Magnetorheological Fluid Vibration Absorber for Suppressing Chatter of Intelligent Spindle of Industrial Robot and Design Method

Abstract

The present invention discloses a magnetorheological fluid vibration absorber and design method for suppressing chatter of an intelligent spindle of an industrial robot. The magnetorheological fluid vibration absorber includes: a shear plate, an excitation coil, a mass block, silicone rubber, magnetorheological fluid, a cylinder, and a guide assembly. The magnetorheological fluid vibration absorber is installed on the intelligent spindle housing at the end of the industrial robot. The magnetorheological fluid vibration absorber can change its own natural frequency by adjusting the magnetic field intensity, so that the industrial robot's processing chatter frequency is the same as the absorber's natural frequency, thereby achieving the effect of suppressing chatter. A Gaussian process regression model is used to predict the modal parameters of different industrial robots at different processing points. Subsequently, the predicted modal parameters are used as input to obtain a transfer matrix for calculating the chatter frequency. Finally, utilizing the controllable storage modulus of the magnetorheological fluid in its pre-yield region, a magnetorheological fluid vibration absorber and design method for suppressing chatter of the intelligent spindle of an industrial robot are designed.

Description

Technical Field

[0001] The present invention belongs to the field of intelligent manufacturing, and in particular relates to a magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot and a design method thereof. Background Art

[0002] The flexibility and low cost of industrial robots make them a promising candidate for machining complex and large structures. However, CNC machine tools remain the mainstay of machining. This is primarily due to the low stiffness of industrial robots, which can cause chatter even at relatively small depths of cut. Therefore, suppressing chatter during industrial robot machining has become a significant issue in the manufacturing industry. Various passive and active strategies have been investigated to suppress chatter during industrial robot machining.

[0003] Passive vibration suppression strategies do not require external energy input and are relatively simple to operate. Passive vibration suppression strategies can be divided into two categories: (1) enhancing the structural stiffness of industrial robots. (2) installing various types of passive absorbers and dampers. Since parameters such as stiffness and damping are uncontrollable, passive vibration suppression strategies are difficult to use in actual industrial robot processing.

[0004] Active suppression strategies overcome the shortcomings of passive chatter suppression by enabling control based on machining signals. Chatter generated during robotic machining is primarily caused by dynamic forces, so various force sensing technologies and active control strategies can be used to suppress chatter. Active suppression strategies primarily rely on force control to avoid chatter, which results in low productivity. Furthermore, the complexity, high cost, and high energy consumption of active suppression strategies hinder their practical application. In contrast, semi-active vibration absorbers hold great promise for addressing chatter in robotic machining.

[0005] Semi-active suppression strategies require less external energy and can control the absorption of chatter energy by varying the input energy. The chatter frequency can vary significantly with different rotational speeds and machining postures during industrial robot machining. Therefore, it is necessary to design a magnetorheological fluid vibration absorber with a controllable natural frequency to suppress chatter generated by industrial robot machining. Summary of the Invention

[0006] In order to solve the technical problems raised in the above background technology, the present invention aims to provide a magnetorheological fluid vibration absorber and design method for suppressing the vibration of the intelligent spindle of an industrial robot. The natural frequency is controllable, providing a solution for achieving high-precision and efficient processing of industrial robots.

[0007] To achieve the above technical objectives, the technical solution of the present invention is as follows: a magnetorheological fluid vibration absorber for suppressing vibration of an intelligent spindle of an industrial robot, comprising a mass block, a magnetorheological fluid storage modulus component, a housing and silicone rubber; the mass block and the silicone rubber are both annular, the end face of the mass block is connected to the silicone rubber, four bosses are evenly arranged around the mass block, and the bosses are connected to the magnetorheological fluid storage modulus component; the mass block, the magnetorheological fluid storage modulus component and the silicone rubber are arranged in the housing; the magnetorheological fluid storage modulus component includes magnetorheological fluid, a cylinder, a shear sheet, an excitation coil and a guide assembly,

[0008] The mass block is connected to the shear piece through a piston rod, the cylinder is filled with magnetorheological fluid, the shear piece is set in the magnetorheological fluid, and an excitation coil is installed on one side of the cylinder; the cylinder is connected to the guide assembly, the guide assembly is guided along the axial direction of the main shaft, the shear piece moves radially along the main shaft, the excitation coil is connected to the power supply, and the excitation coil generates a magnetic field perpendicular to the shear piece.

[0009] The mass block, silicone rubber, shear sheet, and magnetorheological fluid in the absence of a magnetic field provide the fundamental frequency of the magnetorheological fluid vibration absorber. The storage modulus of the magnetorheological fluid in the pre-yield zone varies significantly with the magnetic field intensity, providing the magnetorheological fluid vibration absorber with a wider frequency shift characteristic. The current of the excitation coil is controlled to make the natural frequencies of the magnetorheological fluid vibration absorber in the x and y directions consistent with the flutter frequency. The shear sheet in the magnetorheological fluid vibration absorber operates in the pre-yield zone of the magnetorheological fluid.

[0010] The fundamental frequency of the magnetorheological fluid vibration absorber is adjusted by changing the thickness and contact area of the silicone rubber.

[0011] The mass block is fixed to the silicone rubber through an adhesive, and the mass block is fixed to the silicone rubber for shear motion.

[0012] One end of the piston rod is connected to the shear piece, and the other end is provided with a connecting plate, and the mass block is connected to the connecting plate through screws.

[0013] The displacement of the shear plate when vibrating is in the micrometer and millimeter range.

[0014] The cylinder is made of non-electromagnetic shielding material and is provided with a liquid inlet which is sealed during operation.

[0015] The guide assembly includes a rolling assembly, a splint and a fixed seat. The fixed seat is connected to the shell. A channel for the splint to pass through is set on the fixed seat. The rolling assembly is set at the position where the channel contacts the splint. The guide assembly in the x direction can only move in the y direction, and the guide assembly in the y direction can only move in the x direction. The rolling friction of the rolling assemblies in the x and y directions has a negligible effect on the natural frequency of the magnetorheological fluid vibration absorber. The rolling assembly uses linear bearings or universal balls.

[0016] The design method of the magnetorheological fluid vibration absorber for suppressing chatter of the intelligent spindle of an industrial robot according to the present invention comprises the following steps:

[0017] S1, through modal tests, the modal parameters of the industrial robot milling system in the x and y directions of the tool tip at discrete processing points are obtained respectively;

[0018] S2, using the obtained modal parameters and the corresponding posture of the industrial robot to establish a Gaussian process regression model to predict the modal parameters of the industrial robot in the x and y directions at the remaining processing points;

[0019] S3, using the predicted modal parameters of the processing point to calculate the chatter frequency of the robot under different processing parameters;

[0020] S4, design the silicone rubber thickness and contact area, mass of the mass block, size of the shear piece, excitation coil parameters, magnetorheological fluid model, and cylinder parameters according to the vibration frequency obtained at each processing point;

[0021] S5, using a sinusoidal frequency sweep signal to test the frequency shift characteristics of the magnetorheological fluid vibration absorber, and obtain the relationship between the excitation coil current and the natural frequency of the magnetorheological fluid vibration absorber;

[0022] S6, the magnetorheological fluid vibration absorber is installed on the spindle housing of the industrial robot, and the acceleration sensor is also installed in the x and y directions of the housing;

[0023] S7, planning in advance the change of the magnetorheological fluid vibration absorber current in the x and y directions over time according to the vibration frequency obtained in S3 and the processing parameters of the industrial robot;

[0024] S8, by analyzing the frequency domain components of the acceleration signal, the performance of the magnetorheological fluid vibration absorber in suppressing the machining chatter of the industrial robot is obtained;

[0025] S9, collecting acceleration signals as parameters for optimizing the magnetorheological fluid vibration absorber, and repeating S4 to S9 to obtain the optimized magnetorheological fluid vibration absorber.

[0026] The natural frequency of the magnetorheological fluid vibration absorber in a single direction is:

[0027]

[0028] in is the storage modulus of the magnetorheological fluid, G0, G ∞ , α1 and α2 are empirical constants, B is the magnetic field strength generated by the excitation coil, M is the mass of the mass block and the shear plate, A1=4l1w1 is the area of the shear plate, h1 is the distance between the shear plate and the upper wall of the cylinder, G2 is the shear modulus of the silicone rubber, is the contact area between the silicone rubber and the mass block, and h2 is the thickness of the silicone rubber.

[0029] Compared with the prior art, the present invention has at least the following beneficial effects:

[0030] Compared with existing inventions, the present invention designs a magnetorheological fluid vibration absorber and design method for suppressing the flutter of an intelligent spindle of an industrial robot. In view of the fact that the flutter frequency of an industrial robot varies with the processing posture and processing parameters and the variation range is large, the vibration displacement generated by the industrial robot processing is small, and the magnetorheological fluid has a large stiffness controllability in the pre-yield zone. A magnetorheological fluid vibration absorber with a larger frequency shift range is designed. The storage modulus of the magnetorheological fluid in the pre-yield zone varies greatly with the magnetic field intensity, providing the magnetorheological fluid vibration absorber with a wider frequency shift characteristic. The magnetic field intensities in the x and y directions can respectively control the storage modulus in their respective directions, so that the magnetorheological fluid vibration absorber has different natural frequencies in the x and y directions. In view of the difference between industrial robots and machine tools, that is, the large difference in modal parameters in the x and y directions, structures with controllable magnetic field intensities in the x and y directions are designed respectively to achieve different natural frequencies of the magnetorheological fluid vibration absorber in the x and y directions. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a schematic diagram of an industrial robot and a magnetorheological fluid vibration absorber.

[0032] Figure 2 This is a top view of the magnetorheological fluid vibration absorber.

[0033] Figure 3 This is a front view of the magnetorheological fluid vibration absorber.

[0034] Figure 4 It is a top view of the magnetorheological fluid storage modulus component.

[0035] Figure 5 This is a front view of the magnetorheological fluid storage modulus component.

[0036] Figure 6 It is a guide component.

[0037] In the accompanying drawings, among them: 1. magnetorheological fluid vibration absorber; 2. main shaft; 3. industrial robot; 11. mass block; 12. magnetorheological fluid storage modulus component; 121. magnetorheological fluid; 122. shear piece; 123. cylinder; 124. excitation coil; 13. housing; 14. silicone rubber; 15. guide assembly; 151. bearing; 152. splint; 153. fixing seat. DETAILED DESCRIPTION

[0038] The following is a further description of specific embodiments of the present invention in conjunction with the accompanying drawings. It should be noted that the description of these embodiments is intended to facilitate understanding of the present invention and does not constitute a limitation of the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0039] As core equipment in intelligent manufacturing, industrial robots hold great promise for machining large, complex parts, such as space capsules, aircraft skins, and ship propeller blades. However, their low precision and poor rigidity are bottlenecks hindering their ability to achieve high-precision and efficient machining.

[0040] refer to Figure 1 、 Figure 2 、 Figure 3 The present invention provides a magnetorheological fluid vibration absorber for suppressing vibration of an intelligent spindle of an industrial robot, comprising a mass block 11, a magnetorheological fluid storage modulus component 12, a shell 13 and a silicone rubber 14; the mass block 11 and the silicone rubber 14 are both annular, the end face of the mass block 11 is connected to the silicone rubber 14, four bosses are evenly arranged around the mass block 11, and the bosses are connected to the magnetorheological fluid storage modulus component 12; the mass block 11, the magnetorheological fluid storage modulus component 12 and the silicone rubber 14 are arranged in the shell 13; the magnetorheological fluid storage modulus component 12 includes a magnetorheological fluid 121, a cylinder 123, a shear sheet 122, an excitation coil 124 and a guide Component 15, the mass block 11 is connected to the shear piece 122 through the piston rod, the cylinder 123 is filled with magnetorheological fluid 121, the shear piece 122 is set in the magnetorheological fluid 121, and an excitation coil 124 is installed on one side of the cylinder 123; the x and y directions of the magnetorheological fluid vibration absorber 1 can have the same natural frequency or different natural frequencies; the cylinder 123 is connected to the guide component 15, the guide component 15 is guided along the axial direction of the main shaft, the shear piece 122 moves radially along the main shaft, the excitation coil 124 is connected to the power supply, the excitation coil 124 generates a magnetic field perpendicular to the shear piece 122, and the magnetorheological fluid vibration absorber 1 is set on the outside of the main shaft 2.

[0041] The mass 11, silicone rubber 14, shear plate 122, and magnetorheological fluid 121 in the absence of a magnetic field provide the fundamental frequency of the magnetorheological fluid vibration absorber 1. The shear plate 122 in the magnetorheological fluid vibration absorber 1 operates in the pre-yield region of the magnetorheological fluid 121. The storage modulus of the magnetorheological fluid 121 in the pre-yield region varies significantly with the magnetic field intensity, providing the magnetorheological fluid vibration absorber 1 with a wider frequency shift characteristic. The current in the excitation coil 124 is controlled so that the natural frequencies of the magnetorheological fluid vibration absorber 1 in the x and y directions are consistent with the flutter frequency.

[0042] The fundamental frequency of the magnetofluid variable vibration absorber 1 is adjusted by changing the thickness and contact area of the silicone rubber 14 .

[0043] The mass block 11 is fixed to the silicone rubber 14 by an adhesive, and the mass block 11 is fixed to the silicone rubber 14 for shear motion.

[0044] One end of the piston rod is connected to the shear piece 122 , and the other end is provided with a connecting plate, and the mass block 11 is connected to the connecting plate via screws.

[0045] The displacement of the shear piece 122 during vibration is in the order of micrometers and millimeters.

[0046] The cylinder barrel 123 is made of non-electromagnetic shielding material and has a liquid inlet, which is sealed during operation.

[0047] refer to Figure 3 and Figure 6 The guide assembly 15 includes a rolling assembly, a splint 152 and a fixed seat 153. The fixed seat 153 is connected to the shell. A channel for the splint 152 to pass through is set on the fixed seat 153. The rolling assembly is set at the position where the channel contacts the splint 152, wherein the guide assembly in the x direction can only move in the y direction, and the guide assembly in the y direction can only move in the x direction; the rolling friction of the rolling assemblies in the x and y directions has a negligible effect on the natural frequency of the magnetorheological fluid vibration absorber; the rolling assembly adopts bearings or universal balls; the magnetic field strength in the x and y directions is controlled separately, and different magnetic field strengths make the magnetorheological fluid have different storage moduli in the pre-yield zone; the ability to generate different natural frequencies in the x and y directions is suitable for situations where the modal parameters of the industrial robot in the x and y directions are relatively different.

[0048] The present invention designs a magnetorheological fluid vibration absorber for suppressing the chatter of an intelligent spindle of an industrial robot and a design method thereof to suppress the chatter generated during processing by the industrial robot, thereby achieving high-precision and high-efficiency processing of the industrial robot.

[0049] This embodiment takes commonly used industrial robot milling as an example to illustrate the magnetorheological fluid vibration absorber and design method designed by the present invention, and the steps are as follows.

[0050] Step 1: Obtain the modal parameters of the industrial robot at some discrete processing points through hammer modal tests. The modal parameters include natural frequency, damping ratio, and mass. The modal parameters are a priori Gaussian distribution related to the posture of the industrial robot 3.

[0051] Step 2: Give a Gaussian kernel function, where the hyperparameters in the Gaussian kernel function are obtained by optimizing the maximum likelihood estimation.

[0052] Step 3: The modal parameters of the processing point to be predicted and the measured modal parameters form a joint prior distribution, and then the modal parameters of the predicted processing point are obtained using Bayesian reasoning.

[0053] Step 4: Use the predicted modal parameters to establish the robot-milling system dynamics model and obtain the transfer matrix according to the semi-discrete method.

[0054] Step 5: Obtain the dominant frequency in the milling chatter of the industrial robot 3 by analyzing the characteristic roots of the transfer matrix.

[0055] Step 6: Based on the vibration frequencies obtained at various processing points, specific parameters of each component of the magnetorheological fluid vibration absorber 1 are given.

[0056] Step 7. The guide assembly 15 includes a bearing 151, a clamping plate 152 and a fixing seat 153 to ensure that the mass block can move in any direction. The guide assembly 15 in the x direction can only move in the y direction, and the guide assembly 15 in the y direction can only move in the x direction, ensuring that the x and y directions are subjected to the shear force generated by the silicone rubber 14 and the magnetorheological fluid 121 to the greatest extent.

[0057] Step 8: The proposed magnetorheological fluid vibration absorber 1 can have different natural frequencies in the x and y directions by changing the magnetic field strength in the x and y directions. The natural frequency of the magnetorheological fluid vibration absorber 1 in a single direction can be expressed as:

[0058]

[0059] in is the storage modulus of the magnetorheological fluid 121, G0, G ∞ , α1 and α2 are empirical constants, B is the magnetic field strength generated by the excitation coil 24, M is the mass sum of the mass block 11 and the shear piece 122, A1=4l1w1 is the area sum of the shear piece 122, h1 is the distance between the shear piece 122 and the upper wall of the cylinder 123, G2 is the shear modulus of the silicone rubber 14, is the contact area between the silicone rubber 14 and the mass block 11 , and h2 is the thickness of the silicone rubber 14 .

[0060] Step 9. Take an iron plate and install it on the two linear bearings supported by the vibration table. The magnetorheological fluid vibration absorber 1 is directly fixed on the iron plate. The magnetorheological fluid vibration absorber 1 is excited by a sinusoidal sweep signal. An acceleration sensor is installed on the upper end of the magnetorheological fluid vibration absorber 1 for data acquisition. A DC power supply is used to provide a current with a step size of 0.2A to the excitation coil 124 to test the frequency shift performance of the magnetorheological fluid vibration absorber 1.

[0061] Step 10: Obtain a fitting curve based on the relationship between the current and the natural frequency of the magnetorheological fluid vibration absorber 1 to provide a basis for subsequent control of the natural frequency of the magnetorheological fluid vibration absorber 1 .

[0062] Step 11. The magnetorheological fluid vibration absorber 1 is installed on the main shaft 2 housing. Before the industrial robot 3 performs processing, the dominant frequency of the chatter occurring at different processing points is predicted by the predicted modal parameters. The change of the current of the magnetorheological fluid vibration absorber 1 over time is planned in advance according to the chatter frequency and the processing parameters of the industrial robot 3, so that the industrial robot 3 can control the natural frequency of the magnetorheological fluid vibration absorber 1 to be equal to the dominant frequency of the chatter at each processing point, thereby ensuring the effect of chatter suppression.

Claims

1. A magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot, characterized in that: The invention comprises a mass block (11), a magnetorheological fluid storage modulus component (12), a housing (13) and a silicone rubber (14); the mass block (11) and the silicone rubber (14) are both annular, the end face of the mass block (11) is connected to the silicone rubber (14), four bosses are evenly arranged around the mass block (11), and the bosses are connected to the magnetorheological fluid storage modulus component (12); the mass block (11), the magnetorheological fluid storage modulus component (12) and the silicone rubber (14) are arranged in the housing (13); the magnetorheological fluid storage modulus component (12) comprises a magnetorheological fluid (121), a cylinder (123), a shearing piece (122), an excitation coil (124) and a guide assembly (15), The mass block (11) is connected to the shearing piece (122) through a piston rod, the cylinder (123) is filled with magnetorheological fluid (121), the shearing piece (122) is arranged in the magnetorheological fluid (121), and an excitation coil (124) is installed on one side of the cylinder (123); the cylinder (123) is connected to the guide assembly (15), the guide assembly (15) is guided along the axial direction of the main shaft, the shearing piece (122) moves along the radial direction of the main shaft, the excitation coil (124) is connected to a power supply, and the excitation coil (124) generates a magnetic field perpendicular to the shearing piece (122).

2. The magnetorheological fluid vibration absorber for suppressing chatter of the intelligent spindle of an industrial robot according to claim 1 is characterized in that: The mass block (11), the silicone rubber (14), the shear piece (122) and the magnetorheological fluid (121) in the absence of a magnetic field provide a basic frequency. The storage modulus of the magnetorheological fluid (121) in the pre-yield region varies greatly with the magnetic field intensity, providing a wider frequency shift characteristic for the magnetorheological fluid vibration absorber (1). The current of the excitation coil (124) is controlled so that the natural frequencies in the x and y directions are consistent with the vibration frequency. The shear piece (122) works in the pre-yield region of the magnetorheological fluid (121).

3. The magnetorheological fluid vibration absorber for suppressing chatter of the intelligent spindle of an industrial robot according to claim 1, characterized in that: The fundamental frequency is adjusted by changing the thickness and contact area of the silicone rubber (14).

4. The magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot according to claim 1, characterized in that: The mass block (11) is fixed on the silicone rubber (14) through an adhesive, and the mass block (11) is fixed on the silicone rubber (14) for shearing motion.

5. The magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot according to claim 1, characterized in that: One end of the piston rod is connected to the shearing piece (122), and the other end is provided with a connecting plate, and the mass block (11) is connected to the connecting plate via screws.

6. The magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot according to claim 1, characterized in that: The displacement of the shearing piece (122) during vibration is in the order of micrometers and millimeters.

7. The magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot according to claim 1, characterized in that: The cylinder barrel (123) is made of non-electromagnetic shielding material. A liquid inlet is provided on the cylinder barrel (123), and the liquid inlet is sealed during operation.

8. The magnetorheological fluid vibration absorber for suppressing chatter of an intelligent spindle of an industrial robot according to claim 1, characterized in that: The guide assembly (15) comprises a rolling assembly, a clamping plate (152) and a fixed seat (153), wherein the fixed seat (153) is connected to the housing, a channel for the clamping plate (152) to pass through is provided on the fixed seat (153), and the rolling assembly is provided at a position where the channel contacts the clamping plate (152), wherein the guide assembly in the x direction can only move in the y direction, and the guide assembly in the y direction can only move in the x direction; the rolling friction of the rolling assemblies in the x and y directions has a negligible effect on the natural frequency of the magnetorheological fluid vibration absorber; and the rolling assembly adopts a linear bearing or a universal ball.

9. The design method of the magnetorheological fluid vibration absorber for suppressing chatter of the intelligent spindle of an industrial robot according to any one of claims 1 to 8, characterized in that: The following steps are involved: S1, through modal tests, the modal parameters of the industrial robot milling system in the x and y directions of the tool tip at discrete processing points are obtained respectively; S2, using the obtained modal parameters and the corresponding posture of the industrial robot to establish a Gaussian process regression model to predict the modal parameters of the industrial robot in the x and y directions at the remaining processing points; S3, using the predicted modal parameters of the processing point to calculate the chatter frequency of the robot under different processing parameters; S4, design the silicone rubber thickness and contact area, mass of the mass block, size of the shear piece, excitation coil parameters, magnetorheological fluid model, and cylinder parameters according to the vibration frequency obtained at each processing point; S5, using a sinusoidal frequency sweep signal to test the frequency shift characteristics of the magnetorheological fluid vibration absorber, and obtain the relationship between the excitation coil current and the natural frequency of the magnetorheological fluid vibration absorber; S6, the magnetorheological fluid vibration absorber is installed on the spindle housing of the industrial robot, and the acceleration sensor is also installed in the x and y directions of the housing; S7, planning in advance the change of the magnetorheological fluid vibration absorber current in the x and y directions over time according to the vibration frequency obtained in S3 and the processing parameters of the industrial robot; S8, by analyzing the frequency domain components of the acceleration signal, the performance of the magnetorheological fluid vibration absorber in suppressing the machining chatter of the industrial robot is obtained; S9, collecting acceleration signals as parameters for optimizing the magnetorheological fluid vibration absorber, and repeating S(4) to S(9) to obtain the optimized magnetorheological fluid vibration absorber.

10. The magnetorheological fluid vibration absorber design method for suppressing chatter of an industrial robot intelligent spindle according to claim 1, characterized in that: The natural frequency of the magnetorheological fluid vibration absorber (1) in a single direction is: in is the storage modulus of the magnetorheological fluid (121), G0, G ∞ , α1 and α2 are empirical constants, B is the magnetic field strength generated by the excitation coil (124), M is the mass sum of the mass block (11) and the shear plate (122), A1=4l1w1 is the area sum of the shear plate (122), h1 is the distance between the shear plate (122) and the upper wall of the cylinder (123), G2 is the shear modulus of the silicone rubber (14), is the contact area between the silicone rubber (14) and the mass block (11), and h2 is the thickness of the silicone rubber (14).

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