阅读说明：本技术 单稳态磁悬浮式减振装置及其磁悬浮力的计算方法 (Monostable magnetic suspension type vibration damper and calculation method of magnetic suspension force thereof ) 是由 冷永刚 陈潇雨 苏徐昆 孙帅令 张雨阳 许俊杰 于 2021-08-12 设计创作，主要内容包括：本发明公开了一种悬臂梁振动单稳态磁悬浮式减振装置及其磁悬浮力的计算方法,其中包括单稳态磁悬浮式减振装置、磁悬浮力的计算步骤等。装置结构是：基座立柱固定有悬臂梁,悬臂梁的自由端固定有质量块及紧固件,由磁悬浮管和三块磁铁构成磁悬浮式动力吸振器。磁悬浮管上下两端设有两个固定的磁铁,在固定磁铁之间设有第三块磁铁,通过设置磁铁的极性,使第三块磁铁悬浮在磁悬浮管中。同时给出了磁感应强度、两块磁铁间的磁力、以及磁悬浮合力的计算方法。本发明只需设置三块磁铁的尺寸、两块固定磁铁间距即可实现对悬臂梁的减振。为非线性动力吸振器的应用,提供了新的技术方法,减小了吸振器所占据的体积,有利于减振效果的提升。(The invention discloses a cantilever beam vibration monostable magnetic suspension type vibration damper and a calculation method of magnetic suspension force thereof, wherein the method comprises the steps of monostable magnetic suspension type vibration damper, calculation of magnetic suspension force and the like. The device structure is: the base column is fixed with a cantilever beam, the free end of the cantilever beam is fixed with a mass block and a fastener, and the magnetic suspension type dynamic vibration absorber is composed of a magnetic suspension tube and three magnets. Two fixed magnets are arranged at the upper end and the lower end of the magnetic suspension tube, a third magnet is arranged between the fixed magnets, and the third magnet is suspended in the magnetic suspension tube by setting the polarity of the magnets. And simultaneously, a method for calculating the magnetic induction intensity, the magnetic force between the two magnets and the magnetic suspension resultant force is provided. The invention can realize the vibration reduction of the cantilever beam only by setting the sizes of three magnets and the distance between two fixed magnets. The method provides a new technical method for the application of the nonlinear dynamic vibration absorber, reduces the occupied volume of the vibration absorber and is beneficial to improving the vibration reduction effect.)

1. Monostable magnetic levitation type vibration damping device comprising: base, cantilever beam, quality piece and fastener, magnet and magnetic suspension pipe, characterized by: a cantilever beam (2) is fixed on the inner side of an upright column of a base (1), a mass block and a fastener (3) are fixed at the free end of the cantilever beam, a monostable magnetic suspension type dynamic vibration absorber is formed by a magnetic suspension tube (5) and three magnets, two magnets with the same specification are respectively fixed at the upper end and the lower end of the magnetic suspension tube, a third movable magnet (4-3) is placed between a first magnet (4-1) and a second magnet (4-2), and the polarities of the first magnet and the second magnet are respectively repellent to the third magnet through the set polarities of the magnets, so that the third magnet is suspended in the magnetic suspension tube.

2. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: the fixed positions of the first and second magnets are changed, and the distance between the two magnets can be changed, so that the monostable magnetic suspension type dynamic vibration absorber with different nonlinear characteristics is obtained.

3. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: and fixedly connecting the center of the monostable magnetic suspension type dynamic vibration absorber with a fastening piece at the free end of the cantilever beam to form the monostable magnetic suspension type vibration damping device for the vibration of the cantilever beam.

4. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: the three magnets are all cylindrical in shape.

5. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: the base is L-shaped, and a left upright post of the base is vertical to the bottom surface of the base.

6. A method of calculating a magnetic levitation force in a monostable magnetic levitation type vibration damping device according to claim 1, characterized by: the monostable magnetic levitation force is determined by the following calculation steps:

(1) taking the interaction between the first and third magnets as an example, the magnetic induction B generated by the first magnet is first calculated:

wherein r ═ x-r_Acosθ)²+(y-r_Asinθ)²-(z-z₁)²)^1/2，l_AAnd r_AIs the height and radius of the first magnet, μ₀And M_AThe vacuum magnetic permeability and the magnetization intensity of the first magnet are respectively, the origin of a space coordinate system is established at the geometric center of the first magnet, i, j and k are unit vectors in the directions of x, y and z, and the magnetic induction intensity is expressed as a vector:

B＝B_ii+B_jj+B_kk (2)

(2) calculating the mutual magnetic force between the first magnet and the third magnet, namely the ampere force of the magnetizing current on the surface of the third magnet in the magnetic field generated by the magnetizing current on the surface of the first magnet, wherein the magnetic force F expression between the two magnets is as follows:

in the formula (I), the compound is shown in the specification,l_Band r_BThe height and radius of the third magnet, d is the distance between two magnets, M_BThe magnetization of the third magnet;

(3) calculating the third magnet oscillator receivingThe resultant force of the first and second magnets is fixed, and the distance between the first and second magnets is l, so that the distance between the third magnet oscillator and the upper second magnet is l-d, and the second and first magnets have the same magnetic induction intensity, and according to formula (3), the resultant force of magnetic suspension without considering the influence of gravity is F_mThe tabular form is:

in the formulad and x₁、x₂The relationship of (1) is:

x₁displacement of the free end of the cantilever beam relative to its equilibrium position, x₂Is the displacement of the third magnet relative to the balance position of the cantilever beam,

(4) considering the gravity of the third magnet itself, the actual resultant force is calculated as:

F_mg＝-(F_m-m₂g) (6)。

7. the monostable magnetic levitation type vibration damping device according to claim 1 and the method for calculating the magnetic levitation force in the monostable magnetic levitation type vibration damping device according to claim 6, wherein: the optimal magnet spacing in the vibration absorber is determined by the following calculation steps:

(1) substituting the parameters of the three magnets and the magnet spacing into the formulas (1) to (6) to obtain monostable magnetic levitation force corresponding to different magnet spacings,

(2) and performing numerical fitting on the nonlinear magnetic levitation force corresponding to the distance between the first and second magnets, wherein when the distance between the first and second magnets is 30mm, 40mm, 50mm and 60mm, the mean square error during fitting is controlled to be below 1.7%, and an approximate fitting formula corresponding to each distance is obtained:

F_mg30≈3.697×10¹⁸×(x₂-x₁)⁹-2.511×10¹⁴×(x₂-x₁)⁷+2.499×10¹⁰×(x₂-x₁)⁵+8.706×10⁵×(x₂-x₁)³+55.01×(x₂-x₁)+0.02 (7)

F_mg402.375×10¹⁷×(x₂-x₁)⁹-5.675×10¹³×(x₂-x₁)⁷+7.278×10⁹×(x₂-x₁)⁵-7.958×10⁴×(x₂-x₁)³+16.23×(x₂-x₁)+0.02 (8)

F_mg50≈3.053×10¹⁶×(x₂-x₁)⁹-1.553×10¹³×(x₂-x₁)⁷+3.217×10⁹×(x₂-x₁)⁵-1.839×10⁵×(x₂-x₁)³+8.891×(x₂-x₁)+0.02 (9)

F_mg60≈5.782×10¹⁵×(x₂-x₁)⁹-5.058×10¹²×(x₂-x₁)⁷+1.631×10⁹×(x₂-x₁)⁵-1.746×10⁵×(x₂-x₁)³+7.613×(x₂-x₁)+0.02 (10)

(3) when coefficient of linear term k_oWhen the following formula is approximately satisfied, the damping effect can be optimal:

in the formula m₂The mass of the third magnet;

(4) calculated by equation (11)To obtain k_oValues corresponding to the linear term coefficients in the equations (7) to (10), and if a certain value of the first and second magnet pitches is found, the linear term coefficient and k in the fitting equation corresponding to the certain value are found_oIf the values are approximately equal, the spacing is the optimal magnet spacing, and if the linear term coefficients are all found to be equal to k_oIf the value difference is large, the step (5) is carried out;

(5) and (5) continuously thinning the distance between the first magnet and the second magnet, and repeating the steps (1) to (4) to finally obtain the optimal magnet distance.

8. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: the cantilever beam is made of a silicon steel sheet; nd is selected as the material of the three magnets₂Fe₁₄B; the magnetic suspension pipe is made of acrylic plastic, and the size of the cantilever beam is as follows: length 52mm, width 15.5mm, thickness 0.3 mm; the sizes of the first and third magnets are as follows: the length is 4mm multiplied by the diameter is 6 mm; the dimensions of the third magnet are: the length is 10mm multiplied by the diameter is 6 mm; the magnetic suspension tube has the following dimensions: the outer diameter is 8.02mm, the inner ring diameter is 6.1mm, and the length is 68 mm; the distance between the vibration absorber and the optimal magnet is 42 mm.

9. The monostable magnetic levitation type vibration damping device according to claim 1, wherein: the monostable magnetic suspension type dynamic vibration absorber adopts a completely closed mode, and the smoothness degree of the inner wall of the magnetic suspension pipe is not processed.

Technical Field

The invention belongs to the technical field of electromechanics, and particularly relates to a vibration damping device capable of absorbing and dissipating vibration energy and providing reaction force for a main structure cantilever beam.

Background

Vibration is a phenomenon commonly existing in mechanical engineering, and not only can influence the normal work of a structure, but also can cause damage to the structure. Among numerous vibration structures, cantilever beams are widely used due to the characteristics of simple structure, convenience in operation, strong flexibility and the like. However, the general cantilever beam structure is easily affected by vibration, so that the stress performance is poor, and the attention of researchers is paid to how to weaken the vibration of the cantilever beam. The vibration damping mode of the cantilever beam structure mainly comprises active vibration damping, semi-active vibration damping and passive vibration damping. Both active damping and semi-active damping require external input energy, and the control system is relatively complex, so that the application range of the system is limited. The passive vibration damping mode has a simple structure, is convenient to operate, does not need external input energy, and can achieve a better vibration damping effect only by adding a mass-spring system or pasting a damping material on the cantilever beam system.

The non-linear Energy Sink (NES) technology is named by foreign researchers in the beginning of the 21 st century, and the structure of the NES is similar to that of a linear dynamic vibration absorber, and a mass-spring system is added on the main structure. The damping mechanism is that the main structure is provided with a counterforce capable of inhibiting the vibration of the main structure, and meanwhile, partial energy consumption is absorbed, so that the aim of damping is achieved. The difference is that NES has strong nonlinear rigidity, and can resonate with any main structure as long as the external excitation amplitude reaches a certain strength, so that the vibration reduction frequency band is widened, and the NES has the characteristic of tracking the natural frequency of the main structure.

Research in this field has mainly focused on cubic stiffness NES in recent years, and theoretical research has been developed more fully. However, in practical application, the cubic stiffness NES is difficult to realize perfectly, and researchers can only fit the cubic stiffness by approximation. For example, a piecewise linear stiffness method proposed by a professional researcher adopts micro cantilever beams with different stiffnesses to sequentially support the mass blocks, and the overall stiffness of the system changes along with the increase of the displacement of the mass blocks, so that the system can be approximately fitted to be cubic stiffness. Unfortunately, the impact between the mass block and different micro cantilever beams can greatly affect the overall vibration damping effect of the system, which is not beneficial to the improvement of the vibration damping effect. But also the overall damping of the system is penalized when the non-linear character of NES is too pronounced.

Magnets, a material that more readily achieves nonlinear characteristics, have come into the sight of NES researchers. In the conventional vibration damping technology, the performance of the magnet can be used for vibration isolation, such as a magnetic suspension train, a magnetic suspension workbench and the like. The design of bistable NES has been realized through the arrangement of rectangle magnet to the NES researcher, but this kind of structure often is oversize, reforms transform too much to the major structure, has very big restriction to the application environment, is unfavorable for generally adapting, therefore its damping effect still has great promotion space. Against this background, the present invention creates a vibration damping device that absorbs and dissipates vibrational energy and provides a reaction force to the main structural cantilever.

Disclosure of Invention

In order to compensate the technical defects of the nonlinear energy trap, the invention aims to: a cantilever beam vibration monostable magnetic suspension type vibration damper and a calculation method of magnetic suspension force thereof are provided.

The magnetic suspension type dynamic vibration absorber in the device consists of three magnets and a magnetic suspension pipe, and the center of the vibration absorber is fixedly connected to the tail end of the cantilever beam, so that the aim of damping the cantilever beam can be fulfilled. Meanwhile, under a certain condition, the vibration absorber has the technical characteristic of linear dynamic vibration attenuation, the sensitivity of the nonlinear dynamic vibration absorber to external excitation amplitude is reduced, and the vibration attenuation characteristic of combination of linear rigidity and nonlinear rigidity is realized.

The technical scheme of the invention comprises three parts, namely: the method comprises the steps of calculating the monostable magnetic levitation force in the vibration absorber and calculating the optimal magnet spacing in the vibration absorber.

Monostable magnetic levitation type vibration damping device comprising: base, cantilever beam, quality piece and fastener, magnet, magnetic suspension pipe etc.. The structure composition is as follows: a cantilever beam is fixed on the inner side of an upright post of the base, a mass block and a fastener are fixed at the free end of the cantilever beam, and a monostable magnetic suspension type dynamic vibration absorber is formed by a magnetic suspension pipe and three magnets. Two magnets with the same specification are respectively fixed at the upper end and the lower end of the magnetic suspension tube, a third movable magnet is placed between the first magnet and the second magnet, and the polarities of the first magnet and the second magnet are respectively repelled with the third magnet through the set polarities of the magnets, so that the third magnet is suspended in the magnetic suspension tube.

The relative positions of the two fixed magnets are changed, so that the distance between the first magnet and the second magnet can be changed, and the monostable magnetic suspension type dynamic vibration absorber with different nonlinear characteristics is obtained.

The magnetic levitation force in the monostable magnetic levitation type vibration damping device is determined by the following calculation steps:

(1) taking the interaction between the first and third magnets as an example, the magnetic induction B generated by the first magnet is first calculated:

wherein r ═ x-r_Acosθ)²+(y-r_Asinθ)²-(z-z₁)²)^1/2，l_AAnd r_AIs the height and radius of the first magnet; mu.s₀And M_ARespectively the vacuum magnetic permeability and the magnetization intensity of the first magnet; the origin of the spatial coordinate system is established at the geometric center of the first magnet. i. j and k are unit vectors in x, y and z directions respectively, and magnetic induction is expressed as a vector:

B＝B_ii+B_jj+B_kk (2)

(2) calculating the mutual magnetic force between the first magnet and the third magnet, namely the ampere force of the magnetizing current on the surface of the third magnet in the magnetic field generated by the magnetizing current on the surface of the first magnet, wherein the magnetic force F expression between the two magnets is as follows:

in the formula (I), the compound is shown in the specification,l_Band r_BThe height and radius of the third magnet, d is the distance between two magnets, M_BThe magnetization of the third magnet.

(3) And calculating the resultant force of the third magnet oscillator under the action of the first magnet and the second magnet, wherein the first magnet and the second magnet are fixed, the distance between the two magnets is set to be l, the distance between the third magnet oscillator and the upper end of the second magnet is set to be l-d, and the second magnet and the first magnet have the same magnetic induction intensity. According to equation (3), the resultant force of magnetic levitation without taking the influence of gravity into account_mThe tabular form is:

in the formulad and x₁、x₂The relationship of (1) is:

x₁displacement of the free end of the cantilever beam relative to its equilibrium position, x₂Is the displacement of the third magnet relative to the equilibrium position of the cantilever beam.

(4) Considering the gravity of the third magnet itself, the actual resultant force is calculated as:

F_mg＝-(F_m-m₂g) (6)。

the optimal magnet spacing in the vibration absorber is determined by the following calculation steps:

(1) substituting the parameters of the three magnets and the magnet spacing into the formulas (1) to (6) to obtain monostable magnetic levitation force corresponding to different magnet spacings;

(2) and performing numerical fitting on the nonlinear magnetic levitation force corresponding to the distance between the first magnet and the second magnet, wherein when the distance between the first magnet and the second magnet is 30mm, 40mm, 50mm and 60mm, the mean square error (RMSE) during fitting is controlled to be below 1.7%.

The monostable magnetic suspension type vibration damper for cantilever beam vibration has the working principle that an elastic cantilever beam-mass block structure is used as a vibration damped object, one end of the cantilever beam is fixed with a base, and the other end is used as a free end and connected with a magnetic suspension dynamic vibration absorber. When the cantilever beam vibrates from the outside, the cantilever beam vibrates up and down, and the monostable magnetic suspension type dynamic vibration absorber on the free end of the cantilever beam also moves along with the cantilever beam to excite the suspension magnet to provide a reaction force for restraining the vibration of the cantilever beam, and absorb and consume part of energy to achieve the aim of damping the cantilever beam with the main structure.

The technical key points of the invention are as follows: the three magnets are all cylindrical in shape, and the third fast magnet is suspended in the magnetic suspension tube by arranging the polarities of the magnets. The nonlinear characteristic of monostable magnetic levitation force in the vibration absorber can be changed by adjusting the distance between the two fixed magnets so as to adapt to the vibration reduction of cantilever beams with different main body structures and reduce the sensitivity to external excitation amplitude. The center of the vibration absorber is fixedly connected with the free end of the cantilever beam, so that the vibration absorber provides reaction force for the cantilever beam and plays roles of vibration suppression and vibration absorption.

The invention has the characteristics and beneficial effects that: compared with a conventional nonlinear energy trap, the magnetic suspension type vibration absorber only comprises three cylindrical magnets and one magnetic suspension tube, and vibration reduction on the cantilever beam can be realized only by simply setting the geometric dimension of the three magnets and the distance between the two fixed magnets. On one hand, the vibration reduction method simplifies the complexity of work such as system design, dynamic analysis, debugging and installation and the like, and provides a new technical method for the design and application of the nonlinear dynamic vibration absorber. On the other hand, the space volume of the vibration absorber is reduced, namely the influence on the normal work of the main body structure is reduced, and the miniaturization design of the nonlinear vibration absorber is facilitated. In addition, the nonlinear dynamic vibration absorber can show the vibration reduction characteristic of the linear dynamic vibration absorber under certain conditions, and the improvement of the vibration reduction effect is facilitated.

Drawings

FIG. 1 is a schematic diagram of the principle and structure of the device of the present invention.

Fig. 2 is a schematic diagram of three magnet arrangement structures of a monostable magnetic suspension type dynamic vibration absorber.

FIG. 3 is a graph showing the effect of monostable magnetic levitation force corresponding to different magnet pitches.

Fig. 4 is a potential function diagram of the implementation effect of the monostable magnetic suspension type dynamic vibration absorber corresponding to different magnet intervals.

FIG. 5 is the amplitude-frequency characteristic curve of the device when the magnet spacing is 40mm in the embodiment.

Detailed Description

The technical solution of the present invention is further described below by way of examples with reference to the accompanying drawings. It should be noted that, although the drawings in the specification describe the embodiments, the embodiments are only illustrative and not restrictive. The materials and dimensional parameters of the various elements may be varied without departing from the spirit of the invention and the scope of the claims, all of which are within the scope of the invention.

The technical scheme comprises a monostable magnetic suspension type vibration reduction system structure, a connection mode and a connection position of a vibration absorber and a cantilever beam, a calculation step of monostable magnetic suspension force in the vibration absorber and a calculation method of optimal magnet spacing in the vibration absorber.

With reference to fig. 1 and fig. 2, the monostable magnetic suspension type vibration damping device for cantilever beam vibration has the following structure: a cantilever beam 2 is fixed on the inner side of the upright post of the base 1, and a mass block and a fastener 3 are fixed at the free end of the cantilever beam. The monostable magnetic suspension type dynamic vibration absorber is composed of a magnetic suspension tube 5 and three magnets. Two magnets with the same specification are respectively fixed at the upper end and the lower end of the magnetic suspension tube, a third movable magnet 4-3 is placed between the first magnet 4-1 and the second magnet 4-2, and the polarities of the magnets are set so that the polarities of the first magnet and the second magnet are respectively repellent to the third magnet, and the third magnet is suspended in the magnetic suspension tube.

The S pole of the first magnet is opposite to the S pole of the third magnet, and the N pole of the second fast magnet is opposite to the N pole of the third magnet.

The relative position of the two fixed magnets is changed, and the distance l between the first magnet and the second magnet can be changed, so that the monostable magnetic suspension type dynamic vibration absorber with different nonlinear characteristics is obtained.

The center of the monostable magnetic suspension type dynamic vibration absorber is fixedly connected with a fastener at the free end of the cantilever beam to form the monostable magnetic suspension type vibration damping device for the vibration of the cantilever beam. The three magnets are all cylindrical, the base is L-shaped, and the upright column on the left side of the base is vertical to the bottom surface of the base.

As a specific example. The cantilever beam is made of a silicon steel sheet; nd is selected as the material of the three magnets₂Fe₁₄B; acrylic plastic is selected as the material of the magnetic suspension pipe, and the size of the cantilever beam is as follows: 52mm in length, 15.5mm in width and 0.3mm in thickness. The sizes of the first and third magnets are as follows: length 4mm x diameter 6 mm. The dimensions of the third magnet are: length 10mm x diameter 6 mm. The magnetic suspension tube has the following dimensions: outer diameter of 8.02mm and inner ring diameter₂6.1mm and 68mm in length. The vibration absorber corresponds to an optimal magnet spacing of 42 mm.

The magnetic levitation force in the monostable magnetic levitation type vibration damping device is determined by the following calculation steps:

(1) taking the interaction between the third and the first magnets as an example, the magnetic induction B generated by the first magnet is first calculated:

wherein r ═ x-r_Acosθ)²+(y-r_Asinθ)²-(z-z₁)²)^1/2，l_AAnd r_AIs the height and radius of the first magnet in m. Mu.s₀And M_ARespectively the vacuum permeability and the magnetization of the first magnet, M_AThe unit is A/m. The origin of a space coordinate system is established at the geometric center of the first magnet, i, j and k are unit vectors in the directions of x, y and z respectively, and the magnetic induction intensity is expressed as a vector:

B＝B_ii+B_jj+B_kk (2)

B_i、B_j、B_kthe units are all T (Tesla).

(2) Calculating the mutual magnetic force between the first magnet and the third magnet, namely the ampere force of the magnetizing current on the surface of the third magnet in the magnetic field generated by the magnetizing current on the surface of the first magnet, wherein the magnetic force F expression between the two magnets is as follows:

in the formula (I), the compound is shown in the specification,l_Band r_BThe height and radius of the third magnet, d is the distance between the two magnets, and the unit is m. M_BThe magnetization of the third magnet; the unit is A/m.

(3) And calculating the resultant force of the third magnet oscillator under the action of the first magnet and the second magnet. Since the first and second magnets are fixed and the distance between the two magnets is l, the distance between the third magnet oscillator and the upper second magnet is l-d, the second and first magnets have the same magnetic induction intensity, and according to the formula (3), the magnetic suspension resultant force without considering the influence of gravity, F_mThe tabular form is:

in the formulad and x₁、x₂The relationship of (1) is:

x₁displacement of the free end of the cantilever beam relative to its equilibrium position, x₂The displacement of the third magnet relative to the balance position of the cantilever beam is m.

(4) Considering the gravity of the third magnet itself, the actual resultant force is calculated as:

F_mg＝-(F_m-m₂g) (6)

in the formula m₂The unit is the mass of the third magnet in kg. Due to the gravity of the magnet, the balance position of the magnet oscillator is deviated from the center of the suspension tube and is close to the first fixed magnet.

After the geometric dimension of the magnet is determined, a monostable magnetic suspension force curve can be calculated according to the expressions (1) to (6).

Determining natural circular frequency omega of cantilever beam by experiment_rIt was 80.42 rad/s. Determining an optimal magnet spacing for the shock absorber by:

(1) and (3) substituting the three magnet parameters and the magnet spacing into the formulas (1) to (6) to obtain the monostable magnetic levitation force corresponding to different magnet spacings, as shown in the attached figure 3. Different magnet spacings correspond to different curve characteristics.

(2) Respectively carrying out numerical fitting on the set nonlinear magnetic levitation force with the distance between the two fixed magnets being 30mm, 40mm, 50mm and 60mm, and controlling the mean square error (RMSE) to be below 1.7% during fitting to obtain an approximate fitting formula corresponding to each distance:

F_mg30≈3.697×10¹⁸×(x₂-x₁)⁹-2.511×10¹⁴×(x₂-x₁)⁷+2.499×10¹⁰x(x₂-x₁)⁵+8.706×10⁵×(x₂-x₁)³+55.01×(x₂-x₁)+0.02 (7)

F_mg40≈2.375×10¹⁷×(x₂-x₁)⁹-5.675×10¹³×(x₂-x₁)⁷+7.278×10⁹×(x₂-x₁)⁵-7.958×10⁴×(x₂-x₁)³+16.23×(x₂-x₁)+0.02 (8)

F_mg50≈3.053×10¹⁶×(x₂-x₁)⁹-1.553×10¹³×(x₂-x₁)⁷+-3.217×10⁹×(x₂-x₁)⁵-1.839×10⁵×(x₂-x₁)³+8.891×(x₂-x₁)+0.02 (9)

F_mg60≈5.782×10¹⁵×(x₂-x₁)⁹-5.058×10¹²×(x₂-x₁)⁷+1.631×10⁹×(x₂-x₁)⁵-1.746×10⁵×(x₂-x₁)³+7.613×(x₂-x₁)+0.02 (10)

and when the external excitation amplitude is within a certain range, and the displacement of the levitation magnet is smaller than 1mm, substituting the displacement of the scale into equations (7) to (10). It is not difficult to find that the non-linear term magnetic force in the magnetic levitation force fitting type is small, and the numerical value of each non-linear term magnetic force is different from that of a linear term magnetic force by at least one order of magnitude. Therefore, the linear first-order magnetic force plays a key role in the vibration of the suspension magnet in each magnetic force type, and the effect of the nonlinear magnetic force is very small and can be ignored. This characteristic of magnetic levitation force indicates that the monostable magnetic levitation type vibration absorber has the vibration damping characteristic of the linear dynamic vibration absorber under certain conditions.

(3) When coefficient of linear term k_oWhen the following formula is approximately satisfied, the damping effect can be optimal:

for m in the examples₂It was 0.0021 kg.

(4) K is calculated by equation (11)_oThe value was 13.58. In accordance with the linear term coefficients in the expressions (7) to (10), it was found that when the magnet pitch was 40mm, the linear term coefficient was 16.23, and k was calculated_oThe values are the closest, which means that the damper with the magnet pitch of 40mm has the best damping effect among the four magnet pitches, but 40mm still does not achieve the best damping effect. From the expressions (7) to (10), it can be found that the smaller the magnet pitch, the larger the linear term coefficient; the larger the magnet spacing, the smaller the linear term coefficient. Therefore, it can be seen that the optimum magnet spacing should be between 40mm and 50 mm.

(5) And (5) continuously thinning the magnet spacing between 40mm and 50mm, and repeating the steps (1) to (4) to finally obtain the optimal magnet spacing of 42 mm. The corresponding fitting is as follows:

F_mg42≈1.506×10¹⁷×(x₂-x₁)⁹-4.307×10¹³×(x₂-x₁)⁷+6.098×10⁹×(x₂-x₁)⁵-1.258×10⁵×(x₂-x₁)³+13.77×(x₂-x₁)+0.02 (12)

the coefficient of the linear term is 13.77.

Fig. 4 is a potential function diagram of the implementation effect of the monostable magnetic suspension type dynamic vibration absorber corresponding to different magnet intervals. The influence of the gravity of the levitation magnet makes the potential function shape no longer symmetrical. It is evident from the figure that the potential function presents only one potential well regardless of the variation in the magnet spacing. Along with the increase of the magnet spacing, the width of the potential well is increased, the change of the bottom of the potential function is more gradual, and the change is related to the linear term coefficient in the fitting formula of the monostable magnetic levitation force.

In the embodiment, the monostable magnetic suspension type dynamic vibration absorber adopts a completely closed mode, and the smoothness degree of the inner wall of the magnetic suspension pipe is not treated (no substances such as a lubricant, a friction agent and the like are added). The damping coefficient was experimentally determined to be about 0.25.

FIG. 5 is an amplitude-frequency characteristic curve of the device in the example when the magnet pitch is 40 mm. As can be seen from the curve trend in the figure, the monostable magnetic suspension type vibration reduction mode has no nonlinear characteristics, the curve change is smooth and is similar to that of a linear dynamic vibration absorber, the curve amplitude near the natural frequency is obviously lower than that of a linear cantilever beam without a magnetic suspension type dynamic vibration absorber, the vibration reduction effect is good, and the vibration reduction percentage is 79.95%.