WO2023125172A1 - 局部非线性系统振动响应的快速预测方法 - Google Patents
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- the invention relates to the technical field of mechanical engineering, in particular to a fast prediction method for the vibration response of a local nonlinear system.
- a local nonlinear system In engineering practice, mechanical structures with local friction or percussion pairs are ubiquitous. For the convenience of expression, this kind of mechanical structure with friction or percussion at a specific local position is called a local nonlinear system. .
- a local nonlinear system For example: during the driving process of the car, under the excitation of uneven road surface or powertrain vibration excitation, between the body and the instrument assembly, door, seat and other adjacent components, the interior of the instrument assembly, door, seat and other components adjacent to each other Some local knocks or frictions often occur between parts, and the vehicle or its specific subsystems can be regarded as a local nonlinear system at this time; the frictional contact between the well wall and the drill bit makes the drill bit in the drilling operation It can be equivalent to a local nonlinear system. In short, local nonlinear systems are numerous in engineering practice.
- the SAR-Line or E-Line method is mainly used to assess the risk of abnormal noise in the structure of the vehicle and parts. This type of method does not directly examine the friction pair or knock The actual motion or dynamic behavior of the pair also ignores the influence of these nonlinear links on the vibration response of the structure, and calculates the vibration response of the entire structure as a linear system, and according to the relative vibration displacement, velocity and other information of the pre-selected risk position node pair Conduct a noise risk assessment.
- This type of method avoids the calculation of the vibration response of the nonlinear system, takes less time and can quickly obtain the abnormal sound risk assessment results of the automobile structure, but the evaluation accuracy and effectiveness are very poor, and it is difficult to meet the actual engineering needs of automobile product development . So far, the development of abnormal noise performance of automobiles has had to rely more on empirical design and post-prototype tuning.
- the present invention makes full use of the local nonlinear characteristics of the abnormal sound prediction problem, that is, in almost all engineering practical problems, where it is necessary to analyze and predict the abnormal sound vibration, the machine structure or system itself can be regarded as a linear system, rather than a nonlinear system.
- the percussion kinematic pair or friction kinematic pair only exists in one or several places, and the equivalent model is proposed to deal with these nonlinear links separately, and the nonlinear mapping relationship between the percussion force or friction force and its kinematic input is established, and the The percussion force or friction force is regarded as the external excitation force of the original mechanical structure or system, so that the abnormal sound vibration response of the original mechanical structure or system can still be calculated according to the superposition principle of the linear system, so the effect of local nonlinearity is considered , and make full use of the linear nature of the original machine structure, making it possible to quickly predict the vibration response of abnormal noise, so as to better meet the actual engineering needs of the research and development of abnormal noise performance of automobiles and other products.
- 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 requirement of quickly predicting the structural vibration response of a local nonlinear system in engineering practice.
- the present invention provides a fast prediction method for the vibration response of a local nonlinear system, which splits the whole system into a main linear system part and a local nonlinear link, and treats each local nonlinear link as an equivalent model, so as to Express the nonlinear mapping relationship between each local friction force or percussion force and friction motion or percussion motion, and then realize the rapid prediction of the forced vibration response of the whole system through combined calculation;
- the external excitation that causes the forced vibration response of the entire system is called the primary excitation, and the friction force and percussion force generated by each local nonlinear link are called the secondary excitation, and the main linear system is solved by using mature algorithms and superposition principles.
- the vibration response under the joint action of the secondary excitation of the local nonlinear link so as to realize the rapid prediction of the vibration response of the entire local nonlinear system;
- Step 1 Set the initial state of the local nonlinear system.
- the main linear system can be set to be in a static equilibrium position, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
- Step 2 Calculate the primary vibration response time history of the subject linear system under external excitation (primary excitation force);
- Step 3 For each local nonlinear link, use the corresponding equivalent model, and calculate the friction force or percussion force history based on the primary vibration response history of the main linear system in each specific part;
- Step 4 Use the history of friction force or percussion force 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.
- the equivalent model of each local nonlinear link is used as the feedback link of the main linear system, that is, the vibration response of each specific part in the main linear system is instantly fed back as a secondary excitation force through the nonlinear equivalent model, making it comparable to the primary
- the excitation together constitutes the excitation input of the main linear system, so as to obtain the vibration response under the joint action of the two excitations.
- Step 1 Set the initial state of the local nonlinear system.
- the main linear system can be set to be in a static equilibrium position, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero;
- 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 (that is, the equivalent model feedback output value at the previous moment) are jointly superimposed as the main linear system at the current moment
- the excitation input is calculated to calculate the vibration response at the next moment
- Step 3 For each local nonlinear link, use the corresponding equivalent model, and calculate the corresponding friction force or percussion force at the next moment based on the vibration response of the main linear system at each specific part at the next moment, that is, each specific local nonlinear The secondary excitation force of the feedback output of the equivalent model of the link;
- Step 4 Advance one time step, return to Step 2, and so on, the total vibration response history of the local nonlinear system under external excitation can be obtained.
- any existing mature numerical calculation method suitable for linear vibration systems can be used to solve the forced vibration response time history of the main linear system part, including but not limited to finite element method, time domain method, frequency domain method, etc.
- any existing mature numerical calculation method suitable for the time-domain recursive solution of the linear vibration system can be used, including but not limited to Wilson- ⁇ method, Newmark- ⁇ 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.
- the beneficial effect of the present invention is that: compared with the existing abnormal sound vibration analysis or abnormal sound risk assessment technology (such as SAR-Line, E-Line, etc.), the present invention considers the abnormal sound source or the friction pair and the knocking pair Therefore, the estimation of the risk of structural abnormal sound caused by the external excitation is based more fully, and the prediction accuracy of the structural vibration response is higher (the structural vibration response calculated by the existing method does not include any friction pairs, knocking, etc. side effects).
- the existing abnormal sound vibration analysis or abnormal sound risk assessment technology such as SAR-Line, E-Line, etc.
- the present invention makes full use of the local nonlinear characteristics of the system, and processes the local nonlinear and main linear system parts separately, so that the large-scale vibration system differential equations can be solved It is still a linear system solution, which avoids the time-consuming large-scale solution of nonlinear differential equations. Therefore, mature linear system solution methods can be used, and the solution efficiency is much higher than that of complete nonlinear differential equations. This makes the precise numerical calculation of complex engineering problems such as the analysis and evaluation of automobile abnormal noise possible, which is of great significance for improving the development level and efficiency of automobile abnormal noise performance.
- Fig. 1 is the block diagram of the combination calculation flow chart of serial mode
- Fig. 2 is a combined calculation flow diagram of the feedback mode
- Fig. 3 is the top view of experiment site figure
- Fig. 4 is the side view of experiment site figure
- Fig. 5 is the schematic diagram of the fixed end of the cantilever beam of the experimental field diagram
- Fig. 6 is a schematic diagram of 2 points-friction side effect points of the experimental field diagram
- Fig. 7 is a schematic diagram of point 1 - excitation point of the exciter
- Fig. 8 is a schematic diagram of 3 points-response test points
- Fig. 9 is a finite element model diagram of a cantilever beam structure
- Figure 10 is a time-domain comparison diagram of the combined calculation results in a serial mode
- Fig. 11 is a frequency domain comparison diagram of the combined calculation results in the serial mode
- Fig. 12 is a time-domain comparison diagram of the combined calculation results of the feedback mode
- Fig. 13 is a frequency domain comparison chart of the combined calculation results of the feedback methods.
- the main part of the system is still a linear system, which still satisfies the principle of superposition.
- the knocking force or frictional force existing in each locality is regarded as the secondary excitation force
- the present invention decomposes the whole local nonlinear system into the main linear system and local nonlinear links, and uses equivalent models to process each local nonlinear link to express the relationship between each local friction force or percussion force and its relative motion.
- the nonlinear mapping relationship; the equivalent local nonlinear system is obtained by combining the nonlinear equivalent model with the main linear system, and the external excitation is applied to it, and the local knocking, friction and other secondary excitations can be obtained. Vibration response.
- Preliminary step 1 Establish corresponding nonlinear equivalent models for each local nonlinear link of the system.
- the input of the equivalent model is the normal relative velocity at the percussion point of two adjacent parts, and the output is the normal percussion force acting on the percussion point;
- the friction pair the equivalent model The input of is the tangential relative displacement and velocity at the frictional contact of two adjacent parts, and the output is the tangential frictional force acting on the frictional contact.
- Preliminary step 2 For the linear system part of the main body, establish its dynamic analysis model and solve its inherent problems to obtain the undamped natural frequency and main vibration shape of the main linear system.
- the finite element model of the system is established.
- Establish node pairs where knocking or friction may occur in the adjacent member even if the finite element elements on the adjacent member of the part are collinear in the normal direction on the adjacent surface or the node position of the contact surface, so that knocking can be extracted in subsequent calculations.
- the normal or tangential relative motion relationship of the pair or friction pair For the forced vibration response of the subject linear system, it can be boiled down to solving the following dynamic equations.
- 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 respectively
- 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 vector of the system
- ⁇ n is the unimpeded If the degree of freedom of the subject linear system is n, then all the above-mentioned vectors are n-dimensional vectors, and all matrices are n ⁇ n-dimensional matrices.
- any mature algorithm can be used to solve the matrix eigenvalue problem of (2) to obtain the required natural frequencies and main vibration shapes of the first several orders.
- Step 1 Set the initial state of the local nonlinear system.
- the main linear system can be set to be in a static equilibrium position, so the initial secondary excitation force output by the equivalent model of each local nonlinear link is zero.
- each local nonlinear link is treated with an equivalent model, and the output of each equivalent model, that is, each percussion force or friction force, is regarded as a secondary excitation force acting on the main linear system. Therefore, the local nonlinear
- the forced vibration response of the system under external excitation is equivalent to solving the following equation:
- Q(t) is the n-dimensional quadratic excitation force vector.
- the secondary excitation force usually only exists in a small number of percussion or friction parts, so most of the elements in the secondary excitation force vector are always zero, and only a small number of elements are non-zero.
- the magnitude of the force is given by the corresponding equivalent model respectively.
- the present invention provides two different combined calculation methods, and performs combined calculation 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 the combined calculation method in serial mode, as shown in Figure 1, 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, and then superimposes the vibration response of the main system under the two excitations Vibration response, so as to obtain the overall vibration response of the whole system;
- the other is the combined calculation method of feedback mode, as shown in Figure 2, this method treats each local nonlinear equivalent model as the feedback link of the main linear system, the main The linear system produces 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 acts on the main linear system in feedback, thereby affecting its vibration response at the next moment , so that the total vibration response of the whole system can be obtained through such
- Step 2 Calculate 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 the primary vibration response.
- This step is in fact solving equation (1). Since it is a linear system solution, any mature algorithm can be used.
- the modal superposition method can be used to first decouple equation (1) by using the weighted orthogonality of the main vibration mode, and the damping matrix can be decoupled by ignoring the off-diagonal elements after matrix transformation. Then, the forced response time history in the first m-order modal coordinates is obtained by direct integration method, and then the response time history in generalized coordinates is obtained by modal superposition, that is, the solution of 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 by using harmonic response calculation and superposition principle.
- Step 3 For each local nonlinear link, use the corresponding equivalent model, and calculate the history of friction force or percussion force based on the primary vibration response history of the main linear system in each specific part.
- This step obtains the secondary excitation force vector Q(t).
- the local nonlinear link is located inside the system, that is, the two components where the knocking or friction occurs are both part of the system, for example, friction or knocking occurs between the r-th node of the A component and the s-th node of the B component.
- the secondary excitation force should appear in pairs, equal in size and opposite in direction, and act on the r-th node and s-th node on components A and B respectively; when the local nonlinear link is located at the boundary of the system, that is, the local and external If the fixture is knocked or rubbed, the secondary excitation force should be applied to the corresponding part or node of the component.
- Step 4 Use the history of friction force or percussion force obtained in Step 3 as the secondary excitation force to act on the main linear system alone, and calculate the secondary vibration response history;
- this step is 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 in step 2, any mature solution algorithm for the forced response of the linear system can be used, so I won’t go into details here.
- 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.
- the combined calculation method of the above serial method is convenient 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.
- the secondary excitation force output by the nonlinear equivalent model depends on the vibration response of the main linear system.
- the output of the model produces coupling effects.
- the present invention proposes a combined calculation method of feedback modes with smaller calculation errors (see FIG. 2 ), and its subsequent specific calculation method is as follows.
- Step 2 The value of the external excitation (primary excitation force) at the current time i and the value of the nonlinear secondary excitation force at the current time i (that is, the equivalent model feedback output value at the previous time) are jointly superimposed as the current time i
- the excitation input of the subject linear system is used to calculate the vibration response of the system at the next time (i+1).
- 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.
- the method of mode superposition and mode truncation can be used to express the solution of equation (3) as the superposition of forced responses under the first
- the forced response in the coordinates can be solved by the impulse response method, which is expressed as the convolution integral of the system impulse response function and the excitation force in the modal coordinates; the dynamic condensation method can also be used to reduce the scale or order of the model or equation (3). times, and then use the direct integration method (such as Wilson- ⁇ method, Newmark- ⁇ method, etc.) to solve the polycondensed equation to obtain the time-domain numerical solution of equation (3).
- the direct integration method such as Wilson- ⁇ method, Newmark- ⁇ method, etc.
- Step 3 For each local nonlinear link, use the corresponding equivalent model, and calculate the corresponding friction force at the next moment (i+1) or The percussion force is the secondary excitation force output by the equivalent model feedback of each specific local nonlinear link.
- This step calculates and obtains 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 the previous description, 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 knock pair or friction pair node pair respectively.
- the total vibration response history of the local nonlinear system under external excitation can be obtained, that is, the time domain of equation (3) Numerical Solution.
- Fig. 3-Fig. 8 are the pictures of the experimental site.
- the 960mm-long aluminum square cross-section long rod is fixed on the bench by the steel bracket at the end to form a cantilever beam structure; the marks 1, 2 and 3 in the figure represent respectively The exciter excitation position, friction side effect position and vibration response observation position or reference point; the vibrator applies an external excitation (primary excitation) to the cantilever beam at point 1 to force the entire structure to vibrate; at point 2 through an additional
- the bracket is equipped with a friction pair, that is, the cantilever beam is in pre-press contact with the external fixed boundary at this place.
- the cantilever beam responds to lateral bending vibration under the action of the vibrator, friction will be generated at this place to form a secondary excitation force, which is a Typical nonlinear link: under the excitation of the vibrator, the vibration response of the beam is affected by the primary excitation force and the secondary excitation force at the same time.
- the acceleration sensor is used to observe the vibration response at the reference point 3, in order to compare the prediction of the present invention The value is consistent with the measured value.
- the hexahedron unit is selected to divide the mesh, and the finite element model is established, as shown in Figure 9.
- the modal experiment was compared. 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.
- rms_a' is the root mean square value of acceleration predicted by simulation in the whole time history
- rms_a is the root mean square value of acceleration measured by experiment.
- the program is compiled according to the above-mentioned method, and the system mass matrix and stiffness matrix can be assembled according to the finite element model, the system damping matrix can be defined according to the material internal resistance, the excitation force vector can be imported or customized, and the nonlinear link equivalent can be imported or customized model, thereby obtaining the differential equations of motion of the analyzed object, as shown in equation (3), and then solve the equations by the method of the present invention.
- the specific calculation process is as follows:
- the dynamic force of the model input changes in real time; considering the actual size of the contact surface of the friction pair, it covers far more than one node of the cantilever beam unit, so the friction force can be considered to act equivalently on a node at the center of the friction surface, or Distribute the friction force to each node covered by the friction surface, and the difference between the two is very small.
- N represents the normal pressure acting on the friction pair expressed by this node; Indicates the displacement of the node; Indicates the speed of the node; Indicates the acceleration of the node; j indicates the node number; the superscript 0 indicates the vibration response under one excitation.
- the method of mode superposition and mode truncation can be used in the calculation here to reduce the number of differential equations that need to be directly integrated, thereby improving the calculation speed.
- the order of state truncation is appropriate, and the calculation accuracy is hardly affected.
- the vibration response time history of each node includes the displacement, velocity and acceleration time history of each node.
- the calculation process of the above example uses the finite element model to express the main linear system, and uses the step-by-step integration method to solve the differential equations of the linear system.
- 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, differential equations, or impulse response function matrices, Various forms such as frequency response function matrix; at the same time, it does not limit the solution method of the partial forced response of the main linear system. Any method that can obtain the time history of the vibration response of the linear system is feasible.
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Abstract
局部非线性系统振动响应的快速预测方法，属于机械工程领域，包括：将整个系统拆分为主体线性系统部分和局部非线性环节，并把各个局部非线性环节分别处理为等效模型，以表达各个局部摩擦力或敲击力与摩擦运动或敲击运动之间的非线性映射关系，并实现局部非线性环节中摩擦力或敲击力的快速准确预测，进而通过组合计算实现整个系统的强迫振动响应快速预测。与现有的异响振动分析或异响风险评估技术(例如 SAR-Line、E-Line 等)相比，本方法考虑了异响源或摩擦副、敲击副的实际作用，因而对其在外部激励下所引起的结构异响风险的预估，根据更充分，结构振动响应预测精度更高。
Description
本申请要求于2021年12月27日提交中国专利局、申请号为202111616968.3、发明名称为“局部非线性系统振动响应的快速预测方法”的中国专利申请的优先权，其全部内容通过引用结合在本申请中。
本发明涉及机械工程技术领域，特别是涉及一种局部非线性系统振动响应的快速预测方法。
在工程实践中，局部带有摩擦副或敲击副的机械结构是普遍存在的，为了方便表述，后文将这种在特定局部位置发生了摩擦或敲击的机械结构称为局部非线性系统。例如：在汽车行驶过程中，路面不平激励或动力总成振动激励下，车身与仪表总成、车门、座椅等相邻部件之间，仪表总成、车门、座椅等部件的内部相邻零件之间，都常常发生若干局部的敲击或摩擦，此时汽车或者其特定子系统便可看作是局部非线性系统；井壁与钻头之间存在的摩擦接触使得钻井作业中的钻头也可以等效为局部非线性系统。总之，局部非线性系统在工程实际中不胜枚举。
机械结构振动对产品的工作性能和使用寿命有着重要的影响，振动以及它所辐射的噪声还恶化了人们的生存环境，尤其是敲击与摩擦等导致的结构异响，会大幅降低使用者对机器性能品质的评价。因此异响性能一直是表征产品品质的关键指标，在汽车行业尤其受到重视。在产品设计阶段，准确预测产品的异响性能进而优化设计方案避免出现异响，便可减少后期异响问题的调校工作，进而节约开发成本和周期。目前，计算复杂机械结构或系统的异响振动响应，尚没有成熟的高效算法可供采用，主要原因是非线性系统不适用叠加原理，采用直接积分法求解高维非线性非齐次微分方程组，计算耗时太长，现有算法难以实用于汽车等复杂系统的异响振动响应分析预测。实际上，在汽车工业中主要采用SAR-Line或E-Line方法对汽车整车及零部件结构中可能的异响点做 出异响风险评估，该类方法并不直接考察摩擦副或敲击副的实际运动或动力学行为，也忽略了这些非线性环节对结构振动响应的影响，而将整个结构按照线性系统计算振动响应，并根据预选的风险部位节点对的相对振动位移、速度等信息进行异响风险评估。这类方法避免了非线性系统的振动响应计算，所花时间代价较小，能快速得到汽车结构的异响风险评估结果，但评估精度与有效性很差，难以满足汽车产品研发的工程实际需要。迄今为止，汽车异响性能开发，还不得不更多地依赖经验设计和后期样车调校。
为此，本发明充分利用异响预测问题的局部非线性特征，即几乎所有工程实际问题中，凡需要分析预测异响振动的情形，机器结构或系统本身都可以看成线性系统，而非线性仅仅存在于一处或若干处的敲击运动副或摩擦运动副，提出利用等效模型来单独处理这些非线性环节，建立起敲击力或摩擦力与其运动输入的非线性映射关系，并把敲击力或摩擦力看成原机械结构或系统的外部激励力，从而使原机械结构或系统的异响振动响应仍然可以按线性系统采用叠加原理来计算，于是既考虑了局部非线性的作用，又充分利用了原机器结构的线性性质，使异响振动响应的快速预测成为可能，以更好满足汽车等产品异响性能研发的工程实际需求。
发明内容
本发明的目的在于提供局部非线性系统振动响应的快速预测方法，以满足工程实际中提出的快速预测局部非线性系统结构振动响应的需求。
为了达到上述目的，本发明提供局部非线性系统振动响应的快速预测方法，将整个系统拆分为主体线性系统部分和局部非线性环节，并把各个局部非线性环节分别处理为等效模型，以表达各个局部摩擦力或敲击力与摩擦运动或敲击运动之间的非线性映射关系，进而通过组合计算实现整个系统的强迫振动响应快速预测；
把引起整个系统强迫振动响应的外部激励称为一次激励，把各个局部非线性环节产生的摩擦力和敲击力称为二次激励，并借助成熟算法和叠加原理求解主体线性系统在一次激励和局部非线性环节的二次激励共同作用下的振动响应，从而实现整个局部非线性系统振动响应的快速预测；
为了获得主体线性系统在一次激励和二次激励共同作用下的振动响应，提供了主体线性系统与非线性等效模型的两种组合计算方法，即串行方式的组合计算方法和反馈方式的组合计算方法。
进一步，串行依次计算一次激励引起的主体线性系统振动响应、非线性等效模型产生的二次激励、二次激励引起的主体线性系统振动响应，然后再叠加主体线性系统的两次振动响应，从而获得整个局部非线性系统在外部激励下总的振动响应，该方法的具体计算步骤如下：
步骤一：设定局部非线性系统的初始状态，一般可设定主体线性系统处于静平衡位置静止状态，因此各个局部非线性环节的等效模型输出的初始二次激励力为零；
步骤二：计算外部激励(一次激励力)下主体线性系统的一次振动响应时间历程；
步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性系统在各个特定局部的一次振动响应历程计算摩擦力或敲击力历程；
步骤四：将步骤三得到的摩擦力或敲击力历程作为二次激励力单独作用于主体线性系统，计算二次振动响应历程；
步骤五：求二次振动响应历程与一次振动响应历程的代数和，即为整个局部非线性系统在外部激励下总的振动响应。
进一步，将各个局部非线性环节的等效模型均作为主体线性系统的反馈环节，即将主体线性系统中各个特定局部的振动响应通过非线性等效模型即时反馈为二次激励力，使之与一次激励共同组成主体线性系统的激励输入，从而获得两次激励共同作用下的振动响应，该方法的具体计算步骤如下：
步骤一：设定局部非线性系统的初始状态，一般可设定主体线性系统处于静平衡位置静止状态，因此各个局部非线性环节的等效模型输出的初始二次激励力为零；
步骤二：将外部激励(一次激励力)在当前时刻的值和非线性二次激励力在当前时刻的值(即上一时刻的等效模型反馈输出值)，共同叠加作为当前时刻主体线性系统的激励输入，计算出下一时刻振动响应；
步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性 系统在各个特定局部的下一时刻振动响应计算下一时刻的相应摩擦力或敲击力，即各个特定局部非线性环节的等效模型反馈输出的二次激励力；
步骤四：前进一个时间步长，返回步骤二，如此往复循环，便可得到局部非线性系统在外部激励下的总振动响应历程。
进一步，其中针对主体线性系统部分的强迫振动响应时间历程求解，可采用任何适用于线性振动系统的现有成熟数值计算方法，包括且不限于有限元法、时域法、频域法等。
进一步，其中针对步骤二中主体线性系统部分的强迫振动响应求解，可采用任何适用于线性振动系统时域递推求解的现有成熟数值计算方法，包括但不限于Wilson-θ法、Newmark-β法、中心差分法等，还可结合模态截断、动力缩聚等降阶方法进一步提高计算效率。
本发明的有益效果在于：与现有的异响振动分析或异响风险评估技术(例如SAR-Line、E-Line等)相比，本发明考虑了异响源或摩擦副、敲击副的实际作用，因而对其在外部激励下所引起的结构异响风险的预估，根据更充分，结构振动响应预测精度更高(现有方法计算的结构振动响应根本没有包含任何摩擦副、敲击副的作用)。
与现有的非线性系统振动响应计算方法相比，本发明充分利用了系统局部非线性的特点，将局部非线性与主体的线性系统部分分别处理，从而使大规模的振动系统微分方程组求解仍然是线性系统求解，避免了耗时的大规模非线性微分方程组求解，因此，可以利用成熟的线性系统求解方法，在求解效率上大大高于完全的非线性微分方程组的求解。这使得汽车异响分析评估等复杂工程问题的精确数值计算成为可能，对提升汽车异响性能开发水平和效率具有重要意义。
说明书附图
为了更清楚地说明本发明实施例或现有技术中的技术方案，下面将对实施例中所需要使用的附图作简单地介绍，显而易见地，下面描述中的附图仅仅是本发明的一些实施例，对于本领域普通技术人员来讲，在不付出创造性劳动性的前提下，还可以根据这些附图获得其他的附图。
图1是串行方式的组合计算流程框图；
图2是反馈方式的组合计算流程框图；
图3是实验现场图的俯视图；
图4是实验现场图的侧视图；
图5是实验现场图的悬臂梁固定端示意图；
图6是实验现场图的2点-摩擦副作用点示意图；
图7是1点-激振器激励点示意图；
图8是3点-响应测试点示意图；
图9是悬臂梁结构的有限元模型图；
图10是串行方式组合计算结果的时域对比图；
图11是串行方式组合计算结果的频域对比图；
图12是反馈方式组合计算结果的时域对比图；
图13是反馈方式组合计算结果的频域对比图。
下面通过具体实施方式进一步详细说明。
在局部非线性系统中，除了各个局部存在的敲击或摩擦部分外，系统的主体部分还是线性系统，仍然满足叠加原理，如果把各个局部存在的敲击力或摩擦力看成二次激励力，就可以采用任何适用于线性系统的快速算法来计算系统的强迫振动响应；而对于各个局部存在的敲击或摩擦非线性关系，可以采用等效模型来分别表达，只要获得各个敲击副、摩擦副的相对运动输入，就能获得相应的敲击力、摩擦力输出，也就是作用于主体系统的二次激励力。因此，本发明将整个局部非线性系统分解为主体的线性系统和局部的非线性环节，并采用等效模型处理各局部非线性环节，以表达各局部摩擦力或敲击力与其相对运动之间的非线性映射关系；将非线性等效模型与主体线性系统组合而得到等效的局部非线性系统，对其施加外部激励，便可获得包含局部敲击、摩擦等二次激励共同作用下的振动响应。
本发明计算局部非线性系统强迫振动响应的具体实施方式如下：
预备步骤1：针对系统的各个局部非线性环节，分别建立相应的非线性等效模型。例如，针对敲击副，等效模型的输入为两个相邻件敲击点处的法向相 对速度，输出为作用于敲击点处的法向敲击力；针对摩擦副，等效模型的输入为两个相邻件摩擦接触处的切向相对位移和速度，输出为作用于摩擦接触处的切向摩擦力。
预备步骤2：针对主体的线性系统部分，建立其动力学分析模型，并求解其固有问题，获得主体线性系统的无阻尼固有频率和主振型。
例如采用有限元法进行数值计算，则建立该系统的有限元模型，为了适应后续的局部非线性模型组合计算，需要针对潜在的局部敲击副或摩擦副，对其划分单元时，需要在相邻构件可能发生敲击或摩擦的部位建立节点对，即使该部位相邻构件上的有限元单元在相邻面或接触面的节点位置在法线方向上共线，以便后续计算中提取敲击副或摩擦副的法向或切向的相对运动关系。对于主体线性系统的强迫振动响应，可以归结为求解如下的动力学方程。
而求解主体线性系统的无阻尼固有频率和主振型，则可以归结为求解如下的矩阵特征值问题。
其中M、C、K分别为系统的质量矩阵、阻尼矩阵和刚度矩阵，
X分别为系统的加速度向量、速度向量和位移向量，F(t)为作用于系统的外部激励力向量，可以是时间t的任意函数，A为系统的主振型向量，ω
n系统的无阻尼固有频率，如果主体线性系统的自由度为n，则上述所有向量皆为n维向量，所有矩阵皆为n×n维矩阵。为了采用模态叠加法求解强迫振动响应，通常仅需要求出系统的前m阶固有频率及主振型，且m＜＜n。在此，可采用任何成熟的算法求解(2)式的矩阵特征值问题，获得所需的前若干阶固有频率及主振型。
步骤一：设定局部非线性系统的初始状态，一般可设定主体线性系统处于静平衡位置静止状态，因此各个局部非线性环节的等效模型输出的初始二次激励力为零。
把各个局部非线性环节分别采用等效模型来处理，并把各个等效模型的输出，即各个敲击力或摩擦力，视为作用于主体线性系统的二次激励力，因此，局部非线性系统在外部激励下的强迫振动响应等效归结为求解下述方程：
其中Q(t)为n维二次激励力向量。鉴于系统的局部非线性特性，通常二次激励力仅存在于少量的敲击或摩擦部位，因此二次激励力向量中的绝大部分元素均恒为零，仅少部分元素非零，其激励力大小由相应等效模型分别给出。
本发明提供两种不同的组合计算方式，进行主体线性系统与各个局部的非线性等效模型的组合计算，以获得整个局部非线性系统在外部激励下的总的振动响应。一种是串行方式的组合计算方法，如图1所示，该方法依次计算外部激励下主体线性系统的振动响应、二次激励及其引起的主体系统振动响应，然后再叠加两次激励的振动响应，从而获得整个系统总的振动响应；另一种是反馈方式的组合计算方法，如图2所示，该方法将各个局部的非线性等效模型处理为主体线性系统的反馈环节，主体线性系统在外部激励下产生振动响应，从而得到各个局部非线性等效模型的相对运动输入，进而得到二次激励力输出，它再反馈作用于主体线性系统，进而影响其下一时刻的振动响应，如此循环作用，便获得整个系统总的振动响应。
显然，依据组合计算方式的不同，则后续的计算方法有所不同。
a.串行方式
若采用串行方式的组合计算方法，其后续的具体计算方法如下。
步骤二：计算主体线性系统在外部激励(一次激励力)作用下的振动响应时间历程，可简称为一次振动响应。
该步骤事实上是求解方程(1)。由于是线性系统的求解，可以采用任何成熟的算法。例如，可以采用模态叠加法，先利用主振型加权正交性质将方程(1)解耦，对其中的阻尼矩阵可采用忽略矩阵变换后的非对角元的近似处理方式实现解耦，然后利用直接积分法求出前m阶模态坐标下的强迫响应时间历程，然后再进行模态叠加获得广义坐标下的响应时间历程，即方程(1)的解。如果一次激励力是周期函数，则可以将其展开成傅里叶级数，则利用谐响应计算和叠加原理可更快获得方程(1)的解。
步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性系统在各个特定局部的一次振动响应历程计算摩擦力或敲击力历程。
该步骤获得二次激励力向量Q(t)。注意，当局部非线性环节位于系统内部， 即发生敲击或摩擦的两个构件均属于系统的组成部分，例如甲构件第r节点处与乙构件第s节点处之间发生摩擦或敲击，则二次激励力应成对出现，大小相等、方向相反，并分别作用于甲乙构件上的第r节点与第s节点；当局部非线性环节位于系统边界处，即系统某构件的局部与外部固定物发生敲击或摩擦，则二次激励力应施加于该构件相应部位或节点上。
步骤四：将步骤三得到的摩擦力或敲击力历程作为二次激励力单独作用于主体线性系统，计算二次振动响应历程；
该步骤事实上是求解主体线性系统在二次激励下的响应，相当于求解方程(1)，但其中的激励力向量换成了Q(t)。类似步骤二的求解，可采用线性系统强迫响应的任何成熟求解算法，此不赘述。
步骤五：求二次振动响应历程与一次振动响应历程的代数和，即为整个局部非线性系统在外部激励下总的振动响应。
该步骤即基于线性系统的叠加原理完成了方程(3)的求解，它等效于整个局部非线性系统在外部激励下的响应求解。
b.反馈方式
上述串行方式的组合计算方法，虽然方便实现，但忽略了非线性等效模型与主体线性系统之间的相互耦合作用，必然导致额外的计算误差。事实上，非线性等效模型输出的二次激励力取决于主体线性系统的振动响应，二次激励力一旦产生必然立刻作用于主体线性系统，也必然改变其振动响应，进而对非线性等效模型的输出产生耦合影响。为此，本发明提出具有更小计算误差的反馈方式组合计算方法(参见图2)，其后续的具体计算方法如下。
步骤二：将外部激励(一次激励力)在当前时刻i的值和非线性二次激励力在当前时刻i的值(即上一时刻的等效模型反馈输出值)，共同叠加作为当前时刻i主体线性系统的激励输入，计算出系统在下一时刻(i+1)的振动响应。
该步骤计算方程(3)的解，本质上是线性系统的强迫响应时域求解，可以采用任何成熟的现有线性系统求解方法。例如，针对大规模的线性系统时域求解，可以采用模态叠加与模态截断的方式，将方程(3)的解表达为前若干阶模态坐标下强迫响应的叠加，而在各模态坐标下的强迫响应可以采用脉冲响应法求解，即表达为模态坐标下的系统脉冲响应函数与激励力的卷积积分；也可以 采用动力缩聚法来降低模型或方程(3)的规模或阶次，然后再用直接积分法(例如Wilson-θ法、Newmark-β法等)求解缩聚后的方程，从而获得方程(3)的时域数值解。
步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性系统在各个特定局部的下一时刻(i+1)振动响应计算下一时刻(i+1)的相应摩擦力或敲击力，即各个特定局部非线性环节的等效模型反馈输出的二次激励力。
该步骤计算获得各个局部非线性环节的等效模型的反馈输出，即二次激励力向量Q(t)在下一时刻的取值，它将用于方程(3)下一时刻的递推计算。注意，与之前所述一致，当局部非线性环节位于系统内部时，二次激励力总是成对出现，分别作用于敲击副或摩擦副节点对的两个节点上。
步骤四：令i＝i+1，前进一个时间步长，返回步骤二，如此往复循环，便可得到局部非线性系统在外部激励下的总振动响应历程，即获得方程(3)的时域数值解。
实施例：
以带有局部摩擦副的悬臂梁结构为例，说明本发明的实施方法和适用性。图3-图8是实验现场图，长为960mm的铝制正方形截面长杆在端部通过钢制支架固定在台架上，形成了悬臂梁结构；图中标识1、2、3点分别表示激振器激励位置、摩擦副作用位置和振动响应观测位置或参考点；激振器在1点处给悬臂梁施加外部激励(一次激励)，使整个结构发生强迫振动；在2点处通过额外的支架设置了摩擦副，即悬臂梁在该处与外部固定边界预压接触，当悬臂梁在激振器作用下发生横向弯曲振动响应时，该处将产生摩擦，形成二次激励力，是一个典型的非线性环节；悬臂梁在激振器激励下，梁的振动响应同时受一次激励力和二次激励力影响，试验中利用加速度传感器在参考点3观测振动响应，用以对比本发明预测值与实测值的一致性。
针对上述悬臂梁实验系统，选择六面体单元划分网格，建立有限元模型，如图9所示。为了验证有限元模型的正确性，进行模态实验对比，表1列出了前四阶模态结果，可以看出，有限元计算的模态频率与实验得到的模态频率的相对误差不到3％，说明，所建立的有限元模型基本可以表达真实悬臂梁的动力学特性。
表1有限元计算模态频率与实验模态频率对比
定义加速度均方根值误差百分比ε，以定量评价振动响应预测效果：
式中，rms_a′为仿真预测的加速度在整个时间历程内的均方根值，rms_a为实验测得的加速度均方根值。
将上述有限元模型导出至自编的专门计算程序。该程序按前述方法编制，可根据有限元模型组集系统质量矩阵、刚度矩阵，可根据材料内阻定义系统阻尼矩阵，可导入或自定义激励力向量，可导入或自定义非线性环节等效模型，从而获得分析对象的运动微分方程组，形如方程(3)所示，然后按本发明的方法进行方程求解。具体计算过程如下：
a.按串行方式的组合计算方法进行计算
1)导入位置2的摩擦副等效模型。为了获得该等效模型，我们针对摩擦副的特定材料对样件在专门的试验机上进行往复摩擦试验，然后利用试验数据辨识了等效模型的参数。该等效模型的输入为摩擦副正压力、摩擦副切向运动相对位移、相对速度及相对加速度，输出为摩擦力；在本实例中，摩擦副的正压力为一个给定常量，摩擦副的相对运动即为悬臂梁在点2的位移、速度及加速度，皆为瞬态变化量，摩擦副的摩擦力作用在悬臂梁点2位置，总是阻碍悬臂梁的横向运动，是随着等效模型输入量即时变化的动态力；考虑到摩擦副接触面的实际大小，它覆盖的悬臂梁单元节点远不止一个，可以考虑将摩擦力等效作用在摩擦面中心位置的一个节点上，也可以将摩擦力分散作用在摩擦面覆盖的各个节点上，两者的差异很小。
2)将激振器激励力f
1作用于悬臂梁1点附近的8个节点处，借助Newmark-β算法计算有限元模型的一次振动响应历程X
0，并注意系统的初始状态定义为静平衡位置静止状态；在本算例中，施加的激振器激励为谐波力，可以根据 谐响应计算方法更快获得振动响应的时间历程，两者在数值结果上差异不大。为了不失一般性，我们实际采用的是Newmark-β算法。
3)以上步计算得到的悬臂梁在2点附近的81个节点的一次振动响应历程，包括位移、速度和加速度时间历程，作为各节点摩擦力模型
的输入，从而计算得到作用于这81个节点的摩擦力历程q
f＝[q
p+1，q
p+2，…，q
p+81]
T。其中，N表示作用于该节点表达的摩擦副上的正压力；
表示节点的位移；
表示节点的速度；
表示节点的加速度；j表示节点编号；上标0表示一次激励下的振动响应。
4)借助Newmark-β算法，计算有限元模型的二次振动响应时间历程X
1，二次激励力即上步计算获得的摩擦力历程q
f＝[q
p+1，q
p+2，…，q
p+81]
T，它作用于悬臂梁在2点附近的81个节点上。
5)求一次振动响应历程X
0和二次振动响应历程X
1的代数和，即为整个带摩擦副悬臂梁结构在激振器激励下总的振动响应，X＝X
0+X
1。
提取悬臂梁3点位置相应结点总振动响应的加速度时间历程，与实验中相应位置实测得到的振动加速度时间历程对比，结果如图10所示，预测结果与实验结果的整体变化趋势基本一致，但波形区别较大；图11给出了两个信号的自功率谱密度对比，可以看出，计算结果与实验结果的奇次谐波分量基本重合，但是缺少了偶次谐波分量；表2是定量对比结果，串行方式的组合计算结果相对误差达到24％，说明该方法对局部非线性系统的振动响应预测精度还不够高，非线性环节与主体线性系统间的耦合作用不容忽视，但胜在容易实现，尤其是对线性系统的振动响应计算部分，甚至可以借助现有软件实现，无需自己编程。
b.按反馈方式的组合计算方法进行计算
1)导入位置2的摩擦副等效模型。该步骤与前述方法一致，此不赘述。
2)定义悬臂梁初始状态为在静平衡位置处于静止状态，即X
(0)＝0，
显然2点位置在初始时刻的摩擦力亦为零，即设
是为对应于摩擦面的所有单元节点上的摩擦力；此处各随时间变化的物理量用上标i表示时刻，初始时刻为i＝0。
3)求解当前时刻主体线性系统在一次激励和二次激励作用下的振动响应， 即求解方程(3)。将当前时刻的激振器激励力
(作用与悬臂梁点1处的8个节点)和摩擦副等效模型反馈的摩擦力
(作用于悬臂梁点2处摩擦面覆盖的81个节点，
共同作为有限元模型的输入，借助Newmark-β算法计算i+1时刻有限元模型的振动响应X
(i+1)、
注意计算时间步长(相邻两个时刻的时间差)不宜取得过大。为了进一步提高计算效率，尤其是主体线性系统的规模较大时，此处的计算可以采用模态叠加与模态截断的方法，缩减需要进行直接积分的微分方程数量，从而提高计算速度，只要模态截断的阶次合适，计算精度几乎不受影响。
4)计算摩擦副等效模型的反馈输出，即以上步计算得到的悬臂梁点2附近摩擦面覆盖的81个节点在i+1时刻的振动响应
作为摩擦副等效模型的输入，可求得在i+1时刻作用于这81个节点的摩擦力，即：
5)前进一个时间步长，令i＝i+1，返回3)继续计算，如此往复循环，可完成方程(3)的数值求解，便得到带摩擦副悬臂梁结构在激振器激励下总的振动响应时间历程，包括各个节点的位移、速度和加速度时间历程。
提取悬臂梁参考点3位置总的振动响应历程，并与相同位置实验实测加速度时间历程对比，结果如图12所示，可见仿真预测曲线与实验曲线几乎重合；图13给出了二者的自功率谱密度对比，可以看出，计算结果的奇偶次谐波分量都存在，且与实验结果高度一致。这说明反馈方式的组合计算方法能够正确反映局部非线性系统实际存在的摩擦副与主体线性系统之间的耦合作用与影响关系，对非线性的振动响应预测具有相当高的精度。表2给出了仿真预测信号与实测信号的均方根值对比结果，可见，反馈方式的组合计算结果相对误差仅0.725％，远比串行方式的组合计算结果精确。
表2参考点3的振动响应预测结果与实验结果对比
上述实例计算过程利用有限元模型表达主体线性系统，并借助逐步积分法 求解线性系统微分方程组。但必须指出两点，本发明的局本非线性系统振动响应求解方法，并不限制其中主体线性系统的数学表达方式，可以采用微分方程组，也可以采用差分方程组，或者脉冲响应函数矩阵、频率响应函数矩阵等多种形式；同时也不限制主体线性系统部分强迫响应的求解方法，任何能获得线性系统振动响应时间历程的方法都是可行的。
需要提前说明的是，在本发明中，除非另有明确的规定和限定，术语“安装”、“相连”、“连接”、“固定”等术语应做广义理解，例如，可以是固定连接，也可以是可拆卸连接，或一体地连接；可以是直接相连，也可以通过中间媒介间接相连，可以是两个元件内部的连通。对于本领域的普通技术人员而言，可以根据具体情况理解上述术语在本发明中的具体含义。
以上所述的仅是本发明的实施例，方案中公知的具体结构及特性等常识在此未作过多描述。应当指出，对于本领域的技术人员来说，在不脱离本发明结构的前提下，还可以作出若干变形和改进，这些也应该视为本发明的保护范围，这些都不会影响本发明实施的效果和专利的实用性。本申请要求的保护范围应当以其权利要求的内容为准，说明书中的具体实施方式等记载可以用于解释权利要求的内容。
Claims (5)
- 局部非线性系统振动响应的快速预测方法，其特征在于，将整个系统拆分为主体线性系统部分和局部非线性环节，并把各个局部非线性环节分别处理为等效模型，以表达各个局部摩擦力或敲击力与摩擦运动或敲击运动之间的非线性映射关系，进而通过组合计算实现整个系统的强迫振动响应快速预测；把引起整个系统强迫振动响应的外部激励称为一次激励，把各个局部非线性环节产生的摩擦力和敲击力称为二次激励，并借助成熟算法和叠加原理求解主体线性系统在一次激励和局部非线性环节的二次激励共同作用下的振动响应，从而实现整个局部非线性系统振动响应的快速预测；为了获得主体线性系统在一次激励和二次激励共同作用下的振动响应，提供了主体线性系统与非线性等效模型的两种组合计算方法，即串行方式的组合计算方法和反馈方式的组合计算方法。
- 根据权利要求1所述的局部非线性系统振动响应的快速预测方法，其特征在于，串行依次计算一次激励引起的主体线性系统振动响应、非线性等效模型产生的二次激励、二次激励引起的主体线性系统振动响应，然后再叠加主体线性系统的两次振动响应，从而获得整个局部非线性系统在外部激励下总的振动响应，该方法的具体计算步骤如下：步骤一：设定局部非线性系统的初始状态，设定主体线性系统处于静平衡位置静止状态，因此各个局部非线性环节的等效模型输出的初始二次激励力为零；步骤二：计算外部激励下主体线性系统的一次振动响应时间历程；步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性系统在各个特定局部的一次振动响应历程计算摩擦力或敲击力历程；步骤四：将步骤三得到的摩擦力或敲击力历程作为二次激励力单独作用于主体线性系统，计算二次振动响应历程；步骤五：求二次振动响应历程与一次振动响应历程的代数和，作为整个局部非线性系统在外部激励下总的振动响应。
- 根据权利要求1所述的局部非线性系统振动响应的快速预测方法，其特征在于，将各个局部非线性环节的等效模型均作为主体线性系统的反馈环节，即将主体线性系统中各个特定局部的振动响应通过非线性等效模型即时反馈 为二次激励力，使之与一次激励共同组成主体线性系统的激励输入，从而获得两次激励共同作用下的振动响应，该方法的具体计算步骤如下：步骤二：将外部激励在当前时刻的值和非线性二次激励力在当前时刻的值，共同叠加作为当前时刻主体线性系统的激励输入，计算出下一时刻振动响应；步骤三：针对各个局部非线性环节，采用相应等效模型，并基于主体线性系统在各个特定局部的下一时刻振动响应计算下一时刻的相应摩擦力或敲击力，得到各个特定局部非线性环节的等效模型反馈输出的二次激励力；步骤四：前进一个时间步长，返回步骤二，如此往复循环，便可得到局部非线性系统在外部激励下的总振动响应历程。
- 根据权利要求2所述的局部非线性系统振动响应的快速预测方法，其特征在于，其中针对主体线性系统部分的强迫振动响应时间历程求解，采用有限元法、时域法或频域法实现。
- 根据权利要求3所述的局部非线性系统振动响应的快速预测方法，其特征在于，其中针对步骤二中主体线性系统部分的强迫振动响应求解，采用结合降价方法的Wilson-θ法、Newmark-β法或中心差分法实现，所述降价方法为模态截断或动力缩聚。
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| CN112528411A (zh) * | 2020-12-10 | 2021-03-19 | 中国运载火箭技术研究院 | 一种基于模态减缩的几何非线性结构噪声振动响应计算方法 |
| CN114201903A (zh) * | 2021-12-27 | 2022-03-18 | 重庆大学 | 局部非线性系统振动响应的快速预测方法 |
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2021
- 2021-12-27 CN CN202111616968.3A patent/CN114201903A/zh active Pending
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- 2022-12-21 WO PCT/CN2022/140526 patent/WO2023125172A1/zh unknown
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| CN111207897A (zh) * | 2020-02-23 | 2020-05-29 | 西安理工大学 | 一种基于非线性分离的局部非线性因素定位检测方法 |
| CN112528411A (zh) * | 2020-12-10 | 2021-03-19 | 中国运载火箭技术研究院 | 一种基于模态减缩的几何非线性结构噪声振动响应计算方法 |
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| CN117350135A (zh) * | 2023-12-04 | 2024-01-05 | 华东交通大学 | 一种混合能量收集器的频带拓展方法及系统 |
| CN117350135B (zh) * | 2023-12-04 | 2024-03-08 | 华东交通大学 | 一种混合能量收集器的频带拓展方法及系统 |
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