Electromagnetic Force Control Method of Magnetic Disk Type Negative Stiffness Electromagnetic Actuator

Abstract

The present disclosure is an electromagnetic force control method of a magnetic disk type negative stiffness electromagnetic actuator. The present disclosure relates to the technical field of vibration control. According to the actually required static bearing capacity, the present disclosure determines the positive stiffness k of a mechanical spring required for a magnetic disk type quasi-zero stiffness vibration isolator; and establishes an electromagnetic force mathematical model of a single electromagnet under a condition of magnetic unsaturation. The present disclosure aims at the magnetic disk type quasi-zero stiffness vibration isolator and takes the coil current as an input control variable, so that the electromagnetic force and displacement of the negative stiffness electromagnetic actuator have a linear relationship, thereby changing the non-linear nature of a vibration isolation system, avoiding the multi-stable phenomenon caused by the non-linear electromagnetic force during working, and eliminating complex dynamic behaviors such as jumping when the whole vibration isolator works. Complex sensors and control systems are not needed, and implementation manners are simple and convenient.

Claims

1. A method of electromagnetic force control of a magnetic disk type negative stiffness electromagnetic actuator, comprising the following steps: step 1: according to a static bearing capacity, selecting a mechanical spring for providing positive stiffness of a vibration isolation system, and determining positive stiffness of the spring; step 2: establishing an electromagnetic force expression under magnetic unsaturation, and determining an electromagnetic attraction force of an electromagnet; step 3: enabling a negative stiffness electromagnetic actuator to be symmetrical up and down, and establishing an electromagnetic force-displacement expression according to the negative stiffness electromagnetic actuator under a condition of magnetic unsaturation; step 4: analyzing the electromagnetic force-displacement expression, and determining an expression about a current passing into upper and lower coils, a structural parameter and a displacement, a magnitude of ideal constant value negative stiffness when the negative stiffness electromagnetic actuator takes the current as an input control variable of a system; step 5: establishing a vibration isolation system dynamic formula, and determining a vibration response according to the electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator under ideal conditions; and step 6: according to the vibration response and based on the expression about the current, the structural parameter and the displacement, the magnitude of the ideal constant value negative stiffness, determining a control current required for meeting a linear electromagnetic force-displacement relationship of the negative stiffness electromagnetic actuator.

2. The method according to claim 1, wherein the step 2 comprises: establishing an electromagnetic force expression under magnetic unsaturation, determining an electromagnetic attraction force of an electromagnet, and representing the electromagnetic attraction force of the electromagnet by the following formulas: $F_{mag1}=\frac{1}{2}\frac{{\Phi }_1^2}{{\mu }_0{S}_{\mathrm{in}}}+\frac{1}{2}\frac{{\Phi }_1^2}{{\mu }_0{S}_{out}},{\Phi }_1=\frac{N_1{I}_{c1}}{R_{\mathrm{total}1}},S_{out}=\frac{\pi }{4}\left(d_6^2-d_5^2\right),S_{\mathrm{in}}=\frac{\pi }{4}\left(r_4^2-r_3^2\right),$ wherein F.sub.mag1 represents an electromagnetic attraction force of electromagnet 1; R.sub.total1 represents a total reluctance of the electromagnet 1; Φ.sub.1 represents a magnetic flux of the electromagnet 1; μ.sub.0 represents permeability of vacuum; S.sub.in represents an equivalent cross-sectional area of an inner magnetic pole; S.sub.out represents an equivalent cross-sectional area of an outer magnetic pole; and I.sub.c1 represents a current passing into coil 1; and determining the total reluctance of the electromagnet 1 according to an air gap reluctance of armature 1 and a reluctance of the armature 1, and representing the total reluctance of the electromagnet 1 by the following formulas: $R_{\mathrm{total}1}=R_{\mathrm{gap}11}+R_{gap12}+R_{arm}+R_{\mathrm{iron}},R_{\mathrm{iron}}=\frac{l_b}{{\mu }_1{S}_{\mathrm{in}}}+\frac{l_b}{{\mu }_1{S}_{out}}+\frac{l_a}{{\mu }_1{S}_1},R_{\mathrm{arm}}=\frac{l_8}{2{\mu }_1{S}_{\mathrm{in}}}+\frac{l_8}{2{\mu }_1{S}_{out}}+\frac{l_d}{{\mu }_1{S}_2},R_{gap11}=\frac{x_1}{{\mu }_0{S}_{\mathrm{in}}},R_{\mathrm{gap}12}=\frac{x_1}{{\mu }_0{S}_{out}},S_1=\frac{\pi }{16}\left(l_1-l_2\right)\left(l_3+l_4+l_5+l_6\right),S_2=\frac{\pi }{4}{l}_8\left(l_3+l_4+l_5+l_6\right),l_a=\frac{l_5+l_6-l_3-l_4}{4},l_b=\frac{l_1+l_2}{2},$ wherein S.sub.1 represents an equivalent cross-sectional area of a magnetic circuit in a horizontal direction of iron core 1; S.sub.2 represents an equivalent cross-sectional area of a magnetic circuit in a horizontal direction of the armature 1; x.sub.1 represents an air gap of the electromagnet 1; μ.sub.1 represents a magnetic permeability of materials of the iron core 1, iron core 2, the armature 1 and armature 2; R.sub.gap11 represents an air gap reluctance corresponding to an inner magnetic pole of the armature 1; R.sub.gap12 represents an air gap reluctance corresponding to an outer magnetic pole of the armature 1; R.sub.arm represents the reluctance of the armature 1; R.sub.iron represents a reluctance of the iron core 1; N.sub.1 represents the number of turns of a coil of the electromagnet 1; l.sub.1 represents a height of the iron core 1; l.sub.2 represents a height of a coil of the electromagnet 1; l.sub.3 represents an inner diameter of the armature 1; l.sub.4 represents an inner diameter of an coil of the electromagnet 1; l.sub.5 represents an outer diameter of the coil of the electromagnet 1; l.sub.6 represents an outer diameter of the iron core 1; l.sub.7 represents an outer diameter of the armature 1; and l.sub.8 represents a height of the armature 1.

3. The method according to claim 1, wherein the step 3 comprises: according to a structure that the negative stiffness electromagnetic actuator is symmetrical up and down, establishing an electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator under the condition of magnetic unsaturation, determining a magnitude of an electromagnetic attraction force of the negative stiffness electromagnetic actuator under ideal conditions, and representing an electromagnetic attraction force under ideal conditions by the following formulas: $F_{\mathrm{mag}}=F_{\mathrm{mag}1}-F_{\mathrm{mag2}}=2N_1^2{I}_{c1}^2{\mu }_0{S}_{\mathrm{in}}{S}_{out}\left(S_{\mathrm{in}}+S_{out}\right)\left[\frac{1}{{\left(\mathrm{ax}+b\right)}^2}-\frac{1}{{\left(\mathrm{ax}-b\right)}^2}\right],F_{\mathrm{mag}2}=\frac{1}{2}\frac{{\Phi }_2^2}{{\mu }_0{S}_{\mathrm{in}}}+\frac{1}{2}\frac{{\Phi }_2^2}{{\mu }_0{S}_{out}},a=2\left(S_{\mathrm{in}}+S_{out}\right),b=\left(\frac{l_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2g\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right),$ wherein R.sub.mag2 represents an electromagnetic attraction force of electromagnet 2; Φ.sub.2 represents a magnetic flux of the electromagnet 2; x represents a displacement of a negative stiffness spring; N.sub.2 represents the number of turns of a coil of the electromagnet 2; I.sub.c1 represents a current passing into coil 1, and a current I.sub.c2 passing into coil 2 is equal to the current I.sub.c1; μ.sub.r represents a magnitude of a relative magnetic conductivity of materials of iron core 1, iron core 2, armature 1 and armature 2; a represents an intermediate calculation variable about S.sub.in and S.sub.out; and b represents an intermediate calculation variable about l.sub.8, l.sub.a, l.sub.b, S.sub.in, S.sub.out, S.sub.1, S.sub.2 and μ.sub.r; and according to the number of turns of the coil and the current of the electromagnet 2 and the total reluctance of the electromagnet 2, determining the magnetic flux of the electromagnet 2, and representing the magnetic flux of the electromagnet 2 by the following formulas: ${\Phi }_2=\frac{N_2{I}_2}{R_{\mathrm{total}2}},R_{\mathrm{total}2}=R_{gap21}+R_{gap22}+R_{arm}+R_{\mathrm{iron}},R_{\mathrm{gap}22}=\frac{x_2}{{\mu }_0{S}_{out}},$ wherein R.sub.gap21 represents an air gap reluctance corresponding to an inner magnetic pole of the armature 2, and R.sub.gap22 represents an air gap reluctance corresponding to an outer magnetic pole of the armature 2.

4. The method according to claim 1, wherein the step 4 comprises: step 4.1: determining an electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator, and making the electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator about the structural parameter and the control current equal to an electromagnetic force-displacement expression about ideal negative stiffness: $F_{\mathrm{mag}}=2N_1^2{I}_{c1}^2{\mu }_0{S}_{\mathrm{in}}{S}_{out}\left(S_{\mathrm{in}}+S_{out}\right)\left[\frac{1}{{\left(ax+b\right)}^2}-\frac{1}{{\left(ax-b\right)}^2}\right]=k_{m}x,F_{mag}=k_{m}x,$ wherein k.sub.m represents a magnitude of the ideal constant value negative stiffness; and step 4.2: analyzing the electromagnetic force-displacement expression, and determining the expression about currents passing into upper and lower coils, the displacement, the magnitude of the ideal constant value negative stiffness and the structural parameter when the negative stiffness electromagnetic actuator takes the current as the input control variable of the system: $I_{c1}=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}\sqrt{{\left(ax+b\right)}^2{\left(\mathrm{ax}-b\right)}^2},ax+b=\left(\frac{r_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2g+2x\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right)>0,b-\alpha x=\left(\frac{r_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2\left(g-x\right)\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right)>0,\mathrm{ax}-b<0,I_{c1}=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2-\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{x}^2=I_1-I_2^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,I_2=\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}}x,$ wherein I.sub.c1 represents a current passing into coil 1, and a current I.sub.c2 passing into the coil 2 is equal to the current I.sub.c1; and μ.sub.r represents a magnitude of a relative magnetic conductivity of materials of iron core 1, iron core 2, armature 1 and armature 2.

5. The method according to claim 1, wherein the step 5 comprises: step 5.1: establishing a vibration isolation system dynamic formula, and substituting an ideal electromagnetic force form into a dynamic mathematical model of a vibration isolation system by the following formulas:
m{umlaut over (x)}.sub.t+c({dot over (x)}.sub.t−{dot over (x)}.sub.e)+k.sub.m(x.sub.t−x.sub.e)+k(x.sub.t−x.sub.e)=0,
x.sub.e=X.sub.e cos(w.sub.et),
x=x.sub.t−x.sub.e, wherein m represents a vibration-isolated mass; {umlaut over (x)}.sub.t represents a vibration response acceleration of a vibration-isolated object; {dot over (x)}.sub.t represents a vibration response speed of the vibration-isolated object; x.sub.e represents a system excitation; {dot over (x)}.sub.e represents a system excitation speed; c represents a system damping; x.sub.t represents a vibration response of the vibration-isolated object; X.sub.e represents an excitation amplitude; and x.sub.e represents an excitation frequency; step 5.2: under ideal conditions, when a magnitude of an absolute value of negative stiffness is the same as that of positive stiffness and an overall dynamic stiffness of the system is zero, that is, k.sub.m+k=0, representing a dynamic mathematic model of the vibration isolation system by the following formula: ${\stackrel{\mathrm{.Math.}}{x}}_t+\frac{c}{m}{\stackrel{.}{x}}_t=\frac{-c{X}_e{w}_e}{m}\sin \left(w_{e}t\right),$ determining a displacement response, and representing a vibration response by the following formulas: $x_{t11}=C_1+C_2{e}^{\frac{-c}{m}t},C_1=-X_e,C_2=\frac{X_e{m}^2{w}_e^2}{m^2{w}_e^2+c^2},x_{t12}=C_3\cos \left(w_{e}t\right)+C_4\sin \left(w_{e}t\right)=\sqrt{C_3^2+C_4^2}\sin \left[w_{e}t+\arctan \left(\frac{C_3}{C_4}\right)\right],C_3=\frac{X_e{c}^2}{m^2{w}_e^2+c^2},C_4=\frac{c{X}_{e}m{w}_e}{m^2{w}_e^2+c^2},$ wherein x.sub.t11 represents a solution corresponding to a free vibration response; x.sub.t12 represents a particular solution corresponding to a forced vibration response; C.sub.1 represents coefficient 1 of a particular solution corresponding to the free vibration response; C.sub.2 represents coefficient 2 of the particular solution corresponding to the free vibration response; C.sub.3 represents coefficient 1 of a particular solution corresponding to the forced vibration response; and C.sub.4 represents coefficient 2 of the particular solution corresponding to the forced vibration response; and step 5.3: when stiffness of the negative stiffness mechanism is positive stiffness, that is, k.sub.m+k>0, representing a dynamic mathematic model of the vibration isolation system by the following formula: ${\stackrel{\mathrm{.Math.}}{x}}_t+\frac{c}{m}{\stackrel{.}{x}}_t+\frac{k_m+k}{m}{x}_t=\frac{k_m+k}{m}{X}_e\cos \left(w_{e}t\right)-\frac{c}{m}{X}_e{w}_e\sin \left(w_{e}t\right),$ determining a displacement response, and representing the displacement response by the following formulas: $x_t=x_{t21}+x_{t22},x_{t21}=C_7{e}^{\frac{-c}{2m}t}\cos \left(\sqrt{\frac{k_m+k}{m}}t\right)+C_8{e}^{\frac{-c}{2m}t}\sin \left(\sqrt{\frac{k_m+k}{m}-\frac{c^2}{4m}}t\right),C_7=\frac{-X_e\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},C_8=\frac{-c{X}_e\left\{2m^2{w}_e^4+\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]\right\}}{\sqrt{4m\left(k_m+k\right)+c^2}\left[{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2\right]},x_{t22}=C_9\cos \left(w_{e}t\right)+C_{10}\sin \left(w_{e}t\right)=\sqrt{C_9^2+C_{10}^2}\sin \left[w_{e}t+\arctan \left(\frac{C_9}{C_{10}}\right)\right],C_9=\frac{X_e\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},C_{10}=\frac{cm{X}_e{w}_e^3}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},$ wherein x.sub.t21 represents a solution corresponding to a free vibration response; x.sub.t22 represents a particular solution corresponding to a forced vibration response; C.sub.7 represents coefficient 1 of a particular solution corresponding to the free vibration response; C.sub.8 represents coefficient 2 of the particular solution corresponding to the free vibration response; C.sub.9 represents coefficient 1 of a particular solution corresponding to the forced vibration response; and C.sub.10 represents coefficient 2 of the particular solution corresponding to the forced vibration response.

6. The method according to claim 1, wherein the step 6 comprises: under ideal conditions, when a magnitude of a value of negative stiffness is the same as that of positive stiffness, that is, k.sub.m−−k, representing the control current I.sub.c1(I.sub.c2) by the following formulas: $I_{c1}=I_1-{\left(I_{21}+I_{22}+I_{231}-I_{232}\right)}^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,\begin{array}{c}{I}_2=I_{21}+I_{22}+I_{231}-I_{232}\\ =\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{in}{S}_{out}}}}\\ =C_{0}x=C_0\left(x_t-x_e\right)=C_0{x}_{t11}+C_0{x}_{t12}-C_0{x}_e\\ =C_0{C}_1+C_0{C}_2{e}^{\frac{-c}{m}t}+C_0\sqrt{C_3^2+C_4^2}\sin \\ \left[w_{e}t+\arctan \left(\frac{C_3}{C_4}\right)\right]-C_0{X}_e\cos \left(w_{e}t\right)\end{array},U_{c1}+R_1{I}_{22}+L_1\frac{d{I}_{22}}{dt}=0,{\left(\frac{c}{m}\right)}^2={\left(\frac{R_1}{2L_1}\right)}^2=\frac{1}{L_1{C}_1^{\prime }},U_{c1}\left(0\right)=\left(\frac{c}{m}-R_1\right)C_0{C}_2,$ wherein C.sub.1′ represents a resistance value of a capacitor; R.sub.1 represents a sum of a resistance value of a circuit and a resistance value of a coil; L.sub.1 represents a sum of an inductance of the circuit and an inductance of the coil; I.sub.1 represents control current 1; I.sub.2 represents control current 2; I.sub.21 represents component 1 of the control current 2; I.sub.22 represents component 2 of the control current 2; I.sub.231 represents component 3 of the control current 2; I.sub.232 represents component 4 of the control current 2; C.sub.0 represents a linear coefficient between the control current I.sub.2 and a working displacement x; U.sub.c1 represents a real-time voltage of capacitor 1; and U.sub.c1(0) represents a voltage before the capacitor 1 starts to work; and when the magnitude of the value of the negative stiffness is less than that of the positive stiffness, that is, k.sub.m+k>0, representing the control current I.sub.c1(I.sub.c2) by the following formulas: $I_{c1}=I_1-I_2^{\mathrm{\prime 2}}=I_1-{\left(I_{21}^{\prime }+I_{22}^{\prime }+I_{231}^{\prime }-I_{232}^{\prime }\right)}^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,\begin{array}{c}{I}_2^{\prime }=C_0^{\prime }\left(x_t-x_e\right)=C_0^{\prime }{x}_{t21}+C_0^{\prime }{x}_{t22}-C_0^{\prime }{x}_e\\ =C_0^{\prime }{C}_7{e}^{\frac{-c}{2m}t}\cos \left(\sqrt{\frac{k_m+k}{m}}t\right)+\\ C_0^{\prime }{C}_8{e}^{\frac{-c}{2m}t}\sin \left(\sqrt{\frac{k_m+k}{m}-\frac{c^2}{4m^2}}t\right)+\\ C_0^{\prime }\sqrt{C_9^2+C_{10}^2}\sin \left[w_{e}t+\arctan \left(\frac{C_9}{C_{10}}\right)\right]-C_0^{\prime }{X}_e\cos \left(w_{e}t\right)\\ =I_{21}^{\prime }+I_{22}^{\prime }+I_{231}^{\prime }-I_{232}^{\prime }\end{array},C_0^{\prime }=\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}},U_{c2}+R_2{I}_{22}^{\prime }+L_2\frac{{\mathrm{dI}}_{22}^{\prime }}{dt}=0,\frac{c}{2m}=\frac{R_2}{2L_2},\frac{k_m+k}{m}=\frac{1}{L_2{C}_2^{\prime }}-\frac{R_2^2}{4L_2^2},U_{c2}\left(0\right)=\left(\frac{L_{2}c}{2m}-R_2\right)C_0^{\prime }{C}_7,U_{c3}+R_3{I}_{23}^{\prime }+L_3\frac{{\mathrm{dI}}_{23}^{\prime }}{dt}=0,\frac{c}{2m}=\frac{R_3}{2L_3},\frac{k_m+k}{m}-\frac{c^2}{4m^2}=\frac{1}{L_3{C}_3^{\prime }}-\frac{R_3^2}{4L_3^2},U_{c3}\left(0\right)=-\sqrt{\frac{k_m+k}{m}}{C}_0^{\prime }{C}_8,$ wherein C.sub.2′ represents a resistance value of capacitor 2; R.sub.2 represents a sum of a resistance value of circuit 2 and a resistance value of a coil; L.sub.2 represents a sum of an inductance of the circuit 2 and an inductance of the coil; C.sub.3′ represents a resistance value of capacitor 3; R.sub.3 represents a sum of a resistance value of circuit 3 and the resistance value of the coil; L.sub.3 represents a sum of an inductance of the circuit 3 and the inductance of the coil; I.sub.1 represents control current 1; I.sub.2′ represents control current 2′; I.sub.21′ represents component 1 of the control current 2′; I.sub.22′ represents component 2 of the control current 2′; I.sub.231′ represents component 3 of the control current 2′; I.sub.232′ represents component 4 of the control current 2′; C.sub.0′ represents a linear coefficient between the control current I.sub.2′ and a working displacement x; U.sub.c2 represents a real-time voltage of the capacitor 2; U.sub.c2(0) represents a voltage before the capacitor 2 starts to work; U.sub.c3 represents a real-time voltage of the capacitor 3; and U.sub.c3(0) represents a voltage before the capacitor 3 starts to work.

Description

BRIEF DESCRIPTION OF FIGURES

[0046] FIG. 1 is a flow diagram of steps of an electromagnetic force control method of a magnetic disk type negative stiffness actuator.

[0047] FIG. 2 is a schematic structural diagram of a magnetic disk type quasi-zero stiffness vibration isolator.

[0048] FIG. 3 is a schematic diagram of a magnetic circuit model of a single electromagnet.

[0049] FIG. 4 is a schematic diagram of a negative stiffness electromagnetic actuator.

[0050] FIG. 5 is a schematic diagram of a vibration isolation system model.

[0051] FIG. 6 is a schematic diagram of a control circuit in which a magnitude of a value of the negative stiffness electromagnetic actuator is equal to that of the positive stiffness (k.sub.m=−k).

[0052] FIG. 7 is a schematic diagram of a control circuit in which a magnitude of a value of the negative stiffness electromagnetic actuator is less than that of the positive stiffness (k.sub.m+k>0).

DETAILED DESCRIPTION

[0053] The present disclosure is described in detail below with reference to specific embodiments.

Specific Embodiment I

[0054] As shown in FIG. 1 to FIG. 7, the present disclosure provides an electromagnetic force control method of the magnetic disk type negative stiffness electromagnetic actuator. The electromagnetic force control method of the magnetic disk type negative stiffness electromagnetic actuator includes the following steps:

[0055] Step 1: according to the static bearing capacity, selecting a mechanical spring for providing positive stiffness of a vibration isolation system, and determining the positive stiffness of the spring.

[0056] Step 2: establishing an electromagnetic force expression under magnetic unsaturation, and determining an electromagnetic attraction force of an electromagnet.

[0057] The step 2 is specifically as follows:

[0058] establishing an electromagnetic force expression under magnetic unsaturation, determining an electromagnetic attraction force of an electromagnet, and representing the electromagnetic attraction force of the electromagnet by the following formulas:

[00013]$F_{\mathrm{mag}1}=\frac{1}{2}\frac{{\Phi }_1^2}{{\mu }_0{S}_{\mathrm{in}}}+\frac{1}{2}\frac{{\Phi }_1^2}{{\mu }_0{S}_{\mathrm{out}}},{\Phi }_1=\frac{N_1{I}_{c1}}{R_{\mathrm{total}1}},S_{out}=\frac{\pi }{4}\left(d_6^2-d_5^2\right),S_{\mathrm{in}}=\frac{\pi }{4}\left(r_4^2-r_3^2\right),$

[0059] wherein F.sub.mag1 represents an electromagnetic attraction force of an electromagnet 1; R.sub.total1 represents a total reluctance of the electromagnet 1; Φ.sub.1 represents a magnetic flux of the electromagnet 1; μ.sub.0 represents permeability of vacuum; S.sub.in represents an equivalent cross-sectional area of an inner magnetic pole; S.sub.out represents an equivalent cross-sectional area of an outer magnetic pole; and I.sub.c1 represents a current passing into the coil 1; and

[0060] determining the total reluctance of the electromagnet 1 according to an air gap reluctance of an armature 1 and a reluctance of the armature 1, and representing the total reluctance of the electromagnet 1 by the following formulas:

[00014]$R_{\mathrm{total}1}=R_{\mathrm{gap}11}+R_{gap12}+R_{arm}+R_{\mathrm{iron}},R_{\mathrm{iron}}=\frac{l_b}{{\mu }_1{S}_{\mathrm{in}}}+\frac{l_b}{{\mu }_1{S}_{out}}+\frac{l_a}{{\mu }_1{S}_1},R_{\mathrm{arm}}=\frac{l_8}{2{\mu }_1{S}_{\mathrm{in}}}+\frac{l_8}{2{\mu }_1{S}_{out}}+\frac{l_d}{{\mu }_1{S}_2},R_{gap11}=\frac{x_1}{{\mu }_0{S}_{\mathrm{in}}},R_{\mathrm{gap}12}=\frac{x_1}{{\mu }_0{S}_{out}},S_1=\frac{\pi }{16}\left(l_1-l_2\right)\left(l_3+l_4+l_5+l_6\right),S_2=\frac{\pi }{4}{l}_8\left(l_3+l_4+l_5+l_6\right),l_a=\frac{l_5+l_6-l_3-l_4}{4},l_b=\frac{l_1+l_2}{2},$

[0061] wherein S.sub.1 represents an equivalent cross-sectional area of a magnetic circuit in a horizontal direction of an iron core 1; S.sub.2 represents an equivalent cross-sectional area of a magnetic circuit in a horizontal direction of the armature 1; x.sub.1 represents an air gap of the electromagnet 1; μ.sub.1 represents a magnetic permeability of materials of the iron core 1, an iron core 2, the armature 1 and an armature 2; R.sub.gap11 represents an air gap reluctance corresponding to an inner magnetic pole of the armature 1; R.sub.gap12 represents an air gap reluctance corresponding to an outer magnetic pole of the armature 1; R.sub.arm represents the reluctance of the armature 1; R.sub.iron represents a reluctance of the iron core 1; N.sub.1 represents the number of turns of a coil of the electromagnet 1; l.sub.1 represents a height of the iron core 1; l.sub.2 represents a height of a coil of the electromagnet 1; l.sub.3 represents an inner diameter of the armature 1; l.sub.4 represents an inner diameter of the coil of the electromagnet 1; l.sub.5 represents an outer diameter of the coil of the electromagnet 1; l.sub.6 represents an outer diameter of the iron core 1; l.sub.7 represents an outer diameter of the armature 1; and l.sub.8 represents a height of the armature 1.

[0062] Step 3: enabling a negative stiffness electromagnetic actuator to be symmetrical up and down, establishing an electromagnetic force-displacement expression according to the negative stiffness electromagnetic actuator under a condition of magnetic unsaturation, and determining an electromagnetic attraction force of the negative stiffness electromagnetic actuator under ideal conditions.

[0063] The step 3 is specifically as follows:

[0064] according to a structure that the negative stiffness electromagnetic actuator is symmetrical up and down, establishing an electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator under the condition of magnetic unsaturation, determining an electromagnetic attraction force of the negative stiffness electromagnetic actuator under ideal conditions, and representing an electromagnetic attraction force under ideal conditions by the following formulas:

[00015]$F_{\mathrm{mag}}=F_{\mathrm{mag}1}-F_{\mathrm{mag}2}=2N_1^2{I}_{c1}^2{\mu }_0{S}_{\mathrm{in}}{S}_{out}\left(S_{\mathrm{in}}+S_{out}\right)\left[\frac{1}{{\left(\mathrm{ax}+b\right)}^2}-\frac{1}{{\left(\mathrm{ax}-b\right)}^2}\right],F_{\mathrm{mag}2}=\frac{1}{2}\frac{{\Phi }_2^2}{{\mu }_0{S}_{\mathrm{in}}}+\frac{1}{2}\frac{{\Phi }_2^2}{{\mu }_0{S}_{out}},a=2\left(S_{\mathrm{in}}+S_{out}\right),b=\left(\frac{l_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2g\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right),$

[0065] wherein F.sub.mag2 represents an electromagnetic attraction force of an electromagnet 2; Φ.sub.2 represents a magnetic flux of the electromagnet 2; x represents a displacement of a negative stiffness spring; N.sub.2 represents the number of turns of a coil of the electromagnet 2; I.sub.2 represents currents passing into a electromagnet 2; μ.sub.r represents a magnitude of a relative magnetic conductivity of materials of an iron core 1, an iron core 2, an armature 1 and an armature 2; a represents an intermediate calculation variable about S.sub.in and S.sub.out; and b represents an intermediate calculation variable about l.sub.8, l.sub.a, l.sub.b, S.sub.in, S.sub.out, S.sub.1, S.sub.2 and μ.sub.r;

[0066] determining the magnetic flux of the electromagnet 2 according to the number of turns of the coil and the current of the electromagnet 2 and the total reluctance of the electromagnet 2, and representing the magnetic flux of the electromagnet 2 by the following formulas:

[00016]${\Phi }_2=\frac{N_2{I}_2}{R_{\mathrm{total}2}},R_{\mathrm{total}2}=R_{gap21}+R_{gap22}+R_{arm}+R_{\mathrm{iron}},R_{\mathrm{gap}22}=\frac{x_2}{{\mu }_0{S}_{out}},$

[0067] wherein R.sub.gap21 represents an air gap reluctance corresponding to an inner magnetic pole of the armature 2, and R.sub.gap22 represents an air gap reluctance corresponding to an outer magnetic pole of the armature 2.

[0068] Step 4: analyzing the electromagnetic force-displacement expression, and determining an expression about currents passing into upper and lower coils about a displacement, a magnitude of ideal constant value negative stiffness and a structural parameter when the negative stiffness electromagnetic actuator takes the current as an input control variable of a system.

[0069] The step 4 is specifically as follows:

[0070] Step 4.1: determining an electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator, and making the electromagnetic force-displacement expression of the negative stiffness electromagnetic actuator equal to an electromagnetic force-displacement relationship of the negative stiffness electromagnetic actuator:

[00017]$F_{\mathrm{mag}}=2N_1^2{I}_{c1}^2{\mu }_0{S}_{\mathrm{in}}{S}_{out}\left(S_{\mathrm{in}}+S_{out}\right)\left[\frac{1}{{\left(ax+b\right)}^2}-\frac{1}{{\left(ax-b\right)}^2}\right]=k_{m}x,F_{mag}=k_{m}x,$

[0071] wherein k.sub.m represents the magnitude of the ideal constant value negative stiffness.

[0072] Step 4.2: analyzing the electromagnetic force-displacement expression, and determining the expression about currents passing into upper and lower coils about the displacement, the magnitude of the ideal constant value negative stiffness and the structural parameter when the negative stiffness electromagnetic actuator takes the current as the input control variable of the system:

[00018]$I_{c1}=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}\sqrt{{\left(ax+b\right)}^2{\left(ax-b\right)}^2},ax+b=\left(\frac{r_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2g+2x\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right)>0,b-\mathrm{ax}=\left(\frac{r_8}{{\mu }_r}+\frac{2l_b}{{\mu }_r}+2\left(g-x\right)\right)\left(S_{\mathrm{in}}+S_{out}\right)+\frac{2l_a}{{\mu }_r}{S}_{\mathrm{in}}{S}_{out}\left(\frac{1}{S_1}+\frac{1}{S_2}\right)>0,\mathrm{ax}-b<0,I_{c1}=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2-\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{x}^2=I_1-I_2^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,I_2=\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}}x,$

[0073] wherein I.sub.c1(I.sub.c2=I.sub.c1) represents currents passing into the coil 1 and the coil 2 at the same time.

[0074] Step 5: establishing a vibration isolation system dynamic formula, and determining a vibration response according to the electromagnetic attraction force of the negative stiffness electromagnetic actuator under ideal conditions.

[0075] The step 5 is specifically as follows:

[0076] Step 5.1: establishing a vibration isolation system dynamic formula, and substituting an ideal electromagnetic force form into a dynamic mathematical model of a vibration isolation system by the following formulas:

m{umlaut over (x)}+c({dot over (x)}.sub.t−{dot over (x)}.sub.e)+k.sub.m(x.sub.t−x.sub.e)+k(x.sub.t−x.sub.e)=0,

x.sub.e=X.sub.e cos(w.sub.et),

x=x.sub.t−x.sub.e,

[0077] wherein m represents a vibration-isolated mass; {umlaut over (x)}.sub.t represents a vibration response acceleration of a vibration-isolated object; {dot over (x)}.sub.t represents a vibration response speed of the vibration-isolated object; x.sub.e represents a system excitation; {dot over (x)}.sub.e represents a system excitation speed; c represents a system damping; x.sub.t represents a vibration response of the vibration-isolated object; X.sub.e represents an excitation amplitude; and w.sub.e represents an excitation frequency.

[0078] Step 5.2: under ideal conditions, when a magnitude of an absolute value of negative stiffness is the same as that of positive stiffness and an overall dynamic stiffness of the system is zero, that is, k.sub.m+k=0, representing a dynamic mathematic model of the vibration isolation system by the following formula:

[00019]${\stackrel{\mathrm{.Math.}}{x}}_t+\frac{c}{m}{\stackrel{.}{x}}_t=\frac{-c{X}_e{w}_e}{m}\sin \left(w_{e}t\right),$

[0079] determining a displacement response, and representing a vibration response by the following formulas:

[00020]$x_t=x_{t11}+x_{t12}$$x_{t11}=C_1+C_2{e}^{\frac{-c}{m}t},C_1=-X_e,C_2=\frac{X_e{m}^2{w}_e^2}{m^2{w}_e^2+c^2},x_{t12}=C_3\cos \left(w_{e}t\right)+C_4\sin \left(w_{e}t\right)=\sqrt{C_3^2+C_4^2}\sin \left[w_{e}t+\arctan \left(\frac{C_3}{C_4}\right)\right],C_3=\frac{X_e{c}^2}{m^2{w}_e^2+c^2},C_4=\frac{c{X}_{e}m{w}_e}{m^2{w}_e^2+c^2},$

[0080] wherein x.sub.t11 represents a solution corresponding to a free vibration response; x.sub.t12 represents a particular solution corresponding to a forced vibration response; C.sub.1 represents a particular solution coefficient 1 corresponding to the free vibration response; C.sub.2 represents a particular solution coefficient 2 corresponding to the free vibration response; C.sub.3 represents a particular solution coefficient 1 corresponding to the forced vibration response; and C.sub.4 represents a particular solution coefficient 2 corresponding to the forced vibration response.

[0081] Step 5.3: when stiffness of the negative stiffness mechanism is positive stiffness, that is, k.sub.m+k>0, representing a dynamic mathematic model of the vibration isolation system by the following formula:

[00021]${\stackrel{\mathrm{.Math.}}{x}}_t+\frac{c}{m}{\stackrel{.}{x}}_t+\frac{k_m+k}{m}{x}_t=\frac{k_m+k}{m}{X}_e\cos \left(w_{e}t\right)-\frac{c}{m}{X}_e{w}_e\sin \left(w_{e}t\right),$

[0082] determining a displacement response, and representing the displacement response by the following formulas:

[00022]$x_t=x_{t21}+x_{t22},x_{t21}=C_7{e}^{\frac{-c}{2m}t}\cos \left(\sqrt{\frac{k_m+k}{m}}t\right)+C_8{e}^{\frac{-c}{2m}t}\sin \left(\sqrt{\frac{k_m+k}{m}-\frac{c^2}{4m^2}}t\right),C_7=\frac{-X_e\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},C_8=\frac{-c{X}_e\left\{2m^2{w}_e^4+\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]\right\}}{\sqrt{4m\left(k_m+k\right)+c^2}\left[{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2\right]},x_{t22}=C_9\cos \left(w_{e}t\right)+C_{10}\sin \left(w_{e}t\right)=\sqrt{C_9^2+C_{10}^2}\sin \left[w_{e}t+\arctan \left(\frac{C_9}{C_{10}}\right)\right],C_9=\frac{X_e\left[\left(k_m+k\right)\left(k_m+k-m{w}_e^2\right)+c^2{w}_e^2\right]}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},C_{10}=\frac{cm{X}_e{w}_e^3}{{\left(k_m+k-m{w}_e^2\right)}^2+c^2{w}_e^2},$

[0083] wherein x.sub.t21 represents a solution corresponding to a free vibration response; x.sub.t22 represents a particular solution corresponding to a forced vibration response; C.sub.7 represents a particular solution coefficient 1 corresponding to the free vibration response; C.sub.8 represents a particular solution coefficient 2 corresponding to the free vibration response; C.sub.9 represents a particular solution coefficient 1 corresponding to the forced vibration response; and C.sub.10 represents a particular solution coefficient 2 corresponding to the forced vibration response.

[0084] Step 6: according to the vibration response and based on the expression about currents about a displacement, a magnitude of ideal constant value negative stiffness and a structural parameter, determining a control current required for meeting a linear electromagnetic force-displacement relationship of a negative stiffness electromagnetic actuator.

[0085] The step 6 is specifically as follows:

[0086] under ideal conditions, when a magnitude of a value of negative stiffness is the same as that of positive stiffness, that is, k.sub.m=−k, representing the control current I.sub.c1(I.sub.c2) by the following formulas:

[00023]$I_{c1}=I_1-{\left(I_{21}+I_{22}+I_{231}-I_{232}\right)}^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,I_2=I_{21}+I_{22}+I_{231}-I_{232}=\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}}=C_{0}x=C_0\left(x_t-x_e\right)=C_0{x}_{t11}+C_0{x}_{t12}-C_0{x}_e=C_0{C}_1+C_0{C}_2{e}^{\frac{-c}{m}t}+C_0\sqrt{C_3^2+C_4^2}\sin \left[w_{e}t+\arctan \left(\frac{C_3}{C_4}\right)\right]-C_0{X}_e\cos \left(w_{e}t\right),U_{c1}+R_1{I}_{22}+L_1\frac{d{I}_{22}}{dt}=0,{\left(\frac{c}{m}\right)}^2={\left(\frac{R_1}{2L_1}\right)}^2=\frac{1}{L_1{C}_1^{\prime }},U_{c1}\left(0\right)=\left(\frac{c}{m}-R_1\right)C_0{C}_2,$

[0087] wherein C.sub.1′ represents a resistance value of a capacitor; R.sub.1 represents a sum of a resistance value of a circuit and a resistance value of a coil; L.sub.1 represents an inductance of a circuit 1; I.sub.1 represents a control current 1; I.sub.2 represents a control current 2; I.sub.21 represents a component 1 of the control current 2; I.sub.22 represents a component 2 of the control current 2; I.sub.231 represents a component 3 of the control current 2; I.sub.232 represents a component 4 of the control current 2; C.sub.0 represents a linear coefficient between the control current I.sub.2 and a working displacement x; U.sub.c1 represents a real-time voltage of a capacitor 1; and U.sub.c1(0) represents a voltage before the capacitor 1 starts to work; and

[0088] when the magnitude of the value of the negative stiffness is less than that of the positive stiffness, that is, k.sub.m+k>0, representing the control current I.sub.c1(I.sub.c2) by the following formulas:

[00024]$I_{c1}=I_1-I_2^{\mathrm{\prime 2}}=I_1-{\left(I_{21}^{\prime }+I_{22}^{\prime }+I_{231}^{\prime }-I_{232}^{\prime }\right)}^2,I_1=\frac{1}{2a{N}_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}{b}^2,\begin{array}{c}{I}_2^{\prime }=C_0^{\prime }\left(x_t-x_e\right)=C_0^{\prime }{x}_{t21}+C_0^{\prime }{x}_{t22}-C_0^{\prime }{x}_e\\ =C_0^{\prime }{C}_7{e}^{\frac{-c}{2m}t}\cos \left(\sqrt{\frac{k_m+k}{m}}t\right)+\\ C_0^{\prime }{C}_8{e}^{\frac{-c}{2m}t}\sin \left(\sqrt{\frac{k_m+k}{m}-\frac{c^2}{4m^2}}t\right)+\\ C_0^{\prime }\sqrt{C_9^2+C_{10}^2}\sin \left[w_{e}t+\arctan \left(\frac{C_9}{C_{10}}\right)\right]-\\ C_0^{\prime }{X}_e\cos \left(w_{e}t\right)\\ =I_{21}^{\prime }+I_{22}^{\prime }+I_{231}^{\prime }-I_{232}^{\prime }\prime \end{array},C_0^{\prime }=\sqrt{\frac{a}{2N_1}\sqrt{\frac{-k_m}{{\mu }_{0}b{S}_{\mathrm{in}}{S}_{out}}}},U_{c2}+R_2{I}_{22}^{\prime }\prime =+L_2\frac{{\mathrm{dI}}_{22}^{\prime }}{dt}=0,\frac{c}{2m}=\frac{R_2}{2L_2},\frac{k_m+k}{m}=\frac{1}{L_2{C}_2^{\prime }}-\frac{R_2^2}{4L_2^2},U_{c2}\left(0\right)=\left(\frac{L_{2}c}{2m}-R_2\right)C_0^{\prime }{C}_7,U_{c3}+R_3{I}_{23}^{\prime }+L_3\frac{{\mathrm{dI}}_{23}^{\prime }}{dt}=0,\frac{c}{2m}=\frac{R_3}{2L_3},\frac{k_m+k}{m}-\frac{c^2}{4m^2}=\frac{1}{L_3{C}_3^{\prime }}-\frac{R_3^2}{4L_3^2},U_{c3}\left(0\right)=-\sqrt{\frac{k_m+k}{m}}{C}_0^{\prime }{C}_8,$

[0089] wherein C.sub.2′ represents a resistance value of a capacitor 2; R.sub.2 represents a sum of a resistance value of a circuit 2 and a resistance value of a coil; L.sub.2 represents an inductance of the circuit 2; C.sub.3′ represents a resistance value of a capacitor 3; R.sub.3 represents a sum of a resistance value of a circuit 3 and the resistance value of the coil; L.sub.3 represents an inductance of the circuit 3; I.sub.1 represents a control current 1; I.sub.2′ represents a control current 2′; I.sub.21′ represents a component 1 of the control current 2′; I.sub.22′ represents a component 2 of the control current 2′; I.sub.231′ represents a component 3 of the control current 2′; I.sub.232′ represents a component 4 of the control current 2′; C.sub.0′ represents a linear coefficient between the control current I.sub.2′; and a working displacement x; U.sub.c2 represents a real-time voltage of the capacitor 2; U.sub.c2(0) represents a voltage before the capacitor 2 starts to work; U.sub.c3 represents a real-time voltage of the capacitor 3; and U.sub.c3(0) represents a voltage before the capacitor 3 starts to work.