CN112818494A - Functional gradient flow pipe modal and response analysis method based on differential quadrature method - Google Patents
Functional gradient flow pipe modal and response analysis method based on differential quadrature method Download PDFInfo
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Abstract
The invention discloses a functional gradient flow pipe mode and response analysis method based on a differential integration method, which comprises the steps of firstly establishing a functional gradient flow pipe vibration control differential equation, and then carrying out discrete format construction on the functional gradient flow pipe vibration control differential equation by applying the idea of the differential integration method to obtain a vibration equation and a discrete structure of boundary conditions; then, analyzing the natural frequency and solving the array type to further obtain the natural vibration mode of the continuous system; and finally, solving a control equation obtained by dispersing by using a differential product-solving method in a time domain by using a time domain analysis method and a numerical iteration method to obtain the dynamic response of the system. The invention applies the differential quadrature method to the dynamic response solving and calculating method of the functional gradient flow transmission pipeline, and can solve the problem of analyzing the inherent property, the inherent vibration mode and the dynamic response of the pipeline quickly and efficiently.
Description
Technical Field
The invention belongs to the technical field of fluid dynamics, and particularly relates to a method for analyzing inherent properties and dynamic response of a pipeline.
Background
The research on the dynamics mechanism of the fluid transmission pipeline is always the focus of academia, and the numerical method and means for analyzing the dynamics model are also the focus of attention of people all the time. Firstly, a motion equation or a control equation of the system needs to be obtained, the solution can be carried out after the motion equation is derived, and various numerical methods provided by the academic community mainly comprise a Galerkin method, a finite element method, a transfer matrix method, a differential product method and the like. The Galerkin method adopts a weak form corresponding to a differential equation, discretizes a high-order partial differential equation and reduces the order of the high-order partial differential equation into a low-order ordinary differential equation set, but the Galerkin method for analyzing the nonlinear model of the flow transmission pipeline needs to adopt 4-order or more modal truncation, so that enough precision can be ensured; the finite element method has enough high precision but slow operation speed; the transfer matrix method uses a matrix to describe the relationship between output and input in a multi-input multi-output linear system, is suitable for the calculation of a chain structure, but has a complicated calculation process; the differential integration method uses the function values of the selected nodes to perform weighted summation to represent the function values and derivatives of all the functions in the universe, and further performs the description of the function values and the derivatives of each point by representing the weight coefficients through reasonable trial functions.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a functional gradient flow pipe modal and response analysis method based on a differential integration method, firstly, a functional gradient flow pipe vibration control differential equation is established, then, the idea of the differential integration method is applied to carry out discrete format construction on the functional gradient flow pipe vibration control differential equation, and a discrete structure of a vibration equation and a boundary condition is obtained; then, analyzing the natural frequency and solving the array type to further obtain the natural vibration mode of the continuous system; and finally, solving a control equation obtained by dispersing by using a differential product-solving method in a time domain by using a time domain analysis method and a numerical iteration method to obtain the dynamic response of the system. The invention applies the differential quadrature method to the dynamic response solving and calculating method of the functional gradient flow transmission pipeline, and can solve the problem of analyzing the inherent property, the inherent vibration mode and the dynamic response of the pipeline quickly and efficiently.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: establishing a differential equation for controlling vibration of the functionally gradient fluid delivery pipeline with two fixed branches at two ends;
the outer diameter of the functional gradient flow transmission pipeline is d, the wall thickness is h, the length is L, the flow velocity of internal fluid is u, the axial direction of the pipeline is taken as an x axis, and the radial direction of the section of the pipeline is taken as a z axis;
selecting a volume fraction function as an exponential function form V1By dimensionless quantities
Expressed as:
wherein a represents a parameter for adjusting the distribution of the exponential type body integral number function;
the functional gradient flow transmission pipeline consists of two materials, namely a material 1 and a material 2; volume fraction function of material 1 and material 2 is represented by V0(x) And VL(x) Shows, let V0(x)+VL(x) 1, the elastic modulus E, the density rho and the axial thermal expansion coefficient alpha of the functionally graded flow transmission pipelinexRespectively expressed as:
E＝E(x)＝V0(x)E0+VL(x)EL (2)
ρ＝ρ(x)＝V0(x)ρ0+VL(x)ρL (3)
αx＝ρ(x)＝V0(x)αx0+VL(x)αxL (4)
wherein E is0And ELRespectively representing the elastic moduli, ρ, of the materials 1 and 20And ρLDenotes the density, alpha, of material 1 and material 2, respectivelyx0And alphaxLRespectively representing the thermal expansion coefficients of material 1 and material 2;
the stress balance of the pipeline on the z axis is as follows:
wherein m isp＝ρ(x)ApRepresents the unit length mass of the functional gradient flow transmission pipeline, ApRepresenting the cross-sectional area of the functional gradient flow transmission pipeline; w represents the lateral displacement of the neutral plane of the pipe, P represents the fluid force per unit length, i.e. the normal force,
consists of three terms of centrifugal force, Coriolis force and inertia force, mf＝ρ(x)AfDenotes the mass per unit length of the fluid, AfRepresents the cross-sectional area of the fluid; q represents the shear force on the interface of the functional gradient flow transmission pipeline; f (x, t) Nx＝Nm+NtIndicating axial force, NmAnd NtRespectively representing the axial force brought by mechanical action and the axial force brought by temperature change; θ represents a turning angle;
the mid-point bending moment balance at the right end of the interface is as follows:
wherein M represents a bending moment;
assuming an initial axial force N in a functionally gradient fluid delivery conduitm＝0，
For acting on functional gradients due to temperature changesAxial force on the flow delivery pipe, v is the poisson ratio, and Δ T represents the temperature change; obtaining corners by Euler-Bernoulli Beam theory
The bending moment M is as follows:
wherein I represents the second moment of the cross section of the pipeline;
then equation (5) is expressed as:
the formula (8) is a vibration control equation of the functional gradient flow transmission pipeline;
according to the boundary condition of two fixed branches:
wherein w (x, t) represents the value of the transverse displacement at the position where the axial coordinate is x and the time coordinate is t;
introducing dimensionless quantities:
the simplified control equation obtained by bringing formula (10) into formula (8) is:
the boundary conditions are then expressed as:
wherein Y (X, tau) represents the value of the dimensionless transverse displacement at the position where the dimensionless axial coordinate is X and the dimensionless time coordinate is tau;
step 2: constructing a vibration control equation discrete format of the functional gradient flow transmission pipeline;
according to the differential quadrature method, the derivative value of the function can be approximated by a weighted linear sum of the function values, and the Fung node is selected to obtain a corresponding weighting coefficient matrix:
wherein x (i), i ═ 1,2, …, N0When represents an internal node, i ═ N0+1, …, where N denotes the boundary node and N is the total number of nodes;
consider the continuous differentiable one-dimensional function f (x) over the interval [ a, b ], the derivative of f (x) being expressed as:
wherein L isnFor a linear differential operator, n represents the order of the differential,
representing a weighting coefficient, xjIs N mutually different nodes a ═ x1<x2<…<xNB is the coordinate value of the jth node;
the accuracy of equation (15) depends on the number of nodes and the weighting coefficients; the weighting coefficients for the first derivative are:
wherein lj(x) Is a LagA Langi interpolation function;
the higher order weight coefficients are calculated by the recurrence formula (17):
the vibration control equation (11) of the functionally gradient delivery conduit is rewritten as the following discrete form:
where subscript b represents a boundary point, d represents an inter-block interior point,
and
a block matrix, Y, representing the global stiffness matrix divided by internal nodes d and boundary nodes b, respectivelydAnd YbRespectively representing dimensionless lateral vibration displacements of the internal node d and the boundary node b,
and
a blocking matrix respectively representing the overall damping matrix divided by the internal node d and the boundary node b,
and
a block matrix respectively representing an overall quality matrix divided by an internal node d and a boundary node b;
and step 3: analyzing natural frequency and solving array type;
let the solution of equation (18) have the form:
wherein
Is the amplitude of the functionally graded fluid delivery conduit, and Re (ω) represents the vibration frequency of the system;
substitution of formula (19) for formula (18) and elimination
Then, the generalized eigenvalue problem is obtained:
wherein
A quality matrix representing the system is shown,
a damping matrix representing the system is shown,
is the stiffness matrix of the system;
the discrete governing equation (18) is written as the equivalent of:
substituting equation (19) into equation (21) yields a discrete displacement eigenmode:
namely, the secondary eigenvalue problem of the formula (20) is converted into a primary eigenvalue problem; wherein omega is a complex characteristic value of the system and is used for analyzing stability, the imaginary part of omega represents the damping of each order of natural vibration mode, and the real part of omega represents the natural frequency of vibration; carrying out Lagrange interpolation fitting on the discrete displacement intrinsic vibration mode to obtain the intrinsic vibration mode of the continuous system;
and 4, step 4: a time domain analysis method;
rewriting equation (18) to the form:
wherein the content of the first and second substances,
in the time region of t-t + Δ t, the Newmark integration method adopts the following iterative format, namely:
wherein phi and chi are parameters determined according to the requirements of integration precision and stability respectively;
displacement solution { eta } of time t + delta t in Newmark integration methodt+ΔtIs obtained by satisfying the equation of motion at time t + Δ t, i.e., by equation (26):
obtained from formula (25):
bringing formula (27) into formula (24), and bringing the resulting formula into formula (26) with formula (24) to obtain a formula of { η }t,
Calculating { eta }t+ΔtThe recurrence formula of (c):
carrying out iterative solution on the initial conditions to obtain a dimensionless displacement response { eta } and a dimensionless speed response of the system
And dimensionless acceleration response
According to the definition of axial stress in a nominal stress method, the relation between the shear stress and node displacement is obtained as follows:
wherein M '(x) is a bending moment of a cross section at an x-coordinate, Q' (x) is a shear force on the cross section at the x-coordinate,
Iyis the moment of inertia of the cross-section to the neutral axis y, z is the distance from the median plane in the height direction, b is the width of the cross-section at the desired shear stress,
the area A of the outer part at the required shear stress*Static moment on the neutral axis, i.e.:
the above process is the dynamic response analysis of the functional gradient fluid transmission pipeline.
The invention has the following beneficial effects:
the invention provides a calculation process of the natural frequency of the pipeline vibration under the given boundary condition, the natural vibration mode corresponding to the natural frequency, the time domain displacement response of the transverse vibration under the given boundary condition and the initial condition and the time domain stress response based on a differential quadrature method under the condition that the equivalent material property changes along the length direction of the pipeline, and provides effective kinetic theoretical reference for the design and the application of the axial functional gradient pipeline.
Drawings
FIG. 1 is a schematic view of a piping model of the process of the present invention.
FIG. 2 is a diagram of the infinitesimal force analysis of the method of the present invention.
FIG. 3 is a schematic illustration of the calculation of the static moment according to the method of the present invention.
FIG. 4 is a flow chart of the calculation of the method of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
A functional gradient flow pipe modal and response analysis method based on a differential quadrature method comprises the following steps:
step 1: establishing a differential equation for controlling vibration of the functionally gradient fluid delivery pipeline with two fixed branches at two ends;
as shown in fig. 1, the outer diameter of the functionally gradient fluid delivery pipe is d, the wall thickness is h, the length is L, the flow velocity of the internal fluid is u, the axial direction of the pipe is taken as an x axis, and the radial direction of the section of the pipe is taken as a z axis;
selecting a volume fraction function as an exponential function form V1By dimensionless quantities
Expressed as:
wherein a represents a parameter for adjusting the distribution of the exponential type body integral number function;
the functional gradient flow transmission pipeline consists of two materials, namely a material 1 and a material 2; volume fraction function of material 1 and material 2 is represented by V0(x) And VL(x) Shows, let V0(x)+VL(x) 1, the elastic modulus E, the density rho and the axial thermal expansion coefficient alpha of the functionally graded flow transmission pipelinexRespectively expressed as:
E＝E(x)＝V0(x)E0+VL(x)EL (2)
ρ＝ρ(x)＝V0(x)ρ0+VL(x)ρL (3)
αx＝ρ(x)＝V0(x)αx0+VL(x)αxL (4)
wherein E is0And ELRespectively representing the elastic moduli, ρ, of the materials 1 and 20And ρLDenotes the density, alpha, of material 1 and material 2, respectivelyx0And alphaxLRespectively representing the thermal expansion coefficients of material 1 and material 2;
the stress balance of the pipeline on the z axis is as follows:
wherein m isp＝ρ(x)ApRepresents the unit length mass of the functional gradient flow transmission pipeline, ApRepresenting the cross-sectional area of the functional gradient flow transmission pipeline; w represents the lateral displacement of the neutral plane of the pipe, P represents the fluid force per unit length, i.e. the normal force,
consists of three terms of centrifugal force, Coriolis force and inertia force, mf＝ρ(x)AfDenotes the mass per unit length of the fluid, AfRepresents the cross-sectional area of the fluid; q represents the shear force on the interface of the functional gradient flow transmission pipeline; f (x, t) Nx＝Nm+NtIndicating axial force, NmAnd NtRespectively representing the axial force brought by mechanical action and the axial force brought by temperature change; θ represents a turning angle;
the mid-point bending moment balance at the right end of the interface is as follows:
wherein M represents a bending moment;
assuming an initial axial force N in a functionally gradient fluid delivery conduitm＝0，
V is the poisson's ratio, and Δ T represents the temperature change, which is the axial force acting on the functional gradient delivery conduit due to the temperature change; obtaining corners by Euler-Bernoulli Beam theory
The bending moment M is as follows:
wherein I represents the second moment of the cross section of the pipeline;
then equation (5) is expressed as:
the formula (8) is a vibration control equation of the functional gradient flow transmission pipeline;
according to the boundary condition of two fixed branches:
wherein w (x, t) represents the value of the transverse displacement at the position where the axial coordinate is x and the time coordinate is t;
introducing dimensionless quantities:
the simplified control equation obtained by bringing formula (10) into formula (8) is:
the boundary conditions are then expressed as:
wherein Y (X, tau) represents the value of the dimensionless transverse displacement at the position where the dimensionless axial coordinate is X and the dimensionless time coordinate is tau;
step 2: constructing a vibration control equation discrete format of the functional gradient flow transmission pipeline;
according to the differential quadrature method, the derivative value of the function can be approximated by a weighted linear sum of the function values, and the Fung node is selected to obtain a corresponding weighting coefficient matrix:
wherein x (i), i ═ 1,2, …, N0When represents an internal node, i ═ N0+1, …, where N denotes the boundary node and N is the total number of nodes;
consider the continuous differentiable one-dimensional function f (x) over the interval [ a, b ], the derivative of f (x) being expressed as:
wherein L isnFor a linear differential operator, n represents the order of the differential,
representing a weighting coefficient, xjIs N mutually different nodes a ═ x1<x2<…<xNB is the coordinate value of the jth node;
the accuracy of equation (15) depends on the number of nodes and the weighting coefficients; the weighting coefficients for the first derivative are:
wherein lj(x) Is a lagrange interpolation function;
the higher order weight coefficients are calculated by the recurrence formula (17):
the vibration control equation (11) of the functionally gradient delivery conduit is rewritten as the following discrete form:
where subscript b represents a boundary point, d represents an inter-block interior point,
and
a block matrix, Y, representing the global stiffness matrix divided by internal nodes d and boundary nodes b, respectivelydAnd YbRespectively representing dimensionless lateral vibration displacements of the internal node d and the boundary node b,
and
a blocking matrix respectively representing the overall damping matrix divided by the internal node d and the boundary node b,
and
a block matrix respectively representing an overall quality matrix divided by an internal node d and a boundary node b;
and step 3: analyzing natural frequency and solving array type;
let the solution of equation (18) have the form:
wherein
Is the amplitude of the functionally graded fluid delivery conduit, and Re (ω) represents the vibration frequency of the system;
substitution of formula (19) for formula (18) and elimination
Then, the generalized eigenvalue problem is obtained:
wherein
A quality matrix representing the system is shown,
a damping matrix representing the system is shown,
is the stiffness matrix of the system;
the discrete governing equation (18) is written as the equivalent of:
substituting equation (19) into equation (21) yields a discrete displacement eigenmode:
the second order eigenvalue problem of the formula (20) is converted into a first order eigenvalue problem, a complex eigenvalue and an eigenvector of the system can be obtained by solving, the damping and the frequency of the order natural vibration can be obtained from the eigenvalue, and the natural vibration mode of the discrete system corresponding to the order natural frequency can be obtained from the eigenvector corresponding to the eigenvalue; wherein omega is a complex characteristic value of the system and is used for analyzing stability, the imaginary part of omega represents the damping of each order of natural vibration mode, and the real part of omega represents the natural frequency of vibration; carrying out Lagrange interpolation fitting on the discrete displacement intrinsic vibration mode to obtain the intrinsic vibration mode of the continuous system;
and 4, step 4: a time domain analysis method;
rewriting equation (18) to the form:
wherein the content of the first and second substances,
in the time region of t-t + Δ t, the Newmark integration method adopts the following iterative format, namely:
wherein phi and chi are parameters determined according to the requirements of integration precision and stability respectively;
displacement solution { eta } of time t + delta t in Newmark integration methodt+ΔtIs obtained by satisfying the equation of motion at time t + Δ t, i.e., by equation (26):
obtained from formula (25):
bringing formula (27) into formula (24), and bringing the resulting formula into formula (26) with formula (24) to obtain a formula of { η }t，
Calculating { eta }t+ΔtThe recurrence formula of (c):
carrying out iterative solution on the initial conditions to obtain a dimensionless displacement response { eta } and a dimensionless speed response of the system
And dimensionless acceleration response
According to the definition of axial stress in a nominal stress method, the relation between the shear stress and node displacement is obtained as follows:
wherein M '(x) is a bending moment of a cross section at an x-coordinate, Q' (x) is a shear force on the cross section at the x-coordinate,
Iyis the moment of inertia of the cross-section to the neutral axis y, z is the distance from the midplane in the height direction, b is the width of the cross-section at the desired shear stress, as shown in FIG. 3;
the area A of the outer part at the required shear stress*Static moment on the neutral axis, i.e.:
the above process is the dynamic response analysis of the functional gradient fluid transmission pipeline.
The specific embodiment is as follows:
referring to fig. 4, a flowchart of a method for calculating a dynamic response of a functionally gradient fluid delivery pipeline based on a differential integration method is provided, where the method includes: the system comprises six modules of description of functional gradient materials, infinitesimal stress analysis, establishment of a vibration control equation, construction of a discrete structure, selection of an iteration format and solution of dynamic response.
In the first module, a functionally graded material is described by selecting a suitable volume fraction function, including a representation of its volume fraction, system modulus of elasticity, system density, and system coefficient of thermal expansion. In the second module, the pipe structure infinitesimal shown in fig. 2 is taken to perform stress analysis on the pipe structure infinitesimal, including the balance of force and the balance of moment, so as to obtain a vibration control equation of the pipe. And in the third module, according to the thought of a differential integration method, selecting proper nodes and test functions to establish a weight function of the pipeline and discretizing a control equation.
In the fourth module, the determinant of the coefficient matrix of the discrete structure is taken as zero to calculate the eigenvalue and the eigenvector thereof, the imaginary part of the eigenvalue is the natural frequency, and the eigenvector is the natural vibration mode. And selecting an iteration format in a fifth module, expressing and solving the dimensionless displacement, speed and acceleration at the moment of t + delta t, and approximating the result to a system displacement response after iteration to certain precision. And in the sixth module, the dynamic response of the system can be obtained according to the relation between the stress and the strain and the node force and the node moment.
Claims (1)
1. A functional gradient flow tube modal and response analysis method based on a differential quadrature method is characterized by comprising the following steps:
step 1: establishing a differential equation for controlling vibration of the functionally gradient fluid delivery pipeline with two fixed branches at two ends;
the outer diameter of the functional gradient flow transmission pipeline is d, the wall thickness is h, the length is L, the flow velocity of internal fluid is u, the axial direction of the pipeline is taken as an x axis, and the radial direction of the section of the pipeline is taken as a z axis;
selecting a volume fraction function as an exponential function form V1By dimensionless quantities
Expressed as:
wherein a represents a parameter for adjusting the distribution of the exponential type body integral number function;
the functional gradient flow transmission pipeline consists of two materials, namely a material 1 and a material 2; volume fraction function of material 1 and material 2 is represented by V0(x) And VL(x) Shows, let V0(x)+VL(x) 1, the elastic modulus E, the density rho and the axial thermal expansion coefficient alpha of the functionally graded flow transmission pipelinexRespectively expressed as:
E＝E(x)＝V0(x)E0+VL(x)EL (2)
ρ＝ρ(x)＝V0(x)ρ0+VL(x)ρL (3)
αx＝ρ(x)＝V0(x)αx0+VL(x)αxL (4)
wherein E is0And ELRespectively representing the elastic moduli, ρ, of the materials 1 and 20And ρLDenotes the density, alpha, of material 1 and material 2, respectivelyx0And alphaxLRespectively representing the thermal expansion coefficients of material 1 and material 2;
the stress balance of the pipeline on the z axis is as follows:
wherein m isp＝ρ(x)ApRepresents the unit length mass of the functional gradient flow transmission pipeline, ApRepresenting the cross-sectional area of the functional gradient flow transmission pipeline; w represents the lateral displacement of the neutral plane of the pipe, P represents the fluid force per unit length, i.e. the normal force,
consists of three terms of centrifugal force, Coriolis force and inertia force, and mf is rho (x) AfDenotes the mass per unit length of the fluid, AfRepresents the cross-sectional area of the fluid; q represents the interface of the functionally graded fluid transmission pipelineShearing force; f (x, t) Nx＝Nm+NtIndicating axial force, NmAnd NtRespectively representing the axial force brought by mechanical action and the axial force brought by temperature change; θ represents a turning angle;
the mid-point bending moment balance at the right end of the interface is as follows:
wherein M represents a bending moment;
assuming an initial axial force N in a functionally gradient fluid delivery conduitm＝0，
V is the poisson's ratio, and Δ T represents the temperature change, which is the axial force acting on the functional gradient delivery conduit due to the temperature change; obtaining corners by Euler-Bernoulli Beam theory
The bending moment M is as follows:
wherein I represents the second moment of the cross section of the pipeline;
then equation (5) is expressed as:
the formula (8) is a vibration control equation of the functional gradient flow transmission pipeline;
according to the boundary condition of two fixed branches:
wherein w (x, t) represents the value of the transverse displacement at the position where the axial coordinate is x and the time coordinate is t;
introducing dimensionless quantities:
the simplified control equation obtained by bringing formula (10) into formula (8) is:
the boundary conditions are then expressed as:
Y(0，τ)＝Y(1，τ)＝0
wherein Y (X, tau) represents the value of the dimensionless transverse displacement at the position where the dimensionless axial coordinate is X and the dimensionless time coordinate is tau;
step 2: constructing a vibration control equation discrete format of the functional gradient flow transmission pipeline;
according to the differential quadrature method, the derivative value of the function can be approximated by a weighted linear sum of the function values, and the Fung node is selected to obtain a corresponding weighting coefficient matrix:
wherein x (i), i ═ 1,2, …, N0When represents an internal node, i ═ N0+1, …, where N denotes the boundary node and N is the total number of nodes;
consider the continuous differentiable one-dimensional function f (x) over the interval [ a, b ], the derivative of f (x) being expressed as:
wherein L isnFor a linear differential operator, n represents the order of the differential,
representing a weighting coefficient, xjIs N mutually different nodes a ═ x1＜x2＜…＜xNB is the coordinate value of the jth node;
the accuracy of equation (15) depends on the number of nodes and the weighting coefficients; the weighting coefficients for the first derivative are:
wherein lj(x) Is a lagrange interpolation function;
the higher order weight coefficients are calculated by the recurrence formula (17):
the vibration control equation (11) of the functionally gradient delivery conduit is rewritten as the following discrete form:
where subscript b represents a boundary point, d represents an inter-block interior point,
and
a block matrix, Y, representing the global stiffness matrix divided by internal nodes d and boundary nodes b, respectivelydAnd YbRespectively representing dimensionless lateral vibration displacements of the internal node d and the boundary node b,
and
a blocking matrix respectively representing the overall damping matrix divided by the internal node d and the boundary node b,
and
a block matrix respectively representing an overall quality matrix divided by an internal node d and a boundary node b;
and step 3: analyzing natural frequency and solving array type;
let the solution of equation (18) have the form:
wherein
Is the amplitude of the functionally graded fluid delivery conduit, and Re (ω) represents the vibration frequency of the system;
substitution of formula (19) for formula (18) and elimination
Then, the generalized eigenvalue problem is obtained:
wherein
A quality matrix representing the system is shown,
a damping matrix representing the system is shown,
is the stiffness matrix of the system;
the discrete governing equation (18) is written as the equivalent of:
substituting equation (19) into equation (21) yields a discrete displacement eigenmode:
namely, the secondary eigenvalue problem of the formula (20) is converted into a primary eigenvalue problem; wherein omega is a complex characteristic value of the system and is used for analyzing stability, the imaginary part of omega represents the damping of each order of natural vibration mode, and the real part of omega represents the natural frequency of vibration; carrying out Lagrange interpolation fitting on the discrete displacement intrinsic vibration mode to obtain the intrinsic vibration mode of the continuous system;
and 4, step 4: a time domain analysis method;
rewriting equation (18) to the form:
wherein the content of the first and second substances,
in the time region of t-t + Δ t, the Newmark integration method adopts the following iterative format, namely:
wherein phi and chi are parameters determined according to the requirements of integration precision and stability respectively;
displacement solution { eta } of time t + delta t in Newmark integration methodt+ΔtIs obtained by satisfying the equation of motion at time t + Δ t, i.e., by equation (26):
obtained from formula (25):
bringing formula (27) into formula (24), and bringing the resulting formula into formula (26) with formula (24) to obtain a formula of { η }t，
Calculating { eta }t+ΔtThe recurrence formula of (c):
carrying out iterative solution on the initial conditions to obtain a dimensionless displacement response { eta } and a dimensionless speed response of the system
And dimensionless acceleration response
According to the definition of axial stress in a nominal stress method, the relation between the shear stress and node displacement is obtained as follows:
wherein M '(x) is a bending moment of a cross section at an x-coordinate, Q' (x) is a shear force on the cross section at the x-coordinate,
Iyis the moment of inertia of the cross-section to the neutral axis y, z is the distance from the median plane in the height direction, b is the width of the cross-section at the desired shear stress,
the area A of the outer part at the required shear stress*Static moment on the neutral axis, i.e.:
the above process is the dynamic response analysis of the functional gradient fluid transmission pipeline.
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