Method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robotic milling

Abstract

The present application belongs to the technical field of milling processing, and discloses a parameter setting method of a tuned mass damper for suppressing low-frequency chatter in robot milling. The method comprises the following steps: S1, when low-frequency chatter in robot milling occurs, identifying the dominant mode of low-frequency and side-frequency vibration in the low-frequency chatter; S2, placing the tuned mass damper for low-frequency and side-frequency vibration on the last link of the robot, calculating the position with the maximum amplitude when the low-frequency and side-frequency vibration is dominated by the mode of vibration, and taking the position as the optimal position of the tuned mass damper to suppress the dominant mode; S3, obtaining the modal mass, damping and stiffness of the low-frequency and side-frequency vibration of the robot at the optimal position under the dominant mode, respectively, setting the mass of the tuned mass damper under low-frequency and side-frequency vibration, and calculating the optimal stiffness and optimal damping coefficient of the tuned mass damper in the low-frequency and side-frequency vibration. Through the present application, the problem of vibration suppression of low-frequency chatter in robot milling is solved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field related to milling processing, and more particularly, to a tuned mass damper parameter setting method for suppressing low-frequency chatter in robot milling. BACKGROUND

[0002] Robot milling chatter is a major obstacle to the improvement of processing efficiency, especially low-frequency chatter caused by the weak rigidity of the robot body structure, which often leads to large-amplitude vibration of the robot end and a sharp increase in cutting force, forcing the processing to be interrupted and easily causing damage to the workpiece and the robot processing system.

[0003] Robot milling low-frequency chatter can be avoided to some extent through processing planning. Cen et al. (2017) of the Georgia Institute of Technology in the United States used a conservative congruence transformation stiffness model to minimize the angle between the average cutting force direction and the direction of the maximum principal rigidity of the robot, so as to avoid low-frequency chatter. He et al. (2020) of Northeastern University avoided low-frequency chatter by selecting the processing feed direction. Another method is to suppress low-frequency chatter by applying additional force to the robot system through a specific device. Nguyen et al. (2020) of the Georgia Institute of Technology in the United States solved the optimal control problem of the linear quadratic regulator, obtained the attitude-related controller gain, and suppressed the chatter by controlling the drive joint. Xiao et al. (2020) of Tianjin University proposed a robot servo support milling active vibration suppression scheme.

[0004] Tuned mass damper is a vibration absorber composed of auxiliary mass, damping and spring, which uses its own resonance to react to the main structure, and can achieve good dynamic flexibility suppression effect with small mass ratio, which is a very potential new scheme for robot milling low-frequency chatter suppression. In order to maximize the suppression effect of TMD on robot low-frequency chatter, the present application considers the dynamic characteristics of the robot body and the characteristics of low-frequency chatter, and provides a TMD optimal adjustment strategy for robot milling low-frequency chatter suppression. Through the present application, the vibration suppression effect of TMD on robot milling low-frequency chatter can be maximized. SUMMARY

[0005] In view of the above defects or improvement needs of the prior art, the present application provides a tuned mass damper parameter setting method for suppressing low-frequency chatter in robot milling, which solves the problem of low-frequency chatter suppression in robot milling processing.

[0006] To achieve the above-mentioned purpose, according to the present application, a tuned mass damper parameter setting method for suppressing low-frequency chatter in robot milling is provided, characterized in that the method comprises the following steps:

[0007] S1 When low-frequency chatter occurs in robot milling, identify the dominant mode of low-frequency and side-frequency vibration in the low-frequency chatter.

[0008] S2 places the low-frequency and side-frequency tuned mass damper on the last link of the robot, calculates the position with the maximum amplitude when the low-frequency and side-frequency vibrations are calculated to dominate the modal vibration, and takes the position as the optimal position of the tuned mass damper to suppress the dominant mode;

[0009] S3 respectively acquires the modal mass, damping and stiffness of the low-frequency and side-frequency vibrations of the robot at the optimal position point under the dominant mode, sets the mass of the tuned mass damper under the low-frequency and side-frequency vibrations, and calculates the optimal stiffness and optimal damping coefficient of the tuned mass damper in the low-frequency and side-frequency vibrations.

[0010] Further preferably, in step S1, the dominant mode of the low-frequency vibration is calculated according to the following relationship:

[0011]

[0012] wherein L is the order of the dominant mode of the low-frequency vibration in the low-frequency chatter, M is the highest modal order of the robot under consideration, f L is the Lth modal frequency of the robot, f chatter is the frequency with the maximum amplitude identified from the displacement amplitude spectrum of the milling vibration signal.

[0013] Further preferably, in step S1, the dominant mode of the side-frequency vibration is calculated according to the following relationship:

[0014]

[0015]

[0016] wherein H1 and H2 are the orders of the dominant mode of the side-frequency vibration, n is the spindle speed, is the H1th modal frequency of the robot, is the H2th modal frequency of the robot, and M is the highest modal order of the robot under consideration.

[0017] Further preferably, in step S2, the optimal position of the low-frequency tuned mass damper is calculated according to the following relationship:

[0018]

[0019] wherein p L is the optimal position point of the low-frequency TMD, p (r) is a variable, representing the amplitude of the point p (r) on the rth link of the robot under the ith modal shape of the robot.

[0020] ​Further preferably, in step S2, the optimal position of the sideband tuned mass damper is calculated according to the following relationship:

[0021]

[0022]

[0023] in, and is the optimal position point of the sideband TMD, and are the lower points p of the robot’s H1-order and H2-order modal vibration modes respectively. (r) The amplitude at .

[0024] Further preferably, the installation direction of the optimal position is the direction of the normalized unit vector of the amplitude vector.

[0025] Further preferably, in step S3, the optimal stiffness of the low-frequency vibration is calculated according to the following relationship:

[0026]

[0027] in, and They are the L-th order mode of the robot at p L Point x L Modal mass and modal stiffness in the direction, m L is the modal mass of a given low-frequency TMD, k L is the optimal stiffness of low-frequency TMD.

[0028] Further preferably, in step S3, the optimal damping coefficient of the low-frequency vibration is calculated according to the following relationship:

[0029]

[0030] Among them, c L is the optimal damping coefficient of low-frequency TMD.

[0031] Further preferably, in step S3, the optimal modal stiffness of the sideband is calculated according to the following relationship:

[0032]

[0033]

[0034] in, and is the optimal modal stiffness of sideband TMD1 and sideband TMD2, and is the modal mass of the side frequency TMD1 and the side frequency TMD2, and is the optimal damping ratio of the side frequency TMD1 and the side frequency TMD2, f L is the Lth order modal frequency of the robot.

[0035] Further preferably, in step S3, the optimal damping coefficient of the side frequency is calculated according to the following relationship:

[0036]

[0037]

[0038] wherein, and is the optimal damping coefficient of the side frequency TMD1 and the side frequency TMD2.

[0039] Overall, compared with the prior art, the above technical solutions conceived by the present application have the following beneficial effects:

[0040] 1. The present application greatly improves the suppression effect of the TMD on the low-frequency chatter of the robot milling by suppressing chatter from two aspects of low frequency and side frequency respectively, increases the material removal rate of the robot without chatter, and improves the processing efficiency.

[0041] 2. The present application first determines whether low-frequency chatter occurs from the vibration signal, in the case of low-frequency chatter, determines the robot dominant mode of low-frequency vibration and side-frequency vibration according to the vibration signal, solves the optimal installation position and optimal installation direction of the TMD according to the vibration mode of the dominant mode, sets the modal mass of the TMD, and then calculates the optimal structural parameters of the TMD according to the modal parameters of the robot at the optimal installation position in the optimal installation direction, to maximize the vibration suppression effect. BRIEF DESCRIPTION OF DRAWINGS

[0042] Figure 1 is a flowchart of the setting method of the tuned mass damper for suppressing low-frequency chatter of robot milling according to the preferred embodiment of the present application;

[0043] Figure 2 is a schematic diagram of the side frequency signal of the low-frequency chatter of robot milling according to the preferred embodiment of the present application. (Milling amplitude spectrum schematic diagram) DETAILED DESCRIPTION

[0044] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and not to limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0045] A parameter setting method of a tuned mass damper for suppressing low-frequency chatter in robot milling, first determines whether low-frequency chatter occurs through vibration signals in robot milling, when low-frequency chatter occurs, identifies the dominant mode of low-frequency and side frequency vibration in low-frequency chatter; then, calculates the optimal position and direction of low-frequency TMD and side frequency TMD, and finally calculates the optimal structural parameters of low-frequency TMD and side frequency TMD. The specific steps are as follows:

[0046] S1 dominant mode identification

[0047] The present application is aimed at low-frequency chatter in robot milling, i.e. modal coupling chatter of the robot body, so first determine whether low-frequency chatter occurs from the vibration signals.

[0048] When low-frequency chatter occurs in robot milling, in addition to self-excited vibration at the low-frequency mode and forced vibration at the spindle rotation frequency, there are also significant vibration amplitudes at the distance from the mode frequency on both sides of the rotation frequency, which is called side frequency signal, as shown in Figure 2 The side frequency signal is derived from the modulation effect between the low-frequency vibration signal and the forced vibration signal. At the same time, the side frequency vibration also modulates the forced vibration to generate new low-frequency vibration. Therefore, not only does the low-frequency mode dynamic flexibility suppress low-frequency chatter, but also the suppression of side frequency vibration can suppress low-frequency chatter.

[0049] After determining the existence of low-frequency chatter, the dominant mode of low-frequency and side frequency vibration in low-frequency chatter is identified by the following steps.

[0050] Let the robot modal number be i, and the highest modal order considered be M, then i=1, 2, …, M. Let the frequency of the i-th mode be f i When low-frequency chatter occurs, the chatter frequency is very close to the robot modal frequency, and the amplitude at the chatter frequency is the largest. Therefore, first identify the frequency with the largest amplitude in the displacement amplitude spectrum of the milling vibration signal, denoted as f chatter Let the order of the dominant mode of low-frequency vibration in low-frequency chatter be L, then:

[0051]

[0052] Let the spindle speed be n, and the order of the dominant mode of side frequency vibration be H1 and H2 respectively, then:

[0053]

[0054]

[0055] S2 TMD optimal position and direction calculation

[0056] Let the serial number of the last link of the robot be r, and the r-link mode shape under mode i be X (r,i) In this embodiment, it is a 6x1 vector containing 3-dimensional translation and 3-dimensional rotation mode shapes, and the reference coordinate system is the robot base coordinate system CS0(X0-Y0-Z0). X (r,i) The specific solving method of X (r) may refer to the existing method, which will not be repeated in this application. Then under the i-th mode shape of the robot, the amplitude of the point p on the r-link can be calculated by the following formula:

[0057]

[0058] wherein is a 3x1 amplitude vector representing the amplitude size in the X0, Y0, and Z0 directions. is a coordinate matrix representing:

[0059]

[0060] wherein respectively represent the three-direction coordinates of the point p (r) .

[0061] Let the optimal position point of the low-frequency TMD be p L , and the optimal position points of the side-frequency TMD be and Then:

[0062]

[0063] respectively substitute p L , into formula (4) to calculate the amplitude vector, and the normalized unit vector of the amplitude vector is the optimal installation direction of the TMD at the three points, which are respectively denoted as x L ,

[0064] S3 TMD optimal structure parameter calculation

[0065] Let the modal mass, damping, and stiffness of the robot in the x L direction at the p L point of the L-th mode be and Given the modal mass m L, then according to the classical TMD optimal tuning calculation method, the optimal stiffness k of the low-frequency TMD is L and the optimal damping coefficient c L They are:

[0066]

[0067] The low-frequency TMD parameters obtained according to formula (7) can achieve the optimal suppression effect on the dynamic flexibility of the robot mode L.

[0068] The following describes how to calculate the optimal parameters for the sideband TMD. It's important to note that the configuration methods for low-frequency TMD and sideband TMD differ. When the TMD damping ratio approaches 0, the peak frequency response of the main system is significantly reduced (approaching 0), but the peak frequency response increases on either side (the smaller the TMD damping ratio, the greater the increase). Low-frequency TMD aims to reduce the modal dynamic compliance of the robot, so a certain amount of damping is required to achieve optimal tuning. If the low-frequency TMD damping ratio is too low, the increased dynamic compliance on either side of the main system's natural frequency may induce new low-frequency chatter.

[0069] In contrast, the sideband TMD only targets the dynamic flexibility at the sideband. Since the sideband variation range is very small, the damping ratio of the sideband TMD can be very small to achieve a better vibration suppression effect. However, since the low-frequency mode has a certain bandwidth, the damping ratio of the sideband TMD cannot be too small to avoid inducing self-excited vibration of the dynamic flexibility on both sides of the low-frequency modal frequency. The lower limit of the damping ratio of the sideband TMD depends on the width of the amplitude-frequency curve of the low-frequency mode. The half-power bandwidth of the robot's Lth-order mode can be estimated as:

[0070] Δf L ≈2ξ L f L (8)

[0071] where ξ L is the damping ratio of the Lth-order mode, expressed as:

[0072]

[0073] Note that the H1-order mode of the robot is Pointed The modal stiffness and modal mass in the direction are and The corresponding TMD is recorded as the sideband TMD1, and its modal mass, stiffness, and damping ratio are recorded as Note that the H2-order mode of the robot is Pointed The modal stiffness and modal mass in the direction are and The corresponding TMD is denoted as side frequency TMD2, whose modal mass, stiffness and damping ratio are denoted as The optimal parameter calculation method of side frequency TMD1 is introduced below.

[0074] The frequency band in which the modal dynamic flexibility of the main structure is reduced by more than half after the action of the side frequency TMD is called the effective frequency band of the side frequency TMD, and its bandwidth is approximately proportional to the square root of the mass ratio and the square root of the damping ratio The effective bandwidth of side frequency TMD1 is given by the following empirical formula:

[0075]

[0076] The self-excited vibration of the dynamic flexibility on both sides of the low-frequency modal frequency should be avoided, so:

[0077]

[0078] The minimum value that satisfies should be taken to achieve the optimal vibration suppression effect.

[0079] In summary, given the modal mass and of side frequency TMD1 and side frequency TMD2, the optimal damping ratio is as follows:

[0080]

[0081] The optimal modal stiffness of the side frequency TMD is:

[0082]

[0083] The optimal damping coefficient of the side frequency TMD is: and

[0084]

[0085] Those skilled in the art will readily understand that the above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.​​

Claims

1. A method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling, characterized in that: The parameter setting method includes the following steps: S1. When low-frequency chatter occurs during robot milling, identifying the dominant modes of low-frequency and side-frequency vibrations in the low-frequency chatter; S2 places the low-frequency and side-frequency tuned mass dampers on the last link of the robot and calculates the position where the amplitude of the low-frequency and side-frequency vibrations is the largest when they vibrate in the dominant mode. This position is used as the optimal position of the tuned mass damper to suppress the dominant mode. S3 obtains the modal mass, damping, and stiffness of the robot at the optimal position of the low-frequency and side-frequency vibrations under the dominant mode, sets the mass of the tuned mass damper under low-frequency and side-frequency vibrations, and calculates the optimal modal stiffness and optimal damping coefficient of the tuned mass damper in low-frequency and side-frequency vibrations; The optimal stiffness of the low-frequency vibration is calculated according to the following relationship: in, and The robot's L The first mode is Pointed modal mass and modal stiffness in the direction, is the modal mass of a given low-frequency TMD, is the optimal stiffness of low-frequency TMD; The optimal damping coefficient of the low-frequency vibration is calculated according to the following relationship: in, is the optimal damping coefficient of low-frequency TMD; The optimal modal stiffness of the sideband is calculated according to the following relationship: in, and is the optimal modal stiffness of sideband TMD1 and sideband TMD2, and is the modal mass of sideband TMD1 and sideband TMD2, and is the optimal damping ratio of sideband TMD1 and sideband TMD2, For the robot L The first modal frequency; The optimal damping coefficient of the sideband is calculated according to the following relationship: in, and is the optimal damping coefficient of sideband TMD1 and sideband TMD2.

2. The method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling according to claim 1, characterized in that: In step S1, the dominant mode of the low-frequency vibration is calculated according to the following relationship: in, L is the order of the dominant mode of low-frequency vibration in low-frequency flutter, M is the highest modal order of the robot under consideration, For the robot L The first modal frequency, is the frequency with the largest amplitude identified from the displacement amplitude spectrum of the milling vibration signal.

3. A method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling according to claim 1 or 2, characterized in that: In step S1, the dominant mode of the sideband vibration is calculated according to the following relationship: in, H 1 and H 2 is the order of the dominant mode of the sideband vibration, n is the spindle speed, For the robot H 1st order modal frequency, For the robot H 2nd order modal frequency, M is the highest modal order of the robot under consideration.

4. A method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling according to claim 1 or 2, characterized in that: In step S2, the optimal position of the low-frequency tuned mass damper is calculated according to the following relationship: in, is the optimal position point of low-frequency TMD, Is a variable, representing the robot r Connecting rod up point, For the robot i Lower point of the first mode shape The amplitude at .

5. The method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling according to claim 4, characterized in that: In step S2, the optimal position of the sideband tuned mass damper is calculated according to the following relationship: in, and is the optimal position point of the sideband TMD, and The robot H 1st order sum H Lower point of the 2nd-order mode shape The amplitude at .

6. The method for setting parameters of a tuned mass damper for suppressing low-frequency chatter in robot milling according to claim 5, characterized in that: The installation direction of the optimal position is the direction of the normalized unit vector of the amplitude vector.

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