Eddy Current Damping Calculation Model and Calculation Method for Cylindrical Nonlinear Eddy Current Damping

Abstract

The present invention discloses an accurate calculation model for eddy current damping and a calculation method for cylindrical nonlinear eddy current damping. The technical solution consists of two parts: the calculation of eddy current damping and the calculation of cylindrical nonlinear eddy current damping. This model takes into account the non-uniform distribution characteristics of the magnetic induction intensity in space and cancels the assumption that the conductor length is infinitely long in the current model. Two coordinate systems are established in the model, enabling the model itself to intuitively show the influence of the position of the cylindrical permanent magnet on the eddy current damping coefficient. Based on the above model, a calculation method for cylindrical nonlinear eddy current damping is proposed by changing the thickness of the non-magnetic conductive metal tube. It provides a new idea for the design of nonlinear damping in the field of dynamic vibration absorbers, especially for the design of the nonlinear damping required to eliminate the high-branch response in nonlinear dynamic vibration absorbers.

Description

Technical Field

[0001] The present invention belongs to the technical field of machinery, and particularly relates to a calculation model applicable to the eddy current damping of a dynamic vibration absorber. Background Art

[0002] A dynamic vibration absorber (DVA) is a vibration damping and energy dissipation device. Attaching it to a device with large vibrations can effectively reduce its vibrations. In the development process of DVA, the first to appear was the undamped linear dynamic vibration absorber (LDVA). The undamped LDVA mainly consists of a mass block and a linear stiffness element (spring), and its structure is very simple. When the main structure is subjected to a single excitation frequency, the optimized undamped LDVA can reduce the vibration of the main structure to zero by providing a reaction force to the main structure. Due to the linear stiffness characteristics of the undamped LDVA itself, its vibration damping effect is not affected by the excitation amplitude. However, when the excitation frequency changes, the effect of the undamped LDVA will quickly fail. In other words, the vibration damping frequency band of the undamped LDVA is very narrow, and the vibration damping effect has a strong selectivity for the excitation frequency.

[0003] In order to broaden the vibration damping frequency band of LDVA and reduce the sensitivity of the vibration damping effect to a narrow frequency band, researchers added a damping element on the basis of the spring-mass, so that while the LDVA provides a reaction force to the main structure, the energy transmitted from the main structure to the LDVA is dissipated through damping. Such an LDVA is called a damped LDVA or a tuned mass damper (TMD). The presence of the damping element enables the TMD to retain the advantages of the undamped LDVA (the vibration damping effect is not affected by the excitation amplitude) while broadening the vibration damping frequency band. However, the presence of the damping element also makes the structure of the TMD more complex, occupies a large space and is difficult to miniaturize, which greatly limits the application range of the TMD.

[0004] Therefore, researchers turned their attention to the Nonlinear Dynamic Vibration Absorber (NLDVA). Different from the LDVA, the NLDVA has strong non-linear stiffness characteristics and its own non-constant natural vibration frequency. When the excitation frequency changes, the NLDVA can have resonance capture with the main structure, so the NLDVA has a wider vibration reduction frequency band. At the beginning of the 21st century, an NLDVA called the Nonlinear Energy Sink (NES) was proposed. The structure of the NES is similar to that of the traditional LDVA - it consists of a spring-mass-damper system. When the excitation amplitude reaches a certain threshold, targeted energy transfer occurs between the main structure and the NES, and energy is quickly and unidirectionally transferred from the main structure to the NES and dissipated through the damping element in the NES, achieving the purpose of reducing the vibration of the main structure. However, when the excitation amplitude is less than this threshold, the NES consumes less energy and the vibration reduction effect is poor; when the excitation amplitude is greater than this threshold, the system will also have a high-branch response, causing the NES to fail quickly. Therefore, when the excitation amplitude is large, using non-linear damping can effectively delay or avoid the occurrence of high-branch responses and improve the stability of the NES.

[0005] Through the analysis of the vibration reduction characteristics of different DVAs, it can be seen that damping plays a very important role in the vibration reduction process of DVAs. Traditional DVAs usually use rubber dampers or hydraulic dampers as damping elements. As a high-damping material, rubber can consume energy more effectively, but the aging of rubber makes the life of the damper very short. Hydraulic dampers rely on mechanical friction to consume energy and require working fluids. Wear and liquid leakage will inevitably occur during long-term use, which will also reduce the service life of the damper and increase the maintenance cost. In addition, it is very difficult to adjust the damping of the above dampers in the later stage of the use of DVAs. In contrast, eddy current dampers have the advantages of non-contact, frictionless, simple structure, high reliability, etc., and can be widely applied to fields such as aerospace, building bridges, and military equipment. Moreover, the performance of eddy current dampers will not decrease over time. At the same time, due to their non-contact characteristics, eddy current dampers will not have additional stiffness similar to that of traditional viscous damping or viscoelastic damping due to material shear deformation.

[0006] The principle of the eddy current damper is to generate eddy current damping by the interaction between a magnet and a non-magnetic conductive metal. For the study of the cylindrical eddy current damper, assuming that the eddy current damping coefficient is a constant and the conductor tube is infinitely long, and assuming that the magnetic induction intensity distribution around the magnet is approximately uniform, the eddy current damping coefficient is approximately calculated. The eddy current damping coefficient given by this simplified approximate calculation model ignores the strong relationship between the value of the eddy current damping coefficient and the position of the magnet, and the calculation result has a large error. Therefore, the current simplified eddy current damping calculation model (and its coefficient calculation result) cannot accurately reflect the actual eddy current damping coefficient.

[0007] Under this technical background, the present invention proposes an accurate calculation method for eddy current damping, which not only gives the magnitude of the eddy current damping coefficient, but also intuitively gives the relationship between the eddy current damping coefficient and the position of the cylindrical permanent magnet in the model. According to the proposed accurate calculation model of eddy current damping, a design method for non-linear eddy current damping is proposed, providing a new idea for the design of non-linear damping. Summary of the Invention

[0008] The object of the present invention is to propose an accurate calculation model for eddy current damping and a calculation method for cylindrical non-linear eddy current damping to achieve the accurate calculation of the eddy current damping coefficient and the design of non-linear damping.

[0009] When a regular cylindrical permanent magnet moves in a cylindrical non-magnetic conductive metal tube, since the non-magnetic conductive metal tube is moving in a cutting magnetic induction line relative to the cylindrical permanent magnet, eddy currents will be generated on the surface of the non-magnetic conductive metal tube. The eddy currents generate a magnetic field opposite to the direction of the external magnetic field, providing an eddy current damping force for the cylindrical permanent magnet to hinder the movement of the cylindrical permanent magnet. The magnitude of the eddy current damping force is related not only to the movement speed of the cylindrical permanent magnet but also to the eddy current damping coefficient. The magnitude of the eddy current damping coefficient is related to the magnetic performance parameters, size parameters (radius, thickness) of the cylindrical permanent magnet, and size parameters (length, inner diameter, outer diameter) of the non-magnetic conductive metal tube.

[0010] As the basis of the present invention, the assumed conditions before modeling of the eddy current damping calculation model are: the eddy currents generated on the surface of the non-magnetic conductive metal tube in the actual situation are non-uniformly distributed; it is assumed that the current only moves along the conductor plane perpendicular to the axis of the non-magnetic conductive metal tube; the cylindrical permanent magnet only moves inside the non-magnetic conductive metal tube, and the length of the non-magnetic conductive metal tube should be greater than the length of the cylindrical permanent magnet; the movement trajectory of the cylindrical permanent magnet always follows the central axis of the cylindrical permanent magnet and the central axis of the non-magnetic conductive metal tube, without considering the offset of the cylindrical permanent magnet, the inner diameter and outer diameter of the non-magnetic conductive metal tube remain unchanged, and the thickness of the non-magnetic conductive metal tube is a constant. Based on the above assumptions, the present invention includes the following steps:

[0011] (1) Establish a Cartesian coordinate system at the center of the cylindrical permanent magnet. The cylindrical permanent magnet moves along the z-axis. According to the equivalent magnetization current model, calculate the magnetic induction intensity B of the cylindrical permanent magnet:

[0012]

[0013] Where: r′ is the distance between any point in space and any point on the surface current loop of the cylindrical permanent magnet, r′ = ((x - r A cosθ) 2 + (y - r A sinθ) 2 - (z - z1) 2 ) 1 / 2 , l A and r A are the height and radius of the cylindrical permanent magnet, μ0 and M A are the vacuum permeability and the magnetization intensity of the cylindrical permanent magnet respectively, and i, j, k are the unit vectors in the x, y, and z directions. Simplify the expression of the magnetic induction intensity B:

[0014] B = B i i + B j j + B k k (2)

[0015] The cylindrical permanent magnet moves along the z-axis, and a current is generated in the non-magnetic conductive metal tube. The magnetic field direction generated by the induced current in the non-magnetic conductive metal tube is opposite to the magnetic field direction of the cylindrical permanent magnet, hindering the movement of the cylindrical permanent magnet and generating eddy current damping. According to Faraday's law of electromagnetic induction, the induced electromotive force generated by any micro-element segment with a length of Δz in the non-magnetic conductive metal tube is equal to the sum of the voltage drops around the circumference of the non-magnetic conductive metal tube. The electromotive force ΔE of the Δz segment is:

[0016] ΔE = B j Lv (3)

[0017] v is the moving speed of the cylindrical permanent magnet. L is the equivalent circumference of the non-magnetic conductive metal tube, and its calculation formula is:

[0018] L = 2πr (4)

[0019]

[0020] r is the equivalent radius of the non-magnetic conductive metal tube, r n is the inner radius of the non-magnetic conductive metal tube, r w is the outer radius of the non-magnetic conductive metal tube, B j is the magnetic induction intensity in the y direction generated by the N pole of the cylindrical permanent magnet. According to formulas (1) to (2), the expression of B j is:

[0021]

[0022] As can be seen from formula (4), B j is related to the spatial coordinates (x, y, z) and is non-uniformly distributed over the thickness of the non-magnetic conductive metal tube, that is, the value of B j is different for any value in the x direction.

[0023] (2) Correct B j by dividing the interval [r n , r w into p micro-element segments on average (the value of p is large enough), that is, the values in the interval become r n，0 (r n ), r n，1 , r n，2 , …… r n，p-2 , r n，p-1 , r n，p (r w ). Substitute these values into formula (6) respectively to obtain the corresponding B j，0 …… B j，p . For example, the B n，p (r w ) corresponding to is: j，p is:

[0024]

[0025] According to the root mean square value formula, calculate the root mean square value B j，0 …… B j，p : jRMS :

[0026]

[0027] After correction, formula (3) becomes:

[0028] ΔE c = B jRMS Lv (9)

[0029] ΔE c is the electromotive force of the corrected Δz segment. Then the current Δi generated by the Δz segment is:

[0030]

[0031] R Δz is the equivalent resistance of the non-magnetic conductive metal tube of the Δz segment and is calculated by the following formula:

[0032]

[0033] ΔS = ΔrΔz (12)

[0034] In Equation (11), ΔR Δz is the resistance of the radial Δr micro-element segment within the non-magnetic conductive metal tube for the Δz segment. ρ is the resistivity of the non-magnetic conductive metal tube, ΔS is the equivalent cross-sectional area of the non-magnetic conductive metal tube, Δr is the micro-element segment of the non-magnetic conductive metal tube in the thickness direction for integration, and Δz is the micro-element segment of the non-magnetic conductive metal tube in the length direction. Substitute Equation (12) into (11):

[0035]

[0036]

[0037] After the integration calculation of Equation (14), the following equation is obtained:

[0038]

[0039] The force ΔF exerted by the N pole of the cylindrical permanent magnet on the Δz segment Δz is:

[0040] ΔF Δz = B jRMS ΔiL (16)

[0041] Then the force F exerted by the N pole of the cylindrical permanent magnet on a section of the non-magnetic conductive metal tube N is:

[0042] F N = ∫ΔF Δz (17)

[0043] (3) Establish another Cartesian coordinate system (O p -x p y p z p ) at the center of the initial end of the non-magnetic conductive metal tube. Assume that the position of the cylindrical permanent magnet within the non-magnetic conductive metal tube at this time is (0, 0, z p ). After arrangement, we get:

[0044]

[0045] In Equation (18), l N (z p ) is the length of the non-magnetic conductive metal tube affected by the N pole of the cylindrical permanent magnet, and F N (z p ) is the eddy current damping force corresponding to the N pole of the cylindrical permanent magnet. The values of both will change with the change of the position z p of the cylindrical permanent magnet, and F N (z p) is proportional to the first power of the moving speed v of the cylindrical permanent magnet, which fully conforms to the assumption of linear viscous damping in structural dynamics.

[0046] The interaction between the S pole of the cylindrical permanent magnet and the non-magnetic conductive metal tube is similar to that of the N pole, but the length of the non-magnetic conductive metal tube affected by the S pole of the cylindrical permanent magnet is l T -l N (z p ),l T is the total length of the non-magnetic conductive metal tube. Therefore, the eddy current damping force F S (z p ) acting on the S pole of the cylindrical permanent magnet is:

[0047]

[0048] The total eddy current damping force F C (z p ) acting on the non-magnetic conductive metal tube as a whole is:

[0049] F C (z p )=F N (z p )+F S (z p ) (20)

[0050] Let the eddy current damping coefficient be c m . According to the relationship between the linear damping force and the damping coefficient:

[0051] F C =c m v (21)

[0052] The calculation formula for the eddy current damping coefficient can be obtained as:

[0053]

[0054] Through formula (22), the eddy current damping coefficient c p corresponding to any position z m (z p ) of the cylindrical permanent magnet in the non-magnetic conductive metal tube can be calculated. It should be noted here that when integrating B jRMS in the interval [0, l N (z p )], the value of B jRMS is related to the interval position. In other words, the magnetic induction intensity of the cylindrical permanent magnet is also non-uniformly distributed in the z direction.

[0055] The eddy current damping precise calculation model proposed by the present invention can intuitively reflect the relationship between the eddy current damping coefficient and the position of the cylindrical permanent magnet. On this basis, while keeping the inner diameter of the non-magnetic conductive metal tube unchanged and changing the outer diameter of the non-magnetic conductive metal tube, when the cylindrical permanent magnet is at different positions inside the non-magnetic conductive metal tube, the corresponding thickness of the non-magnetic conductive metal tube is different, and the eddy current damping force received by the cylindrical permanent magnet is also different. Based on this, a design or calculation method for non-linear eddy current damping is proposed.

[0056] Based on the eddy current damping calculation model of the present invention, the calculation of the cylindrical non-linear eddy current damping is then carried out. The assumed conditions are: the inner diameter r of the non-magnetic conductive metal tube n remains unchanged, and the outer diameter r w changes (the other assumed conditions except the thickness are the same as those of the eddy current damping calculation model). When the cylindrical permanent magnet is at different positions inside the non-magnetic conductive metal tube, the corresponding thickness of the non-magnetic conductive metal tube is different. Based on this, a calculation method for the cylindrical non-linear eddy current damping is proposed, including the following steps:

[0057] (1) Set the functional relationship between the outer radius r of the non-magnetic conductive metal tube w and the position z of the cylindrical permanent magnet p :

[0058] r w (z p ) = f(z p ) (23)

[0059] Formula (5) becomes:

[0060]

[0061] (2) Assume that the position of the cylindrical permanent magnet is z p , and the interval [r n , r w (z p )] needs to be evenly divided into p micro-element segments (the value of p is large enough). The values within the interval become r n，0 (r n ), r n，1 , r n，2 , …… r n，p-2 , r n，p-1 , r n，p (r w (z p ))). Substitute these values into formula (6) respectively to obtain the corresponding B j，0 …… B j，p . For example, the B n，p (r w (z p )) corresponding to j，p is:

[0062]

[0063] Equation (25) is the same in form as Equation (7), but the r in Equation (7) n，p always remains unchanged, and the r in Equation (25) n，p or rather r w (z p ) has a value related to the position z of the cylindrical permanent magnet p . After that, continue to calculate the root mean square value B j，0 ... B j，p , and perform the calculations of Equations (9) to (13). When performing the integration operation of Equation (14), the upper integration limit becomes r jRMS (z w ): p )

[0064]

[0065] After the integration calculation, the following equation is obtained:

[0066]

[0067] The subsequent derivation process is the same as that of Equations (16) to (21), and finally the calculation formula for the cylindrical non-linear eddy current damping coefficient is:

[0068]

[0069] According to Equations (17) to (20), first calculate the eddy current damping forces exerted on the N pole and S pole of the cylindrical permanent magnet by the non-magnetic conductive metal tube respectively, and then perform a linear addition calculation to obtain the resultant eddy current damping force on the cylindrical permanent magnet.

[0070] The key technologies of the present invention are as follows: the magnet is cylindrical in shape, and the non-magnetic conductive metal tube is cylindrical in shape. By equating the non-magnetic conductive metal tube to a coil of the same material, a coordinate system is established at the center of the cylindrical permanent magnet, and the interactions between the N pole and S pole of the cylindrical permanent magnet and the non-magnetic conductive metal tube (conductor coil) are calculated separately. Then, the magnetic induction intensity in the thickness direction of the non-magnetic conductive metal tube is reasonably equated (the magnetic induction intensity is non-uniformly distributed in space), and the resistance of the non-magnetic conductive metal tube (conductor coil) is calculated by integration. Moreover, by establishing a second coordinate system on the non-magnetic conductive metal tube, the relationship between the position of the cylindrical permanent magnet and the effective action length of the non-magnetic conductive metal tube is established (the overall length of the non-magnetic conductive metal tube is finite), so that the model itself can intuitively show the influence of the position of the cylindrical permanent magnet on the eddy current damping coefficient, and thus calculate the magnitude of the eddy current damping coefficient corresponding to different positions of the cylindrical permanent magnet inside the non-magnetic conductive metal tube. Keeping the inner diameter of the non-magnetic conductive metal tube unchanged and changing the outer diameter of the non-magnetic conductive metal tube, the outer diameter of the non-magnetic conductive metal tube corresponding to different positions of the cylindrical permanent magnet inside the non-magnetic conductive metal tube is different. When the cylindrical permanent magnet moves to different positions, it will be affected by different eddy current damping coefficients, realizing the design of non-linear eddy current damping.

[0071] The features and beneficial effects of the present invention are as follows: (1) Compared with the current eddy current damping calculation model, this calculation model takes into account the non-uniform distribution characteristics of the magnetic induction intensity in space and cancels the assumption that the conductor length is infinite in the traditional model. It can accurately calculate the magnitude of the eddy current damping coefficient when the cylindrical permanent magnet is in different positions, and the model itself can intuitively show the influence of the position of the cylindrical permanent magnet on the eddy current damping coefficient. (2) This model is closer to the actual interaction situation between the cylindrical permanent magnet and the non-magnetic conductive metal tube in practical applications. Based on the eddy current damping calculation model, the calculation method of non-linear eddy current damping only needs to adjust the outer diameter of the non-magnetic conductive metal tube to achieve the design of non-linear eddy current damping, with clear principle, simple structure and easy implementation. (3) It provides a new idea for the non-linear damping design in the field of dynamic vibration absorbers, especially for the non-linear damping design required for non-linear dynamic vibration absorbers to eliminate high-branch responses. (4) It is particularly emphasized that the eddy current damping calculation and the non-linear eddy current damping calculation method are not limited to the application of magnetic dynamic vibration absorbers. For non-magnetic dynamic vibration absorbers, only by replacing the original mass block with a cylindrical permanent magnet of the same mass and adding a non-magnetic conductive metal tube outside, the present invention is also applicable. Description of the Drawings

[0072] Appendix Figure 1 is a schematic diagram of the eddy current damping calculation model (the outer diameter of the non-magnetic conductive metal tube remains unchanged).

[0073] Appendix Figure 2 is the calculation result diagram of the eddy current damping calculation model (the outer diameter of the non-magnetic conductive metal tube remains unchanged).

[0074] Appendix Figure 3 is the schematic diagram of the segmented non-linear eddy current damping design method.

[0075] Appendix Figure 4 is the calculation result diagram of the segmented non-linear eddy current damping design method.

[0076] Appendix Figure 5 is the schematic diagram of the continuous non-linear eddy current damping design method.

[0077] Appendix Figure 6 is the calculation result diagram of the continuous non-linear eddy current damping design method. Specific implementation manner

[0078] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings and through embodiments. It should be noted that although the accompanying drawings of the specification describe the embodiments, this implementation manner is merely illustrative and not restrictive. Without departing from the spirit of the present invention and the scope protected by the claims, the materials, shape parameters, and size parameters of each component can also be changed, and these all fall within the protection scope of the present invention.

[0079] The technical solution of the present invention consists of two parts, namely: the calculation model of eddy current damping and the calculation method of cylindrical non-linear eddy current damping.

[0080] Combined with Appendix Figure 1 , in the accurate calculation model of eddy current damping, the cylindrical permanent magnet 1 moves along the z direction at a speed v inside the non-magnetic conductive metal tube 2. It is assumed that the induced current generated inside the non-magnetic conductive metal tube only moves along the conductor plane perpendicular to the axis of the non-magnetic conductive metal tube, that is, the non-magnetic conductive metal tube is equivalent to a conductor coil of the same length, and the induced current moves along the conductor coil. The cylindrical permanent magnet only moves inside the non-magnetic conductive metal tube and does not exceed the two ends of the non-magnetic conductive metal tube. This also means that the length of the non-magnetic conductive metal tube is longer than the length of the cylindrical permanent magnet, but the length of the non-magnetic conductive metal tube is limited. Moreover, when the cylindrical permanent magnet moves, the central axis of the cylindrical permanent magnet always follows the central axis of the non-magnetic conductive metal tube, and the offset of the cylindrical permanent magnet is not considered.

[0081] The material of the cylindrical permanent magnet can be selected from neodymium iron boron, ferrite, alnico, or samarium cobalt, and the magnetic performance parameters of the cylindrical permanent magnet are not limited. The material of the non-magnetic conductive metal tube can be selected from copper, aluminum, magnesium, zinc, etc. There are no special restrictions on the size parameters of the cylindrical permanent magnet and the non-magnetic conductive metal tube. There are also no special restrictions on the length of the cylindrical permanent magnet, the diameter of the cylindrical permanent magnet, the inner diameter, length, and outer diameter of the non-magnetic conductive metal tube. The outer diameter of the non-magnetic conductive metal tube needs to be selected according to different design methods.

[0082] As a specific embodiment 1, the material of the cylindrical permanent magnet is selected as Nd2Fe 14 B, the material of the non-magnetic conductive metal tube is selected as aluminum, and the size of the cylindrical permanent magnet is: length 10 mm × diameter 5.97 mm. The size of the non-magnetic conductive metal tube is: outer diameter , inner diameter , length 60 mm.

[0083] Calculate the corresponding eddy current damping coefficient when the cylindrical permanent magnet 1 is at different positions in the non-magnetic conductive metal tube 2 through the eddy current damping calculation model:

[0084] Establish a Cartesian coordinate system (O-xyz) at the center of the cylindrical permanent magnet as shown in the appendix Figure 1 , and the cylindrical permanent magnet moves along the z-axis. According to the magnetization current theory, calculate the magnetic induction intensity B of the cylindrical permanent magnet:

[0085]

[0086] Among them, r′ is the distance between any point in space and any point on the surface current loop of the cylindrical permanent magnet. r′ = ((x - r A cosθ) 2 +(y - r A sinθ) 2 -(z - z1) 2 ) 1 / 2 , l A and r A are the height and radius of the cylindrical permanent magnet, with the unit of m. μ0 and M A are the vacuum magnetic permeability and the magnetization intensity of the cylindrical permanent magnet respectively, and M A has the unit of A / m. i, j, k are the unit vectors in the x, y, z directions respectively. Simplify the expression of the magnetic induction intensity B:

[0087] B = B i i + B j j + B k k (2)

[0088] B i 、Bj , B k The units are all T (tesla).

[0089] According to Faraday's law of electromagnetic induction, the induced electromotive force generated by any infinitesimal segment with a length of Δz in a non-magnetic conductive metal tube is equal to the sum of the voltage drops around the circumference of the non-magnetic conductive metal tube. The electromotive force ΔE of the Δz segment is obtained as follows:

[0090] ΔE = B j Lv (3)

[0091] The unit of ΔE is V (volt), and v is the moving speed of the cylindrical permanent magnet, with the unit of m / s (meter per second).

[0092] L is the equivalent circumference of the non-magnetic conductive metal tube, with the unit of m. Its calculation formula is:

[0093] L = 2πr (4)

[0094]

[0095] r is the equivalent radius of the non-magnetic conductive metal tube, r n is the inner radius of the non-magnetic conductive metal tube, r w is the outer radius of the non-magnetic conductive metal tube, and the units are all m (meter). B j is the magnetic induction intensity in the y direction generated by the N pole of the cylindrical permanent magnet. According to formulas (1) to (2), the expression of B j is:

[0096]

[0097] Correct B j . Divide the interval [r n , r w into p infinitesimal segments on average (the value of p is large enough), that is, the values in the interval become r n，0 (r n ), r n，1 , r n，2 , …… r n，p-2 , r n，p-1 , r n，p (r w ). Substitute the above values into formula (6) respectively to obtain the corresponding B j，0 …… B j，p .

[0098] For example, the B n，p corresponding to r w (r j，p ) is:

[0099]

[0100] According to the root mean square value formula, for B j，0 ……B j，p Find the root mean square value B jRMS :

[0101]

[0102] Therefore, formula (3) is corrected to:

[0103] ΔE c = B jRMS Lv (9)

[0104] ΔE c is the electromotive force of the corrected Δz segment, with the unit of V (volt). Then the current Δi generated by the Δz segment is:

[0105]

[0106] The unit of Δi is A (ampere). R Δz is the equivalent resistance of the non-magnetic conductive metal tube in the Δz segment, with the unit of Ω (ohm). It is calculated by the following formula:

[0107]

[0108] ΔS = ΔrΔz (12)

[0109] In formula (11), ΔR Δz is the resistance of the radial Δr micro-element segment in the non-magnetic conductive metal tube in the Δz segment, with the unit of Ω (ohm). ρ is the resistivity of the non-magnetic conductive metal tube, with the unit of Ω / m (ohm / meter). ΔS is the equivalent cross-sectional area of the non-magnetic conductive metal tube, with the unit of m 2 (square meter). Δr is the micro-element segment of the non-magnetic conductive metal tube in the thickness direction for integration, with the unit of m (meter). Δz is the micro-element segment of the non-magnetic conductive metal tube in the length direction, with the unit of m. Substitute formula (12) into (11):

[0110]

[0111]

[0112] After the integral calculation of formula (14), the following formula is obtained:

[0113]

[0114] The force ΔF (unit: N (newton)) exerted on the Δz segment by the N pole of the cylindrical permanent magnet is: Δz (unit: N (newton)) is:

[0115] ΔFΔz = B jRMS ΔiL(16)

[0116] Then the force F exerted by the N pole of the cylindrical permanent magnet on a non-magnetic conductive metal tube N (in N (Newton)) is:

[0117] F N = ∫ΔF Δz (17)

[0118] Establish a Cartesian coordinate system (O p -x p y p z p ) at the center of the initial end of the non-magnetic conductive metal tube. Assume that the position of the cylindrical permanent magnet in the non-magnetic conductive metal tube at this time is (0, 0, z p ). After arrangement, we get:

[0119]

[0120] In formula (18), l N (z p ) is the length of the non-magnetic conductive metal tube affected by the N pole of the cylindrical permanent magnet, in m (meter). F N (z p ) is the eddy current damping force corresponding to the N pole of the cylindrical permanent magnet, in N (Newton). The values of both will change with the change of the position z p of the cylindrical permanent magnet. It can be seen that F N (z p ) is proportional to the first power of the moving speed v of the cylindrical permanent magnet, which fully conforms to the assumption of linear viscous damping in structural dynamics. The interaction between the S pole of the cylindrical permanent magnet and the non-magnetic conductive metal tube is similar to that of the N pole, but the length of the non-magnetic conductive metal tube affected by the S pole of the cylindrical permanent magnet is l T -l N (z p )(i.e., the l Figure 1 in S ), and l T is the total length of the non-magnetic conductive metal tube, in m (meter). Therefore, the eddy current damping force F S (z p )(in N (Newton)) exerted by the S pole of the cylindrical permanent magnet is:

[0121]

[0122] Therefore, the total eddy current damping force F C (z p )(in N (Newton)) exerted on the non-magnetic conductive metal tube as a whole is:

[0123] F C (z p ) = F N (z p ) + F S (z p ) (20)

[0124] Let the eddy current damping coefficient be c m , with the unit of Ns / m (Newton·second / meter). According to the relationship between the damping force and the damping coefficient:

[0125] F C = c m v (21)

[0126] The calculation formula for the eddy current damping coefficient can be obtained as:

[0127]

[0128] According to formulas (17) - (20), first calculate the eddy current damping forces acting on the N - pole and S - pole of the cylindrical permanent magnet and the non - magnetic conductive metal tube respectively, and then perform a linear addition calculation to obtain the resultant eddy current damping force on the cylindrical permanent magnet.

[0129] When the position z p of the cylindrical permanent magnet changes, the value of the eddy current damping coefficient c m (z p ) will change accordingly. Through formula (22), the eddy current damping coefficient c p corresponding to any position z m (z p ) of the cylindrical permanent magnet in the non - magnetic conductive metal tube can be calculated. It should be noted here that when B jRMS is integrated in the interval [0, l N (z p )], the value of B jRMS is related to the interval position. In other words, the magnetic induction intensity of the cylindrical permanent magnet is also non - uniformly distributed in the z - direction.

[0130] Since the cylindrical permanent magnet is always moving inside the non - magnetic conductive metal tube and does not exceed the two ends of the non - magnetic conductive metal tube, according to the length of the non - magnetic conductive metal tube and the length of the cylindrical permanent magnet in Example 1, the range of z p is [5mm, 55mm]. When the value of z p is 5mm, the value of c m (5) is 0.1566 Ns / m; when the value of z p is 10mm, the value of c m (10) is 0.2102 Ns / m; when the value of z pWhen the value is 20 mm, c m (20) has a value of 0.2107 Ns / m; when z p When the value is 30 mm, c m (30) has a value of 0.2107 Ns / m; when z p When the value is 50 mm, c m (50) has a value of 0.2102 Ns / m; when z p When the value is 55 mm, c m (55) has a value of 0.1566 Ns / m. The remaining result values are as shown in the appendix Figure 2 as follows.

[0131] Combined with the appendix Figure 3 , in the non-linear eddy current damping design method, first consider the segmented non-linear eddy current damping design method. The cylindrical permanent magnet 1 moves along the z-direction at a speed v inside the non-magnetic conductive metal tube 2. Assume that the induced current generated inside the non-magnetic conductive metal tube only moves along the conductor plane perpendicular to the axis of the non-magnetic conductive metal tube, that is, the non-magnetic conductive metal tube is equivalent to a conductor coil of the same length, so that the induced current moves along the conductor coil. The cylindrical permanent magnet only moves inside the non-magnetic conductive metal tube and does not extend beyond the two ends of the non-magnetic conductive metal tube. This also means that the length of the non-magnetic conductive metal tube is longer than the length of the cylindrical permanent magnet, but the length of the non-magnetic conductive metal tube is limited. Moreover, when the cylindrical permanent magnet moves, the central axis of the cylindrical permanent magnet always follows the central axis of the non-magnetic conductive metal tube, and the offset of the cylindrical permanent magnet is not considered.

[0132] As a specific embodiment 2, the material of the cylindrical permanent magnet is selected as Nd2Fe 14 B, the material of the non-magnetic conductive metal tube is selected as aluminum, and the size of the cylindrical permanent magnet is: length 10 mm × diameter 5.97 mm. The size of the non-magnetic conductive metal tube is: inner diameter length 60 mm. The outer radius r w of the non-magnetic conductive metal tube and the position z p of the cylindrical permanent magnet satisfy the following conditions:

[0133]

[0134] The calculation process of the segmented non-linear eddy current damping is roughly the same as that of Embodiment 1. It can be seen from formula (23) that the outer radius r w of the non-magnetic conductive metal tube becomes a function of the position z p of the cylindrical permanent magnet, and formula (5) in the specific embodiment l becomes:

[0135]

[0136] Assume that the position of the cylindrical permanent magnet is zp , then the interval [r n , r w (z p )] is evenly divided into p micro-segments (the p value is large enough), that is, the value in the interval becomes r n，0 (r n ), r n，1 , r n，2 , ...r n，p-2 , r n，p-1 , r n，p (r w (z p )). Substitute the above values ​​into formula (4) to obtain the corresponding B j，0 ……B j，p For example, n，p (r w (z p )) corresponding to B j，p for:

[0137]

[0138] Formula (25) is the same as formula (7) in ① in form, but r in formula (7) is n，p Always remain unchanged, r in formula (25) n，p Or r w (z p ) and the position z of the cylindrical permanent magnet p Then continue to B j，0 ……B j，p Find the RMS value B jRMS , and perform the calculations of formulas (9) to (13) in ①. When performing the integral operation of formula (14), the upper limit of the integral becomes r w (z p ):

[0139]

[0140] After integral calculation, we get the following formula:

[0141]

[0142] The subsequent derivation process is consistent with formulas (16) to (21) in ①. Finally, the calculation formula for the nonlinear eddy current damping coefficient is:

[0143]

[0144] Since the cylindrical permanent magnet always moves in the non-magnetic conductive metal tube and does not exceed the two ends of the non-magnetic conductive metal tube, according to the length of the non-magnetic conductive metal tube and the length of the cylindrical permanent magnet in Example 2, zp ranges from [5 mm, 55 mm]. When z p is 5 mm, c m (5) is 0.2376 Ns / m; when z p is 10 mm, c m (10) is 0.3194 Ns / m; when z p is 20 mm, c m (20) is 0.2659 Ns / m; when z p is 30 mm, c m (30) is 0.2117 Ns / m; when z p is 50 mm, c m (50) is 0.3194 Ns / m; when z p is 55 mm, c m (55) is 0.2376 Ns / m. The remaining result values are as shown in the appendix Figure 4 .

[0145] Combined with the appendix Figure 5 , consider the continuous non-linear eddy current damping design method. The cylindrical permanent magnet 1 moves along the z-direction at a speed v inside the non-magnetic conductive metal tube 2. It is assumed that the induced current generated inside the non-magnetic conductive metal tube only moves along the conductor plane perpendicular to the axis of the non-magnetic conductive metal tube, that is, the non-magnetic conductive metal tube is equivalent to a conductor coil of the same length, so that the induced current moves along the conductor coil. The cylindrical permanent magnet only moves inside the non-magnetic conductive metal tube and does not exceed the two ends of the non-magnetic conductive metal tube. This also means that the length of the non-magnetic conductive metal tube is longer than the length of the cylindrical permanent magnet, but the length of the non-magnetic conductive metal tube is limited. Moreover, when the cylindrical permanent magnet moves, the central axis of the cylindrical permanent magnet always follows the central axis of the non-magnetic conductive metal tube, and the offset of the cylindrical permanent magnet is not considered.

[0146] As a specific embodiment 3, in the calculation of the cylindrical segmented non-linear eddy current damping, the material of the cylindrical permanent magnet is selected as Nd2Fe 14 B, the material of the non-magnetic conductive metal tube is selected as aluminum, and the size of the cylindrical permanent magnet is: length 10 mm × diameter 5.97 mm. The size of the non-magnetic conductive metal tube is: inner diameter , length 60 mm. The outer radius r w and the position z p of the cylindrical permanent magnet satisfy the following conditions:

[0147]

[0148] The subsequent calculation process of the continuous non-linear eddy current damping is the same as the foregoing.

[0149] Since the cylindrical permanent magnet always moves inside the non-magnetic conductive metal tube and does not extend beyond the two ends of the non-magnetic conductive metal tube, according to the length of the non-magnetic conductive metal tube and the length of the cylindrical permanent magnet in Embodiment 3, z p ranges from [5 mm, 55 mm]. When z p is 5 mm, c m (5) is 0.2199 Ns / m; when z p is 10 mm, c m (10) is 0.2862 Ns / m; when z p is 20 mm, c m (20) is 0.2510 Ns / m; when z p is 30 mm, c m (30) is 0.2326 Ns / m; when z p is 50 mm, c m (50) is 0.2862 Ns / m; when z p is 55 mm, c m (55) is 0.2199 Ns / m. The remaining result values are as shown in the appendix Figure 6 .

[0150] This damping design method is not limited to the embodiments mentioned in the text. The material and magnetic performance parameters of the cylindrical permanent magnet are not restricted; the dimensional parameters of the cylindrical permanent magnet and the metal tube are also not restricted. For example: the length of the cylindrical permanent magnet is 5 mm - 100 mm, and the diameter is 4 mm - 50 mm. The inner diameter of the metal tube is 4.1 mm - 50.1 mm, and the length is 10 mm - 1000 mm. The parameters selected in the embodiments are just arbitrary ones

[0151] Moreover, this damping design method significantly reduces the size of the vibration absorber, lowers the requirement for the installation space, and has a wider application range

Claims

1. Calculation method for constructing eddy current damping calculation model, characterized in that: Assumptions before modeling: The eddy current generated on the surface of the non-magnetic conductive metal tube in the actual situation is non-uniformly distributed; it is assumed that the current only moves along the conductor plane perpendicular to the axis of the non-magnetic conductive metal tube; the cylindrical permanent magnet only moves inside the non-magnetic conductive metal tube, and the length of the non-magnetic conductive metal tube should be greater than the length of the cylindrical permanent magnet; the movement trajectory of the cylindrical permanent magnet always follows the central axis of the cylindrical permanent magnet and the central axis of the non-magnetic conductive metal tube, without considering the offset of the cylindrical permanent magnet, the inner diameter and outer diameter of the non-magnetic conductive metal tube remain unchanged, and the thickness of the non-magnetic conductive metal tube is a constant. Based on the above assumptions, the calculation includes the following steps: (1) Establish a Cartesian coordinate system at the center of the cylindrical permanent magnet. The cylindrical permanent magnet moves along the z-axis. According to the equivalent magnetization current model, calculate the magnetic induction intensity B of the cylindrical permanent magnet: where: r′ is the distance between any point in space and any point on the surface current loop of the cylindrical permanent magnet, r′ = ((x - r A cosθ) 2 +(y - rAsinθ) 2 -(z - z) 2 ) 1 / 2 , l A is the height of the cylindrical permanent magnet, r A is the radius of the cylindrical permanent magnet, μ0 and M A are the vacuum permeability and the magnetization intensity of the cylindrical permanent magnet respectively, i, j, k are the unit vectors in the x, y, z directions respectively, and simplify the expression of the magnetic induction intensity B: B = B i + B j j + B k k (2) The cylindrical permanent magnet moves along the z-axis, and a current is generated inside the non-magnetic conductive metal tube. The magnetic field direction generated by the induced current inside the non-magnetic conductive metal tube is opposite to the magnetic field direction of the cylindrical permanent magnet, hindering the movement of the cylindrical permanent magnet and generating eddy current damping. According to Faraday's law of electromagnetic induction, the induced electromotive force generated by any micro-element segment with a length of Δz in the non-magnetic conductive metal tube is equal to the sum of the voltage drops around the circumference of the non-magnetic conductive metal tube. The electromotive force ΔE of the Δz segment is: ΔE = B j Lv (3) v is the moving speed of the cylindrical permanent magnet, and L is the equivalent circumference of the non-magnetic conductive metal tube. Its calculation formula is: L＝2πr (4) r is the equivalent radius of the non-magnetic conductive metal tube, r n is the inner radius of the non-magnetic conductive metal tube, r w is the outer radius of the non-magnetic conductive metal tube, B j is the magnetic induction intensity in the y direction generated by the N pole of the cylindrical permanent magnet. According to formulas (1) to (2), B j The expression of is: As can be seen from Equation (4), B j is related to the spatial coordinates (x, y, z) and is non-uniformly distributed over the thickness of the non-magnetic conductive metal tube, that is, the value of B j is different for any value in the x-direction. (2) Correct B j Modify it by evenly dividing the interval [r n , r w into p micro - segments, where the value of p is large enough, that is, the values in the interval become r n，0 (r n ), r n，1 , r n，2 , …… r n，p-2 , r n，p-1 , r n,p (r w ). Substitute these values into formula (6) respectively to obtain the corresponding B j,0 …… B j,p , r n,p (r w ) - corresponding B j,p is: According to the root mean square value formula, for B j，0 ……B j,p Find the root mean square value B jRMS : Formula (3) after correction is: ΔE c = B jRMS Lv (9) ΔE c is the electromotive force of the corrected Δz segment, and the current Δi generated by the Δz segment is: R Δz is the equivalent resistance of the non-magnetic conductive metal tube for the Δz section, which is calculated by the following formula: ΔS = ΔrΔz (12) In Equation (11), ΔR Δz is the resistance of the radial Δr micro-element segment within the non-magnetic conductive metal tube in the Δz segment. ρ is the resistivity of the non-magnetic conductive metal tube, ΔS is the equivalent cross-sectional area of the non-magnetic conductive metal tube, Δr is the micro-element segment of the non-magnetic conductive metal tube in the thickness direction for integration, and Δz is the micro-element segment of the non-magnetic conductive metal tube in the length direction. Substitute Equation (12) into (11): After the integral calculation of formula (14), the following formula is obtained: The Δz segment is subjected to the force ΔF of the N pole of the cylindrical permanent magnet Δz which is ΔF Δz = B jRMS ΔiL (16) Then the force F exerted by the N pole of the cylindrical permanent magnet on a section of non-magnetic conductive metal tube N is as follows: (3) Establish another Cartesian coordinate system (O p -x p y p z p ) at the center of the initial end of the non-magnetic conductive metal tube. Assume that the position of the cylindrical permanent magnet in the non-magnetic conductive metal tube at this time is (0, 0, z p ). After arrangement, we get: In formula (18), l N (z p ) is the length of the non-magnetic conductive metal tube under the action of the N pole of the cylindrical permanent magnet, and F N (z p ) is the eddy current damping force corresponding to the N pole of the cylindrical permanent magnet. The numerical values of both will change with the change of the position z p of the cylindrical permanent magnet. F N (z p ) is proportional to the first power of the moving speed v of the cylindrical permanent magnet, which fully conforms to the assumption of linear viscous damping in structural dynamics. The interaction between the S pole of the cylindrical permanent magnet and the non-magnetic conductive metal tube is similar to that of the N pole, but the length of the non-magnetic conductive metal tube affected by the S pole of the cylindrical permanent magnet is l T -l N (z p ),l T is the total length of the non-magnetic conductive metal tube. Therefore, the eddy current damping force F S (z p ) is as follows: The eddy current damping force F acting on the entire non-magnetic conductive metal tube C (z p ) is as follows: F c (z p ) = F N (z p ) + F S (z p ) (20) Let the eddy current damping coefficient be c m , according to the relationship between the linear damping force and the damping coefficient: F C = c m v (21) The calculation formula for the eddy current damping coefficient can be obtained as: Through formula (22), the eddy current damping coefficient c corresponding to any position z of the cylindrical permanent magnet inside the non-magnetic conductive metal tube can be calculated p corresponding to the eddy current damping coefficient c m (z p ).

2. The calculation method for constructing the eddy current damping calculation model according to claim 1, characterized in that: According to the above formulas (17) to (20), first calculate the eddy current damping forces received by the N pole and S pole of the cylindrical permanent magnet interacting with the non-magnetic conductive metal tube respectively, and then perform a linear addition calculation to obtain the resultant eddy current damping force received by the cylindrical permanent magnet.

3. The calculation method for constructing the eddy current damping calculation model according to claim 1, characterized in that: The material of the cylindrical permanent magnet is neodymium iron boron, or ferrite, or alnico, or samarium cobalt, and the magnetic performance parameters of the cylindrical permanent magnet are not restricted. The material of the non-magnetic conductive metal tube is copper, or aluminum, or magnesium, or zinc. The size parameters of the cylindrical permanent magnet and the non-magnetic conductive metal tube are not restricted either. There are no special restrictions on the length of the cylindrical permanent magnet, the diameter of the cylindrical permanent magnet, the inner diameter, length, and outer diameter of the non-magnetic conductive metal tube.

4. The calculation method for constructing the eddy current damping calculation model according to claim 1, characterized in that: The material of the cylindrical permanent magnet is selected as neodymium iron boron, and the material of the non-magnetic conductive metal tube is selected as aluminum; the size of the cylindrical permanent magnet is: length 10mm × The size of the non-magnetic conductive metal tube with a diameter of 5.97 mm is: outer diameter It is estimated according to the length of the non-magnetic conductive metal tube and the length of the cylindrical permanent magnet that z p ranges from [5 mm, 55 mm]. When z p has a value of 5 mm, c m (5) has a value of 0.1566 Ns / m; when z p has a value of 10 mm, c m (10) has a value of 0.2102 Ns / m; when z p has a value of 20 mm, c m (20) has a value of 0.2107 Ns / m; when z p has a value of 30 mm, c m (30) has a value of 0.2107 Ns / m; when z p has a value of 50 mm, c m (50) has a value of 0.2102 Ns / m; when z p has a value of 55 mm, c m (55) has a value of 0.1566 Ns / m.

5. A calculation method for cylindrical non-linear eddy current damping based on the eddy current damping calculation model according to claim 1, with the assumption that: the inner diameter r of the non-magnetic conductive metal tube n remains unchanged, and the outer diameter r w changes, and its characteristics are: When the cylindrical permanent magnet is at different positions inside the non-magnetic conductive metal tube, the corresponding thickness of the non-magnetic conductive metal tube is different. Based on this, a design method for cylindrical non-linear eddy current damping is proposed, including the following steps: (1) Set the functional relationship between the outer radius r of the non-magnetic conductive metal tube w and the position z of the cylindrical permanent magnet p : r w (z p ) = f(z p ) (23) Formula (5) becomes: (2) Assume the position of the cylindrical permanent magnet is z p , the interval [r n , r w (z p )] needs to be evenly divided into p micro - element segments, and the values in the interval become r n，0 (r n ), r n，1 , r n，2 , …… r n，p-2 , r n，p-1 , r n，p (r w (z p ))). Substitute these values into formula (6) respectively to obtain the corresponding B j，0 …… B j，p , r n，p (r w (z p )) - corresponding B j，p is: Equation (25) has the same form as Equation (7), but the r in Equation (7) n，p always remains unchanged, and the value of r in Equation (25) n,p is related to the position z of the cylindrical permanent magnet p . After that, continue to calculate the root mean square value of B j,0 ... B j，p , and perform the calculations of Equations (9) to (13). When performing the integration operation of Equation (14), the upper limit of integration becomes r jRMS (z w ) p : After integral calculation, the following formula is obtained: The subsequent derivation process is the same as that of formulas (16) to (21). Finally, the calculation formula for the cylindrical non-linear eddy current damping coefficient is:

6. The calculation method of the cylindrical non-linear eddy current damping based on the eddy current damping calculation model according to claim 5, characterized in that: The material of the cylindrical permanent magnet is selected as neodymium iron boron, and the material of the non-magnetic conductive metal tube is selected as aluminum; the dimensions of the cylindrical permanent magnet are: length 10 mm × diameter 5.97 mm, and the dimensions of the non-magnetic conductive metal tube are: The outer radius r of the non-magnetic conductive metal tube w and the position z of the cylindrical permanent magnet p meet the following conditions: It is estimated based on the length of the non-magnetic conductive metal tube and the length of the cylindrical permanent magnet that: z p ranges from [5 mm, 55 mm]. When z p is 5 mm, c m (5) is 0.2376 Ns / m; when z p is 10 mm, c m (10) is 0.3194 Ns / m; when z p is 20 mm, c m (20) is 0.2659 Ns / m; when z p is 30 mm, c m (30) is 0.2117 Ns / m; when z p is 50 mm, c m (50) is 0.3194 Ns / m; when z p is 55 mm, c m (55) is 0.2376 Ns / m.

7. The calculation method of the cylindrical non-linear eddy current damping based on the eddy current damping calculation model according to claim 5, characterized in that: The material of the cylindrical permanent magnet is selected as neodymium iron boron, and the material of the non-magnetic conductive metal tube is selected as aluminum; the size of the cylindrical permanent magnet is: length 10 mm × diameter 5.97 mm, and the size of the non-magnetic conductive metal tube is: The outer radius r of the non-magnetic conductive metal tube w and the position z of the cylindrical permanent magnet p meet the following conditions: It is estimated based on the length of the non-magnetic conductive metal tube and the length of the cylindrical permanent magnet that: z p ranges from [5 mm, 55 mm]. When z p has a value of 5 mm, c m (5) has a value of 0.2199 Ns / m; when z p has a value of 10 mm, c m (10) has a value of 0.2862 Ns / m; when z p has a value of 20 mm, c m (20) has a value of 0.2510 Ns / m; when z p has a value of 30 mm, c m (30) has a value of 0.2326 Ns / m; when z p has a value of 50 mm, c m (50) has a value of 0.2862 Ns / m; when z p has a value of 55 mm, c m (55) has a value of 0.2199 Ns / m.

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