A method for identifying non-linear parameters of structural clearances
By establishing a nonlinear system model of the structural gap, and utilizing sinusoidal frequency sweep excitation and particle swarm optimization algorithm, combined with frequency domain identification method and harmonic balance method, the problem of accuracy and precision in nonlinear parameter identification was solved, and efficient gap nonlinear parameter identification was achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2022-11-16
- Publication Date
- 2026-05-26
AI Technical Summary
Existing technologies struggle to accurately and reliably identify nonlinear structural parameters, especially when engineering structures exhibit strong nonlinearity. Frequency domain identification methods lack sufficient accuracy, and traditional methods lack clear physical meaning.
By establishing a mathematical model of the nonlinear system of the structural gap, the frequency response function is obtained using sinusoidal frequency sweep excitation. The nonlinear equivalent stiffness of the system is extracted by combining the harmonic balance method and the particle swarm algorithm. The gap characteristic parameters are identified by using a frequency domain identification method and curve fitting.
It enables faster and more accurate identification of gap nonlinear parameters globally, improving identification precision and accuracy. It is applicable to single-mode estimation and avoids local optima problems.
Smart Images
Figure CN115730517B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of structural technology, and specifically relates to a method for identifying nonlinear parameters of structural gaps. Background Technology
[0002] Modal parameter identification is a crucial step in structural testing modeling. It involves using optimization algorithms with minimal error criteria to determine the modal parameters of a structural system from measured data. One method, known as time-domain identification, directly identifies system vibration characteristic parameters from input and output time-domain data based on motion differential equations, state equations, difference equations, and impulse response function models. This method directly identifies system modal parameters from measured response signals, eliminating the need for FFT transformation, reducing data transformation errors, and improving identification accuracy. Conversely, another method, based on accurate estimation of the frequency response function, uses the frequency domain expansion of the modal model as the identification formula and employs methods such as least squares with auxiliary variables, residual methods, iterative methods, and Taylor expansions to estimate system modal parameters locally or globally. This method is more intuitive, has clear physical concepts, and offers high identification accuracy, making it more suitable for single-mode estimation.
[0003] Currently, most mainstream methods in the frequency domain identification of modal parameters, such as the rational fractional orthogonal polynomial method, the least squares complex frequency method, and the frequency domain subspace method, only focus on linear systems. Weak nonlinearities in the system are often ignored or treated as uncertainties, partly because accurately and reliably identifying nonlinear parameters is quite difficult. Frequency domain identification methods for nonlinear modal parameters mainly employ frequency response function identification methods. The basic idea is to obtain the harmonic solution of the equation of motion using methods such as averaging, substitute it into the differential equation of motion, integrate it over the system amplitude and frequency, then establish a linear relationship between the target parameter and frequency or time using wavelet transform or natural logarithm methods, and finally obtain the corresponding parameters by fitting using methods such as least squares.
[0004] In the prior art:
[0005] 1. All actual engineering structures are nonlinear to some extent, and the nonlinearity cannot be completely ignored. Treating it as an uncertainty will also bring a lot of errors.
[0006] 2. The target parameters identified by methods such as least squares are mostly classical objects such as cubic stiffness or square damping terms. Their differential equations are simple, and the analysis results show that the stiffness and displacement amplitude, and the damping and velocity amplitude are linear relationships, lacking variation. The selection of nonlinear terms in actual situations is too simplistic. More target parameter identification methods should be explored.
[0007] 3. If the structure exhibits strong nonlinearity, the frequency response function obtained through FFT technology is related to the type and magnitude of the excitation and contains significant uncertainty. Its accuracy is insufficient, and it lacks clear physical meaning, making it difficult to accurately and reliably identify nonlinear parameters. In the process of identifying system nonlinear parameters using the measured frequency response function, the quality requirements for the frequency response function are very high; therefore, the traditional steady-state sinusoidal frequency sweep method should be used. Summary of the Invention
[0008] To overcome the shortcomings of existing technologies, this invention provides a method for identifying nonlinear parameters of structural gaps. First, a mathematical model of the nonlinear structural gap system is established. Then, the frequency response function of the nonlinear structural gap system under sinusoidal sweep excitation is obtained through the system's input and output responses. In the frequency response function, a pair of symmetrical displacement points exist at any displacement response amplitude to define the same displacement frequency response function. By defining the real and imaginary parts of the displacement frequency response function through the symmetrical displacement FRF points at each given displacement response amplitude, the system's natural frequency related to the displacement response amplitude can be obtained. Next, the expression for the system's nonlinear equivalent stiffness is extracted through model simulation analysis, and the system's equivalent stiffness curve is further obtained through model simulation data. Finally, based on the particle swarm optimization algorithm, curve fitting is performed between the equivalent stiffness expression obtained by integration and the equivalent stiffness curve obtained from model simulation to obtain the gap characteristic parameters. This invention can identify gap nonlinear parameters more quickly and accurately on a global scale.
[0009] The technical solution adopted by this invention to solve its technical problem includes the following steps:
[0010] Step 1: Establish a mathematical model for the nonlinear system with structural gaps;
[0011] The target vibration system is established as a single-degree-of-freedom nonlinear system subjected to simple harmonic forces, and sinusoidal sweep frequency excitation is used. The differential equation of motion is:
[0012]
[0013] Where m represents the mass coefficient, c represents the damping coefficient, k represents the stiffness coefficient, x represents the displacement, g(x) represents the gap nonlinear term, and f(t) is the sinusoidal sweep frequency excitation.
[0014] The gap nonlinear term is expressed as:
[0015]
[0016] Where x represents displacement, k s d represents the nonlinear stiffness coefficient, and d represents the clearance.
[0017] Step 2: Obtain the frequency response function of the nonlinear system with structural gap under sinusoidal sweep excitation by using the system input and output responses, i.e., the relationship curve between displacement amplitude and frequency;
[0018] Step 3: Analyze the system frequency response function. For each given displacement response amplitude, define the real and imaginary parts of the displacement frequency response function using a pair of symmetrical displacement frequency response function FRF points:
[0019]
[0020] Where H d1 H d2 Let R1, R2, I1, I2 be the frequency response function of a pair of symmetrical points, where R1, R2, I1, I2 are the real and imaginary parts, respectively.
[0021] Furthermore, the system's natural frequency related to the displacement amplitude is obtained:
[0022]
[0023] in ω1 and ω2 are the squares of the natural frequency, and ω1 and ω2 are the corresponding frequencies of a pair of symmetrical points.
[0024] Step 4: Extract the expression for the nonlinear equivalent stiffness of the system through simulation analysis of the model, and further obtain the equivalent stiffness curve of the system through model simulation data;
[0025] Based on the harmonic balance method, a first-order approximate solution of the steady-state displacement response is taken. Substituting the values, we further expand the gap nonlinearity term and obtain the integral formula for solving the equivalent stiffness from the equivalent stiffness characteristics. Substituting the gap nonlinearity term, we obtain the equivalent stiffness expression of the system:
[0026]
[0027] Substitute the system's natural frequency and the real and imaginary parts of its frequency response into the formula for calculating the system's equivalent stiffness:
[0028] k e ′ q =mω r (X) 2
[0029] Step 5: Based on the particle swarm optimization algorithm, the gap characteristic parameters are obtained by curve fitting between the equivalent stiffness expression obtained by integration and the equivalent stiffness curve obtained by model simulation.
[0030] Step 5-1: Initialize a group of random particles. Each particle has only two attributes: velocity and position. The entire process of particle swarm iteration is the process of gradually approaching the optimal position. The particle position represents the target parameter value, i.e., the gap parameter value.
[0031] Step 5-2: Set the fitness value to evaluate the quality of the particle position; set the fitness value to the deviation of the curve from the target curve under this gap parameter, and the deviation is calculated by the function deviation value at each point;
[0032] Step 5-3: For each particle, compare its fitness value with the fitness value of the optimal position. If the fitness value of the particle is greater than that of the optimal position, set it as the current optimal position and update the individual optimal position and the individual optimal fitness value.
[0033] Step 5-4: Update the group's optimal position and optimal fitness value based on the individual's optimal position;
[0034] Step 5-5: Update the velocity according to the particle swarm optimization velocity update formula, as follows:
[0035]
[0036] Where k is the number of iterations; ω is the inertia weight; c1 is the individual learning factor; c2 is the group learning factor; r1 and r2 are random numbers in the interval [0,1] to increase the randomness of the search; It is the d-th dimension velocity vector of particle i in the k-th iteration; It is the d-th dimension position vector of particle i in the k-th iteration; It is the best historical position of particle i in the d-th dimension during the k-th iteration; It represents the best historical position of the group in the d-th dimension during the k-th iteration;
[0037] Step 5-6: If the termination condition is not met, proceed to step 5-3.
[0038] Preferably, the mathematical model of the structural gap nonlinear system adopts the dead zone model.
[0039] The beneficial effects of this invention are as follows:
[0040] This invention, based on a frequency domain identification method and combined with the harmonic balance method, theoretically derives a formula for calculating the equivalent stiffness of the gap nonlinear system, thus broadening the scope of classical nonlinear identification objects. The frequency domain identification method is intuitive, has clear physical concepts, and high identification accuracy, making it more suitable for single-mode estimation. Combining this with traditional sinusoidal frequency sweep excitation for gap nonlinear system identification allows for accurate and rapid determination of the system's gap nonlinear stiffness. The particle swarm optimization algorithm's fast convergence speed and leapfrog nature ensure it avoids getting trapped in local optima, thereby achieving faster and more accurate global identification of gap nonlinear parameters. Attached Figure Description
[0041] Figure 1This is a flowchart of the method of the present invention.
[0042] Figure 2 Frequency response function curve of the system in this embodiment of the invention.
[0043] Figure 3 The inherent frequency curve of the system in this embodiment of the invention.
[0044] Figure 4 Equivalent stiffness curve of the system in this embodiment of the invention.
[0045] Figure 5 The parameter identification flowchart of the method of this invention is based on the particle swarm optimization algorithm. Detailed Implementation
[0046] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0047] This invention applies the harmonic balance method to the equations of motion, enabling the gap nonlinear system to be approximately described by a linear variable-parameter model. An expression for the gap nonlinear stiffness under this model is derived theoretically. Based on a method for identifying nonlinear modal parameters using the frequency response function, and combined with the particle swarm optimization algorithm, a method for identifying gap nonlinear parameters in non-classical nonlinear objects is provided.
[0048] This invention proposes a method for identifying the stiffness parameters of nonlinear gap characteristic structures. The overall design is as follows:
[0049] 1. First, a mathematical model of the nonlinear system with structural gaps is established. There are three commonly used mathematical models for gap nonlinearity. The first is the dead-zone model, which mainly considers the change in transmitted torque due to position changes. As the relative position of the drive and load changes, the force output to the load changes proportionally to the position difference. When the difference between the two is less than the corresponding amplitude of the gap, the force output to the load becomes zero. The second is the hysteresis model, which mainly uses displacement change as a reference for velocity. For classic gear transmissions, due to the existence of gaps, when the gears mesh, their velocities are the same; when the gears disengage, the velocity at the load end is zero. The third is the collision model, based on the principle of momentum conservation after a collision, describing the collision process of a purely rigid system, and is suitable for purely rigid systems. This invention selects the dead-zone model to study gap nonlinearity.
[0050] Establish a target vibration system, selecting a single-degree-of-freedom nonlinear system subjected to a simple harmonic force, and choosing a sinusoidal sweep frequency excitation. Its motion differential equation is:
[0051]
[0052] The gap nonlinearity term selected in this invention is expressed as:
[0053]
[0054] 2. Obtain the frequency response function of the system under sinusoidal sweep excitation by analyzing the system's input and output responses, i.e., the relationship curve between displacement amplitude and frequency. This provides basic information for the next step of determining the natural frequency, and also allows observation of the "frequency shift" phenomenon caused by nonlinear factors.
[0055] The frequency response function of a system with added gapped nonlinear terms is as follows: Figure 2 As shown;
[0056] 3. By defining the real and imaginary parts of the displacement frequency response function for a pair of symmetrical displacement FRF points for each given displacement response amplitude, the natural frequency related to the displacement response amplitude can be obtained.
[0057] A schematic diagram of the system's natural frequency is shown below. Figure 3 As shown;
[0058] At this time, the following should be noted:
[0059] 1) For the initially obtained frequency response function, it is impossible to obtain strictly symmetrical points on its curve, so appropriate interpolation is required.
[0060] 2) The changing natural frequency and damping obtained from different points are due to the nonlinear "frequency shift" phenomenon.
[0061] 3) The change in natural frequency reflects the nonlinear stiffness factor contained in the nonlinear system.
[0062] 4. Extract the expression for the nonlinear equivalent stiffness of the system through simulation analysis of the model, and further obtain the equivalent stiffness of the system through model simulation data.
[0063] The equivalent stiffness of the system can be obtained by substituting information such as the system's natural frequency and the real and imaginary parts of the frequency response into the formula. The equivalent stiffness identified by the system can be compared with the equivalent stiffness calculated by theoretical analysis. Figure 4 As shown in the figure, the horizontal axis represents the displacement response amplitude, and the vertical axis represents the equivalent stiffness value. The red curve represents the model's identification result, while the blue curve represents the theoretically calculated equivalent stiffness result.
[0064] Based on the harmonic balance method, we take the first approximate solution of the steady-state displacement response, further expand the nonlinear terms, and obtain the integral formula for solving the equivalent stiffness from the equivalent stiffness characteristics. Substituting the gap nonlinear terms into the solution, we can obtain its equivalent stiffness expression.
[0065]
[0066] 5. Based on the particle swarm optimization algorithm, the gap characteristic parameters are obtained by curve fitting between the equivalent stiffness expression obtained by integration and the equivalent stiffness curve obtained by model simulation.
[0067] 1) Initialize a group of random particles. Each particle has only two attributes: velocity and position. The entire process of particle swarm iteration is a process of gradually approaching the optimal position. The particle position represents the target parameter value, i.e., the gap parameter value.
[0068] 2) Set the fitness value to evaluate the quality of the particle position. In this method, the fitness value is set as the deviation of the curve from the target curve under the gap parameter. The deviation is calculated by the function deviation value at each point.
[0069] 3) For each particle, compare its fitness value with the fitness value of the optimal position. If it is better, set it as the current optimal position and update the individual optimal position and the individual optimal fitness value.
[0070] 4) Update the group's optimal position and optimal fitness value based on the individual's optimal position.
[0071] 5) Update the speed according to the formula.
[0072] 6) If the termination condition is not met, proceed to step 3).
[0073] The flowchart of the particle swarm optimization algorithm is as follows: Figure 5 As shown:
[0074] 6. Multiple identification experiments were conducted by changing different gap parameters, and the gap parameter identification results are shown in Table 1.
[0075] Table 1. Results of nonlinear gap parameter identification
[0076]
[0077] As can be seen from Table 1, the error in the identification results of nonlinear gap parameters is small, and the identification of nonlinear gap parameters has been achieved.
[0078] 7. The overall flowchart of this invention is as follows: Figure 1 As shown.
[0079] This invention uses a frequency domain identification method to identify the function description of the gap nonlinear modal parameters. While a time domain identification method can also be used to directly identify the modal parameters of the response signal in the time domain, the frequency domain method is more intuitive, has clearer physical concepts, and higher accuracy, making it more suitable for single-mode estimation. This invention processes the modal parameters of a sinusoidally excited nonlinear system through harmonic linearization. Small perturbation linearization or feedback linearization methods can also be used, but small perturbation linearization has a relatively large error, and feedback linearization requires specific conditions to be met.
Claims
1. A method for identifying structural gap nonlinearity parameters, characterized by, Includes the following steps: Step 1: Establish a mathematical model for the nonlinear system with structural gaps; The target vibration system is established as a single-degree-of-freedom nonlinear system subjected to simple harmonic forces, and sinusoidal sweep frequency excitation is used. The differential equation of motion is: wherein, denotes the mass coefficient, denotes the damping coefficient, denotes the stiffness coefficient, denotes the displacement, denotes the gap nonlinearity term, is the sinusoidal sweep excitation received; The gap nonlinear term is expressed as: wherein represents a displacement, represents a non-linear stiffness coefficient, represents a gap; Step 2: Obtain the frequency response function of the nonlinear system with structural gap under sinusoidal sweep excitation by using the system input and output responses, i.e., the relationship curve between displacement amplitude and frequency; Step 3: Analyze the system frequency response function. For each given displacement response amplitude, define the real and imaginary parts of the displacement frequency response function using a pair of symmetrical displacement frequency response function FRF points: in For a pair of symmetrical points, the frequency response function Information consisting of the real and imaginary parts; Furthermore, the system's natural frequency related to the displacement amplitude is obtained: in The square of the natural frequency. These are the corresponding frequencies of a pair of symmetrical points; Step 4: Extract the expression for the nonlinear equivalent stiffness of the system through simulation analysis of the model, and further obtain the equivalent stiffness curve of the system through model simulation data; Based on the harmonic balance method, a first-order approximate solution of the steady-state displacement response is taken. Substituting the values, we further expand the gap nonlinearity term and obtain the integral formula for solving the equivalent stiffness from the equivalent stiffness characteristics. Substituting the gap nonlinearity term, we obtain the equivalent stiffness expression of the system: Substitute the system's natural frequency and the real and imaginary parts of its frequency response into the formula for calculating the system's equivalent stiffness: Step 5: Based on the particle swarm optimization algorithm, the gap characteristic parameters are obtained by curve fitting between the equivalent stiffness expression obtained by integration and the equivalent stiffness curve obtained by model simulation. Step 5-1: Initialize a group of random particles. Each particle has only two attributes: velocity and position. The entire process of particle swarm iteration is the process of gradually approaching the optimal position. The particle position represents the target parameter value, i.e., the gap parameter value. Step 5-2: Set the fitness value to evaluate the quality of the particle position; set the fitness value to the deviation of the curve from the target curve under this gap parameter, and the deviation is calculated by the function deviation value at each point; Step 5-3: For each particle, compare its fitness value with the fitness value of the optimal position. If the fitness value of the particle is greater than that of the optimal position, set it as the current optimal position and update the individual optimal position and the individual optimal fitness value. Step 5-4: Update the group's optimal position and optimal fitness value based on the individual's optimal position; Step 5-5: Update the velocity according to the particle swarm optimization velocity update formula, as follows: in It is the number of iterations; It is inertial weight; It is an individual learning factor; It is a group learning factor; It is a random number within the interval [0,1], which increases the randomness of the search; It is a particle In the In the nth iteration A velocity vector of dimension; It is a particle In the In the nth iteration A dimensional position vector; It is a particle In the In the nth iteration The best historical position for Vienna; It is the group in the first In the nth iteration The best historical position for Vienna; Step 5-6: If the termination condition is not met, proceed to step 5-3.
2. The method for identifying nonlinear parameters of structural gaps according to claim 1, characterized in that, The mathematical model of the nonlinear system with structural gaps adopts the dead zone model.