A machine tool working mode test device and method based on cutting load simulation

Abstract

This invention belongs to the field of CNC machine tool performance testing, and discloses a machine tool working mode testing device and method based on cutting load simulation. The machine tool working mode testing device consists of a tool holder module, an axial force loading module, a radial force loading module, and a displacement testing module. The method steps include: constructing the machine tool working mode testing device; force-speed load graded loading; machine tool response data acquisition; working mode parameter identification; and quantitative analysis and modeling of working condition-response. This machine tool working mode testing device and method addresses the problem of difficulty in quantitatively analyzing cutting conditions and machine tool dynamic response. By actively applying static and dynamic forces simulating actual cutting conditions to the machine tool and quantitatively adjusting the magnitude of the applied load, quantitative analysis of machine tool working modes under different cutting conditions is achieved, establishing a quantitative relationship between working condition and response.

Description

Technical Field

[0001] This invention belongs to the field of CNC machine tool performance testing, and relates to a machine tool working mode testing device and method based on cutting load simulation. Background Technology

[0002] Operating modes refer to the inherent vibration characteristics of a structure under service conditions. During actual service, the contact stiffness and interfacial friction characteristics of the joints between machine tool components change with the operating conditions, causing deviations in modal parameters such as natural frequencies and damping ratios compared to results obtained from tests conducted under static conditions. Compared to traditional experimental modes, operating modes can more realistically reflect the dynamic characteristics of a machine tool under cutting conditions. Therefore, developing a method for testing the operating modes of a complete machine tool is of significant theoretical and engineering importance for revealing its dynamic behavior during actual cutting and ensuring the stability and reliability of the machining process.

[0003] In 2017, Liu Zhifeng et al. from Beijing University of Technology disclosed an experimental method for identifying the modal parameters of heavy machine tool foundations in patent CN201710329957.4. Through on-site ground vibration testing, they obtained the spectrum diagram of the heavy machine tool foundation, overcoming the shortcomings of insufficient excitation in existing hammer-based methods. In 2019, Wu Qin et al. from Lanzhou University of Technology disclosed a modal testing method for CNC machine tool feed systems based on built-in sensors in patent CN201910005800.5. This method effectively improves the signal-to-noise ratio of the test signals by repeatedly emitting excitation signals to the machine tool using a vibrator and collecting the machine tool vibration response signals using built-in sensors. In 2021, Mao Xinyong et al. from Huazhong University of Science and Technology disclosed a single-point online identification method for the main vibration modes of CNC machine tools under cutting conditions in patent CN202111226606.3. By including the excitation force in the state variables to construct a state space model of the machine tool, the main vibration mode identification can be completed with only a single measurement point, without the need for measurement point arrangement on the entire machine tool. In 2024, Liu Yanqiang et al. from Beijing University of Aeronautics and Astronautics disclosed a modal testing method based on the self-excitation of CNC machine tools in patent CN202411782897.8. By using the self-excitation of the machine tool machining process to measure the modal parameters of the workpiece, the use of a vibrator for excitation is avoided, effectively reducing the testing cost.

[0004] Currently, machine tool operating mode testing mainly falls into two categories: one is conducted under spindle idle conditions, and the other is conducted under actual cutting conditions. However, both methods suffer from the problem of difficulty in quantifying cutting loads, making it impossible to establish a quantitative relationship between cutting loads and machine tool dynamic response. Summary of the Invention

[0005] To address the aforementioned issues, this invention proposes a machine tool working mode analysis method and testing device based on cutting load simulation, which enables quantitative analysis of the dynamic response characteristics of machine tools under different cutting conditions.

[0006] The technical solution of the present invention: A machine tool working mode testing device based on cutting load simulation consists of a tool holder module 1, an axial force loading module 2, a radial force loading module 3, and a displacement testing module 4; Tool holder module 1 is used to connect the machine tool spindle and the machine tool working mode testing device; axial force loading module 2 is used to apply axial force to the machine tool; radial force loading module 3 is used to apply radial force to the machine tool; displacement testing module 4 is used to test the gap between tool holder module 1 and axial force loading module 2 and radial force loading module 3.

[0007] The tool holder module 1 consists of a central rotating shaft 1-1, a thrust disk 1-2, a thrust disk pressure ring 1-3, an upper pressure ring 1-4 for the rotor core 1-5, a lower pressure ring 1-6 for the rotor core 1-6, a displacement detection bushing 1-7, and a bushing end cover 1-8. The thrust disk 1-2, rotor core 1-5, and displacement detection bushing 1-7 are interference-fitted with the central rotating shaft 1-1 from top to bottom. The thrust disk 1-2 is axially fixed by the thrust disk pressure ring 1-3. The two ends of the rotor core 1-5 are axially fixed by the upper pressure ring 1-4 and the lower pressure ring 1-6 for the rotor core, respectively. The displacement detection bushing 1-7 is axially fixed by the bushing end cover 1-8.

[0008] The axial force loading module 2 consists of an axial stator housing 2-1, an upper axial stator core 2-2, an upper axial stator coil 2-3, an upper axial stator cover plate 2-4, a lower axial stator core 2-5, a lower axial stator coil 2-6, a lower axial stator cover plate 2-7, and an axial connection wire interface 2-8. The upper axial stator coil 2-3 is wound around the upper axial stator core 2-2, and the upper axial stator cover plate 2-4 is connected to the upper axial stator core 2-2. The lower axial stator coil 2-6 is wound around the lower axial stator core 2-5, and the lower axial stator cover plate 2-7 is connected to the lower axial stator core 2-5. The upper axial stator core 2-2 and the lower axial stator core 2-5 are fixed on the axial stator housing 2-1, and the axial stator housing 2-1 has an axial connection wire interface 2-8.

[0009] The radial force loading module 3 consists of a radial stator housing 3-1, a radial stator upper pressure ring 3-2, a radial stator core 3-3, eight radial stator coil frames 3-4, eight radial stator coils 3-5, a radial stator lower pressure ring 3-6, a fixed base 3-7, an air-cooling inlet 3-8, and a radial connecting wire interface 3-9. The eight radial stator coils 3-5 are wound around the eight radial stator coil frames 3-4. The radial stator coil frames 3-4 and the radial stator core 3-3 are interference-fitted. The two ends of the radial stator core 3-3 are fixed to the radial stator housing 3-1 through the radial stator upper pressure ring 3-2 and the radial stator lower pressure ring 3-6, respectively. The fixed base 3-7 is connected to the radial stator housing 3-1. The air-cooling inlet 3-8 is located on the side of the radial stator housing 3-1 and is used to cool the machine tool working mode test device based on cutting load simulation.

[0010] The displacement testing module 4 consists of a sensor mounting plate 4-1, a sensor cover plate 4-2, four radial displacement sensors 4-3, two axial displacement sensors 4-4, and four radial displacement sensor pressure plates 4-5. The four radial displacement sensors 4-3 are fixed to the sensor mounting plate 4-1 by the four radial displacement sensor pressure plates 4-5, the two axial displacement sensors 4-4 are fixed to the sensor mounting plate 4-1, and the sensor cover plate 4-2 is fixed to the sensor mounting plate 4-1.

[0011] The arrangement of the tool holder module 1, axial force loading module 2, radial force loading module 3, and displacement testing module 4 is as follows: the displacement testing module 4 is connected to the radial stator housing 3-1 of the radial force loading module 3 via the sensor mounting plate 4-1; the axial force loading module 2 is connected to the radial stator housing 3-1 of the radial force loading module 3 via the axial stator housing 2-1; the tool holder module 1 is connected to the machine tool spindle via the central rotating shaft 1-1; the thrust plate 1-2 of the tool holder module 1 is clearance-fitted with the axial upper stator core 2-2 of the axial force loading module 2; the rotor core 1-5 of the tool holder module 1 is clearance-fitted with the radial stator core 3-3 of the radial force loading module 3; and the displacement detection bushing 1-7 of the tool holder module 1 is clearance-fitted with the sensor mounting plate 4-1 of the displacement testing module 4.

[0012] A method for machine tool working modal analysis based on cutting load simulation includes the following steps: Step 1: Construction of the machine tool working mode testing device; First, connect the central rotating shaft 1-1 to the machine tool spindle, and fix the fixed base 3-7 on the machine tool worktable. By adjusting the coordinates of the machine tool spindle, according to the reading of the axial displacement sensor 4-4, adjust the gap between the thrust plate 1-2 and the axial stator core 2-2 to 0.5mm. According to the reading of the radial displacement sensor 4-3, adjust the gap between the rotor core 1-5 and the radial stator core 3-3 to 0.5mm. Then, arrange three-dimensional acceleration sensors at the end of the machine tool spindle, the spindle box, the column, the bed, and the worktable, ensuring that the coordinate system of the three-dimensional acceleration sensors is consistent with the direction of the machine tool coordinate system. Step 2: Gradual application of force-speed load; Using static force and machine tool spindle speed as two test factors, a static force was applied to the machine tool through a machine tool working mode testing device, while the machine tool spindle speed was adjusted using the machine tool CNC system; the machine tool spindle speed was then set to its limit speed. The static force is divided into three levels: 80%, 50%, and 20%, according to the ultimate bearing capacity. The ±80%, ±50%, and ±20% were divided into 6 levels. A full combination design with two experimental factors was adopted, and a total of 18 groups of experiments were carried out. Step 3: Machine tool response data acquisition; According to the 18 sets of tests designed in step two, the speed of the machine tool spindle was adjusted and the corresponding static force was applied in sequence. Under each set of tests, after the machine tool spindle ran stably and the static force was loaded to the set value, the acceleration response signals at each of the three-dimensional acceleration sensors on the machine tool were collected synchronously, and a total of 18 sets of test data were obtained. The collected acceleration response signals were processed by removing the mean, eliminating the trend term, and bandpass filtering. Step 4: Identification of working modal parameters; First, construct a discrete-time stochastic state-space model of the machine tool: In the formula, This is the system state vector; This is the acceleration response signal vector measured by the triaxial accelerometer; The system state matrix is ​​determined by the current static force. and the speed of the machine tool spindle The determination reflects the stiffness and damping of the machine tool under this test; This is the output matrix; Process noise is the excitation noise of the machine tool spindle rotation during machine tool operation. The noise is the test error of the triaxial accelerometer. Then, for each of the 18 sets of experimental data, the system state matrix was identified from the acceleration response signals measured by the triaxial accelerometer using the random subspace method. and output matrix For the system state matrix Perform eigenvalue decomposition: In the formula, It is a diagonal matrix, where the diagonal elements are discrete-time eigenvalues. λ r r=1,2,3,…,n ; The corresponding eigenvector matrix; using Discrete-time eigenvalues λ r Convert to continuous-time eigenvalues , The sampling time interval; It exists in the form of conjugate complex pairs, let , j The imaginary unit, Then the first r First mode in experiment The natural frequency is Damping ratio is ; Finally, the modal parameter identification results of the 18 sets of experiments were summarized to form a structured dataset. ; Step 5: Quantitative analysis and modeling of operating conditions and responses.

[0013] Based on the structured dataset extracted in step four D static force and the speed of the machine tool spindle As the independent variable, the extracted first r First mode in experiment The natural frequency below or damping ratio As the target response, a second-order polynomial regression model is used to fit the nonlinear mapping relationship between rotational speed, static force, and modal parameters: In the formula, , For constant terms; , , , The coefficients are linear main effects. , , , These are the coefficients for second-order nonlinear effects; , This is the coefficient for the interaction coupling effect between static force and rotational speed; Solving the regression coefficient matrix using the least squares method and ,in , Based on the quantitative relationship model established above, a three-dimensional response surface diagram of rotational speed-static force-modal parameters is plotted to analyze the relationship between the working conditions: force, rotational speed and the response: frequency, damping ratio.

[0014] The beneficial effects of this invention are: 1) This invention proposes a quantitative analysis method for machine tool working modes. This method actively applies static and dynamic forces simulating actual cutting conditions to the machine tool and quantitatively adjusts the applied load to achieve quantitative analysis of the machine tool working modes under different cutting conditions, thus resolving the quantitative relationship between working conditions (force, speed) and response (frequency, damping ratio). 2) Based on the above method, a machine tool working mode testing device was designed. This device adopts a non-contact loading method based on the principle of electromagnetic induction, which can apply controllable static and dynamic forces to the machine tool, ensuring the authenticity of the dynamic response without changing the machine tool structure, while replacing the actual cutting process and reducing material and tool consumption. Attached Figure Description

[0015] Figure 1 This is a flowchart of a machine tool working mode analysis method based on cutting load simulation; Figure 2 This is a schematic diagram of the overall machine tool working mode testing device based on cutting load simulation; Figure 3 This is a schematic diagram of the internal structure of a machine tool working mode testing device based on cutting load simulation; Figure 4 This is a schematic diagram of the displacement testing module of a machine tool working mode testing device based on cutting load simulation.

[0016] In the diagram: 1. Tool holder module; 2. Axial force loading module; 3. Radial force loading module; 4. Displacement testing module; 1-1. Central rotating shaft; 1-2. Thrust disc; 1-3. Thrust disc pressure ring; 1-4. Upper pressure ring of rotor core; 1-5. Rotor core; 1-6. Lower pressure ring of rotor core; 1-7. Displacement detection bushing; 1-8. Bushing end cover; 2-1. Axial stator housing; 2-2. Upper axial stator core; 2-3. Upper axial stator coil; 2-4. Upper axial stator cover plate; 2-5. Lower axial stator core; 2-6. Lower axial stator. 2-7 Coil; 2-7 Axial lower stator cover plate; 2-8 Axial connection line interface; 3-1 Radial stator housing; 3-2 Radial stator upper pressure ring; 3-3 Radial stator core; 3-4 Radial stator coil frame; 3-5 Radial stator coil; 3-6 Radial stator lower pressure ring; 3-7 Fixed base; 3-8 Air-cooled air inlet; 3-9 Radial connection line interface; 4-1 Sensor mounting plate; 4-2 Sensor cover plate; 4-3 Radial displacement sensor; 4-4 Axial displacement sensor; 4-5 Radial displacement sensor pressure plate. Detailed Implementation

[0017] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0018] Taking the working mode test and analysis of a three-axis vertical machining center as an example.

[0019] Step 1: Construction of the machine tool working mode testing device; First, connect the central rotating shaft 1-1 to the machine tool spindle, and fix the fixed base 3-7 on the machine tool worktable. By adjusting the coordinates of the machine tool spindle, according to the reading of the axial displacement sensor 4-4, adjust the gap between the thrust plate 1-2 and the axial stator core 2-2 to 0.5mm. According to the reading of the radial displacement sensor 4-3, adjust the gap between the rotor core 1-5 and the radial stator core 3-3 to 0.5mm. Then, arrange three-dimensional acceleration sensors at the end of the machine tool spindle, the spindle box, the column, the bed, and the worktable, ensuring that the coordinate system of the three-dimensional acceleration sensors is consistent with the direction of the machine tool coordinate system. Step 2: Gradual application of force-speed load; Using static force and machine tool spindle speed as two test factors, a machine tool working mode test device was used to apply static force to the machine tool, while the machine tool CNC system was used to adjust the spindle speed. Based on the actual service conditions of the machine tool under test, the spindle speed was set at three levels: 2000 rpm, 6000 rpm, and 10000 rpm, and the static force was set at six levels: -200 N, +200 N, -500 N, +500 N, -1000 N, and +1000 N. A two-factor full combination design was adopted, and a total of 18 sets of tests were carried out.

[0020] Step 3: Machine tool response data acquisition; According to the 18 sets of tests designed in step two, the spindle speed was adjusted and the corresponding static force was applied in sequence. Under each set of tests, after the machine tool spindle ran stably and the static force was loaded to the set value, the acceleration response signal of each measuring point on the machine tool was collected synchronously. In order to eliminate low-frequency drift and high-frequency noise during the test, the collected acceleration response signal was processed by removing the mean, eliminating the trend term, and bandpass filtering.

[0021] Step 4: Identification of working modal parameters; For the 18 sets of test data obtained in step 3, the natural frequency and damping ratio of the three-axis vertical machining center under different working conditions were identified and summarized into a structured dataset.

[0022] Step 5: Quantitative analysis and modeling of operating conditions and responses; Based on the structured dataset formed in step four, static force and rotational speed are used as independent variables, and the extracted first... j Using the first natural frequency or damping ratio as the target response, a nonlinear mapping relationship between rotational speed, static force, and modal parameters is fitted, and the regression coefficients are solved using the least squares method. β Based on the established mathematical model, three-dimensional response surface plots of rotational speed-static force-natural frequency and rotational speed-static force-damping ratio were plotted respectively.

[0023] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features, and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for machine tool working mode analysis based on cutting load simulation, characterized in that, Includes the following steps: Step 1: Construction of the machine tool working mode testing device; The machine tool working mode testing device based on cutting load simulation consists of a tool holder module (1), an axial force loading module (2), a radial force loading module (3), and a displacement testing module (4); The tool holder module (1) is used to connect the machine tool spindle and the machine tool working mode test device; the axial force loading module (2) is used to apply axial force to the machine tool; the radial force loading module (3) is used to apply radial force to the machine tool; the displacement test module (4) is used to test the gap between the tool holder module (1) and the axial force loading module (2) and the radial force loading module (3); The tool holder module (1) consists of a central rotating shaft (1-1), a thrust disk (1-2), a thrust disk pressure ring (1-3), an upper pressure ring (1-4) for the rotor core, a rotor core (1-5), a lower pressure ring (1-6) for the rotor core, a displacement detection bushing (1-7), and a bushing end cover (1-8). Among them, the thrust disc (1-2), rotor core (1-5), and displacement detection bushing (1-7) are interference-fitted with the central rotating shaft (1-1) from top to bottom. The thrust disc (1-2) is axially fixed by the thrust disc pressure ring (1-3). The two ends of the rotor core (1-5) are axially fixed by the upper pressure ring (1-4) and the lower pressure ring (1-6) of the rotor core, respectively. The displacement detection bushing (1-7) is axially fixed by the bushing end cover (1-8). The axial force loading module (2) consists of an axial stator housing (2-1), an axial upper stator core (2-2), an axial upper stator coil (2-3), an axial upper stator cover plate (2-4), an axial lower stator core (2-5), an axial lower stator coil (2-6), an axial lower stator cover plate (2-7), and an axial connection wire interface (2-8). An upward stator coil (2-3) is wound around an upward stator core (2-2), and an upward stator cover plate (2-4) is connected to the upward stator core (2-2); a downward stator coil (2-6) is wound around a downward stator core (2-5), and a downward stator cover plate (2-7) is connected to the downward stator core (2-5); the upward stator core (2-2) and the downward stator core (2-5) are fixed on an axial stator housing (2-1), and an axial connection wire interface (2-8) is provided on the axial stator housing (2-1); The radial force loading module (3) consists of a radial stator shell (3-1), a radial stator upper pressure ring (3-2), a radial stator core (3-3), eight radial stator coil frames (3-4), eight radial stator coils (3-5), a radial stator lower pressure ring (3-6), a fixed base (3-7), an air-cooled air inlet (3-8), and a radial connecting wire interface (3-9); Eight radial stator coils (3-5) are wound around eight radial stator coil frames (3-4). The radial stator coil frames (3-4) are interference-fitted with the radial stator core (3-3). The two ends of the radial stator core (3-3) are fixed to the radial stator housing (3-1) through the upper radial stator pressure ring (3-2) and the lower radial stator pressure ring (3-6) respectively. The fixed base (3-7) is connected to the radial stator housing (3-1). The air-cooling inlet (3-8) is located on the side of the radial stator housing (3-1) and is used to cool the machine tool working mode test device based on cutting load simulation. The displacement testing module (4) consists of a sensor mounting plate (4-1), a sensor cover plate (4-2), four radial displacement sensors (4-3), two axial displacement sensors (4-4), and four radial displacement sensor pressure plates (4-5). Four radial displacement sensors (4-3) are fixed to the sensor mounting plate (4-1) by four radial displacement sensor pressure plates (4-5), two axial displacement sensors (4-4) are fixed to the sensor mounting plate (4-1), and the sensor cover plate (4-2) is fixed to the sensor mounting plate (4-1). The arrangement relationship of the tool holder module (1), axial force loading module (2), radial force loading module (3) and displacement test module (4) is as follows: the displacement test module (4) is connected to the radial stator housing (3-1) of the radial force loading module (3) through the sensor mounting plate (4-1); the axial force loading module (2) is connected to the radial stator housing (3-1) of the radial force loading module (3) through the axial stator housing (2-1); the tool holder module (1) is connected to the machine tool spindle through the central rotating shaft (1-1); the thrust plate (1-2) of the tool holder module (1) is clearance-fitted with the axial upper stator core (2-2) of the axial force loading module (2); the rotor core (1-5) of the tool holder module (1) is clearance-fitted with the radial stator core (3-3) of the radial force loading module (3); and the displacement detection bushing (1-7) of the tool holder module (1) is clearance-fitted with the sensor mounting plate (4-1) of the displacement test module (4). First, connect the central rotating shaft (1-1) to the machine tool spindle, and fix the fixed base (3-7) on the machine tool worktable. By adjusting the coordinates of the machine tool spindle, according to the reading of the axial displacement sensor (4-4), adjust the gap between the thrust plate (1-2) and the axial stator core (2-2) to 0.5mm. According to the reading of the radial displacement sensor (4-3), adjust the gap between the rotor core (1-5) and the radial stator core (3-3) to 0.5mm. Then, arrange three-dimensional acceleration sensors at the end of the machine tool spindle, the spindle box, the column, the bed, and the worktable, ensuring that the coordinate system of the three-dimensional acceleration sensors is consistent with the direction of the machine tool coordinate system. Step 2: Gradual application of force-speed load; Using static force and machine tool spindle speed as two test factors, a static force was applied to the machine tool through a machine tool working mode testing device, while the machine tool spindle speed was adjusted using the machine tool CNC system; the machine tool spindle speed was then set to its limit speed. The static force is divided into three levels: 80%, 50%, and 20%, according to the ultimate bearing capacity. The ±80%, ±50%, and ±20% were divided into 6 levels. A full combination design with two experimental factors was adopted, and a total of 18 groups of experiments were carried out. Step 3: Machine tool response data acquisition; According to the 18 sets of tests designed in step two, the speed of the machine tool spindle was adjusted and the corresponding static force was applied in sequence. Under each set of tests, after the machine tool spindle ran stably and the static force was loaded to the set value, the acceleration response signals at each of the three-dimensional acceleration sensors on the machine tool were collected synchronously, and a total of 18 sets of test data were obtained. The collected acceleration response signals were processed by removing the mean, eliminating the trend term, and bandpass filtering. Step 4: Identification of working modal parameters; First, construct a discrete-time stochastic state-space model of the machine tool: In the formula, This is the system state vector; This is the acceleration response signal vector measured by the triaxial accelerometer; The system state matrix is ​​determined by the current static force. and the speed of the machine tool spindle The determination reflects the stiffness and damping of the machine tool under this test; This is the output matrix; Process noise is the excitation noise of the machine tool spindle rotation during machine tool operation. The noise is the test error of the triaxial accelerometer. Then, for each of the 18 sets of experimental data, the system state matrix was identified from the acceleration response signals measured by the triaxial accelerometer using the random subspace method. and output matrix For the system state matrix Perform eigenvalue decomposition: In the formula, It is a diagonal matrix, where the diagonal elements are discrete-time eigenvalues. λ r ( r=1, 2, 3,…, n ); The corresponding eigenvector matrix; using Discrete-time eigenvalues λ r Convert to continuous-time eigenvalues , The sampling time interval; It exists in the form of conjugate complex pairs, let , j The imaginary unit, Then the first r First mode in experiment The natural frequency is Damping ratio is ; Finally, the modal parameter identification results of the 18 sets of experiments were summarized to form a structured dataset. ; Step 5: Quantitative analysis and modeling of operating conditions and responses.

2. The machine tool working mode analysis method based on cutting load simulation according to claim 1, characterized in that, The specific implementation process of step five is as follows: Based on the structured dataset extracted in step four D static force and the speed of the machine tool spindle As the independent variable, the extracted first r First mode in experiment The natural frequency below or damping ratio As the target response, a second-order polynomial regression equation is used to establish a quantitative relationship model between rotational speed, static force, and modal parameters: In the formula, , For constant terms; , , , The coefficients are linear main effects. , , , These are the coefficients for second-order nonlinear effects; , This is the coefficient for the interaction coupling effect between static force and rotational speed; Solving the regression coefficient matrix using the least squares method and ,in , Based on the quantitative relationship model established above, a three-dimensional response surface diagram of rotational speed-static force-modal parameters is plotted to analyze the relationship between the working conditions: force, rotational speed and the response: frequency, damping ratio.

Images