A fast prediction method for vibration response of local nonlinear systems

Abstract

The present invention belongs to the field of mechanical engineering, and specifically relates to a method for rapid prediction of vibration response of a local nonlinear system, which divides the entire system into a main linear system part and a local nonlinear link, and processes each local nonlinear link as an equivalent model to express the nonlinear mapping relationship between each local friction force or knocking force and the friction motion or knocking motion, and realizes rapid and accurate prediction of the friction force or knocking force in the local nonlinear link, and then realizes rapid prediction of the forced vibration response of the entire system through combined calculation. Compared with the existing abnormal noise vibration analysis or abnormal noise risk assessment technology, the present invention takes into account the actual role of the abnormal noise source or friction pair and knocking pair, so the estimation of the risk of structural abnormal noise caused by it under external excitation is more sufficient, and the structural vibration response prediction accuracy is higher.

Description

Technical Field

[0001] The invention belongs to the field of mechanical engineering, and in particular relates to a method for quickly predicting vibration response of a local nonlinear system. Background Art

[0002] In engineering practice, mechanical structures with friction pairs or knocking pairs are common. For the convenience of description, this kind of mechanical structure with friction or knocking at a specific local position is called a local nonlinear system. For example, when a car is driving, under the excitation of uneven road surface or vibration of the powertrain, there are often some local knocking or friction between the car body and adjacent parts such as the instrument assembly, door, seat, etc., and between the adjacent parts inside the instrument assembly, door, seat, etc. At this time, the car or its specific subsystem can be regarded as a local nonlinear system; the friction contact between the well wall and the drill bit makes the drill bit in the drilling operation equivalent to a local nonlinear system. In short, local nonlinear systems are too numerous to mention in engineering practice.

[0003] Mechanical structure vibration has an important impact on the working performance and service life of the product. Vibration and the noise it radiates also worsen people's living environment, especially the structural abnormal noise caused by knocking and friction, which will greatly reduce the user's evaluation of the performance quality of the machine. Therefore, abnormal noise performance has always been a key indicator to characterize product quality, and is particularly valued in the automotive industry. In the product design stage, accurately predicting the abnormal noise performance of the product and then optimizing the design scheme to avoid abnormal noise can reduce the adjustment work of the abnormal noise problem in the later stage, thereby saving development costs and cycles. At present, there is no mature and efficient algorithm available for calculating the abnormal noise vibration response of complex mechanical structures or systems. The main reason is that the superposition principle is not applicable to nonlinear systems. The direct integration method is used to solve high-dimensional nonlinear nonhomogeneous differential equations, and the calculation takes too long. The existing algorithm is difficult to apply to the analysis and prediction of abnormal noise vibration response of complex systems such as automobiles. In fact, the SAR-Line or E-Line method is mainly used in the automotive industry to make abnormal noise risk assessments for possible abnormal noise points in the structure of automobiles and parts. This type of method does not directly examine the actual movement or dynamic behavior of the friction pair or the knocking pair, and also ignores the influence of these nonlinear links on the structural vibration response. Instead, the vibration response of the entire structure is calculated according to the linear system, and the abnormal noise risk assessment is performed based on the relative vibration displacement, velocity and other information of the pre-selected risk position node pairs. This type of method avoids the vibration response calculation of the nonlinear system, takes less time and cost, and can quickly obtain the abnormal noise risk assessment results of the automobile structure, but the assessment accuracy and effectiveness are very poor, and it is difficult to meet the actual engineering needs of automobile product research and development. So far, the development of automobile abnormal noise performance has to rely more on empirical design and later prototype adjustment.

[0004] To this end, the present invention makes full use of the local nonlinear characteristics of the abnormal noise prediction problem, that is, in almost all practical engineering problems, in any situation where abnormal noise and vibration need to be analyzed and predicted, the machine structure or system itself can be regarded as a linear system, and the nonlinearity only exists in one or several knocking motion pairs or friction motion pairs. It is proposed to use an equivalent model to separately deal with these nonlinear links, establish a nonlinear mapping relationship between the knocking force or friction force and its motion input, and regard the knocking force or friction force as an external excitation force of the original mechanical structure or system, so that the abnormal noise and vibration response of the original mechanical structure or system can still be calculated according to the superposition principle of the linear system. Therefore, the effect of local nonlinearity is taken into account, and the linear properties of the original machine structure are fully utilized, so that the rapid prediction of the abnormal noise and vibration response becomes possible, so as to better meet the actual engineering needs of the research and development of abnormal noise performance of products such as automobiles. Summary of the invention

[0005] The purpose of the present invention is to provide a method for quickly predicting the vibration response of a local nonlinear system, so as to meet the demand for quickly predicting the vibration response of a local nonlinear system structure proposed in engineering practice.

[0006] In order to achieve the above-mentioned purpose, the present invention provides a method for quickly predicting the vibration response of a local nonlinear system, which divides the entire system into a main linear system part and a local nonlinear link, and processes each local nonlinear link as an equivalent model to express the nonlinear mapping relationship between each local friction force or knocking force and the friction motion or knocking motion, thereby realizing a quick prediction of the forced vibration response of the entire system through combined calculation;

[0007] The external excitation that causes the forced vibration response of the entire system is called primary excitation, and the friction and knocking force generated by each local nonlinear link is called secondary excitation. With the help of mature algorithms and superposition principle, the vibration response of the main linear system under the combined action of primary excitation and secondary excitation of local nonlinear links is solved, thereby realizing the rapid prediction of the vibration response of the entire local nonlinear system.

[0008] 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 the nonlinear equivalent model are provided, namely, the combined calculation method of serial mode and the combined calculation method of feedback mode.

[0009] Furthermore, 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 calculated in series, and then the two vibration responses of the main linear system are superimposed to obtain the total vibration response of the entire local nonlinear system under external excitation. The specific calculation steps of this method are as follows:

[0010] Step 1: Set the initial state of the local nonlinear system. Generally, the main linear system can be set to be in a static equilibrium position and static state, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;

[0011] Step 2: Calculate the time history of the primary vibration response of the main linear system under external excitation (primary excitation force);

[0012] Step 3: For each local nonlinear link, use the corresponding equivalent model and calculate the friction force or knocking force history based on the primary vibration response history of the main linear system in each specific local area;

[0013] Step 4: Use the friction force or knocking force history obtained in step 3 as the secondary excitation force to act on the main linear system alone, and calculate the secondary vibration response history;

[0014] Step 5: Find the algebraic sum of the secondary vibration response history and the primary vibration response history, which is the total vibration response of the entire local nonlinear system under external excitation.

[0015] Furthermore, the equivalent models of each local nonlinear link are used as the feedback link of the main linear system, that is, the vibration response of each specific local part in the main linear system is instantly fed back as a secondary excitation force through the nonlinear equivalent model, so that it and the primary excitation together constitute the excitation input of the main linear system, thereby obtaining the vibration response under the joint action of the two excitations. The specific calculation steps of this method are as follows:

[0016] Step 1: Set the initial state of the local nonlinear system. Generally, the main linear system can be set to be in a static equilibrium position and static state, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;

[0017] Step 2: The value of the external excitation (primary excitation force) at the current moment and the value of the nonlinear secondary excitation force at the current moment (i.e., the equivalent model feedback output value at the previous moment) are superimposed together as the excitation input of the main linear system at the current moment, and the vibration response at the next moment is calculated;

[0018] Step 3: For each local nonlinear link, the corresponding equivalent model is used, and the corresponding friction force or knocking force at the next moment is calculated based on the vibration response of the main linear system at the next moment in each specific local part, that is, the secondary excitation force output by the equivalent model feedback of each specific local nonlinear link;

[0019] Step 4: Go forward one time step and return to step 2. Repeat this cycle to obtain the total vibration response history of the local nonlinear system under external excitation.

[0020] Furthermore, for solving the time history of the forced vibration response of the main linear system part, any existing mature numerical calculation method applicable to the linear vibration system can be used, including but not limited to the finite element method, time domain method, frequency domain method, etc.

[0021] Furthermore, for solving the forced vibration response of the main linear system in step 2, any existing mature numerical calculation method suitable for time-domain recursive solution of linear vibration systems can be used, including but not limited to Wilson-θ method, Newmark-β method, central difference method, etc., and can also be combined with modal truncation, dynamic condensation and other order reduction methods to further improve the calculation efficiency.

[0022] The beneficial effects of the present invention are as follows: compared with the existing abnormal noise vibration analysis or abnormal noise risk assessment technology (such as SAR-Line, E-Line, etc.), the present invention takes into account the actual effects of the abnormal noise source or friction pair and knocking pair, so the estimation of the structural abnormal noise risk caused by it under external excitation is more sufficient, and the prediction accuracy of the structural vibration response is higher (the structural vibration response calculated by the existing method does not include any effect of friction pair or knocking pair at all).

[0023] Compared with the existing nonlinear system vibration response calculation method, the present invention makes full use of the local nonlinear characteristics of the system, and treats the local nonlinearity and the main linear system separately, so that the solution of the large-scale vibration system differential equations is still the linear system solution, avoiding the time-consuming large-scale nonlinear differential equation solution. Therefore, the mature linear system solution method can be used, which is much more efficient than the solution of the complete nonlinear differential equation system. This makes it possible to accurately calculate complex engineering problems such as automobile abnormal noise analysis and evaluation, which is of great significance to improving the level and efficiency of automobile abnormal noise performance development. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 It is a flowchart of the combined calculation process in a serial manner;

[0025] Figure 2 It is a flowchart of the combined calculation process of the feedback method;

[0026] Figure 3 This is a picture of the experimental site;

[0027] Figure 4 It is the finite element model of the cantilever beam structure;

[0028] Figure 5 It is a time domain comparison diagram of the combined calculation results in serial mode;

[0029] Figure 6 It is a frequency domain comparison diagram of the serial combination calculation results;

[0030] Figure 7 It is a time domain comparison diagram of the calculation results of the feedback mode combination;

[0031] Figure 8 It is a frequency domain comparison diagram of the calculation results of the feedback method combination. DETAILED DESCRIPTION

[0032] The following is further explained in detail through specific implementation methods.

[0033] In a local nonlinear system, except for the local knocking or friction parts, the main part of the system is still a linear system and still satisfies the superposition principle. If the local knocking force or friction force is regarded as a secondary excitation force, any fast algorithm suitable for a linear system can be used to calculate the forced vibration response of the system; and for the local knocking or friction nonlinear relationship, an equivalent model can be used to express it separately. As long as the relative motion input of each knocking pair and friction pair is obtained, the corresponding knocking force and friction force output can be obtained, that is, the secondary excitation force acting on the main system. Therefore, the present invention decomposes the entire local nonlinear system into a main linear system and local nonlinear links, and uses an equivalent model to process each local nonlinear link to express the nonlinear mapping relationship between each local friction force or knocking force and its relative motion; the nonlinear equivalent model is combined with the main linear system to obtain an equivalent local nonlinear system, and an external excitation is applied to it, so as to obtain the vibration response under the joint action of secondary excitations such as local knocking and friction.

[0034] The specific implementation method of calculating the forced vibration response of a local nonlinear system according to the present invention is as follows:

[0035] Preliminary step 1: For each local nonlinear link of the system, establish the corresponding nonlinear equivalent model. For example, for the knocking pair, the input of the equivalent model is the normal relative velocity at the knocking point of the two adjacent parts, and the output is the normal knocking force acting at the knocking point; for the friction pair, the input of the equivalent model is the tangential relative displacement and velocity at the friction contact of the two adjacent parts, and the output is the tangential friction force acting at the friction contact.

[0036] Preliminary step 2: For the linear system part of the main body, establish its dynamic analysis model, solve its inherent problems, and obtain the undamped natural frequency and main vibration mode of the main linear system.

[0037] For example, if the finite element method is used for numerical calculation, the finite element model of the system is established. In order to adapt to the subsequent local nonlinear model combination calculation, it is necessary to target the potential local knocking pairs or friction pairs. When dividing the units, it is necessary to establish node pairs at the locations where knocking or friction may occur in adjacent components, even if the node positions of the finite element units on the adjacent components at this location are collinear in the normal direction on the adjacent surfaces or contact surfaces, so as to extract the normal or tangential relative motion relationship of the knocking pair or friction pair in the subsequent calculation. The forced vibration response of the main linear system can be reduced to solving the following dynamic equations.

[0038]

[0039] Solving the undamped natural frequency and main vibration mode of the main linear system can be reduced to solving the following matrix eigenvalue problem.

[0040]

[0041] Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the system, respectively. X is the acceleration vector, velocity vector and displacement vector of the system, F(t) is the external excitation force vector acting on the system, which can be any function of time t, A is the main vibration mode vector of the system, ω n The undamped natural frequency of the system. If the degree of freedom of the main linear system is n, then all the above vectors are n-dimensional vectors, and all matrices are n×n-dimensional matrices. In order to solve the forced vibration response using the modal superposition method, it is usually only necessary to find the first m-order natural frequencies and main vibration modes of the system, and m<<n. Here, any mature algorithm can be used to solve the matrix eigenvalue problem of formula (2) to obtain the required first several orders of natural frequencies and main vibration modes.

[0042] Step 1: Set the initial state of the local nonlinear system. Generally, the main linear system can be set to be in a static equilibrium position and stationary state, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero.

[0043] Each local nonlinear link is treated with an equivalent model, and the output of each equivalent model, that is, each impact force or friction force, is regarded as a secondary excitation force acting on the main linear system. Therefore, the forced vibration response of the local nonlinear system under external excitation is equivalent to solving the following equation:

[0044]

[0045] Where Q(t) is the n-dimensional secondary excitation force vector. In view of the local nonlinear characteristics of the system, the secondary excitation force usually only exists in a small number of knocking or friction parts. Therefore, most of the elements in the secondary excitation force vector are always zero, and only a few elements are non-zero. The magnitude of their excitation force is given by the corresponding equivalent model.

[0046] The present invention provides two different combined calculation methods, which perform combined calculations of the main linear system and each local nonlinear equivalent model to obtain the total vibration response of the entire local nonlinear system under external excitation. One is a serial combined calculation method, such as Figure 1 As shown in , this method calculates the vibration response of the main linear system under external excitation, the secondary excitation and the vibration response of the main system caused by it in sequence, and then superimposes the vibration responses of the two excitations to obtain the total vibration response of the whole system; the other is a combined calculation method of feedback mode, such as Figure 2 As shown in the figure, this method processes each local nonlinear equivalent model as the feedback link of the main linear system. The main linear system generates a vibration response under external excitation, thereby obtaining the relative motion input of each local nonlinear equivalent model, and then obtaining the secondary excitation force output, which then feeds back to the main linear system, thereby affecting its vibration response at the next moment. In this way, the total vibration response of the entire system is obtained through such a cycle.

[0047] Obviously, the subsequent calculation method will be different depending on the combination calculation method.

[0048] a.Serial mode

[0049] If a serial combined calculation method is adopted, the subsequent specific calculation method is as follows.

[0050] Step 2: Calculate the vibration response time history of the main linear system under external excitation (primary excitation force), which can be referred to as the primary vibration response.

[0051] This step is actually to solve equation (1). Since it is a linear system solution, any mature algorithm can be used. For example, the modal superposition method can be used to first decouple equation (1) using the weighted orthogonal property of the main vibration mode. The damping matrix can be decoupled by ignoring the off-diagonal elements after matrix transformation. Then, the forced response time history under the first m-order modal coordinates is obtained by direct integration. Then, modal superposition is performed to obtain the response time history under generalized coordinates, that is, the solution of equation (1). If the primary excitation force is a periodic function, it can be expanded into a Fourier series. Then, the solution of equation (1) can be obtained more quickly by using harmonic response calculation and superposition principle.

[0052] Step 3: For each local nonlinear link, use the corresponding equivalent model and calculate the friction force or impact force history based on the primary vibration response history of the main linear system in each specific local area.

[0053] This step obtains the secondary excitation force vector Q(t). Note that when the local nonlinear link is located inside the system, that is, the two components that are knocked or rubbed are both components of the system, for example, friction or knocking occurs between the rth node of component A and the sth node of component B, then the secondary excitation force should appear in pairs, with equal magnitude and opposite direction, and act on the rth node and the sth node on components A and B respectively; when the local nonlinear link is located at the boundary of the system, that is, a part of a component of the system knocks or rubs against an external fixed object, then the secondary excitation force should be applied to the corresponding part or node of the component.

[0054] Step 4: Use the friction force or knocking force history obtained in step 3 as the secondary excitation force to act on the main linear system alone, and calculate the secondary vibration response history;

[0055] This step is actually to solve the response of the main linear system under the secondary excitation, which is equivalent to solving equation (1), but the excitation force vector is replaced by Q (t). Similar to the solution of step 2, any mature solution algorithm for the forced response of the linear system can be used, which will not be repeated here.

[0056] Step 5: Find the algebraic sum of the secondary vibration response history and the primary vibration response history, which is the total vibration response of the entire local nonlinear system under external excitation.

[0057] This step completes the solution of equation (3) based on the superposition principle of linear systems, which is equivalent to solving the response of the entire local nonlinear system under external excitation.

[0058] b. Feedback method

[0059] Although the above serial combined calculation method is easy to implement, it ignores the mutual coupling between the nonlinear equivalent model and the main linear system, which will inevitably lead to additional calculation errors. In fact, the secondary excitation force output by the nonlinear equivalent model depends on the vibration response of the main linear system. Once the secondary excitation force is generated, it will immediately act on the main linear system and will inevitably change its vibration response, thereby producing a coupling effect on the output of the nonlinear equivalent model. To this end, the present invention proposes a feedback combined calculation method with smaller calculation errors (see Figure 2 ), and the subsequent specific calculation method is as follows.

[0060] Step 2: Superimpose 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 (i.e., the equivalent model feedback output value at the previous moment) as the excitation input of the main linear system at the current moment i, and calculate the vibration response of the system at the next moment (i+1).

[0061] This step calculates the solution of equation (3), which is essentially a time-domain solution of the forced response of the linear system, and any mature existing linear system solution method can be used. For example, for the time-domain solution of large-scale linear systems, the mode superposition and mode truncation method can be used to express the solution of equation (3) as the superposition of the forced response under the first several modal coordinates, and the forced response under each modal coordinate can be solved by the impulse response method, that is, expressed as the convolution integral of the system impulse response function and the excitation force under the modal coordinate; the dynamic condensation method can also be used to reduce the scale or order of the model or equation (3), and then the direct integration method (such as Wilson-θ method, Newmark-β method, etc.) is used to solve the condensed equation, so as to obtain the time-domain numerical solution of equation (3).

[0062] Step 3: For each local nonlinear link, the corresponding equivalent model is used, and the corresponding friction force or impact force at the next moment (i+1) is calculated based on the vibration response of the main linear system at the next moment (i+1) in each specific local part, that is, the secondary excitation force output by the equivalent model feedback of each specific local nonlinear link.

[0063] This step calculates the feedback output of the equivalent model of each local nonlinear link, that is, the value of the secondary excitation force vector Q(t) at the next moment, which will be used for the recursive calculation of equation (3) at the next moment. Note that, consistent with what has been said before, when the local nonlinear link is located inside the system, the secondary excitation force always appears in pairs, acting on the two nodes of the knocking pair or friction pair node pair.

[0064] Step 4: Let i = i + 1, advance one time step, return to step 2, and repeat this cycle to obtain the total vibration response history of the local nonlinear system under external excitation, that is, the time domain numerical solution of equation (3).

[0065] Example:

[0066] The implementation method and applicability of the present invention are explained by taking a cantilever beam structure with a local friction pair as an example. Figure 3It is a picture of the experimental site, in which a 960mm long aluminum square cross-section long rod is fixed on the bench at the end by a steel bracket to form a cantilever beam structure; points 1, 2, and 3 in the figure respectively represent the excitation position of the vibrator, the friction side effect position, and the vibration response observation position or reference point; the vibrator applies external excitation (primary excitation) to the cantilever beam at point 1 to force the entire structure to vibrate; a friction pair is set at point 2 through an additional bracket, that is, the cantilever beam is pre-stressed and contacted with the external fixed boundary at this point, and when the cantilever beam undergoes a lateral bending vibration response under the action of the vibrator, friction will be generated at this point to form a secondary excitation force, which is a typical nonlinear link; when the cantilever beam is excited by the vibrator, the vibration response of the beam is affected by both the primary excitation force and the secondary excitation force. In the experiment, an acceleration sensor is used to observe the vibration response at reference point 3 to compare the consistency between the predicted value of the present invention and the measured value.

[0067] For the cantilever beam experimental system mentioned above, hexahedral units are selected to divide the mesh and a finite element model is established, such as Figure 4 In order to verify the correctness of the finite element model, a modal experiment comparison was carried out. Table 1 lists the first four modal results. It can be seen that the relative error between the modal frequency calculated by the finite element and the modal frequency obtained by the experiment is less than 3%, indicating that the established finite element model can basically express the dynamic characteristics of the real cantilever beam.

[0068] Table 1 Comparison of finite element calculated modal frequencies and experimental modal frequencies

[0069]

[0070] Define the acceleration root mean square error percentage ε to quantitatively evaluate the vibration response prediction effect:

[0071]

[0072] Where rms_a′ is the RMS value of the acceleration predicted by simulation over the entire time history, and rms_a is the RMS value of the acceleration measured experimentally.

[0073] The above finite element model is exported to a self-compiled special calculation program. The program is compiled according to the above method, and can assemble the system mass matrix and stiffness matrix according to the finite element model, define the system damping matrix according to the material internal resistance, import or customize the excitation force vector, and import or customize the nonlinear link equivalent model, so as to obtain the motion differential equation group of the analysis object, as shown in equation (3), and then solve the equation according to the method of the present invention. The specific calculation process is as follows:

[0074] a. Calculate by serial combination calculation method

[0075] 1) Import the equivalent model of the friction pair at position 2. In order to obtain the equivalent model, we conducted reciprocating friction tests on the samples of the specific materials of the friction pair on a special testing machine, and then used the test data to identify the parameters of the equivalent model. The input of the equivalent model is the positive pressure of the friction pair, the relative displacement, relative velocity and relative acceleration of the tangential motion of the friction pair, and the output is the friction force; in this example, the positive pressure of the friction pair is a given constant, and the relative motion of the friction pair is the displacement, velocity and acceleration of the cantilever beam at point 2, all of which are transient changes. The friction force of the friction pair acts on the cantilever beam at point 2, always hindering the lateral motion of the cantilever beam, and is a dynamic force that changes instantly with the input of the equivalent model; considering the actual size of the contact surface of the friction pair, it covers far more than one node of the cantilever beam unit. It can be considered that the friction force acts equivalently on a node at the center of the friction surface, or it can be dispersed to act on each node covered by the friction surface. The difference between the two is very small.

[0076] 2) Apply the excitation force f1 of the vibrator to the eight nodes near point 1 of the cantilever beam, and use the Newmark-β algorithm to calculate the primary vibration response history X of the finite element model. 0 , and note that the initial state of the system is defined as the static equilibrium position state; in this example, the excitation of the exciter applied is a harmonic force, and the time history of the vibration response can be obtained more quickly according to the harmonic response calculation method. The difference between the two in numerical results is not large. In order to maintain generality, we actually use the Newmark-β algorithm.

[0077] 3) The vibration response history of the cantilever beam at 81 nodes near point 2 obtained in the above step, including displacement, velocity and acceleration time history, is used as the friction model of each node. The friction history q acting on these 81 nodes is calculated by inputting f =[q p+1 ,q p+2 ,…,q p+81 ] T .

[0078] 4) Using the Newmark-β algorithm, calculate the secondary vibration response time history X of the finite element model 1 The secondary excitation force is the friction force history q obtained in the previous step. f =[q p+1 ,q p+2 ,…,q p+81 ] T , which acts on 81 nodes of the cantilever beam near point 2.

[0079] 5) Find the vibration response history X 0 And the secondary vibration response history X 1The algebraic sum of is the total vibration response of the entire cantilever beam structure with friction pair under the excitation of the exciter, X = X 0 +X 1 .

[0080] The acceleration time history of the total vibration response of the corresponding nodes at the three positions of the cantilever beam is extracted and compared with the vibration acceleration time history measured at the corresponding positions in the experiment. The results are as follows: Figure 5 As shown in the figure, the overall trend of the predicted results is basically consistent with that of the experimental results, but the waveforms are quite different; Figure 6 The autopower spectral density comparison of the two signals is given. It can be seen that the odd harmonic components of the calculated results are basically consistent with those of the experimental results, but the even harmonic components are missing. Table 2 is the quantitative comparison result. The relative error of the combined calculation results of the serial method reaches 24%, which shows that the vibration response prediction accuracy of this method for local nonlinear systems is not high enough, and the coupling effect between the nonlinear link and the main linear system cannot be ignored. However, it is easy to implement, especially for the vibration response calculation part of the linear system, which can even be implemented with the help of existing software without programming by yourself.

[0081] b. Calculate using the combined calculation method of the feedback method

[0082] 1) Import the equivalent model of the friction pair at position 2. This step is consistent with the above method and will not be repeated here.

[0083] 2) Define the initial state of the cantilever beam as being at rest in the static equilibrium position, that is, Obviously, the friction force at the 2 points is also zero at the initial moment, that is, is the friction force on all unit nodes corresponding to the friction surface; here, the physical quantities that change with time are represented by superscript i, and the initial time is i=0.

[0084] 3) Solve the vibration response of the main linear system under the primary excitation and secondary excitation at the current moment, that is, solve equation (3). (acting on the 8 nodes at point 1 of the cantilever beam) and the friction force fed back by the friction pair equivalent model (Acting on 81 nodes covered by the friction surface at point 2 of the cantilever beam, Together they are used as the input of the finite element model, and the vibration response of the finite element model at time i+1 is calculated using the Newmark-β algorithm. Note that the calculation time step (the time difference between two adjacent moments) should not be too large. In order to further improve the calculation efficiency, especially when the main linear system is large, the calculation here can use the method of modal superposition and modal truncation to reduce the number of differential equations that need to be directly integrated, thereby improving the calculation speed. As long as the order of modal truncation is appropriate, the calculation accuracy is almost unaffected.

[0085] 4) Calculate the feedback output of the equivalent model of the friction pair, that is, the vibration response of the 81 nodes covered by the friction surface near the cantilever beam point 2 calculated in the above step at time i+1 As the input of the friction pair equivalent model, the friction force acting on these 81 nodes at time i+1 can be obtained, that is:

[0086] 5) Go forward one time step, set i=i+1, and return to 3) to continue the calculation. Repeat this cycle to complete the numerical solution of equation (3) and obtain the total vibration response time history of the cantilever beam structure with friction pair under the excitation of the exciter, including the displacement, velocity and acceleration time history of each node.

[0087] The total vibration response history of the cantilever beam reference point 3 is extracted and compared with the experimentally measured acceleration time history at the same position. The results are as follows: Figure 7 As shown, it can be seen that the simulation prediction curve and the experimental curve are almost coincident; Figure 8 The comparison of the auto-power spectrum density of the two is given. It can be seen that both odd and even harmonic components of the calculated results exist and are highly consistent with the experimental results. This shows that the combined calculation method of the feedback method can correctly reflect the coupling and influence relationship between the friction pair and the main linear system that actually exist in the local nonlinear system, and has a very high accuracy in predicting the nonlinear vibration response. Table 2 shows the comparison results of the root mean square value of the simulation prediction signal and the measured signal. It can be seen that the relative error of the combined calculation result of the feedback method is only 0.725%, which is much more accurate than the combined calculation result of the serial method.

[0088] Table 2 Comparison of vibration response prediction results and experimental results of reference point 3

[0089]

[0090] The calculation process of the above example uses the finite element model to express the main linear system, and solves the linear system differential equations with the help of the step-by-step integration method. However, two points must be pointed out that the method for solving the vibration response of the local nonlinear system of the present invention does not limit the mathematical expression of the main linear system, and can use differential equations, difference equations, or impulse response function matrices, frequency response function matrices and other forms; at the same time, it also does not limit the solution method of the partial forced response of the main linear system, and any method that can obtain the time history of the vibration response of the linear system is feasible.

[0091] It should be noted in advance that, in the present invention, unless otherwise clearly specified and limited, the terms "installed", "connected", "connected", "fixed" and the like should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a direct connection, or an indirect connection through an intermediate medium, or it can be the internal communication of two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0092] The above is only an embodiment of the present invention, and the common knowledge such as the known specific structure and characteristics in the scheme is not described in detail here. It should be pointed out that for those skilled in the art, several deformations and improvements can be made without departing from the structure of the present invention, 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 practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.

Claims

1. A fast prediction method for the vibration response of a local nonlinear system, characterized in that: The whole system is divided into the main linear system part and the local nonlinear link, and each local nonlinear link is treated as an equivalent model to express the nonlinear mapping relationship between each local friction force or knocking force and the friction motion or knocking motion, and then the forced vibration response of the whole system is quickly predicted through combined calculation; The external excitation that causes the forced vibration response of the entire system is called primary excitation, and the friction and knocking force generated by each local nonlinear link is called secondary excitation. With the help of mature algorithms and superposition principle, the vibration response of the main linear system under the combined action of primary excitation and secondary excitation of local nonlinear links is solved, thereby realizing the rapid prediction of the vibration response of the entire local nonlinear system. 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 the nonlinear equivalent model are provided, namely, the combined calculation method of serial mode and the combined calculation method of feedback mode. The serial combined calculation method comprises: sequentially calculating the vibration response of the main linear system caused by the primary excitation, the secondary excitation generated by the nonlinear equivalent model, the vibration response of the main linear system caused by the secondary excitation, and then superimposing the two vibration responses of the main linear system, so as to obtain the total vibration response of the entire local nonlinear system under the external excitation; The combined calculation method of the feedback mode includes: taking the equivalent models of each local nonlinear link as the feedback link of the main linear system, that is, instantly feeding back the vibration response of each specific local part in the main linear system as a secondary excitation force through the nonlinear equivalent model, so that it and the primary excitation together constitute the excitation input of the main linear system, thereby obtaining the vibration response under the joint action of the two excitations.

2. The method for rapid prediction of vibration response of a local nonlinear system according to claim 1, characterized in that: The specific calculation steps of the serial combination calculation method are as follows: Step 1: Set the initial state of the local nonlinear system, and set the main linear system to be in a static equilibrium position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero; Step 2: Calculate the time history of the primary vibration response of the main linear system under the primary excitation force; Step 3: For each local nonlinear link, use the corresponding equivalent model and calculate the friction force or knocking force history based on the primary vibration response history of the main linear system in each specific local area; Step 4: Use the friction force or knocking force history obtained in step 3 as the secondary excitation force to act on the main linear system alone, and calculate the secondary vibration response history; Step 5: Find the algebraic sum of the secondary vibration response history and the primary vibration response history, which is the total vibration response of the entire local nonlinear system under external excitation.

3. The method for rapid prediction of vibration response of a local nonlinear system according to claim 1, characterized in that: The specific calculation steps of the combined calculation method of the feedback mode are as follows: Step 1: Set the initial state of the local nonlinear system, and set the main linear system to be in a static equilibrium position, so that the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero; i is used to represent the time, and i=0 represents the initial time; Step 2: Superimpose the value of the primary excitation force at time i and the value of the nonlinear secondary excitation force at time i as the excitation input of the main linear system at the current time, and calculate the vibration response at time i+1; Step 3: For each local nonlinear link, use the corresponding equivalent model, and calculate the corresponding friction force or knocking force at time i+1 based on the vibration response of the main linear system at each specific local time i+1, that is, the secondary excitation force output by the equivalent model feedback of each specific local nonlinear link; Step 4: Go forward one time step and return to step 2. Repeat this cycle to obtain the total vibration response history of the local nonlinear system under external excitation.

4. The method for rapid prediction of vibration response of a local nonlinear system according to claim 2, characterized in that: The forced vibration response time history of the main linear system is solved by using any existing mature numerical calculation method suitable for linear vibration systems, including finite element method, time domain method and frequency domain method.

5. The method for rapid prediction of vibration response of a local nonlinear system according to claim 3, characterized in that: For solving the forced vibration response of the main linear system in step 2, any existing mature numerical calculation method suitable for time-domain recursive solution of linear vibration systems is adopted, including Wilson-θ method, Newmark-β method, and central difference method, combined with modal truncation and dynamic condensation and order reduction methods to further improve the calculation efficiency.

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