CN102606673A - Load-bearing adjustable zero-stiffness electromagnetic vibration isolator and control method thereof - Google Patents

Abstract

The invention discloses a load-bearing adjustable zero-stiffness electromagnetic vibration isolator and a control method thereof. The electromagnetic spring in the vertical direction of the vibration isolator is composed of three permanent magnets and two groups of controllable DC electromagnets, and the polarities of the two magnets are opposite to each other. The repulsive force is generated, and the stiffness of the electromagnetic spring can be adjusted by changing the control current; the horizontal direction is composed of two groups of electromagnetic springs symmetrically distributed left and right, and the stiffness of the electromagnetic spring can be adjusted by changing the control current; when the load capacity of the system changes, it can be controlled by adjusting current, so that the stiffness of the vertical and horizontal electromagnetic springs meets a certain proportional relationship, so that the stiffness of the system at the equilibrium position is zero; The natural frequency is very low, which can realize a wide range of frequency vibration isolation, and has a good low-frequency vibration isolation effect; and has a compact structure, light weight, and easy control.

Description

Load-bearing can harmonize zero stiffness electromagnetism vibration isolator and controlling method thereof

Technical field

The present invention relates to vibrating isolation system, particularly a kind of load-bearing can harmonize zero stiffness electromagnetism vibration isolator and controlling method thereof.

Background technique

Along with the development of society and the raising of quality of life, people require increasingly high to the vibration isolation technique of some equipment.Yet those bad vibration negative effects are very big, and the work of affect people's makes the people produce discomfort, or produced the problem of others.Such as, the protection of the high-precision surveying instrument that during to the high-comfort requirement of automobile, rocket launching aerospace craft is carried, the vibration and noise reducing of underwater navigation submarine etc., all more and more stricter to the requirement of vibration isolation technique.Traditional linear system vibration isolator only exists

(ω, ω _nBe respectively the natural frequency of energizing frequency and system) vibration isolating effect just arranged.Non-linear vibration isolation technique is the necessary ways that break through this isolation frequency restriction.Recent study shows, accurate zero stiffness vibrating isolation system has premium properties such as high static, low dynamic rate, with the natural frequency of this effective reduction system, reaches and carries out effective vibration isolation in low-frequency range and can bear the bigger load of quality simultaneously again.

Existing accurate zero stiffness vibrating isolation system structure is seen Fig. 1 at present; Its defective is only to a certain particular bearer weight design system parameter; If variation has taken place load capacity, vibrating isolation system will depart from original accurate zero stiffness working state, the vibration isolating effect that does not have; In some application (like automotive seat), load capacity changes, and this vibrating isolation system can not be regulated rigidity automatically according to the variation of load capacity.

Summary of the invention

Technical problem to be solved by this invention is, and is not enough to existing technology, and a kind of load-bearing can harmonize zero stiffness electromagnetism vibration isolator and controlling method thereof are provided, and realizes the wide range of frequencies vibration isolation, strengthens the low frequency vibration isolation ability.

For solving the problems of the technologies described above; The technological scheme that the present invention adopted is: a kind of load-bearing zero stiffness electromagnetism vibration isolator of can harmonizing; Comprise by the electromagnetic spring of the electromagnetic spring of device for vibration insutation, base, one group of Vertical direction, two groups of substantially horizontals; The base left and right sides is equipped with external frame, also comprises control circuit, and the electromagnetic spring of substantially horizontal comprises one group of electromagnet and two permanent magnets; Be connected with first permanent magnet of right side substantially horizontal electromagnetic spring through connecting rod by the device for vibration insutation right side; First permanent magnet of right side substantially horizontal electromagnetic spring is connected with second permanent magnet through guide rod, and second permanent magnet is connected with electromagnet, and electromagnet is fixed on the external frame of right side; The electromagnetic spring of Vertical direction comprises two groups of electromagnet and three permanent magnets; Be connected with first group of electromagnet of Vertical direction electromagnetic spring through the loading stand by device for vibration insutation; Three permanent magnets of Vertical direction electromagnetic spring are serially connected between two groups of electromagnet; Connect through guide rod between the permanent magnet, second group of electromagnet is fixed on the base; Be symplex structure by the device for vibration insutation left and right sides; On the loading stand force transducer is installed; The control circuit input end is connected with force transducer, and the control circuit output terminal is connected with electromagnet.

Said control circuit comprises force transducer, A/D modular converter, controller, power conversion module, and force transducer, A/D modular converter, controller, power conversion module connect successively, and power conversion module is connected with electromagnet.

The load-bearing zero stiffness electromagnetism vibration isolator of can harmonizing comprises three groups of electromagnetic springs; One of Vertical direction; Two of substantially horizontals (being symmetrically distributed in the left and right sides); When load capacity changes, at first regulate the size of control electric current in the vertical electromagnetic spring, make system remain on former equipoise (horizontal position).According to accurate zero the condition of rigidity, regulate the size of horizontal electromagnetic spring control electric current then, system is always worked under the accurate zero stiffness state; Change electromagnetic force through regulating electric current; Make electromagnetic force and load capacity change to be complementary and satisfy the mathematical relationship of accurate zero stiffness, keep the accurate zero stiffness anti-vibration performance under the system load change condition, when near the equilibrium position, being done small size vibration by device for vibration insutation; Its dynamic stiffness is very little; The natural frequency of whole system is very low, can realize the wide range of frequencies vibration isolation, and has good low frequency vibration isolation effect.

Can the harmonize controlling method of zero stiffness electromagnetism vibration isolator of said load-bearing is:

1) can harmonize one group of electromagnetic spring of zero stiffness electromagnetism vibration isolator of load-bearing is experimentized, obtains the data that electromagnetic repulsion force and air gap and electric current change, utilize numerical simulation software to carry out curve fitting, obtain approximate formula the data that record:

$F={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-z}{p_2}\right)*\exp \left(\frac{-i}{p_3}\right),$

Wherein: F representes the repulsion that electromagnetic spring produces, and z representes two distances between the permanent magnet, and i representes the size of control electric current in the coil, p ₁, p ₂, p ₃Represent the coefficient relevant with the DC electromagnet coil turn with the permanent magnet model;

2) the load-bearing zero stiffness electromagnetism vibration isolator of can harmonizing is carried out force analysis, the suffered ∑ F that makes a concerted effort of the system that obtains:

$\mathrm{\&Sigma;F}=-2{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-b+\sqrt{a^2-x^2}}{p_2}\right)*\exp \left(\frac{{-i}_1}{p_3}\right)\frac{x}{\sqrt{a^2-x^2}}+{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-c+x}{p_2}\right)*\exp \left(\frac{{-i}_2}{p_3}\right)-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-d-x}{p_2}\right)*\exp \left(\frac{{-i}_3}{p_3}\right),$

Wherein: a representes the length of connecting rod; B representes that electromagnet is to the distance of loading stand in the horizontal electromagnetic spring; Permanent magnet was to the distance between its below permanent magnet in the middle of c represented vertical electromagnetic spring, and d representes that the top permanent magnet is to the distance between the middle permanent magnet in the vertical electromagnetic spring, and x representes the variation distance of the relative initial position of sliding magnet in the vertical electromagnetic spring; I1 representes the control electric current in the horizontal electromagnet, i ₂And i ₃Represent in the vertical electromagnetic spring control electric current in first group of electromagnet and the second group of electromagnet respectively;

3) to 2) in the formula differentiate of making a concerted effort, obtain the global stiffness K of system:

$K=-\frac{2{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-b+\sqrt{a^2-x^2}}{p_2}\right)-\frac{i_1}{p_3}}{\sqrt{a^2-x^2}}-\frac{2p_1{x}^2*\exp (\frac{-b+\sqrt{a^2-x^2}}{p_2}-\frac{i_1}{p_3})}{{(a^2-x^2)}^{\frac{3}{2}}}+,$

$\frac{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp (\frac{-b-x}{p_2}-\frac{i_3}{p_3})}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}+\frac{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp (\frac{-c+x}{p_2}-\frac{i_2}{p_3})}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}+\frac{2p_1{x}^2*\exp (\frac{-b+\sqrt{a^2-x^2}}{p_2}-\frac{i_1}{p_3})}{p_2(a^2-x^2)}$

4) equilibrium position is arranged on horizontal position, controls current i in the Vertical direction electromagnet when obtaining being in the equipoise ₂And i ₃Relation:

$\mathrm{mg}={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-c+x}{p_2}\right)*\exp \left(\frac{{-i}_2}{p_3}\right)-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-d-x}{p_2}\right)*\exp \left(\frac{{-i}_3}{p_3}\right),$

Wherein: mg representes to carry weight of heavy;

5) when the equilibrium position, the derivative of rigidity is zero, obtains formula:

$\frac{\mathrm{<figure-callout\; id="2"\; label="c\; p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">c</figure-callout>}}{p_{}}+\frac{i_2}{p_3}=\frac{\mathrm{<figure-callout\; id="2"\; label="d\; p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">d</figure-callout>}}{p_{}}+\frac{i_3}{p_3},$

6) when the equilibrium position, the minimum value that makes rigidity is zero, obtains accurate zero stiffness condition:

$K_{\min }=\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}*\exp (-\frac{c}{p_2}-\frac{i_2}{p_3})+\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}*\exp (-\frac{d}{p_2}-\frac{i_3}{p_3})-2\frac{p_1}{a}*\exp \left(\frac{-b+a}{p_2}\right)*\exp \left(\frac{{-i}_1}{p_3}\right)=0,$

Wherein: K _MinMinimum value for rigidity;

7) according to step 4)～6) definite control current i ₁, i ₂And i ₃The relation that should satisfy.

Load-bearing of the present invention can be harmonized zero stiffness electromagnetism vibration isolator when changing according to load capacity, automatically adjustment control current i ₁, i ₂And i ₃Size, make the rigidity of vertical and horizontal electromagnetic spring satisfy certain proportionate relationship, be zero thereby make the rigidity of system at equilibrium position place; When near the equilibrium position, being done small size vibration by device for vibration insutation, its dynamic stiffness is very little, and the natural frequency of whole system is very low, can realize the wide range of frequencies vibration isolation, and has good low frequency vibration isolation effect; And compact structure, in light weight, control convenience.

Description of drawings

Fig. 1 is the existing load-bearing zero stiffness spring vibration isolation device structural representation of can not harmonizing;

Fig. 2 is one embodiment of the invention structural representation;

The relation curve that electromagnetic repulsion force that Fig. 3 simulates for one embodiment of the invention and air gap and electric current change;

Fig. 4 is the synthetic schematic representation of the positive and negative stiffness curve of one embodiment of the invention;

Fig. 5 is one embodiment of the invention control circuit for electromagnet structured flowchart;

Wherein:

1,2: permanent magnet; 3: electromagnet; 4: guide rod; 5: connecting rod; 6: by device for vibration insutation; 7: force transducer; 8: the loading stand; 9: control circuit; 10: external frame; 11: base; 12: slide block; 13: oblique spring; 14: damper; 15: uprighting spring.

Embodiment

As shown in Figure 2; One embodiment of the invention comprises by device for vibration insutation 6, the electromagnetic spring of one group of Vertical direction, the electromagnetic spring of two groups of substantially horizontals; Base 11 left and right sides are equipped with external frame 10, also comprise control circuit 9, and the electromagnetic spring of substantially horizontal comprises one group of electromagnet and two permanent magnets; Be connected with first permanent magnet of right side substantially horizontal electromagnetic spring through connecting rod by device for vibration insutation 6 right sides; First permanent magnet of right side substantially horizontal electromagnetic spring is connected with second permanent magnet through guide rod 4, and second permanent magnet is connected with electromagnet, and electromagnet is fixed on the right side external frame 10; The electromagnetic spring of Vertical direction comprises two groups of electromagnet and three permanent magnets; Be connected with first group of electromagnet of Vertical direction electromagnetic spring through loading stand 8 by device for vibration insutation 6 belows; Three permanent magnets of Vertical direction electromagnetic spring are serially connected between two groups of electromagnet; Connect through guide rod 4 between the permanent magnet, second group of electromagnet is fixed on the base 11; Be symplex structure by device for vibration insutation 6 left and right sides; Force transducer 7 is installed on the loading stand; Control circuit 9 input ends are connected with force transducer 7, and control circuit 9 output terminals are connected with electromagnet.

Vertical electromagnetic spring is made up of three Nd-Fe-B permanent magnets and two groups of controlled DC electromagnets, and middle permanent magnet is a sliding magnet, can on guide rod, be free to slide; Two ends are fixed magnet, and polarity is opposite between sliding magnet and the fixed magnet, and the electromagnetic force of generation is a repulsion; End at fixed magnet is provided with iron core, twines enameled cable on it, feeds sense of current through changing; Original magnetic field can be strengthened or weaken,, the size in original magnetic field can be changed through changing the size of electric current; Change the size of equivalent electromagnetism spring rate with this, vertical electromagnetic spring is placed on the neutral position of system, mainly is used for load bearing equipment weight; Sliding magnet and loading stand are fixed together, and can slide through the guide rod easy on and off; Horizontal electromagnetic spring is made up of two opposite polarity Nd-Fe-B permanent magnets and controlled DC electromagnet, is provided with iron core at an end of fixed magnet, twines enameled cable on it, through changing sense of current and size, can equivalence changes the size of electromagnetic spring rigidity; Two groups of substantially horizontal electromagnetic springs are symmetrically distributed in the left and right sides, remain on the same straight line, and electromagnet is fixed on the external frame, and sliding magnet is through connecting rod link to each other with the loading stand of placing weight (hinged).When being added excitation by device for vibration insutation, motion downward or upward, the angle of connecting rod and substantially horizontal can increase gradually, but under the restoring force effect that system provides, amplitude reduces gradually, until tending to balance.

Can harmonize one group of electromagnetic spring of zero stiffness electromagnetism vibration isolator of load-bearing is experimentized, and electromagnetic spring is made up of two permanent magnets and one group of electromagnet, and the permanent magnet trade mark is that (diameter: 30mm, thickness: 4mm), coil turn is 4000 circles to N30.Coil current changes to 2A from-2A in the electromagnet, and step-length is 0.4A.During corresponding current value, changes the distance between two permanent magnets, excursion is from 2mm to 30mm, and step-length is 2mm, obtains the relation of electromagnetic repulsion force and air gap variation under this current value.Change electric current then successively, measure the relation that electromagnetic repulsion force and air gap change under the corresponding current value, finally obtained 11 groups of data of electromagnetic repulsion force and air gap and electric current variation, utilize numerical simulation software to carry out curve fitting, obtain approximate formula the data that record:

$F={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-z}{p_2}\right)*\exp \left(\frac{-i}{p_3}\right)---\left(1\right)$

Wherein: F representes the repulsion that electromagnetic spring produces, and z representes the distance between two permanent magnets, and i representes the size of control electric current in the coil, p ₁=57.26, p ₂=0.0081, p ₃=-3.04, change curve is as shown in Figure 3;

As shown in Figure 2, the ∑ F that makes a concerted effort that whole system receives can be expressed as:

$\mathrm{\&Sigma;F}=-2{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-b+\sqrt{a^2-x^2}}{p_2}\right)*\exp \left(\frac{{-i}_1}{p_3}\right)\frac{x}{\sqrt{a^2-x^2}}+{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-c+x}{p_2}\right)*\exp \left(\frac{{-i}_2}{p_3}\right)-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-d-x}{p_2}\right)*\exp \left(\frac{{-i}_3}{p_3}\right)---\left(2\right)$

Wherein: a representes the length of connecting rod; B representes that electromagnet is to the distance of loading stand in the horizontal electromagnetic spring; Permanent magnet was to the distance between its below permanent magnet in the middle of c represented vertical electromagnetic spring; D representes in the vertical electromagnetic spring top permanent magnet to the distance between the middle permanent magnet, and x representes that the variation of the relative initial position of sliding magnet in the vertical electromagnetic spring representes the control electric current in the horizontal electromagnet, i apart from i1 ₂And i ₃Represent in the vertical electromagnetic spring control electric current in first group of electromagnet and the second group of electromagnet respectively;

The degree of whole system can be through differentiate draws to formula (2):

$K=-\frac{2{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-b+\sqrt{a^2-x^2}}{p_2}\right)-\frac{i_1}{p_3}}{\sqrt{a^2-x^2}}-\frac{2p_1{x}^2*\exp (\frac{-b+\sqrt{a^2-x^2}}{p_2}-\frac{i_1}{p_3})}{{(a^2-x^2)}^{\frac{3}{2}}}+$ (3)

$\frac{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp (\frac{-b-x}{p_2}-\frac{i_3}{p_3})}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}+\frac{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp (\frac{-c+x}{p_2}-\frac{i_2}{p_3})}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}+\frac{2p_1{x}^2*\exp (\frac{-b+\sqrt{a^2-x^2}}{p_2}-\frac{i_1}{p_3})}{p_2(a^2-x^2)}$

The equilibrium position is arranged on horizontal position, and this moment, connecting rod just was in substantially horizontal, and the weight of equipment is born by the Vertical direction electromagnetic spring fully.After weight of equipment is confirmed, control current i in (x=0) Vertical direction electromagnet in the time of can confirming to be in the equipoise according to following formula ₂And i ₃Relation,

$\mathrm{mg}={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-c+x}{p_2}\right)*\exp \left(\frac{{-i}_2}{p_3}\right)-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_1*\exp \left(\frac{-d-x}{p_2}\right)*\exp \left(\frac{{-i}_3}{p_3}\right)---\left(4\right)$

Wherein: mg representes to carry weight of heavy;

When the equipoise, the derivative of rigidity should be zero (minimum position)

$\frac{\mathrm{<figure-callout\; id="2"\; label="c\; p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">c</figure-callout>}}{p_{}}+\frac{i_2}{p_3}=\frac{\mathrm{<figure-callout\; id="2"\; label="d\; p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">d</figure-callout>}}{p_{}}+\frac{i_3}{p_3}---\left(5\right)$

When the equilibrium position, making the system stiffness minimum value is zero, can get accurate zero stiffness condition

$K_{\min }=\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}*\exp (-\frac{c}{p_2}-\frac{i_2}{p_3})+\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000147031260000012.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}*\exp (-\frac{d}{p_2}-\frac{i_3}{p_3})-2\frac{p_1}{a}*\exp \left(\frac{-b+a}{p_2}\right)*\exp \left(\frac{{-i}_1}{p_3}\right)=0---\left(6\right)$

Wherein: K _MinMinimum value for rigidity;

That is to say, for certain load capacity, the control current i in the Vertical direction electromagnet ₂And i ₃Relation confirmed by formula (4), just can obtain i respectively in the substitution formula (5) ₂And i ₃Then according to formula (6), the control current i in the substantially horizontal electromagnet ₁Also just and then confirmed.When load capacity changed, the force transducer that is placed on the loading stand can detect the gravity value that makes new advances, the control current i ₂And i ₃Change i corresponding to load capacity ₁As long as according to i ₂And i ₃Numerical value and change, just can the assurance system satisfy the condition of accurate zero stiffness all the time, that is to say to have reached can the harmonize purpose of zero stiffness system design of load-bearing.

In the vibrating isolation system, horizontal electromagnetic spring produces non-linear negative stiffness (being similar to parabola), its stiffness curve and a, b and i ₁Value is relevant, and after the system design well, physical dimension can not change, can be through regulating i ₁Change rigidity value.Vertical electromagnetic spring produces non-linear positive rigidity, its stiffness curve and c, d, i ₂And i ₃Value is relevant, and after the system design well, physical dimension can not change, can be through regulating i ₂And i ₃Change rigidity value.The rigidity of whole system promptly is both synthetic, as shown in Figure 4.

Fig. 5 is one embodiment of the invention control circuit for electromagnet structured flowchart; Control circuit comprises force transducer, A/D modular converter, AT89S51 single-chip microcomputer, power conversion module; Force transducer, A/D modular converter, controller, power conversion module connect successively, and power conversion module is connected with electromagnet.

Claims (6)

1. A load-bearing adjustable zero-stiffness electromagnetic vibration isolator, comprising a vibration-isolated device, a base, a group of vertical electromagnetic springs, and two groups of horizontal electromagnetic springs, and outer frames are installed on the left and right sides of the base. Its characteristics It also includes a control circuit. The electromagnetic spring in the horizontal direction includes a set of electromagnets and two permanent magnets. The right side of the vibration isolation device is connected to the first permanent magnet of the electromagnetic spring in the horizontal direction on the right through a connecting rod. The first permanent magnet of the electromagnetic spring is connected with the second permanent magnet through the guide rod, the second permanent magnet is connected with the electromagnet, and the electromagnet is fixed on the right outer frame; the electromagnetic spring in the vertical direction includes two sets of electromagnets and The three permanent magnets are connected by the vibration isolation device to the first group of electromagnets of the vertical direction electromagnetic spring through the object stage, and the three permanent magnets of the vertical direction electromagnetic spring are connected in series between the two groups of electromagnets, and the permanent magnets The second group of electromagnets is fixed on the base through the connection of guide rods; the left and right sides of the vibration isolation device are symmetrical in structure; force sensors are installed on the loading platform; the input end of the control circuit is connected to the force sensor, and the output end of the control circuit is connected to the electromagnetic iron connection. 2. The load-bearing adjustable zero-stiffness electromagnetic vibration isolator according to claim 1, wherein the control circuit includes an A/D conversion module, a controller, a power conversion module, an A/D conversion module, and a controller The power conversion modules are connected in sequence, and the power conversion modules are connected with the electromagnet. 3. The load-bearing adjustable zero-stiffness electromagnetic vibration isolator according to claim 2, wherein the controller is an AT89S51 single-chip microcomputer. 4. The load-bearing adjustable zero-stiffness electromagnetic vibration isolator according to claim 1, wherein the permanent magnet is an NdFeB permanent magnet. 5. The load-bearing adjustable zero-stiffness electromagnetic vibration isolator according to claim 1, wherein the electromagnet is a DC electromagnet. 6. the control method of load-bearing adjustable zero-stiffness electromagnetic vibration isolator as claimed in claim 1, is characterized in that, this method is: 1) Conduct experiments on a group of electromagnetic springs of the load-bearing adjustable zero-stiffness electromagnetic vibration isolator to obtain the data of electromagnetic repulsion, air gap and current change, and use numerical simulation software to perform curve fitting on the measured data to obtain an approximate formula : $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; z</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ Among them: F represents the repulsive force generated by the electromagnetic spring, z represents the distance between two permanent magnets, i represents the size of the control current in the coil, p ₁ , p ₂ , p ₃ represent the relationship between the permanent magnet model and the number of turns of the DC electromagnet coil coefficient; 2) Carry out force analysis on the load-bearing adjustable zero-stiffness electromagnetic vibration isolator, and obtain the resultant force ΣF on the system: $\mathrm{<span\; class="notranslate">\; \&Sigma;F</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ Among them: a represents the length of the connecting rod, b represents the distance from the electromagnet in the horizontal electromagnetic spring to the stage, c represents the distance between the permanent magnet in the middle of the vertical electromagnetic spring and the permanent magnet below it, and d represents the upper part of the vertical electromagnetic spring The distance between the permanent magnet and the middle permanent magnet, x represents the change distance of the sliding magnet relative to the initial position in the vertical electromagnetic spring, i1 represents the control current in the horizontal electromagnet, i ₂ and i ₃ represent the first group in the vertical electromagnetic spring respectively control current in the electromagnet and the second set of electromagnets; 3) Deriving the resultant force formula in 2) to obtain the total stiffness K of the system: $\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}}{\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; ,</span>$ $\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}$ 4) Set the balance position at the horizontal position to obtain the vertical electromagnetic The relationship between the control current i ₂ and i ₃ in iron: $\mathrm{<span\; class="notranslate">\; mg</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ Among them: mg represents the weight of the load; 5) In the equilibrium position, the derivative of the stiffness is zero, and the following formula is obtained: $\frac{\mathrm{<span\; class="notranslate">\; c</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; ,</span>$ 6) At the equilibrium position, let the minimum value of the stiffness be zero, and obtain the quasi-zero stiffness condition: ${\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; c</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; a</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\frac{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ Among them: K _min is the minimum value of stiffness; 7) Determine the relationship that the control currents i ₁ , i ₂ and i ₃ satisfy according to steps 4) to 6).

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