calculating the contribution of the mode shape to the total vibration response
(5) Superposing the contribution of each order of vibration mode in the step (4) to the vibration response to obtain the total vibration response of the constraint damping I-beam in the analysis frequency band:
and outputting a finite element grid of the constraint damping structure and a vibration response result file through an output interface of the finite element software, and importing the finite element grid and the vibration response result file into an LMS Virtual Lab (LMS VL) of acoustic calculation software. In the LMS VL, a boundary grid of the constraint damping I-beam is extracted, the vibration response of the constraint damping I-beam is used as an acoustic boundary condition, and the acoustic boundary element solving module in the LMS VL is used for calculating the acoustic response of the constraint damping I-beam.
The working principle and the process of the invention are as follows: firstly, determining the initial natural vibration frequency and the natural vibration mode of the structure under the working condition of not laying a constrained damping layer through modal analysis in a calculation analysis frequency band of sound-vibration response; then, taking the viscoelastic material parameter corresponding to the initial natural vibration frequency of the structure as an initial condition, and performing cyclic modal analysis on the structure by adopting an iteration method until the relative error of the shear modulus or the loss factor of the damping layer material obtained by two adjacent calculations is smaller than a given control value; after the iteration process is finished, calculating the vibration response of the constraint damping structure by using each order of self-vibration modes of the constraint damping structure obtained by calculation as a space substrate and adopting a modal superposition method; and finally, extracting a boundary grid of the constraint damping structure, taking the vibration response of the constraint damping structure as a boundary condition, and calculating the acoustic response of the constraint damping structure by adopting a boundary element method.
The beneficial effects of the invention are as follows: the invention provides a frequency-dependent characteristic-considered sound-vibration response calculation method of a constraint damping structure, which solves the problem that parameters of a viscoelastic damping material change along with frequency in dynamic analysis of the constraint damping structure through an iterative algorithm. The invention can provide technical support for dynamic response analysis of the constrained damping structure and parameter optimization design of the constrained damping structure in various fields.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.
Claims (5)
1. A method for calculating the sound-vibration response of a constrained damping structure considering frequency-dependent characteristics is characterized by comprising the following steps of:
s1: acquiring the initial natural vibration frequency and the natural vibration mode of the structure without laying the constrained damping layer;
s2: establishing a finite element model of a constrained damping structure, taking material parameters corresponding to the initial natural vibration frequency of the structure as iteration initial conditions, performing iteration modal analysis, and entering step S3 after the iteration modal analysis is completed;
s3: the self-vibration mode of the constrained damping structure in the analysis frequency band obtained after iterative modal analysis is used as a space substrate, and the vibration response of the constrained damping structure is calculated based on a modal superposition method;
s4: calculating the acoustic response of the constraint damping structure by taking the vibration response of the constraint damping structure as an acoustic boundary condition;
in the step S2, a constraint damping structure is establishedFinite element model, and initial natural vibration frequency omega of the structure without laying the constraint damping layer m,0 Corresponding damping layer shear modulus G d (ω m,0 ) And material loss factor eta v (ω m,0 ) Performing iterative modal analysis as an iterative initial condition until the relative error | G of the shear modulus of the viscoelastic damping material obtained by two adjacent modal analyses d (ω m,n+1 )-G d (ω m,n )|/G d (ω m,n ) Or relative error | η of loss factor v (ω m,n+1 )-η v (ω m,n )|/η v (ω m,n ) If the amplitude is smaller than the set value, performing next-order iterative modal analysis until all order self-vibration modes of the constrained damping structure in the analysis frequency band are obtained, and performing step S3;
in step S2, the eigenvalue equation obtained by performing iterative modal analysis is:
wherein i is an imaginary unit, [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]Being the imaginary part of the stiffness matrix of the damping layer,
for constraining the self-vibration mode of the damping structure, m is the order of the mode, n is the iteration number, omega m,n Calculating a value of the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the control conditions of the mth order iterative modal analysis are as follows:
wherein epsilon G Is the relative error in shear modulus of the viscoelastic damping material,
is the relative error in the dissipation factor of the viscoelastic damping material,
is an iterative control condition corresponding to the shear modulus of the viscoelastic damping material,
for iterative control conditions corresponding to the dissipation factor of the viscoelastic damping material, G d (ω m,n ) Calculated value of shear modulus, G, of viscoelastic damping material after nth iteration d (ω m,n+1 ) Is the calculated value of the shear modulus, eta, of the viscoelastic damping material after the n +1 iteration v (ω m,n ) Is a calculated value of the loss factor, eta, of the viscoelastic damping material after the nth iteration v (ω m,n+1 ) The calculated value of the loss factor of the viscoelastic damping material after the (n + 1) th iteration is obtained.
2. The method for calculating the acoustic-vibration response of the constrained damping structure considering the frequency-dependent characteristics according to claim 1, wherein in the step S1, the specific method for acquiring the initial natural vibration frequency and the natural vibration mode of the structure without the constrained damping layer is as follows: establishing a structural finite element model without laying a constrained damping layer, and carrying out modal analysis on the structural finite element model to obtain an initial natural vibration frequency omega m,0 And self-vibration mode [ phi ] m,0 And m is the order of the mode shape.
3. The method for calculating the acousto-vibration response of the constrained damping structure considering the frequency-dependent characteristics according to claim 1, wherein in the step S3, the specific method for calculating the vibration response of the constrained damping structure is as follows: and establishing a vibration differential equation of the constraint damping structure, solving generalized coordinates corresponding to all orders of self-vibration modes according to the vibration differential equation of the constraint damping structure, and calculating the vibration response of the constraint damping structure according to the generalized coordinates corresponding to all orders of self-vibration modes.
4. The acoustic-vibration response calculation method of a constrained damping structure considering frequency-varying characteristics according to claim 3, wherein the expression of the vibration differential equation of the constrained damping structure is:
where i is the imaginary unit, { F } is the extrinsic payload vector, [ mu ] m]For constraining the mass matrix of the damping structure, N is the number of the intercepted vibration modes in the analysis frequency band,
mth order vibration mode, x of constrained damping structure m Is a generalized coordinate, [ K ] e ]Is a stiffness matrix of the elastic layer, [ K vR (ω m,n )]Is the real part of the damping layer stiffness matrix, [ K ] vI (ω m,n )]As the imaginary part, ω, of the stiffness matrix of the damping layer m,n Calculating a value of the mth order natural vibration frequency of the constrained damping structure after the nth iteration;
the calculation formula of the vibration response { u } of the constraint damping structure is as follows:
5. the method for calculating the acousto-vibration response of the constrained damping structure with the frequency-dependent characteristic taken into consideration according to claim 1, wherein in the step S4, the concrete method for calculating the acoustic response of the constrained damping structure is as follows: and extracting the boundary grid of the constraint damping structure by using acoustic calculation software, introducing the vibration response { u } of the constraint damping structure as an acoustic boundary condition, and calculating the acoustic response of the constraint damping structure by using the acoustic calculation software.
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