CN116702381B - Equivalent and vibration response calculation method for non-linearity of bolt connection - Google Patents
Equivalent and vibration response calculation method for non-linearity of bolt connection Download PDFInfo
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Abstract
The invention relates to a method for calculating equivalent and vibration response of a nonlinear bolt connection, which comprises the steps of establishing a bolt connection equivalent model, establishing a nonlinear spring damping element on a corresponding point of a connection region of a finite element model of a connected piece, and calculating the connection force of the nonlinear spring damping element to the connected piece; splitting a connected structure into a plurality of substructures, performing matrix partitioning on the substructures, and calculating nonlinear spring damping element connecting force which participates in the connection of equivalent connecting pieces in the substructures; condensing the substructure by adopting a Craig-Bampton substructure method; and carrying out sweep frequency or random vibration response calculation by utilizing the condensed substructure, carrying out iterative calculation of response, and finally obtaining an amplitude-frequency response curve of the nonlinear substructure. The method can realize the rapid calculation of the vibration response of the equivalent nonlinear structure of the bolt.
Description
Technical Field
The invention belongs to the technical field of structural dynamics, and particularly relates to a method for calculating equivalent and vibration response of bolt connection nonlinearity, a computer-readable storage medium and an electronic device.
Background
The bolt connection is a connection mode used in a large amount in a carrier rocket, particularly each cabin section of an rocket body, and a plurality of bolts are usually used for connecting end frames of the cabin sections. A typical bay section connection is shown schematically in figure 1. When the vibration characteristics of the structure are analyzed by utilizing finite element modeling, the bolt connection is generally required to be simplified, and the bolt connection is generally equivalent by adopting linear units such as a beam unit, a spring unit and the like, so that the finally obtained finite element model is a linear model, the vibration characteristics of the structure can be obtained through modal and frequency response calculation, and time domain transient calculation does not need to be consumed in a large amount of time. However, since actual bolting typically involves non-linear factors such as contact, pretension, friction, clearance, etc., particularly for the main joint structure of the cabin segment, these non-linear factors have a more pronounced effect on the vibration characteristics of the overall structure. The linear unit simulation can not achieve better simulation effect at last, and can not be matched with the vibration test better. However, if the nonlinear model is directly built, various nonlinear factors can be considered, but the mode and the frequency response cannot be calculated, so that the vibration characteristic of the structure can be obtained indirectly only through a transient calculation method such as time domain sweep frequency or random vibration, which is time-consuming compared with the method.
Disclosure of Invention
Aiming at the problems that the vibration characteristic of a structure can only be indirectly obtained through a transient computing method such as time domain frequency sweep or random vibration and the like by directly establishing a nonlinear model through bolt connection, and time is very consuming, the invention provides an equivalent simplifying and vibration response computing method considering the nonlinearity of the bolt connection, introduces a nonlinear spring damping element considering a gap, nonlinear stiffness force and nonlinear damping force, and reduces the order of a finite element model by adopting a Craig-Bampton substructure method in order to improve the efficiency of transient computing after introducing nonlinearity.
The invention provides a method for calculating equivalent and vibration response of non-linearity of bolt connection, wherein a bolt connector is connected with a connected structure, and the method comprises the following steps:
establishing a connection equivalent model, namely establishing a nonlinear spring damping element on a connection region corresponding point of a finite element model of a connected piece, and calculating the connection force of the nonlinear spring damping element to the connected piece; the connection equivalent model establishment is specifically a bolt connection equivalent model establishment; the connecting area is specifically the bolt connecting area;
splitting a connected structure into a plurality of substructures, performing matrix partitioning on the substructures, and calculating nonlinear spring damping element connecting force which participates in the connection of equivalent connecting pieces in the substructures;
condensing the substructure by adopting a Craig-Bampton substructure method, and calculating a substructure modal set) Calculating a modal mass matrix in the sub-structure modal spaceModal damping matrixModal stiffness matrixModal force vector；
Sweep frequency or random vibration response calculation is carried out by utilizing the condensed substructures, and interaction force between the connection degrees of freedom of the substructures is calculated according to the last moment state of the connection faces of the substructures in each time step, wherein the states comprise displacement and speed of the connection degrees of freedomAnd performing iterative calculation of the response, and finally obtaining an amplitude-frequency response curve of the nonlinear substructure.
Further, a connection equivalent model is established, nonlinear spring damping elements are established on corresponding points of a connection region of a finite element model of the connected piece, and the connection force of the nonlinear spring damping elements to the connected piece is calculated, wherein the connection equivalent model comprises
Defining two nodes to which the spring damper element is connected, a first connection node and a second connection nodeThe three translational degrees of freedom of the connecting node are respectivelyAndwherein the x direction is the normal direction of the connecting surface; the acting force (connecting force) of the nonlinear spring damping element on the connected piece is obtained through displacement and speed state calculation of three translational degrees of freedom of the connected node, and the connecting force calculating method of the constructed spring damping element is specifically as follows, so as to calculate the connecting force of the first connecting node, and greatly reverse the connecting force of the second connecting node to the connecting force of the first connecting node and the like:
(1)
wherein:
；the rigidity and damping nonlinear orders are natural numbers, the values are selected according to the needs, and the orders are preferably small under the condition of being well matched with the test;
in order to connect the rigidity and the damping coefficient,is an axial connection gap;
the method comprises dividing the connected structure into a plurality of sub-structures, dividing the sub-structures into matrix blocks, and calculating nonlinear spring damping element connecting force participating in equivalent connector connection in the sub-structures, wherein the nonlinear spring damping element connecting force comprises
Defining the set of degrees of freedom in the substructure that do not participate in the equivalent connection (bolting) asThe set of degrees of freedom involved in the connection of equivalent connectors (bolts) is noted asTaking a certain substructure as an example, the vibration equation can be written as follows:
(2)
wherein:
、、、is a block quality matrix;
、、、is a block damping matrix;
、、、is a block stiffness matrix;
an external force vector applied to the non-connection degree of freedom of the substructure;
for the corresponding connection force of the nonlinear spring damping element, for the interconnected substructuresAssuming the presence ofAnd (3) connecting points, namely:
(3)；
the method adopts the Craig-Bampton substructure method to agglomerate the substructure and calculate the substructure modal set) Calculating a modal mass matrix in the sub-structure modal spaceModal damping matrixModal stiffness matrixModal force vectorComprises
The following generalized eigenvalue equation is calculated by fully constraining the degrees of freedom of the connection of the substructures:
(4)
wherein:
a vibration mode matrix is calculated for the characteristic equation;
is a diagonal matrix, the kth element of the diagonal isA characteristic value corresponding to the kth column vibration mode;
for a pair ofPerforming mode cut-off, selecting low-order modeThereby constructing a constrained sub-structure main modality:
(5)
wherein the method comprises the steps ofIs a matrix of 0.
Constructing a constraint mode according to the static equilibrium relation between the connection freedom degree and the internal freedom degree:
(6)
wherein the method comprises the steps ofIs a unit array;
thereby obtaining a sub-structure modal set:
(7)
carrying out modal coordinate transformation on the substructure through the modal set, namely:
(8)
wherein:
the mode coordinates corresponding to the constraint main mode are obtained;
the mode coordinates corresponding to the constraint mode are obtained;
namely, isAnda modal coordinate set is formed;
as can be seen from the modal coordinate transformation, the degrees of freedom of the connection before and after transformation maintain the physical properties, namely；
The substructure vibration equation after coordinate transformation is as follows:
(9)
wherein the method comprises the steps of、、、The modal mass matrix, the modal damping matrix, the modal stiffness matrix and the modal force vector are respectively calculated according to the following formulas:
(10)
wherein, the liquid crystal display device comprises a liquid crystal display device,the subscript L of (2) corresponds to linear, ">The subscript NL of (1) corresponds to the nonlinearity, and these two forces are respectively linear and nonlinear forces, which correspond to the two terms to the left of the equal sign in equation (10), respectively.
The number of degrees of freedom of the vibration equation of the substructure after condensation is the number of degrees of freedom of connection and the cutting modeThe sum of the orders can be greatly condensed on the substructure through modal truncation.
Further, the method also comprises nonlinear parameter correction, and specifically, the parameters of the nonlinear spring damping element are corrected by comparing the amplitude-frequency response curve of the nonlinear substructure with the test result.
Further, the parameters of the nonlinear spring damping element are corrected, when the nonlinear spring damping element is actually used, a linear connection force model with the degradation of the nonlinear spring damping element is firstly adopted for preliminary calculation and correction, nonlinear items are gradually considered for correction, the nonlinear spring damping element specifically comprises a connection gap of 0, and when the connection stiffness damping force is only 1 st order, namely n and m are both 1, the nonlinear spring damping element is the linear connection model.
Specifically, the connecting piece is a bolt-nut connecting piece.
In another aspect, the present invention also provides a computer readable storage medium, where the computer readable storage medium includes a stored program, where the program when run performs the bolting nonlinear equivalent and vibration response calculation method described above.
In yet another aspect, the invention provides an electronic device comprising a memory having a computer program stored therein and a processor configured to perform the bolting nonlinear equivalent and vibration response calculation method by the computer program.
Compared with the prior art, the method of the invention has the following beneficial effects:
the invention provides an equivalent simplifying method considering bolt connection nonlinearity, which introduces a nonlinear spring damping element considering clearance, nonlinear rigidity force and nonlinear damping force, splits a connected structure into a plurality of substructures, completely constrains and calculates a generalized eigenvalue equation by connection degrees of freedom of the substructures, calculates a vibration mode matrix obtained by calculation of the eigenvalue equation, carries out modal truncation on the vibration mode matrix, selects a low-order mode of the vibration mode matrix, thereby constructing a constraint substructure main mode, constructs constraint modes according to a static balance relation between the connection degrees of freedom and internal degrees of freedom, thereby obtaining a substructure mode set, carries out modal coordinate transformation on the substructure through the mode set, obtains a substructure vibration equation after coordinate transformation, and calculates a modal mass matrix, a modal damping matrix, a modal rigidity matrix and a modal force vector. The number of degrees of freedom of the vibration equation of the substructure after condensation is the sum of the number of degrees of freedom of connection and the number of modes of interception, namely the number of modes of a low-order mode, and the substructure can be greatly condensed through the mode interception. In addition, due to the characteristic of Craig-Bampton substructure transformation, the connection degree of freedom still keeps the physical properties, and when the vibration equation after transformation is solved, the transformed coordinates can be directly used for calculating the nonlinear connection force, and the mode coordinates do not need to be returned to a physical space and then the nonlinear connection force is calculated. Therefore, the method can realize the rapid calculation of the vibration response of the equivalent nonlinear structure of the bolt.
Drawings
FIG. 1 is a schematic view of a typical bolting configuration;
FIG. 2 is a schematic diagram of a bolt equivalent model of the present invention;
FIG. 3 is a schematic flow chart of the method of the present invention;
FIG. 4 is a schematic diagram of an example structure of the condensation calculation of the method of the present invention;
FIGS. 5-1 (a) and (b) are diagrams showing the same degree of freedom acceleration response obtained by calculation when the excitation angular velocity of the provided example is 20rad/s, wherein (a) is the result of condensation calculation by the method of the present invention, and (b) is the result of direct calculation; (c) is an overlay of (a) and (b);
FIGS. 5-2 (a) and (b) are, respectively, a condensed calculated phase diagram and a directly calculated phase diagram of the method of the present invention; (c) is an overlay of (a) and (b);
5-3 (a) are results of condensing calculated displacement differences between the first and second connection nodes of the bolt by the method of the present invention, (b) are results of directly calculated displacement differences between the first and second connection nodes of the bolt; (c) is an overlay of (a) and (b);
FIGS. 5-4 (a) are the results of the constructed spring damper element connection force calculated by the method of the present invention by condensation and (b) are the results of the constructed spring damper element connection force calculated directly; (c) is an overlay of (a) and (b);
FIGS. 5-5 (a) show the corresponding calculation results obtained by condensing the calculated different excitation frequencies (14 rand/s-19 rand/s) by the method of the invention, and (b) show the corresponding calculation results obtained by directly calculating the calculated different excitation frequencies (14 rand/s-19 rand/s); (c) is an overlay of (a) and (b); the leftmost columns are excitation circle frequencies (rand/s) -e.g. 14, 15.2, 16.5, 19;
FIGS. 5-6 (a) are graphs showing the results of the same degree of freedom sweep amplitude calculation obtained by condensing the sweep frequency in the method of the present invention, and (b) are graphs showing the results of the same degree of freedom sweep amplitude calculation obtained by direct calculation; (c) is an overlay of (a) and (b);
FIGS. 5-7 (a) are the results of the same degree of freedom displacement bifurcation diagram obtained by condensation calculation in the method of the present invention, and (b) are the results of the same degree of freedom displacement bifurcation diagram obtained by direct calculation; the system has chaos and cycle doubling phenomena when excited by 15-30 rad/s during frequency sweeping.
Wherein, 1-bolt (including nut), 2-bulkhead (21-first end frame, 21 a-first end frame connection face, 22-second end frame, 22 a-second end frame connection face), 3-nonlinear spring damping element (equivalent connection), 31-first connection node, 32-second connection node, 311-first connection node degree of freedom, 322-second connection node degree of freedom.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the present invention without making any inventive effort, are intended to fall within the scope of the present invention.
As shown in the flow chart of FIG. 3, the present embodiment provides a method for calculating the equivalent and vibration response of a nonlinear connection of a connection member, the connection member being a bolt (including a nut), the bolt connecting a connected structure, the connected structure being a bulkhead provided with a first end frame and a second end frame, the bulkhead being connected by the first end frame and the second end frame by bolts, comprising the steps of
Step 1: bolt connection equivalent model establishment
And establishing a nonlinear spring damping element, namely the bolting equivalent model, on the corresponding point of the bolting area of the finite element model of the connected piece, as shown in figure 2. The spring damping element is connected with two first connecting nodes and two second connecting nodes, the first connecting nodes correspond to the first end frames, the second connecting nodes correspond to the second end frames, and the following three translational degrees of freedom of the first connecting nodes and the second connecting nodes are respectively:、(wherein ux 1 And ux 2 The directions are normal directions of the first end frame connecting surface and the second end frame connecting surface respectively). The acting force of the nonlinear spring damping element on the connected piece can be obtained through displacement and speed state calculation of three translational degrees of freedom of the first connecting node and the second connecting node.
The method for calculating the connecting force of the spring damping element is constructed by comprehensively considering the nonlinear effect of the bolt connection, and in the embodiment, the connecting force of the first connecting node is calculated as an example, and the connecting force of the second connecting node is greatly opposite to the connecting force of the first connecting node.
Wherein:
；
in order to connect the rigidity and the damping coefficient,is an axial connection gap.
As can be seen from the above-mentioned coupling force expression of the nonlinear spring element, the nonlinearity is mainly represented in the axial direction of the coupling, i.e., X-direction, whereinI.e., the parameters to be determined for the nonlinear connection, can be identified by test data,the rigidity and damping nonlinear orders are natural numbers, can be selected according to the needs, and the orders are preferably small under the condition of being well matched with the test. When the steps are all taken to be 1 and the connection gap is taken to be 0, the connection can be degraded into a linear spring damped connection.
Step 2: processing of sub-structure finite element models
Splitting the connected structure into a plurality of substructures, wherein a degree of freedom set which does not participate in equivalent bolting in the substructures is recorded asThe degree of freedom set involved in equivalent bolting is noted as. Taking a certain substructure as an example, the vibration equation can be written as follows:
wherein:
、、、is a block quality matrix;
、、、is a block damping matrix;
、、、is a block stiffness matrix;
an external force vector applied to the non-connection degree of freedom of the substructure;
for a corresponding connection force of the non-linear spring damping element, for the interconnected substructures,assuming the presence ofAnd (3) connecting points, namely:
step 3: substructure agglomeration
Condensing the substructure by adopting a Craig-Bampton substructure method, and calculating the following generalized eigenvalue equation by fully restricting the connection freedom degree of the substructure:
(4)
wherein:
a vibration mode matrix is calculated for the characteristic equation;
is a diagonal matrix, the kth element of the diagonal isA characteristic value corresponding to the kth column vibration mode;
for a pair ofPerforming mode cut-off, selecting low-order modeThereby constructing a constrained sub-structure main modality:
wherein the method comprises the steps ofIs a matrix of 0.
Constructing a constraint mode according to the static equilibrium relation between the connection freedom degree and the internal freedom degree:
wherein the method comprises the steps ofIs a unit array.
Thereby obtaining a sub-structure modal set:
(7)
carrying out modal coordinate transformation on the substructure through the modal set, namely:
wherein:
the mode coordinates corresponding to the constraint main mode are obtained;
the mode coordinates corresponding to the constraint mode are obtained;
namely, isAnda modal coordinate set is formed.
As can be seen from the modal coordinate transformation, the degrees of freedom of the connection before and after transformation maintain the physical properties, namely。
The substructure vibration equation after coordinate transformation is as follows:
(9)
wherein the method comprises the steps of、、、The modal mass matrix, the modal damping matrix, the modal stiffness matrix and the modal force vector are respectively calculated according to the following formulas:
the number of degrees of freedom of the vibration equation of the substructure after condensation is the number of degrees of freedom of connection and the cutting modeThe sum of the orders can be greatly condensed on the substructure through modal truncation. In addition, due to the characteristic of Craig-Bampton substructure transformation, the connection degree of freedom still keeps the physical properties, and when the vibration equation after transformation is solved, the transformed coordinates can be directly used for calculating the nonlinear connection force, and the mode coordinates do not need to be returned to a physical space and then the nonlinear connection force is calculated. Therefore, the method can realize the rapid calculation of the vibration response of the equivalent nonlinear structure of the bolt.
Step 4: substructures comprehensive calculation
And carrying out sweep frequency or random vibration response calculation by utilizing the condensed substructure, and calculating interaction force between the connection degrees of freedom of the substructure according to the last moment state (displacement and speed of the connection degrees of freedom) of the connection faces of the substructure in each time step so as to carry out iterative calculation of response. Finally, an amplitude-frequency response curve of the nonlinear substructure is obtained, and parameters of the nonlinear spring damping element are corrected by comparing the amplitude-frequency response curve with test results so as to achieve the effect that simulation can better predict reality.
When in actual use, a nonlinear spring damping element degradation linear connection force model is adopted to perform preliminary calculation and correction, nonlinear items are gradually considered to perform correction, the method specifically comprises the step of taking 0 for a connection gap, and the step of taking 1 for a connection stiffness damping force, namely, taking 1 for both n and m, is the linear connection model.
The model correction specific method is not in the scope of the invention, the invention mainly provides a nonlinear connection model and correspondingly provides a quick and accurate dynamic response calculation method, and the method introduced in the invention has high calculation efficiency, and is convenient and quick to correct parameters of the nonlinear model if needed.
As a calculation case, a series structure (fig. 1 includes two bulkheads provided with end frames, and two bulkheads are connected in series), two substructures are connected in series, namely, two bulkhead substructures are connected in series, and a degree of freedom of connection exists, and the connection is considered to adopt the nonlinear stiffness damping simulation of the invention and to consider a connection gap.
FIG. 4 is a schematic diagram of an exemplary structure of the condensation calculation according to the method of the present invention. The calculation example calculates a structure as shown in fig. 4 (the left side is an illustration of the calculation example, the right side is an annotation of the calculation example), and the whole structure comprises two parts (the upper structure comprises 100 degrees of freedom in series, and the lower structure comprises 200 degrees of freedom in series), and the middle part is connected through the nonlinear spring damping element of the invention. The method is adopted to perform modal aggregation on the substructure (10 degrees of freedom are reserved in a modal space after the two parts of the structure are aggregated), the calculation result is compared with the result obtained by directly calculating the whole structure, and the time domain response, the amplitude-frequency curve, the phase diagram and the bifurcation diagram are compared, so that the method has better energy precision from the analysis result, and therefore, the method can ensure the calculation precision while improving the calculation speed.
The method for calculating the structural aggregation is adopted (the method is called herein in the figure), the structure is calculated after aggregation, and the calculation result is compared with direct calculation.
Calculating working condition I, and calculating fixed frequency excitation response:
FIGS. 5-1 (a) and (b) are diagrams showing the same degree of freedom acceleration response obtained by calculation when the excitation angular velocity of the provided example is 20rad/s, wherein (a) is the result of condensation calculation by the method of the present invention, and (b) is the result of direct calculation; (c) is an overlay of (a) and (b);
FIGS. 5-2 (a) and (b) are, respectively, a condensed calculated phase diagram and a directly calculated phase diagram of the method of the present invention; (c) is an overlay of (a) and (b);
5-3 (a) are results of condensing calculated displacement differences between the first and second connection nodes of the bolt by the method of the present invention, (b) are results of directly calculated displacement differences between the first and second connection nodes of the bolt; (c) is an overlay of (a) and (b);
FIGS. 5-4 (a) are the results of the constructed spring damper element connection force calculated by the method of the present invention by condensation and (b) are the results of the constructed spring damper element connection force calculated directly; (c) is an overlay of (a) and (b);
from the calculation result, the nonlinear equivalent and dynamic response calculation method has higher precision, and the higher precision can be maintained while the calculation speed is improved by condensing the nonlinear model.
The corresponding calculation results obtained by adjusting different excitation frequencies are shown in the table:
FIGS. 5-5 (a) show the corresponding calculation results obtained by condensing the calculated different excitation frequencies (14 rand/s-19 rand/s) by the method of the invention, and (b) show the corresponding calculation results obtained by directly calculating the calculated different excitation frequencies (14 rand/s-19 rand/s); (c) is an overlay of (a) and (b);
from the calculation result, the calculation magnitude is consistent with that of direct calculation, and the nonlinear characteristics of the nonlinear structure obtained by calculation are well consistent.
Calculating working condition II, and calculating sweep frequency:
the comparison of the calculated sweep frequency amplitude values of the two calculation methods with the same degree of freedom is as follows:
FIGS. 5-6 (a) are graphs showing the results of the same degree of freedom sweep amplitude calculation obtained by condensing the sweep frequency in the method of the present invention, and (b) are graphs showing the results of the same degree of freedom sweep amplitude calculation obtained by direct calculation; (c) is an overlay of (a) and (b);
FIGS. 5-7 (a) are the results of the same degree of freedom displacement bifurcation diagram obtained by condensation calculation in the method of the present invention, and (b) are the results of the same degree of freedom displacement bifurcation diagram obtained by direct calculation; the system has chaos and cycle doubling phenomena when excited by 15-30 rad/s during frequency sweeping;
from the calculation result, the nonlinear equivalent and dynamic response calculation method has higher precision, and the higher precision can be maintained while the calculation speed is improved by condensing the nonlinear model.
Claims (7)
1. A method of calculating equivalent and vibrational response of a bolted joint nonlinear, said bolted joint joining a structure to be joined, comprising the steps of:
establishing a connection equivalent model, namely establishing a nonlinear spring damping element on a connection region corresponding point of a finite element model of a connected piece, and calculating the connection force of the nonlinear spring damping element to the connected piece;
splitting a connected structure into a plurality of substructures, performing matrix partitioning on the substructures, and calculating nonlinear spring damping element connecting force which participates in the connection of equivalent connecting pieces in the substructures;
condensing the substructure by adopting a Craig-Bampton substructure method, and calculating a substructure modal set phi= [ phi ] N Φ C ]Calculating a modal mass matrix in the sub-structure modal spaceModal damping matrix->Modal stiffness matrix->Modal force vector->Wherein phi is N To constrain the primary mode of the substructure, phi C Is a constraint mode;
by means of condensed knotsThe structure carries out sweep frequency or random vibration response calculation, and the interaction force between the connection degrees of freedom of the substructures is calculated according to the last moment state of the connection faces of the substructures in each time step, wherein the state comprises displacement and speed of the connection degrees of freedomAnd performing iterative calculation of the response, and finally obtaining an amplitude-frequency response curve of the nonlinear substructure.
2. The method for calculating the equivalent and vibration response of a bolted nonlinear joint according to claim 1, characterized in that
Establishing a connection equivalent model, establishing a nonlinear spring damping element on a connection region corresponding point of a finite element model of a connected piece, and calculating the connection force of the nonlinear spring damping element to the connected piece, wherein the connection equivalent model comprises the following steps of
Three translational degrees of freedom of a first node and a second node of two nodes connected by a defined spring damping element are { u }, respectively x1 ,u y1 ,u z1 } T Sum { u } x2 ,u y2 ,u z2 } T Wherein the x direction is the normal direction of the connecting surface; the acting force of the nonlinear spring damping element on the connected piece is obtained through displacement and speed state calculation of three translational degrees of freedom of the connected node, the connecting force calculation method of the constructed spring damping element is specifically as follows, so as to calculate the connecting force of the first node, and the connecting force of the second node is greatly opposite to the connecting force of the first node, and the like:
wherein:
Δu i ＝u i1 -u i2 ,i=x, y, z; n and m are rigidity and damping nonlinear orders, the values are natural numbers, and the rigidity and damping nonlinear orders are determined according to the requirementThe order is preferably small under the condition that the test can be well matched; k (K) y 、K z 、k xi 、C y 、C z 、c xj For connecting rigidity and damping coefficient, delta is axial connecting gap;
the method comprises dividing the connected structure into a plurality of sub-structures, dividing the sub-structures into matrix blocks, and calculating nonlinear spring damping element connecting force participating in equivalent connector connection in the sub-structures, wherein the nonlinear spring damping element connecting force comprises
And (3) defining a degree of freedom set which does not participate in the connection of the equivalent connecting piece in the substructure as i, and a degree of freedom set which participates in the connection of the equivalent connecting piece as j, wherein a certain substructure is taken as an example, and a vibration equation of the structure can be written as follows:
wherein:
M ii 、M ij 、M ji 、M jj is a block quality matrix;
C ii 、C ij 、C ji 、C jj is a block damping matrix;
K ii 、K ij 、K ji 、K jj is a block stiffness matrix;
F i an external force vector applied to the non-connection degree of freedom of the substructure;
F j for the corresponding connection force of the nonlinear spring damping element, for the interconnected substructuresAssuming that there are k connection points, then:
F j ＝[F 1x F 1y F 1z …F kx F ky F kz ] T (3)；
the substructure is condensed by adopting a Craig-Bampton substructure method, and a substructure modal set phi= [ phi ] is calculated N Φ C ]Calculating a modal mass matrix in the sub-structure modal spaceModal damping matrix->Modal stiffness matrix->Modal force vector->Included
The following generalized eigenvalue equation is calculated by fully constraining the degrees of freedom of the connection of the substructures:
wherein:
a vibration mode matrix is calculated for the characteristic equation;
λ ii is a diagonal matrix, the kth element of the diagonal isA characteristic value corresponding to the kth column vibration mode;
for a pair ofPerforming mode cut-off, selecting low-order mode +.>Thereby constructing a constrained sub-structure main modality:
wherein 0 is jL A matrix of 0;
constructing a constraint mode according to the static equilibrium relation between the connection freedom degree and the internal freedom degree:
wherein I is jj Is a unit array;
thereby obtaining a sub-structure modal set:
Φ＝[Φ N Φ C ] (7)
carrying out modal coordinate transformation on the substructure through the modal set, namely:
wherein:
q iL the mode coordinates corresponding to the constraint main mode are obtained;
q j the mode coordinates corresponding to the constraint mode are obtained;
q is q iL And q j A modal coordinate set is formed;
as can be seen from the modal coordinate transformation, the degree of freedom of connection before and after transformation maintains the physical properties, namely x j ＝q j ；
The substructure vibration equation after coordinate transformation is as follows:
wherein the method comprises the steps ofThe modal mass matrix, the modal damping matrix, the modal stiffness matrix and the modal force vector are respectively calculated according to the following formulas:
the number of degrees of freedom of the vibration equation of the substructure after condensation is the number of degrees of freedom of connection and the cutting modeThe sum of the orders can be greatly condensed on the substructure through modal truncation.
3. The bolting nonlinear equivalent and vibration response calculation method according to claim 2, characterized in that
The method also comprises the step of nonlinear parameter correction, and specifically corrects parameters of the nonlinear spring damping element by comparing an amplitude-frequency response curve of the nonlinear substructure with test results.
4. The method for calculating the equivalent and vibration response of a bolted nonlinear joint according to claim 3, characterized in that
The method comprises the steps of correcting parameters of a nonlinear spring damping element, performing preliminary calculation and correction by adopting a linear connection force model of nonlinear spring damping element degradation in actual use, and gradually considering nonlinear terms to perform correction, wherein the method specifically comprises the step of taking 0 for a connection gap, and the step of taking 1 for a connection stiffness damping force, namely, taking 1 for both n and m, namely, the linear connection model.
5. A method of calculating the equivalent and vibrational response of a bolted joint nonlinear according to any one of claims 1 to 3, wherein
The connecting piece is a bolt and nut connecting piece.
6. A computer-readable storage medium, characterized in that the computer-readable storage medium comprises a stored program, wherein the program when run performs the bolting nonlinear equivalent and vibration response calculation method according to any of the preceding claims 1 to 5.
7. An electronic device comprising a memory and a processor, characterized in that the memory has stored therein a computer program, the processor being arranged to execute the bolting nonlinear equivalent and vibration response calculation method according to any of the claims 1 to 5 by means of the computer program.
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