CN114201903A - Rapid prediction method for vibration response of local nonlinear system - Google Patents
Rapid prediction method for vibration response of local nonlinear system Download PDFInfo
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Abstract
The invention belongs to the field of mechanical engineering, and particularly relates to a method for quickly predicting vibration response of a local nonlinear system. Compared with the existing abnormal sound vibration analysis or abnormal sound risk evaluation technology (such as SAR-Line, E-Line and the like), the method considers the actual effects of an abnormal sound source, a friction pair and a knocking pair, so that the prediction of the structural abnormal sound risk caused by external excitation is more complete, and the prediction precision of the structural vibration response is higher.
Description
Technical Field
The invention belongs to the field of mechanical engineering, and particularly relates to a method for quickly predicting vibration response of a local nonlinear system.
Background
In engineering practice, mechanical structures with local friction or tapping pairs are common, and for convenience of description, such mechanical structures with friction or tapping at specific local positions are hereinafter referred to as local nonlinear systems. For example: during the running process of an automobile, under the excitation of uneven road surface or the vibration excitation of a power assembly, a plurality of local knocking or friction often occurs between an automobile body and adjacent parts such as an instrument assembly, an automobile door and a seat and between adjacent parts inside the instrument assembly, the automobile door and the seat, and at the moment, the automobile or a specific subsystem thereof can be regarded as a local nonlinear system; the frictional contact that exists between the borehole wall and the drill bit allows the drill bit to be equivalent to a locally nonlinear system during drilling operations. In summary, locally nonlinear systems are not well enumerated in engineering practice.
The mechanical structure vibration has important influence on the working performance and the service life of a product, the vibration and the noise radiated by the vibration also worsen the living environment of people, and particularly, the structural abnormal sound caused by knocking, friction and the like can greatly reduce the evaluation of a user on the performance quality of the machine. Therefore, the abnormal sound performance is always a key index for representing the quality of products, and is particularly emphasized in the automobile industry. In the product design stage, the abnormal sound performance of the product is accurately predicted, the design scheme is optimized, abnormal sound is avoided, the adjustment work of the later-stage abnormal sound problem can be reduced, and the development cost and the development period are saved. At present, no mature high-efficiency algorithm is available for calculating abnormal sound vibration response of a complex mechanical structure or system, the main reason is that a nonlinear system is not suitable for a superposition principle, a direct integration method is adopted for solving a high-dimensional nonlinear non-homogeneous differential equation set, the calculation time is too long, and the existing algorithm is difficult to be applied to the abnormal sound vibration response analysis prediction of complex systems such as automobiles and the like. In fact, in the automobile industry, an SAR-Line or E-Line method is mainly adopted to carry out abnormal sound risk assessment on possible abnormal sound points in the whole automobile and part structures of the automobile, the method does not directly consider the actual motion or dynamic behavior of a friction pair or a knocking pair, neglects the influence of nonlinear links on the vibration response of the structure, calculates the vibration response of the whole structure according to a linear system, and carries out abnormal sound risk assessment according to information such as relative vibration displacement and speed of preselected risk part node pairs. The method avoids the vibration response calculation of a nonlinear system, is low in time cost, can quickly obtain the abnormal sound risk evaluation result of the automobile structure, is poor in evaluation precision and effectiveness, and is difficult to meet the actual engineering requirements of automobile product research and development. So far, the development of the abnormal sound performance of the automobile has to rely more on empirical design and later-stage automobile model adjustment.
Therefore, the invention fully utilizes the local nonlinear characteristics of the abnormal sound prediction problem, namely in almost all practical engineering problems, in the situation that the abnormal sound vibration needs to be analyzed and predicted, the machine structure or the system can be regarded as a linear system, and the nonlinearity exists only in a knocking kinematic pair or a friction kinematic pair at one or a plurality of positions, the invention proposes that an equivalent model is utilized to independently process the nonlinear links, establishes the nonlinear mapping relation between the knocking force or the friction force and the motion input of the knocking force or the friction force and takes the knocking force or the friction force as the external excitation force of the original mechanical structure or the system, thereby enabling the abnormal sound vibration response of the original mechanical structure or the system to be calculated by adopting the superposition principle according to the linear system, not only considering the local nonlinear effect, but also fully utilizing the linear property of the original mechanical structure, and enabling the rapid prediction of the abnormal sound vibration response to be possible, so as to better meet the actual engineering requirements of the research and development of the abnormal sound performance of products such as automobiles and the like.
Disclosure of Invention
The invention aims to provide a method for quickly predicting the vibration response of a local nonlinear system so as to meet the requirement of quickly predicting the structural vibration response of the local nonlinear system in engineering practice.
In order to achieve the aim, the invention provides a rapid prediction method of vibration response of a local nonlinear system, which comprises the steps of splitting the whole system into a main linear system part and local nonlinear links, respectively processing each local nonlinear link into an equivalent model to express the nonlinear mapping relation between each local friction force or knocking force and friction motion or knocking motion, and further realizing rapid prediction of forced vibration response of the whole system through combined calculation;
external excitation causing the forced vibration response of the whole system is called primary excitation, friction force and knocking force generated by each local nonlinear link are called secondary excitation, and the vibration response of the main linear system under the combined action of the primary excitation and the secondary excitation of the local nonlinear links is solved by means of a mature algorithm and a superposition principle, so that the quick prediction of the vibration response of the whole local nonlinear system is realized;
in order to obtain the vibration response of the main linear system under the combined action of primary excitation and secondary excitation, two combined calculation methods of the main linear system and a nonlinear equivalent model, namely a combined calculation method in a serial mode and a combined calculation method in a feedback mode, are provided.
Further, serially and sequentially calculating the vibration response of the main linear system caused by primary excitation, the secondary excitation generated by the nonlinear equivalent model and the vibration response of the main linear system caused by secondary excitation, and then superposing the two vibration responses of the main linear system, so as to obtain the total vibration response of the whole local nonlinear system under external excitation, wherein the specific calculation steps of the method are as follows:
the method comprises the following steps: setting the initial state of the local nonlinear system, generally setting the static state of the main linear system at a static balance position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
step two: calculating a primary vibration response time course of the linear system of the main body under external excitation (primary excitation force);
step three: aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating a friction force or knocking force process based on a primary vibration response process of a main linear system at each specific local part;
step four: taking the friction force or knocking force history obtained in the step three as a secondary excitation force to act on the main linear system independently, and calculating a secondary vibration response history;
step five: and (3) calculating the algebraic sum of the secondary vibration response process and the primary vibration response process, namely the total vibration response of the whole local nonlinear system under external excitation.
Further, the equivalent model of each local nonlinear link is used as a feedback link of the main linear system, that is, the vibration response of each specific local in the main linear system is instantly fed back as a secondary excitation force through the nonlinear equivalent model, so that the secondary excitation force and the primary excitation form the excitation input of the main linear system together, and the vibration response under the combined action of the two excitations is obtained, wherein the specific calculation steps of the method are as follows:
the method comprises the following steps: setting the initial state of the local nonlinear system, generally setting the static state of the main linear system at a static balance position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
step two: the value of external excitation (primary excitation force) at the current moment and the value of nonlinear secondary excitation force at the current moment (namely the equivalent model feedback output value at the previous moment) are jointly superposed to be used as the excitation input of the main linear system at the current moment, and the vibration response at the next moment is calculated;
step three: aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating corresponding friction force or knocking force at the next moment based on the vibration response of the main linear system at the next moment of each specific local link, namely, the secondary excitation force fed back and output by the equivalent model of each specific local nonlinear link;
step four: and (5) advancing by a time step, returning to the step two, and repeating the steps to obtain the total vibration response course of the local nonlinear system under external excitation.
Further, where the forced vibration response time history of the main linear system portion is solved, any existing well-established numerical computation method suitable for linear vibration systems may be employed, including and not limited to finite element methods, time domain methods, frequency domain methods, and the like.
Further, for the solution of the forced vibration response of the main linear system part in the second step, any existing mature numerical value calculation method suitable for time domain recursive solution of the linear vibration system can be adopted, including but not limited to a Wilson-theta method, a Newmark-beta method, a central difference method and the like, and the calculation efficiency can be further improved by combining a modal truncation method, a dynamic polycondensation method and other order reduction methods.
The invention has the beneficial effects that: compared with the existing abnormal sound vibration analysis or abnormal sound risk evaluation technology (such as SAR-Line, E-Line and the like), the method considers the actual effects of an abnormal sound source, a friction pair and a knocking pair, so that the prediction accuracy of the structural abnormal sound risk caused by external excitation is higher according to the estimation of the structural abnormal sound risk, and the structural vibration response is higher (the structural vibration response calculated by the existing method does not contain any effects of the friction pair and the knocking pair at all).
Compared with the existing nonlinear system vibration response calculation method, the method fully utilizes the characteristic of local nonlinearity of the system, and respectively processes the local nonlinearity and the linear system part of the main body, so that the solution of the large-scale vibration system differential equation set is still linear system solution, and the time-consuming solution of the large-scale nonlinear differential equation set is avoided. The method makes accurate numerical calculation of complex engineering problems such as automobile abnormal sound analysis and evaluation possible, and has important significance for improving the development level and efficiency of automobile abnormal sound performance.
Drawings
FIG. 1 is a block diagram of a combined computation flow in a serial manner;
FIG. 2 is a block diagram of a combined computation flow for a feedback mode;
FIG. 3 is an experimental field diagram;
FIG. 4 is a finite element model of a cantilever beam structure;
FIG. 5 is a time domain comparison graph of the results of the combined calculations in a serial fashion;
FIG. 6 is a frequency domain comparison graph of the results of the combined calculations in serial fashion;
FIG. 7 is a time domain comparison graph of the combined calculation results in the feedback mode;
fig. 8 is a frequency domain comparison graph of the results of the feedback mode combination calculations.
Detailed Description
The following is a more detailed description of the present invention by way of specific embodiments.
In a local nonlinear system, the main body part of the system is a linear system except for each locally existing knocking or friction part, and still meets the superposition principle, if each locally existing knocking or friction force is regarded as a secondary excitation force, any fast algorithm suitable for the linear system can be adopted to calculate the forced vibration response of the system; and for each locally existing knocking or friction nonlinear relation, an equivalent model can be adopted to respectively express the relation, and corresponding knocking force and friction force output, namely secondary exciting force acting on a main body system, can be obtained as long as relative motion input of each knocking pair and each friction pair is obtained. Therefore, the whole local nonlinear system is decomposed into a main linear system and local nonlinear links, and each local nonlinear link is processed by adopting an equivalent model so as to express the nonlinear mapping relation between each local friction force or knocking force and the relative motion of the local friction force or knocking force; the nonlinear equivalent model and the main linear system are combined to obtain an equivalent local nonlinear system, and external excitation is applied to the equivalent local nonlinear system, so that the vibration response under the combined action of secondary excitation including local knocking, friction and the like can be obtained.
The specific implementation mode of the invention for calculating the forced vibration response of the local nonlinear system is as follows:
a preparation step 1: and respectively establishing corresponding nonlinear equivalent models aiming at each local nonlinear link of the system. For example, for a knocking pair, the input of the equivalent model is the normal relative speed at the knocking points of two adjacent pieces, and the output is the normal knocking force acting on the knocking points; for the friction pair, the input of the equivalent model is the tangential relative displacement and the speed of the friction contact position of two adjacent parts, and the output is the tangential friction force acting on the friction contact position.
A preparation step 2: and aiming at the linear system part of the main body, establishing a dynamic analysis model of the main body, solving the inherent problem of the main body, and obtaining the undamped inherent frequency and the main vibration mode of the linear system of the main body.
For example, a finite element method is adopted for numerical calculation, a finite element model of the system is established, in order to adapt to subsequent local nonlinear model combination calculation, a potential local knocking pair or friction pair needs to be established, when units are divided, node pairs need to be established at positions where knocking or friction may occur on adjacent components, even if the finite element units on the adjacent components at the positions are collinear in the normal direction at the node positions of adjacent surfaces or contact surfaces, so that the normal or tangential relative motion relationship of the knocking pair or the friction pair is extracted in the subsequent calculation. The forced vibration response for a subject linear system can be reduced to solve the following kinetic equations.
Solving the undamped natural frequency and the main vibration mode of the main linear system can be summarized as solving the following matrix eigenvalue problem.
M, C, K are the mass matrix, damping matrix and stiffness matrix of the system,
x is the acceleration vector, the velocity vector and the displacement vector of the system, F (t) is the external excitation force vector acting on the system and can be any function of time t, A is the main mode vector of the system, and omeganThe undamped natural frequency of the system, if the degree of freedom of the subject linear system is n, all the vectors are n-dimensional vectors, and all the matrices are n × n-dimensional matrices. In order to solve the forced vibration response by adopting a modal superposition method, only the first m-order natural frequency and the main mode shape of the system are required to be solved, and m is less than n. Here, any sophisticated algorithm may be used to solve (2)And (4) solving the matrix eigenvalue problem of the formula to obtain the required inherent frequencies and the main vibration modes of the first orders.
The method comprises the following steps: the initial state of the local nonlinear system is set, and the static state of the main linear system at the static equilibrium position can be generally set, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero.
Each local nonlinear link is processed by adopting an equivalent model respectively, and the output of each equivalent model, namely each knocking force or friction force, is regarded as a secondary excitation force acting on the main linear system, so that the forced vibration response of the local nonlinear system under external excitation is equivalently summarized as solving the following equation:
where Q (t) is an n-dimensional quadratic excitation force vector. In view of the local nonlinear characteristics of the system, usually, the secondary excitation force exists only in a small number of knocking or friction parts, so that most elements in the vector of the secondary excitation force are constantly zero, only a small number of elements are nonzero, and the excitation force magnitudes are respectively given by corresponding equivalent models.
The invention provides two different combined calculation modes, and the combined calculation of the main linear system and each local nonlinear equivalent model is carried out to obtain the total vibration response of the whole local nonlinear system under external excitation. One is a serial combined calculation method, as shown in fig. 1, the method sequentially calculates the vibration response of the main linear system under external excitation, secondary excitation and the vibration response of the main system caused by the secondary excitation, and then superposes the vibration responses of the two excitations, thereby obtaining the total vibration response of the whole system; the other is a combined calculation method of a feedback mode, as shown in fig. 2, the method processes each local nonlinear equivalent model as a feedback link of a main linear system, the main linear system generates a vibration response under external excitation, so that relative motion input of each local nonlinear equivalent model is obtained, secondary excitation force output is obtained, the secondary excitation force output is fed back to act on the main linear system, and further the vibration response of the main linear system at the next moment is influenced, so that the total vibration response of the whole system is obtained through the cyclic action.
Obviously, the subsequent calculation method is different according to different combination calculation methods.
a. In serial mode
If a serial combined calculation method is adopted, the following specific calculation method is as follows.
Step two: and calculating the vibration response time history of the main linear system under the action of external excitation (primary excitation force), which can be referred to as primary vibration response.
This step is in fact to solve equation (1). Any sophisticated algorithm may be employed due to the linear system solution. For example, a modal superposition method can be adopted, the equation (1) is decoupled by using the weighted orthogonal property of the principal mode shape, decoupling can be realized on a damping matrix in the damping matrix by adopting an approximate processing mode of neglecting a non-diagonal element after matrix transformation, then a forced response time history under the previous m-order modal coordinate is solved by using a direct integration method, and then modal superposition is carried out to obtain a response time history under the generalized coordinate, namely a solution of the equation (1). If the primary excitation force is a periodic function, it can be expanded into a Fourier series, and the solution of equation (1) can be obtained faster using the harmonic response calculation and superposition principle.
Step three: and aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating the friction force or the knocking force process based on the primary vibration response process of the main linear system at each specific local part.
This step obtains a secondary excitation force vector q (t). Note that when the local nonlinear link is located inside the system, that is, both members with knocking or friction belong to the components of the system, for example, friction or knocking occurs between the r-th node of the member a and the s-th node of the member b, the secondary excitation forces should occur in pairs, have equal magnitude and opposite directions, and act on the r-th node and the s-th node of the member a and the member b respectively; when the local nonlinear link is located at the boundary of the system, namely a part of a certain component of the system is knocked or rubbed with an external fixture, a secondary excitation force is applied to the corresponding part or node of the component.
Step four: taking the friction force or knocking force history obtained in the step three as a secondary excitation force to act on the main linear system independently, and calculating a secondary vibration response history;
this step is in fact a solution to the response of the subject linear system under quadratic excitation, which is equivalent to solving equation (1), but in which the excitation force vector is transformed into q (t). Similar to the solution in step two, any mature solution algorithm of the forced response of the linear system can be adopted, which is not described herein.
Step five: and (3) calculating the algebraic sum of the secondary vibration response process and the primary vibration response process, namely the total vibration response of the whole local nonlinear system under external excitation.
This step completes the solution of equation (3) based on the superposition principle of the linear system, which is equivalent to the response solution of the whole local nonlinear system under external excitation.
b. Feedback mode
Although the combined calculation method in the serial manner is convenient to implement, the mutual coupling effect between the nonlinear equivalent model and the main linear system is neglected, and extra calculation errors are inevitably caused. In fact, the secondary excitation force output by the nonlinear equivalent model depends on the vibration response of the main linear system, and once the secondary excitation force is generated, the secondary excitation force necessarily acts on the main linear system immediately, and the vibration response of the main linear system is also necessarily changed, so that the coupling influence is generated on the output of the nonlinear equivalent model. For this reason, the present invention proposes a feedback-type combined calculation method with smaller calculation errors (see fig. 2), and the following specific calculation method is as follows.
Step two: and (3) superposing the value of the external excitation (primary excitation force) at the current moment i and the value of the nonlinear secondary excitation force at the current moment i (namely the equivalent model feedback output value at the previous moment) together to be used as the excitation input of the main linear system at the current moment i, and calculating the vibration response of the system at the next moment (i + 1).
This step calculates the solution of equation (3), which is essentially a time domain solution of the forced response of a linear system, and any well-established solution method of existing linear systems can be used. For example, for large-scale linear system time domain solution, a mode of mode superposition and mode truncation can be adopted, the solution of the equation (3) is expressed as superposition of forced responses under the previous several orders of mode coordinates, and the forced responses under each mode coordinate can be solved by adopting an impulse response method, namely expressed as convolution integral of a system impulse response function and an excitation force under the mode coordinates; the time domain numerical solution of equation (3) may also be obtained by using a dynamic polycondensation method to reduce the scale or order of the model or equation (3) and then solving the equation after polycondensation by using a direct integration method (e.g., Wilson-theta method, Newmark-beta method, etc.).
Step three: and aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating corresponding friction force or knocking force at the next moment (i +1) based on the vibration response of the main linear system at the next moment (i +1) of each specific local link, namely secondary excitation force fed back and output by the equivalent model of each specific local nonlinear link.
The step of calculating obtains the feedback output of the equivalent model of each local nonlinear link, namely the value of the secondary excitation force vector Q (t) at the next moment, and the feedback output is used for the recursion calculation of the equation (3) at the next moment. Note that consistent with the foregoing, when the locally nonlinear element is located inside the system, the secondary excitation forces always occur in pairs, acting on both nodes of the pair of nodes, either the strike pair or the friction pair, respectively.
Step four: and (3) advancing by a time step length by setting i as i +1, returning to the step two, and repeating the steps to obtain the total vibration response history of the local nonlinear system under external excitation, namely obtaining the time domain numerical solution of the equation (3).
Example (b):
the cantilever beam structure with local friction pair is taken as an example to illustrate the implementation method and the applicability of the invention. FIG. 3 is an experimental field diagram, wherein a 960mm long aluminum square section long rod is fixed on a bench through a steel bracket at the end to form a cantilever beam structure; in the figure, points 1, 2 and 3 respectively represent an excitation position of a vibration exciter, a friction side effect position and a vibration response observation position or a reference point; the vibration exciter applies external excitation (primary excitation) to the cantilever beam at the point 1 to make the whole structure generate forced vibration; a friction pair is arranged at the point 2 through an additional bracket, namely, the cantilever beam is in prepressing contact with an external fixed boundary at the point, when the cantilever beam generates transverse bending vibration response under the action of a vibration exciter, friction is generated at the point to form secondary exciting force, and the secondary exciting force is a typical nonlinear link; under the excitation of a vibration exciter, the vibration response of the cantilever beam is simultaneously influenced by a primary excitation force and a secondary excitation force, and the vibration response is observed at a reference point 3 by using an acceleration sensor in the test to compare the consistency of a predicted value and an actually measured value.
Aiming at the cantilever beam experiment system, hexahedral units are selected to divide the grids, and a finite element model is established, as shown in fig. 4. In order to verify the correctness of the finite element model and compare modal experiments, table 1 lists the results of the first four-order modes, and it can be seen that the relative error between the modal frequency calculated by the finite element and the modal frequency obtained by the experiments is less than 3%, which indicates that the established finite element model can basically express the dynamic characteristics of the real cantilever beam.
TABLE 1 finite element calculation modal frequency vs. Experimental modal frequency
Defining the error percentage epsilon of the mean square root value of the acceleration to quantitatively evaluate the vibration response prediction effect:
in the formula, rms _ a' is the root mean square value of the simulated and predicted acceleration in the whole time history, and rms _ a is the experimentally measured acceleration root mean square value.
And exporting the finite element model to a self-compiled special calculation program. The program is compiled according to the method, a system mass matrix and a stiffness matrix can be assembled according to a finite element model, a system damping matrix can be defined according to the internal resistance of the material, an excitation force vector can be imported or customized, and a nonlinear link equivalent model can be imported or customized, so that a motion differential equation set of an analysis object is obtained, and the equation solution is carried out according to the method disclosed by the invention, wherein the shape of the equation set is shown as equation (3). The specific calculation process is as follows:
a. performing calculations by a combined calculation method in a serial manner
1) The friction pair equivalent model of position 2 is imported. In order to obtain the equivalent model, a reciprocating friction test is carried out on a sample piece on a special testing machine according to specific materials of a friction pair, and then parameters of the equivalent model are identified by using test data. The input of the equivalent model is friction pair positive pressure, friction pair tangential motion relative displacement, relative speed and relative acceleration, and the output is friction force; in this example, the positive pressure of the friction pair is a given constant, the relative motion of the friction pair is the displacement, speed and acceleration of the cantilever beam at the point 2, and all are transient variation, the friction force of the friction pair acts on the cantilever beam point 2 to always block the transverse motion of the cantilever beam, and is a dynamic force which is instantly changed along with the input quantity of the equivalent model; considering the actual size of the contact surface of the friction pair, the cantilever beam unit node covered by the contact surface is far more than one, the friction force can be equivalently acted on one node at the center of the friction surface, and the friction force can be dispersed to act on each node covered by the friction surface, and the difference between the two nodes is small.
2) Exciting force f of vibration exciter1Acting on 8 nodes near 1 point of the cantilever beam, and calculating a primary vibration response course X of the finite element model by means of a Newmark-beta algorithm0And note that the initial state of the system is defined as the static equilibrium position static state; in the present embodiment, the applied excitation of the vibration exciter is a harmonic force, and the time history of the vibration response can be obtained more quickly according to the harmonic response calculation method, and the two are not different in numerical result. Without loss of generality, we actually use the Newmark-beta algorithm.
3) The primary vibration response process of the cantilever beam at 81 nodes near the 2 points, including displacement, speed and acceleration time processes, obtained by the calculation in the above steps is used as a friction force model of each node
So as to calculate the frictional force history q acting on the 81 nodesf＝[qp+1，qp+2，…，qp+81]T。
4) Calculating the secondary vibration response time history X of the finite element model by means of a Newmark-beta algorithm1The second excitation force is the friction force history q obtained by the previous calculationf＝[qp+1，qp+2，…，qp+81]TIt acts on 81 nodes of the cantilever beam near point 2.
5) Calculating a vibration response course X0And a secondary vibration response history X1The algebraic sum of (1) is the total vibration response of the whole cantilever beam structure with the friction pair under the excitation of the vibration exciter, and X is X0+X1。
Extracting acceleration time histories of total vibration responses of nodes corresponding to the positions of the cantilever beam 3 points, and comparing the acceleration time histories with vibration acceleration time histories obtained by actual measurement of corresponding positions in an experiment, wherein the result is shown in figure 5, the whole variation trend of a prediction result is basically consistent with that of an experiment result, but the waveform difference is large; FIG. 6 shows a comparison of the self-power spectral densities of two signals, where it can be seen that the calculated results substantially coincide with the odd harmonic components of the experimental results, but lack the even harmonic components; table 2 shows the result of the quantitative comparison, and the relative error of the combined calculation result in the serial manner reaches 24%, which indicates that the prediction accuracy of the method for the vibration response of the local nonlinear system is not high enough, and the coupling effect between the nonlinear link and the main linear system is not negligible but superior to the realization of the method, especially the calculation part for the vibration response of the linear system, even can be realized by the existing software without programming.
b. Performing calculation according to combined calculation method in feedback mode
1) The friction pair equivalent model of position 2 is imported. This step is consistent with the foregoing method and is not described herein.
2) Defining the initial state of the cantilever beam as being at rest in a static equilibrium position, i.e. at rest
Obviously, the friction force of the 2-point position at the initial moment is also zero, namely, the 2-point position is set
Is the friction force on all the unit nodes corresponding to the friction surface; here, each time-varying physical quantity is represented by a superscript i, and the initial time is 0.
3) And solving the vibration response of the main linear system under the action of the primary excitation and the secondary excitation at the current moment, namely solving an equation (3). Exciting force of vibration exciter at current moment
(action and 8 nodes at cantilever beam point 1) and friction force fed back by friction pair equivalent model
(81 nodes acting on the friction face overlay at cantilever point 2,
the vibration response of the finite element model at the moment i +1 is calculated by means of a Newmark-beta algorithm together as the input of the finite element model
Note that the calculation time step (the time difference between two adjacent time instants) should not be too large. In order to further improve the calculation efficiency, especially when the scale of the main linear system is large, the calculation here can adopt a mode superposition and mode truncation method, and reduce the number of differential equations which need to be directly integrated, thereby improving the calculation speed, and as long as the order of the mode truncation is proper, the calculation precision is hardly influenced.
4) Calculating the feedback output of the friction pair equivalent model, namely the vibration response of 81 nodes covered by the friction surface near the cantilever beam point 2 at the moment of i +1, which is obtained by the calculation in the step
As input to the friction pair equivalent model, the frictional force acting on the 81 nodes at time i +1 can be obtained, that is:
5) and (3) advancing a time step, making i equal to i +1, returning to 3), continuing to calculate, and repeating the steps to complete the numerical solution of the equation (3), so as to obtain the total vibration response time history of the cantilever beam structure with the friction pair under the excitation of the vibration exciter, including the displacement, speed and acceleration time histories of each node.
Extracting the total vibration response history of the 3 positions of the cantilever beam reference point, and comparing the total vibration response history with the actually measured acceleration time history of the experiment at the same position, wherein the result is shown in FIG. 7, and the simulation prediction curve is almost overlapped with the experiment curve; fig. 8 shows a comparison of the self-power spectral densities of the two, and it can be seen that the odd-even sub-harmonic components of the calculation result are both present and highly consistent with the experimental result. The method can accurately reflect the coupling action and influence relation between the friction pair actually existing in the local nonlinear system and the main linear system, and has quite high accuracy on nonlinear vibration response prediction. The comparison result of the root mean square values of the simulation predicted signal and the actual measurement signal is shown in table 2, and it can be seen that the relative error of the combined calculation result of the feedback mode is only 0.725%, which is far more accurate than the combined calculation result of the serial mode.
TABLE 2 comparison of vibration response prediction results with experimental results for reference point 3
The above example calculation procedure utilizes a finite element model to express the subject linear system and solves the linear system differential equations with the help of a stepwise integration method. Two points must be pointed out, the method for solving the vibration response of the local nonlinear system of the invention does not limit the mathematical expression mode of the main linear system, and can adopt a differential equation set, or various forms such as an impulse response function matrix, a frequency response function matrix and the like; meanwhile, the method for solving the partial forced response of the main linear system is not limited, and any method capable of obtaining the vibration response time history of the linear system is feasible.
It should be noted that, unless otherwise explicitly stated or limited, the terms "mounted," "connected," "fixed," and the like are used broadly in the present invention, and may be, for example, fixedly connected, detachably connected, or integrally connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
The foregoing is merely an example of the present invention and common general knowledge of known specific structures and features of the embodiments is not described herein in any greater detail. It should be noted that, for those skilled in the art, without departing from the structure of the present invention, several changes and modifications can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicability of the patent. The scope of the claims of the present application shall be determined by the contents of the claims, and the description of the embodiments and the like in the specification shall be used to explain the contents of the claims.
Claims (5)
1. The method for quickly predicting the vibration response of the local nonlinear system is characterized in that the whole system is divided into a main linear system part and local nonlinear links, each local nonlinear link is processed into an equivalent model respectively to express the nonlinear mapping relation between each local friction force or knocking force and friction motion or knocking motion, and then the forced vibration response of the whole system is quickly predicted through combined calculation;
external excitation causing the forced vibration response of the whole system is called primary excitation, friction force and knocking force generated by each local nonlinear link are called secondary excitation, and the vibration response of the main linear system under the combined action of the primary excitation and the secondary excitation of the local nonlinear links is solved by means of a mature algorithm and a superposition principle, so that the quick prediction of the vibration response of the whole local nonlinear system is realized;
in order to obtain the vibration response of the main linear system under the combined action of primary excitation and secondary excitation, two combined calculation methods of the main linear system and a nonlinear equivalent model, namely a combined calculation method in a serial mode and a combined calculation method in a feedback mode, are provided.
2. The method for rapidly predicting the vibration response of the local nonlinear system according to claim 1, wherein the vibration response of the main linear system caused by the primary excitation, the secondary excitation generated by the nonlinear equivalent model and the vibration response of the main linear system caused by the secondary excitation are serially and sequentially calculated, and then the two vibration responses of the main linear system are superposed, so that the total vibration response of the whole local nonlinear system under the external excitation is obtained, wherein the method comprises the following specific calculation steps:
the method comprises the following steps: setting the initial state of the local nonlinear system, generally setting the static state of the main linear system at a static balance position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
step two: calculating a primary vibration response time course of the linear system of the main body under external excitation (primary excitation force);
step three: aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating a friction force or knocking force process based on a primary vibration response process of a main linear system at each specific local part;
step four: taking the friction force or knocking force history obtained in the step three as a secondary excitation force to act on the main linear system independently, and calculating a secondary vibration response history;
step five: and (3) calculating the algebraic sum of the secondary vibration response process and the primary vibration response process, namely the total vibration response of the whole local nonlinear system under external excitation.
3. The method for rapidly predicting the vibration response of the local nonlinear system according to claim 1, wherein the equivalent model of each local nonlinear link is used as a feedback link of the main linear system, that is, the vibration response of each specific local in the main linear system is instantly fed back to a secondary excitation force through the nonlinear equivalent model, so that the secondary excitation force and the primary excitation form the excitation input of the main linear system, and the vibration response under the joint action of two excitations is obtained, and the method comprises the following specific calculation steps:
the method comprises the following steps: setting the initial state of the local nonlinear system, generally setting the static state of the main linear system at a static balance position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
step two: the value of external excitation (primary excitation force) at the current moment and the value of nonlinear secondary excitation force at the current moment (namely the equivalent model feedback output value at the previous moment) are jointly superposed to be used as the excitation input of the main linear system at the current moment, and the vibration response at the next moment is calculated;
step three: aiming at each local nonlinear link, adopting a corresponding equivalent model, and calculating corresponding friction force or knocking force at the next moment based on the vibration response of the main linear system at the next moment of each specific local link, namely, the secondary excitation force fed back and output by the equivalent model of each specific local nonlinear link;
step four: and (5) advancing by a time step, returning to the step two, and repeating the steps to obtain the total vibration response course of the local nonlinear system under external excitation.
4. The method of claim 2, wherein the solution of the forced vibration response time history of the main linear system part can be implemented by any existing well-established numerical calculation method suitable for linear vibration systems, including but not limited to finite element method, time domain method, frequency domain method, etc.
5. The method for fast predicting the vibration response of the local nonlinear system according to claim 3, wherein for the solution of the forced vibration response of the main linear system part in the step two, any existing mature numerical calculation method suitable for the time domain recursive solution of the linear vibration system can be adopted, including but not limited to Wilson- θ method, Newmark- β method, central difference method, etc., and the calculation efficiency can be further improved by combining with reduced order methods such as modal truncation and dynamic polycondensation.
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