# Magnetic Nonlinear Energy Sink for Vibration Attenuation of Unbalanced Rotor System

Authors: Hongliang Yao; Dasheng Zheng; Bangchun Wen
Journal: Shock and Vibration
 (2017)

---

## Abstract

A novel nonlinear energy sink (NES) consisting of permanent magnetic springs and coil springs is proposed, and the vibration attenuation performance of the NES for unbalanced rotor system is investigated. Firstly, the nonlinearity of the magnet spring is analyzed and the structure of the NES is introduced. Then, the dynamic model of the rotor system with the NES is built, and the responses and stabilities of the system are studied by applying Complexification-averaging method. The strongly modulated responses (SMR) behavior, which is the most important performance characteristic of the NES, is analytically studied by combining Complexification-averaging method and multiscale method and numerically verified by Runge-Kutta method. The results show that the NES is effective in attenuating the vibration of unbalanced rotor, and the SMR occurrence range can be broadened by increasing the nonlinearity of the NES. And also, the NES has better performance over a wider frequency range than the linear absorber.


---

## Body

## 1. Introduction

Excessive vibration of rotor system is an important factor which can lead to instability of rotating machinery. To maintain the safety operation of the rotating machinery, the vibration of rotor system must be controlled. Many methods are proposed to control the vibration of rotor system, such as adjusting stiffness [1] or damping, imposing external forces [2], or reducing the unbalance [3].Besides the above methods, adding dynamic vibration absorbers on the rotor system is a popular vibration suppression method, by which the vibrations of the rotor system can be “absorbed” by absorber. In torsional vibration suppression, centrifugal pendulum vibration absorbers [4] have long been widely used. In lateral vibration attenuation, many studies have also been carried out: [5] proposed a vertical vibration absorber which is mounted on the support of rotating Rayleigh beam; [6] developed a magnetic based dynamic absorber combining negative stiffness with conventional positive stiffness for rotor system vibration suppression; [7] adopted vibration absorber to eliminate the self-vibration of milling process.As the traditional vibration absorbers are only effective in a narrow range of frequencies, researchers began to pay attention to nonlinear energy sinks (NESs), which can react efficiently on the amplitude characteristics of the external forcing in a wide range of frequencies [8]. The mechanism of the nonlinear energy sink is energy pumping and has been studied for a long time, and various types of nonlinear energy sinks are designed and analyzed [9–12]. Nonlinear stiffness is the most important component in all NES structures. Cubic stiffness is widely applied in NES designing [8, 11], and many NESs with nonpolynomial nonlinearity are also presented [13]. Vibro-impact type NES or NES with nonsmooth stiffness also proved to be efficient in controlling the vibration of the primary vibration system [14–16]. Besides, [17] modified the purely cubic stiffness NES and presented a NES with negative linear and nonlinear coupling stiffness components, and the performance of new NES is much improved.In vibration suppression application, the NESs have been applied on beams and cables [18, 19], machine tool [20, 21], steel frames [22], and so on. In rotating systems, many types of NES have been developed and applied. For example, [23] proposed a vertical vibration absorber which is mounted on the supporting of rotating Rayleigh beam; [24] presented nonlinear energy sinks which are mounted on journal bearings in both vertical and horizontal direction; [25] proposed nonlinear energy sink to unbalanced hollow rotor system to reduce the whirling vibration; [26] investigated the performance of a smooth NES to mitigate vibration of a rotating beam under an external force.The above works prove that the NES has shown great prospect in vibration attenuation of rotor vibration. In this paper, a novel NES made up of permanent magnetic spring and coil springs is proposed and its dynamics are investigated analytically and numerically; the performance of the NES and the affecting parameters for SMR behavior occurrence are studied.


## 2. Magnetic Nonlinear Stiffness and Structure of the NES

### 2.1. Nonlinear Stiffness of the Magnetic Spring

In [27], a nonlinear energy sink is presented by using asymmetric permanent magnets, in which the asymmetric nonlinear force is parallel to the normal of the magnet surface. In [6], the authors developed another type of magnetic spring for rotor system. The magnetic spring is made up of ring type magnets, which can be divided into outer magnets and central magnets, as shown in Figure 1. The outer magnets and central magnets are mounted in repulsive interaction, so a repulsive force in radial direction can occur and the central magnet acts as a spring with negative stiffness. Different from [27], the nonlinear force is in radial direction.Figure 1
Structure of the magnetic spring.


### 2.2. Magnet Force and Stiffness Coefficients Approximation

#### 2.2.1. Magnet Force Calculation Using Equivalent Magnetic Charge Method

The repulsive magnet force in radial direction can be calculated by applying equivalent magnetic charge method. It is introduced in detail in [6], so it is only introduced briefly here.According to the equivalent magnetic charge method, the point charge of a pointr2,α in plane 2 (as shown in Figure 1) can be calculated as(1)q2=Brr2dr2dα,where Br is the residual flux density.The point charge of a pointr3,β in plane 3 is(2)q3=Brr3dr3dβ.So the interaction force between pointsr2,α and r3,β is(3)dF→23=Br24πμ0r2r3dr2dαdr3dβr232r→23,where μ0 is the relative permeability of the permanent magnets.When the central magnet moves in the radial direction for a distancee, the interaction force between planes 2 and 3 can be calculated as(4)F23=Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2r3r3cos⁡β-r1cos⁡α-er3cos⁡β-r1cos⁡α-e2+r3sin⁡β-r1sin⁡α2+h23/2dr2dr3dαdβ.The interaction force between plane 2 and planei can be written as(5)F2i=-1i+1Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2riricos⁡β-r2cos⁡α-ericos⁡β-r2cos⁡α-e2+risin⁡β-r2sin⁡α2+h2i23/2dr2dridαdβi=3–6,where h23=h, h24=h+b2, h25=h+b2+B, and h26=2h+b2+B.And also, the interaction force between plane 1 and planei can be written as(6)F1i=-1iBr24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r1riricos⁡β-r1cos⁡α-ericos⁡β-r1cos⁡α-e2+risin⁡β-r1sin⁡α2+b1+h2i23/2dr1dridαdβi=3,4.As the structure is symmetrical, the interaction forces between the right outer magnet and the central magnets are equal to those of the left ones. So the total force between the central magnets and the outer magnets is(7)Fy=2∑i=36F1i+F2i.Equations (4)–(7) can be solved numerically and the relationship between Fy and different e can be obtained. Then the radial stiffness of the negative stiffness magnetic spring can be obtained by (8)km=dFyde.


#### 2.2.2. Approximation of Stiffness Coefficients

Numerical simulations can be carried out based on (4)–(7) and the parameters are shown in Table 1.Table 1
Parameters for numerical simulation of magnetic force and stiffness.
Parameter
Value
R
D
1
50 mm
R
d
1
25 mm
b
1
6 mm
R
D
2
50 mm
R
d
2
25 mm
b
2
6 mm
B
15 mm
μ
0
4
π
×
1
0
-
7 N/A2
B
r
1.34 TAs shown in [6], the nonlinearity of the magnet spring can be ignored when distance h is large. But when h is small, the nonlinearity is rather large.The Taylor-expansion can be applied to study the nonlinearity of the magnet spring, as in [17]. Figure 2 compares the Taylor-expansion results, which are expanded to 3 orders and 10 orders. It can be seen that the 3-order expansion is also very accurate, so 3-order expansion is used in polynomial fitting of the relationship between the total stiffness and the radial displacement. As the stiffness curve is symmetrical, the fitting components in odd number are all zero, so the relationship between the stiffness and the displacement is(9)km=km1+km2e2.Figure 2
Taylor-expansion and stiffness approximation comparison whenh are 5 mm and 3 mm, respectively.
(a)
h
=
5 mm
(b)
h
=
3 mmThe approximated stiffness coefficients of different magnet distances are shown in Table2. It can be seen from Table 2 that the polynomial fitting effect is rather good and km2 is rather large when h is small.Table 2
Parameters for approximate stiffness.
h (mm)
k
m
1 (N/mm)
k
m
2 (N/mm3)
5
-
17.25
0.54
3
-
30.16
1.65
1.8
-
46.49
4.77


### 2.3. Structure of the NES

Assembling the magnet spring and coil springs together, the NES is formed, whose structure is shown in Figure3. The central magnets connect with the bearing pedestal by the coil springs and the bearing pedestal connects with the shaft by a rolling bearing. With the rolling bearing, the NES will not rotate with the shaft together, so the centrifugal force and the gyroscopic effect of the central magnet can be ignored. The outer magnets are fixed on the aluminum plates and can rotate with the shaft.Figure 3
Structure of the NES.The magnetic spring has negative linear stiffness and the coil spring has positive linear stiffness. When choosing the coil springs for appropriate stiffness, the superposition linear stiffness of the NES can be positive but of a small magnitude. For example, whenh=5mm and km1=-46.49N/mm, when choosing the coil spring with total stiffness 47N/mm, the above linear stiffness of the NES is 0.51N/mm, while its nonlinear stiffness is 4.77N/mm3. So the NES has a small linear stiffness and a large nonlinear stiffness.


## 2.1. Nonlinear Stiffness of the Magnetic Spring

In [27], a nonlinear energy sink is presented by using asymmetric permanent magnets, in which the asymmetric nonlinear force is parallel to the normal of the magnet surface. In [6], the authors developed another type of magnetic spring for rotor system. The magnetic spring is made up of ring type magnets, which can be divided into outer magnets and central magnets, as shown in Figure 1. The outer magnets and central magnets are mounted in repulsive interaction, so a repulsive force in radial direction can occur and the central magnet acts as a spring with negative stiffness. Different from [27], the nonlinear force is in radial direction.Figure 1
Structure of the magnetic spring.


## 2.2. Magnet Force and Stiffness Coefficients Approximation

### 2.2.1. Magnet Force Calculation Using Equivalent Magnetic Charge Method

The repulsive magnet force in radial direction can be calculated by applying equivalent magnetic charge method. It is introduced in detail in [6], so it is only introduced briefly here.According to the equivalent magnetic charge method, the point charge of a pointr2,α in plane 2 (as shown in Figure 1) can be calculated as(1)q2=Brr2dr2dα,where Br is the residual flux density.The point charge of a pointr3,β in plane 3 is(2)q3=Brr3dr3dβ.So the interaction force between pointsr2,α and r3,β is(3)dF→23=Br24πμ0r2r3dr2dαdr3dβr232r→23,where μ0 is the relative permeability of the permanent magnets.When the central magnet moves in the radial direction for a distancee, the interaction force between planes 2 and 3 can be calculated as(4)F23=Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2r3r3cos⁡β-r1cos⁡α-er3cos⁡β-r1cos⁡α-e2+r3sin⁡β-r1sin⁡α2+h23/2dr2dr3dαdβ.The interaction force between plane 2 and planei can be written as(5)F2i=-1i+1Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2riricos⁡β-r2cos⁡α-ericos⁡β-r2cos⁡α-e2+risin⁡β-r2sin⁡α2+h2i23/2dr2dridαdβi=3–6,where h23=h, h24=h+b2, h25=h+b2+B, and h26=2h+b2+B.And also, the interaction force between plane 1 and planei can be written as(6)F1i=-1iBr24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r1riricos⁡β-r1cos⁡α-ericos⁡β-r1cos⁡α-e2+risin⁡β-r1sin⁡α2+b1+h2i23/2dr1dridαdβi=3,4.As the structure is symmetrical, the interaction forces between the right outer magnet and the central magnets are equal to those of the left ones. So the total force between the central magnets and the outer magnets is(7)Fy=2∑i=36F1i+F2i.Equations (4)–(7) can be solved numerically and the relationship between Fy and different e can be obtained. Then the radial stiffness of the negative stiffness magnetic spring can be obtained by (8)km=dFyde.


### 2.2.2. Approximation of Stiffness Coefficients

Numerical simulations can be carried out based on (4)–(7) and the parameters are shown in Table 1.Table 1
Parameters for numerical simulation of magnetic force and stiffness.
Parameter
Value
R
D
1
50 mm
R
d
1
25 mm
b
1
6 mm
R
D
2
50 mm
R
d
2
25 mm
b
2
6 mm
B
15 mm
μ
0
4
π
×
1
0
-
7 N/A2
B
r
1.34 TAs shown in [6], the nonlinearity of the magnet spring can be ignored when distance h is large. But when h is small, the nonlinearity is rather large.The Taylor-expansion can be applied to study the nonlinearity of the magnet spring, as in [17]. Figure 2 compares the Taylor-expansion results, which are expanded to 3 orders and 10 orders. It can be seen that the 3-order expansion is also very accurate, so 3-order expansion is used in polynomial fitting of the relationship between the total stiffness and the radial displacement. As the stiffness curve is symmetrical, the fitting components in odd number are all zero, so the relationship between the stiffness and the displacement is(9)km=km1+km2e2.Figure 2
Taylor-expansion and stiffness approximation comparison whenh are 5 mm and 3 mm, respectively.
(a)
h
=
5 mm
(b)
h
=
3 mmThe approximated stiffness coefficients of different magnet distances are shown in Table2. It can be seen from Table 2 that the polynomial fitting effect is rather good and km2 is rather large when h is small.Table 2
Parameters for approximate stiffness.
h (mm)
k
m
1 (N/mm)
k
m
2 (N/mm3)
5
-
17.25
0.54
3
-
30.16
1.65
1.8
-
46.49
4.77


## 2.2.1. Magnet Force Calculation Using Equivalent Magnetic Charge Method

The repulsive magnet force in radial direction can be calculated by applying equivalent magnetic charge method. It is introduced in detail in [6], so it is only introduced briefly here.According to the equivalent magnetic charge method, the point charge of a pointr2,α in plane 2 (as shown in Figure 1) can be calculated as(1)q2=Brr2dr2dα,where Br is the residual flux density.The point charge of a pointr3,β in plane 3 is(2)q3=Brr3dr3dβ.So the interaction force between pointsr2,α and r3,β is(3)dF→23=Br24πμ0r2r3dr2dαdr3dβr232r→23,where μ0 is the relative permeability of the permanent magnets.When the central magnet moves in the radial direction for a distancee, the interaction force between planes 2 and 3 can be calculated as(4)F23=Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2r3r3cos⁡β-r1cos⁡α-er3cos⁡β-r1cos⁡α-e2+r3sin⁡β-r1sin⁡α2+h23/2dr2dr3dαdβ.The interaction force between plane 2 and planei can be written as(5)F2i=-1i+1Br24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r2riricos⁡β-r2cos⁡α-ericos⁡β-r2cos⁡α-e2+risin⁡β-r2sin⁡α2+h2i23/2dr2dridαdβi=3–6,where h23=h, h24=h+b2, h25=h+b2+B, and h26=2h+b2+B.And also, the interaction force between plane 1 and planei can be written as(6)F1i=-1iBr24πμ0∫02π∫02π∫Rd1RD1∫Rd2RD2r1riricos⁡β-r1cos⁡α-ericos⁡β-r1cos⁡α-e2+risin⁡β-r1sin⁡α2+b1+h2i23/2dr1dridαdβi=3,4.As the structure is symmetrical, the interaction forces between the right outer magnet and the central magnets are equal to those of the left ones. So the total force between the central magnets and the outer magnets is(7)Fy=2∑i=36F1i+F2i.Equations (4)–(7) can be solved numerically and the relationship between Fy and different e can be obtained. Then the radial stiffness of the negative stiffness magnetic spring can be obtained by (8)km=dFyde.


## 2.2.2. Approximation of Stiffness Coefficients

Numerical simulations can be carried out based on (4)–(7) and the parameters are shown in Table 1.Table 1
Parameters for numerical simulation of magnetic force and stiffness.
Parameter
Value
R
D
1
50 mm
R
d
1
25 mm
b
1
6 mm
R
D
2
50 mm
R
d
2
25 mm
b
2
6 mm
B
15 mm
μ
0
4
π
×
1
0
-
7 N/A2
B
r
1.34 TAs shown in [6], the nonlinearity of the magnet spring can be ignored when distance h is large. But when h is small, the nonlinearity is rather large.The Taylor-expansion can be applied to study the nonlinearity of the magnet spring, as in [17]. Figure 2 compares the Taylor-expansion results, which are expanded to 3 orders and 10 orders. It can be seen that the 3-order expansion is also very accurate, so 3-order expansion is used in polynomial fitting of the relationship between the total stiffness and the radial displacement. As the stiffness curve is symmetrical, the fitting components in odd number are all zero, so the relationship between the stiffness and the displacement is(9)km=km1+km2e2.Figure 2
Taylor-expansion and stiffness approximation comparison whenh are 5 mm and 3 mm, respectively.
(a)
h
=
5 mm
(b)
h
=
3 mmThe approximated stiffness coefficients of different magnet distances are shown in Table2. It can be seen from Table 2 that the polynomial fitting effect is rather good and km2 is rather large when h is small.Table 2
Parameters for approximate stiffness.
h (mm)
k
m
1 (N/mm)
k
m
2 (N/mm3)
5
-
17.25
0.54
3
-
30.16
1.65
1.8
-
46.49
4.77


## 2.3. Structure of the NES

Assembling the magnet spring and coil springs together, the NES is formed, whose structure is shown in Figure3. The central magnets connect with the bearing pedestal by the coil springs and the bearing pedestal connects with the shaft by a rolling bearing. With the rolling bearing, the NES will not rotate with the shaft together, so the centrifugal force and the gyroscopic effect of the central magnet can be ignored. The outer magnets are fixed on the aluminum plates and can rotate with the shaft.Figure 3
Structure of the NES.The magnetic spring has negative linear stiffness and the coil spring has positive linear stiffness. When choosing the coil springs for appropriate stiffness, the superposition linear stiffness of the NES can be positive but of a small magnitude. For example, whenh=5mm and km1=-46.49N/mm, when choosing the coil spring with total stiffness 47N/mm, the above linear stiffness of the NES is 0.51N/mm, while its nonlinear stiffness is 4.77N/mm3. So the NES has a small linear stiffness and a large nonlinear stiffness.


## 3. Dynamics of the Rotor System with NES

### 3.1. Dynamic Model of the Rotor with the NES

Assuming the rotor system is a Jeffcott rotor, the dynamic model of rotor system with the NES is shown in Figure4, and the dynamic equations can be written as(10)m1x¨1+c-1x˙1+k-1x+k-rθθy+c-2x˙1-x˙2+k-2+k-m1x1-x2+k-m1x1-x23=m1ω2δcos⁡ωtm1y¨1+c-1y˙1+k-1y-k-rθθx+c-2y˙1-y˙2+k-2+k-m1y1-y2+k-m1y1-y23=m1ω2δsin⁡ωtJdθ¨x+ΩJpθ˙y-kθry+kθθθx=0Jdθ¨y-ΩJpθ˙y+kθrx+kθθθy=0m2x¨2-c-2x˙1-x˙2-k-2+k-m1x1-x2-k-m1x1-x23=0m2y¨2-c-2y˙1-y˙2-k-2+k-m1y1-y2-k-m1y1-y23=0,where “·” denotes d/dt, m1, Jd, and Jp are mass, diameter moment of inertia, and polar moment of inertia of the rotor, respectively, c-1 is the damping of the rotor, k-1, k-rθ, k-θr, and k-θθ are stiffness of bending and rotating of the rotor, respectively, m2, c-2, and k-2+k-m are mass, damping, and stiffness of the NES, respectively, δ is the unbalance eccentricity of the rotor, ω is the rotating speed of the rotor system, and Ω is the whirling speed of the rotor system.Figure 4
Dynamic model of the rotor-NES system.When the gyroscopic effect is small and can be ignored, the system is symmetrical and can be simplified. The vibration in radial direction is shown in Figure4 and the dynamic equations can be written as(11)m1x¨1+c-1x˙1+k-1x+c-2x˙1-x˙2+F-nx1-x2=m1ω2δcos⁡ωtm2x¨2-c-2x˙1-x˙2-F-nx1-x2=0,where F-nx1-x2=k-2+k-m1x1-x2+k-m2x1-x23.We assume that(12)ωn=k1m1,τ=ωt.Equations (11) can be simplified to(13)x1′′+εc1x1′+1+σx+εc2x1′-x2′+Fnx1-x2=εAcos⁡τεx2′′-εc2x1′-x2′-Fnx1-x2=0,where “′” denotes d/dτ and ε=m2/m1, c1=c-1/εm1ω, σ=ωn2/ω2-1, c2=c-2/εm1ω, A=δ/ε, Fnx1-x2=k2+km1x1-x2+km2x1-x23, k2=k-2/εm1ω2, km1=k-m1/εm1ω2, and km2=k-m2/εm1ω2.


### 3.2. Responses of the Rotor System with the NES

Complexification-averaging method is applied to analyze the response of the rotor-NES system.We assume that(14)v=x1+εx2w=x1-x2φ1eiτ=v˙+ivφ2eiτ=w˙+iw.Substitute (14) into (13) and extract only the eiτ components; one obtains(15)φ1′+εi+c1-iσφ1-εi+εc1-iεσφ221+ε=εA2φ2′-ε21+ε-c1φ1+iσφ1+ε21+εc2+ε2c2+εc2+c1-iσφ2-iε21+εφ1-φ2-i21+εφ2G1φ22=εA2,where G1φ22=k2+km1+3km2φ22/4.To obtain the steady state responses of the system, equate the time derivatives of (15) to zero (φ˙1=φ˙2=0) and substitute the first equation of (15) into the second equation; one obtains(16)φ10-εi-εc1+iεσφ20+1+εAi+c1-iσ=0φ20-c1+iσ+c2-1+ic1+σ+c1-iσ-1φ20G1φ202+A=0.The second equation of (16) can be changed to(17)G12φ202-2c12-σ+σ2c12+σ-12G1φ202+c1+c22+σ2+c22c12-2σ+σ2c12+σ-12φ202=A2c12+σ-12.The steady state responseφ20 can be obtained from (17), and then the response φ10 can be obtained from the first equation of (16).Saddle-node (SN) and Hopf bifurcations can happen in the responses, and the judgment criteria can be found in [10].


### 3.3. Strongly Modulated Responses of the Rotor System with the NES

The SMR behavior of the system can be investigated by combining Complexification-averaging method and multiscale method.Change (15) to single second-order ODE as follows:(18)4φ2′′+i+c2+εc1-iεσφ2′+ε1-ic1G1φ22+c1c2+ic1+c2+σ-σGφ22+iσc2φ2-2iddτφ2Gφ22=iεA.Introduce multiple scale expansion as(19)φ2=φ2τ0,τ1,…;ddτ=∂∂τ0+ε∂∂τ1+⋯;τk=εkτ,k=0,1,….Substituting (19) into (18) and setting the coefficients of powers of ε equal to zero, the following equations are obtained:(20)∂2φ2∂τ02+∂∂τ012i+c2φ2-i2φ2G1φ22=08∂2φ2∂τ0∂τ1+∂∂τ02c1φ2+2c2φ2-2iσφ2-2iφ2G1φ22+∂∂τ12c2φ2+2iφ2-2iφ2G1φ22+1-ic1G1φ22+c1c2+ic1+c2+σ-σG1φ22+iσc2φ2=iA.Integrate the first equation of (20) and obtain(21)∂φ2∂τ0+12i+c2φ2-i2φ2G1φ22=Cτ1,…,where C is arbitrary function of higher-order time scales.Whenτ0→∞, parameters in τ0 order stay invariant and φ2 reaches an asymptotic equilibrium ϕ, which is(22)i+c2ϕ-iϕG1ϕ2=Cτ1,….Assumingϕτ1=Nτ1eiθτ1, it can be obtained that(23)N2c22+G1N2-12=Cτ12or equivalently(24)Zc22+G1Z-12=Cτ12,where Zτ1=N2τ1.The magnitudeNτ1 can be obtained from (24) and the angle θτ1 can be obtained as(25)θτ1=arg⁡Cτ1-atan⁡1-G1Zc2.If the parameters of the system satisfy (14), the system may give rise to strongly modulated responses. Taking the limit τ0→∞ in the second equation of (19), one can obtain(26)∂∂τ12c2φ2+2iφ2-2iφ2G1φ22+1-ic1G1φ22+c1c2+ic1+c2+σ-σG1φ22+iσc2φ2=iA.It can be written in a more convenient form:(27)iϕ+c2ϕ-iG2ϕ2∂ϕ∂τ1-iG3ϕ2∂ϕ∗∂τ1=PP=12iA-221-ic1G1ϕ2+c1c2+ic1+c2+σ-σGϕ2+iσc2ϕ,where G2ϕ22=k2+km1+3km2φ22/2, G3ϕ22=3km2/4, and G1=G2-G3Φ2.By taking complex conjugate of (27) and assuming ϕτ1=Nτ1eiθτ1, one can obtain(28)∂N∂τ1=-N1-G12c1+c2+A1-G1cos⁡θ+Aλsin⁡θ2c22+1-G22-G32N4∂θ∂τ1=1-G2-G3N2G1-c22+1-G11-G2-G3N21-c22σ+Ac2cos⁡θ-1-G2-G3N2sin⁡θ/N2c22+1-G22-G32N4.The possibility of occurrence of the SMR behavior can be predicted by using phase plane plots of (28). The fold singularities which can provide SMR are those which satisfy numerators = 0 and denominator = 0, which are(29)c22+1-G22-G32N4=0-N1-G12c1+c2+A1-G1cos⁡θ+Aλsin⁡θ=0.We assume that the first pair of folded singularities areN1,θ1 and N1,θ2 and the second pair of folded singularities are N2,θ3 and N2,θ4. N1 and N2 can be obtained from the first equation of (29) that(30)Z1,2=N1,22=81-k2-km1km2±4km22k2+km1-12-3c229km22.θ
1, θ2, θ3, and θ4 can be obtained from the second equation of (30):(31)θ1,2=arcsin⁡-16Ac216A2-4+4k2+4km1+3km2N122+16Ac22±arccos⁡N116c2+-4+4k2+4km1+3km2N122c1+16c22c116A2-4+4k2+4km1+3km2N122+16Ac22;θ3,4=arcsin⁡-16Ac216A2-4+4k2+4km1+3km2N222+16Ac22±arccos⁡N116c2+-4+4k2+4km1+3km2N222c1+16c22c116A2-4+4k2+4km1+3km2N222+16Ac22.From (31), it can be seen that the critical amplitude of A for occurrence of the saddle-node bifurcation is (32)Fcritical1=-16Ac216-4+4k2+4km1+3km2N122+256c22Fcritical2=-16Ac216-4+4k2+4km1+3km2N222+256c22.This is a necessary condition of SMR behavior occurrence, but not a sufficient condition. The accurate existence conditions of SMR behavior can be obtained by constructing one-dimensional map [10], the procedure of which can be divided into 4 steps.(1) The first step is taking a pointθ01 in the interval θ1,θ2 and computing the fast jump from N1,θ01 to Nu,θu, in which Nu can be obtained by(33)-48km2N12+27km22N14+16k23km2N12-2=-48km2Nu2+27km22Nu4+16k23km2Nu2-2and θu can be obtained as(34)θu=θ01+atan⁡12c2km2Nu2-N1216c22+-4+4k2+4km1+3km2N12-4+4k2+4km1+3km2Nu2.(2) The second step is computing the slow branch fromNu,θu to N2,θ02, which can be solved by (28).(3) The third step is computing the fast jump fromN2,θ02 to Nd,θd, in which Nd can be obtained by(35)-48km2N22+27km22N24+16k23km2N22-2=-48km2Nu2+27km22Nu4+16k23km2Nu2-2and θd can be obtained as(36)θd=θ02-atan⁡12c2km2Nu2-N2216c22+-4+4k2+4km1+3km2N22-4+4k2+4km1+3km2Nu2.(4) The fourth step is computing the slow branch fromNd,θd to N1,θ03, which can be solved by (28).Ifθ03 can be mapped into interval [θ1,θ2] after the 4 steps, the SMR behavior is sure to happen. If θ03 cannot be mapped into the interval, the SMR behavior is sure not to happen.


## 3.1. Dynamic Model of the Rotor with the NES

Assuming the rotor system is a Jeffcott rotor, the dynamic model of rotor system with the NES is shown in Figure4, and the dynamic equations can be written as(10)m1x¨1+c-1x˙1+k-1x+k-rθθy+c-2x˙1-x˙2+k-2+k-m1x1-x2+k-m1x1-x23=m1ω2δcos⁡ωtm1y¨1+c-1y˙1+k-1y-k-rθθx+c-2y˙1-y˙2+k-2+k-m1y1-y2+k-m1y1-y23=m1ω2δsin⁡ωtJdθ¨x+ΩJpθ˙y-kθry+kθθθx=0Jdθ¨y-ΩJpθ˙y+kθrx+kθθθy=0m2x¨2-c-2x˙1-x˙2-k-2+k-m1x1-x2-k-m1x1-x23=0m2y¨2-c-2y˙1-y˙2-k-2+k-m1y1-y2-k-m1y1-y23=0,where “·” denotes d/dt, m1, Jd, and Jp are mass, diameter moment of inertia, and polar moment of inertia of the rotor, respectively, c-1 is the damping of the rotor, k-1, k-rθ, k-θr, and k-θθ are stiffness of bending and rotating of the rotor, respectively, m2, c-2, and k-2+k-m are mass, damping, and stiffness of the NES, respectively, δ is the unbalance eccentricity of the rotor, ω is the rotating speed of the rotor system, and Ω is the whirling speed of the rotor system.Figure 4
Dynamic model of the rotor-NES system.When the gyroscopic effect is small and can be ignored, the system is symmetrical and can be simplified. The vibration in radial direction is shown in Figure4 and the dynamic equations can be written as(11)m1x¨1+c-1x˙1+k-1x+c-2x˙1-x˙2+F-nx1-x2=m1ω2δcos⁡ωtm2x¨2-c-2x˙1-x˙2-F-nx1-x2=0,where F-nx1-x2=k-2+k-m1x1-x2+k-m2x1-x23.We assume that(12)ωn=k1m1,τ=ωt.Equations (11) can be simplified to(13)x1′′+εc1x1′+1+σx+εc2x1′-x2′+Fnx1-x2=εAcos⁡τεx2′′-εc2x1′-x2′-Fnx1-x2=0,where “′” denotes d/dτ and ε=m2/m1, c1=c-1/εm1ω, σ=ωn2/ω2-1, c2=c-2/εm1ω, A=δ/ε, Fnx1-x2=k2+km1x1-x2+km2x1-x23, k2=k-2/εm1ω2, km1=k-m1/εm1ω2, and km2=k-m2/εm1ω2.


## 3.2. Responses of the Rotor System with the NES

Complexification-averaging method is applied to analyze the response of the rotor-NES system.We assume that(14)v=x1+εx2w=x1-x2φ1eiτ=v˙+ivφ2eiτ=w˙+iw.Substitute (14) into (13) and extract only the eiτ components; one obtains(15)φ1′+εi+c1-iσφ1-εi+εc1-iεσφ221+ε=εA2φ2′-ε21+ε-c1φ1+iσφ1+ε21+εc2+ε2c2+εc2+c1-iσφ2-iε21+εφ1-φ2-i21+εφ2G1φ22=εA2,where G1φ22=k2+km1+3km2φ22/4.To obtain the steady state responses of the system, equate the time derivatives of (15) to zero (φ˙1=φ˙2=0) and substitute the first equation of (15) into the second equation; one obtains(16)φ10-εi-εc1+iεσφ20+1+εAi+c1-iσ=0φ20-c1+iσ+c2-1+ic1+σ+c1-iσ-1φ20G1φ202+A=0.The second equation of (16) can be changed to(17)G12φ202-2c12-σ+σ2c12+σ-12G1φ202+c1+c22+σ2+c22c12-2σ+σ2c12+σ-12φ202=A2c12+σ-12.The steady state responseφ20 can be obtained from (17), and then the response φ10 can be obtained from the first equation of (16).Saddle-node (SN) and Hopf bifurcations can happen in the responses, and the judgment criteria can be found in [10].


## 3.3. Strongly Modulated Responses of the Rotor System with the NES

The SMR behavior of the system can be investigated by combining Complexification-averaging method and multiscale method.Change (15) to single second-order ODE as follows:(18)4φ2′′+i+c2+εc1-iεσφ2′+ε1-ic1G1φ22+c1c2+ic1+c2+σ-σGφ22+iσc2φ2-2iddτφ2Gφ22=iεA.Introduce multiple scale expansion as(19)φ2=φ2τ0,τ1,…;ddτ=∂∂τ0+ε∂∂τ1+⋯;τk=εkτ,k=0,1,….Substituting (19) into (18) and setting the coefficients of powers of ε equal to zero, the following equations are obtained:(20)∂2φ2∂τ02+∂∂τ012i+c2φ2-i2φ2G1φ22=08∂2φ2∂τ0∂τ1+∂∂τ02c1φ2+2c2φ2-2iσφ2-2iφ2G1φ22+∂∂τ12c2φ2+2iφ2-2iφ2G1φ22+1-ic1G1φ22+c1c2+ic1+c2+σ-σG1φ22+iσc2φ2=iA.Integrate the first equation of (20) and obtain(21)∂φ2∂τ0+12i+c2φ2-i2φ2G1φ22=Cτ1,…,where C is arbitrary function of higher-order time scales.Whenτ0→∞, parameters in τ0 order stay invariant and φ2 reaches an asymptotic equilibrium ϕ, which is(22)i+c2ϕ-iϕG1ϕ2=Cτ1,….Assumingϕτ1=Nτ1eiθτ1, it can be obtained that(23)N2c22+G1N2-12=Cτ12or equivalently(24)Zc22+G1Z-12=Cτ12,where Zτ1=N2τ1.The magnitudeNτ1 can be obtained from (24) and the angle θτ1 can be obtained as(25)θτ1=arg⁡Cτ1-atan⁡1-G1Zc2.If the parameters of the system satisfy (14), the system may give rise to strongly modulated responses. Taking the limit τ0→∞ in the second equation of (19), one can obtain(26)∂∂τ12c2φ2+2iφ2-2iφ2G1φ22+1-ic1G1φ22+c1c2+ic1+c2+σ-σG1φ22+iσc2φ2=iA.It can be written in a more convenient form:(27)iϕ+c2ϕ-iG2ϕ2∂ϕ∂τ1-iG3ϕ2∂ϕ∗∂τ1=PP=12iA-221-ic1G1ϕ2+c1c2+ic1+c2+σ-σGϕ2+iσc2ϕ,where G2ϕ22=k2+km1+3km2φ22/2, G3ϕ22=3km2/4, and G1=G2-G3Φ2.By taking complex conjugate of (27) and assuming ϕτ1=Nτ1eiθτ1, one can obtain(28)∂N∂τ1=-N1-G12c1+c2+A1-G1cos⁡θ+Aλsin⁡θ2c22+1-G22-G32N4∂θ∂τ1=1-G2-G3N2G1-c22+1-G11-G2-G3N21-c22σ+Ac2cos⁡θ-1-G2-G3N2sin⁡θ/N2c22+1-G22-G32N4.The possibility of occurrence of the SMR behavior can be predicted by using phase plane plots of (28). The fold singularities which can provide SMR are those which satisfy numerators = 0 and denominator = 0, which are(29)c22+1-G22-G32N4=0-N1-G12c1+c2+A1-G1cos⁡θ+Aλsin⁡θ=0.We assume that the first pair of folded singularities areN1,θ1 and N1,θ2 and the second pair of folded singularities are N2,θ3 and N2,θ4. N1 and N2 can be obtained from the first equation of (29) that(30)Z1,2=N1,22=81-k2-km1km2±4km22k2+km1-12-3c229km22.θ
1, θ2, θ3, and θ4 can be obtained from the second equation of (30):(31)θ1,2=arcsin⁡-16Ac216A2-4+4k2+4km1+3km2N122+16Ac22±arccos⁡N116c2+-4+4k2+4km1+3km2N122c1+16c22c116A2-4+4k2+4km1+3km2N122+16Ac22;θ3,4=arcsin⁡-16Ac216A2-4+4k2+4km1+3km2N222+16Ac22±arccos⁡N116c2+-4+4k2+4km1+3km2N222c1+16c22c116A2-4+4k2+4km1+3km2N222+16Ac22.From (31), it can be seen that the critical amplitude of A for occurrence of the saddle-node bifurcation is (32)Fcritical1=-16Ac216-4+4k2+4km1+3km2N122+256c22Fcritical2=-16Ac216-4+4k2+4km1+3km2N222+256c22.This is a necessary condition of SMR behavior occurrence, but not a sufficient condition. The accurate existence conditions of SMR behavior can be obtained by constructing one-dimensional map [10], the procedure of which can be divided into 4 steps.(1) The first step is taking a pointθ01 in the interval θ1,θ2 and computing the fast jump from N1,θ01 to Nu,θu, in which Nu can be obtained by(33)-48km2N12+27km22N14+16k23km2N12-2=-48km2Nu2+27km22Nu4+16k23km2Nu2-2and θu can be obtained as(34)θu=θ01+atan⁡12c2km2Nu2-N1216c22+-4+4k2+4km1+3km2N12-4+4k2+4km1+3km2Nu2.(2) The second step is computing the slow branch fromNu,θu to N2,θ02, which can be solved by (28).(3) The third step is computing the fast jump fromN2,θ02 to Nd,θd, in which Nd can be obtained by(35)-48km2N22+27km22N24+16k23km2N22-2=-48km2Nu2+27km22Nu4+16k23km2Nu2-2and θd can be obtained as(36)θd=θ02-atan⁡12c2km2Nu2-N2216c22+-4+4k2+4km1+3km2N22-4+4k2+4km1+3km2Nu2.(4) The fourth step is computing the slow branch fromNd,θd to N1,θ03, which can be solved by (28).Ifθ03 can be mapped into interval [θ1,θ2] after the 4 steps, the SMR behavior is sure to happen. If θ03 cannot be mapped into the interval, the SMR behavior is sure not to happen.


## 4. Numerical Simulations and Verifications

### 4.1. Parameters Definition

The parameters of the magnets are the same as those in Table1. The rotor system is shown in Figure 6. We assume that the length of the shaft l is 370 mm and the diameter is 10 mm and the modulus of elasticity is 2.11×1011Pa. When l0=l/2, the disc is in the middle of the shaft, the gyroscopic effect can be ignored, and the numerical simulation can be carried out based on (11). The bending stiffness of the rotor is 9.87×104N/m. The other parameters of the rotor system are shown in Table 3.Table 3
Parameters for numerical simulation of the rotor system.
Parameter
Value
m
1
1 kg
c
-
1
0 Ns/m
J
d
4
×
1
0
-
4 kg⋅m2
c
-
2
3 Ns/m
m
2
0.1 kg
k
-
1
9.87
×
1
0
4 N/m


### 4.2. Stability Analysis

As the nonlinearity of the magnetic spring and the magnitude of the unbalance eccentricity have decisive effects on the dynamics of the rotor system with NES, three conditions with different nonlinearity and different exciting force are studied, which areh=5mm and δ=0.025mm, h=1.8mm and δ=0.025mm, and h=1.8mm and δ=0.05mm.The frequency responses ofφ20 in these three cases can be obtained by applying (16) and are shown in Figures 7(a), 7(b), and 7(c), respectively. It can be seen in Figure 5 that, in the case of h=5mm and δ=0.025mm, there is only a small range near resonance region that shows SN bifurcation. In the second case, there are two regions that show SN bifurcation, which are 190rad/s<ω<205rad/s and 400rad/s<ω<650rad/s, and there is also a region that can show Hopf bifurcation, which is 360rad/s<ω<650rad/s. In the case of h=1.8mm and δ=0.05mm, there are also two regions of SN bifurcation, which are 150rad/s<ω<220rad/s and ω>550rad/s, and the Hopf bifurcation region is ω>250rad/s.Figure 5
Dynamic model of the rotor-NES system when gyroscopic effect is ignored.Figure 6
Rotor system used for numerical simulation.Figure 7
Frequency response diagrams of the system in different cases.
(a)
h
=
5
m
m and δ=0.025mm
(b)
h
=
1.8
m
m and δ=0.025mm
(c)
h
=
1.8
m
m and δ=0.05mmComparing the three cases, it can be concluded that, with the increase of nonlinearity and magnitude of unbalance eccentricity, the regions of local bifurcations become broader and the possibility of SMR behavior occurring becomes larger.


### 4.3. SMR Analysis

#### 4.3.1.h=5mm and δ=0.025mm

Whenh is 5 mm, the nonlinear stiffness is 0.54 N/mm3; the critical forces under different rotating speeds are shown in Figure 8. It can be seen that Fcritical1 is greater than 0.25 mm in most frequencies, so SMR behavior cannot happen.Figure 8
Critical forces under different exciting frequency whenh=5mm.The phase portrait of the system whenω=314rad/s is shown in Figure 9, from which it can be seen that the phase trajectory flows into a fixed point, which means the system can only reach stable state. This can also be proved by the amplitude-frequency response curves of the system shown in Figure 10, which are numerically obtained by applying Runge-Kutta method. It can be seen from Figure 10 that the amplitudes of the main system and the NES are all very large at resonance region and no SMR behavior happens.Figure 9
Phase portrait of the slow motion of the system whenδ=0.025mm and h=5mm.Figure 10
Amplitude-frequency response curves of the system whenδ=0.025mm and h=5mm.


#### 4.3.2.h=1.8mm and δ=0.025mm

Whenh is 1.8 mm, the nonlinear stiffness is 4.77 N/mm3; the critical forces under different rotating speeds are shown in Figure 11. It can be seen that now A is greater than Fcritical1, so SMR behavior has the possibility of occurring. The phase portrait of the system when ω=314rad/s is shown in Figure 12. It can be seen from the phase portrait that there exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2. This provides the possibility of SMR behavior occurring.Figure 11
Critical forces under different exciting frequency whenh=1.8mm.Figure 12
Phase portrait of the slow motion of the system whenδ=0.025mm and h=1.8mm.Figure13(a) shows the slow motion return map of ω=314rad/s which starts from θ=0rad, and Figure 13(b) shows the one-dimensional map when ω=314rad/s. It can be judged from the one-dimensional map that SMR behavior can happen, as all the points starting from θ1,θ2 are mapped into the same interval.Figure 13
Slow motion return map and one-dimensional map of the system whenω=314rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapBut the SMR behavior can only be sustained in the small range near the resonance region. For example, whenω=200rad/s, the slow motion return map and one-dimensional map are shown in Figure 14. It can be seen from Figure 14(a) that the phase orbit flows to a fixed point and cannot jump to upper folder line. From Figure 14(b), it can be seen that all the points are mapped into a single point which is outside of the interval θ1,θ2; this means no SMR behavior can happen.Figure 14
Slow motion return map and one-dimensional map of the system whenω=200rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure15, from which it can be seen that the SMR behavior can only happen in a small interval near the resonance region.Figure 15
Amplitude-frequency response curves of the system whenδ=0.025mm and h=1.8mm.Figure16 shows the responses of the rotor and the NES when ω=314rad/s. It can be seen from Figures 16(a) and 16(b) that the responses are not in steady state, the amplitude of the response of the rotor system is low, and that of the NES is high, which means the energy is absorbed by the NES.Figure 16
Responses of the rotor and the NES whenω=314rad/s.
(a)
Response of the rotor system
(b)
Response of the NES


#### 4.3.3.h=1.8mm and δ=0.05mm

The phase portrait of the system whenω=314rad/s is shown in Figure 17. It can be seen that the phase portrait is similar to the former case; there also exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2.Figure 17
Phase portrait of the slow motion of the system whenr=0.5mm and h=1.8mm.Figures18(a) and 18(b) are the one-dimensional maps of the system when ω=200rad/s and ω=314rad/s, respectively. It can be seen from the maps that SMR behavior happens at both the two frequencies, which proves that the SMR region is larger than the former case.Figure 18
One-dimensional map of the system whenω=200rad/s and ω=314rad/s.
(a)
ω
=
20
0
r
a
d
/
s
(b)
ω
=
31
4
r
a
d
/
sThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure19, from which it can be seen that the SMR occurrence range is much larger than the former case.Figure 19
Amplitude-frequency response curves of the system whenδ=0.05mm and h=1.8mm.


### 4.4. Effect of Linear Stiffness

The linear stiffnessk-2 can affect the dynamics of the system, so here the effect of k-2 is analyzed. Figures 20(a) and 20(b) are the amplitude-frequency response curves of the rotor and the NES with different k-2, respectively. It can be seen from Figure 20 that the responses of the rotor become lower with the increase of linear stiffness. Although the linear stiffness becomes larger, the NES keeps its effectiveness.Figure 20
Amplitude-frequency response curves of the system with different linear stiffness.
(a)
Response of the rotor system
(b)
Response of the NES


### 4.5. Comparison with Linear Absorber

The comparison of the NES performance with linear absorber is carried out. The parameters of the linear NES are chosen to bem2=0.1kg, k-2=987N/m, and c-2=3Ns/m. The parameter of the NES is h=1.8mm. At δ=0.05mm, the results are shown in Figure 21, in which the amplitudes of the rotor system with NES, with linear absorber, and without NES or absorber are all shown. It can be seen from Figure 21 that both the NES and linear absorber can reduce the amplitude of the rotor system in the resonance region, but the system with linear absorber has two large amplitude areas besides the resonance region, while the system with NES only has one large amplitude area which is in the resonance region. In a very large frequency range, the system with NES has lower vibration amplitude than the system with absorber.Figure 21
Amplitude-frequency response curves of the system with different linear stiffness.


### 4.6. Gyroscopic Effect

Whenl0=l/3, the gyroscopic effect is large and it can affect the dynamics of the system greatly. The numerical simulation can be carried out based on (10). In this case, the stiffness parameters of the rotor system can be obtained ask-1=729/8EI/l3=1.85×105 N/m,k-rθ=k-θr=-81/4EI/l2=-1.52×104 Nm/rad,k-θθ=81/6EI/l=3.76×103 Nm2/rad.Whenh=1.8mm and δ=0.05mm, the responses of the rotor and the NES are analyzed and shown in Figure 22. It can be seen that although the gyroscopic effect changes the natural frequency of the rotor system, the NES can still reduce the amplitude of the rotor system in the resonance region.Figure 22
Amplitude-frequency response curves of the system when considering gyroscopic effect.
(a)
Response of the rotor system inx direction
(b)
Response of the rotor system iny direction


## 4.1. Parameters Definition

The parameters of the magnets are the same as those in Table1. The rotor system is shown in Figure 6. We assume that the length of the shaft l is 370 mm and the diameter is 10 mm and the modulus of elasticity is 2.11×1011Pa. When l0=l/2, the disc is in the middle of the shaft, the gyroscopic effect can be ignored, and the numerical simulation can be carried out based on (11). The bending stiffness of the rotor is 9.87×104N/m. The other parameters of the rotor system are shown in Table 3.Table 3
Parameters for numerical simulation of the rotor system.
Parameter
Value
m
1
1 kg
c
-
1
0 Ns/m
J
d
4
×
1
0
-
4 kg⋅m2
c
-
2
3 Ns/m
m
2
0.1 kg
k
-
1
9.87
×
1
0
4 N/m


## 4.2. Stability Analysis

As the nonlinearity of the magnetic spring and the magnitude of the unbalance eccentricity have decisive effects on the dynamics of the rotor system with NES, three conditions with different nonlinearity and different exciting force are studied, which areh=5mm and δ=0.025mm, h=1.8mm and δ=0.025mm, and h=1.8mm and δ=0.05mm.The frequency responses ofφ20 in these three cases can be obtained by applying (16) and are shown in Figures 7(a), 7(b), and 7(c), respectively. It can be seen in Figure 5 that, in the case of h=5mm and δ=0.025mm, there is only a small range near resonance region that shows SN bifurcation. In the second case, there are two regions that show SN bifurcation, which are 190rad/s<ω<205rad/s and 400rad/s<ω<650rad/s, and there is also a region that can show Hopf bifurcation, which is 360rad/s<ω<650rad/s. In the case of h=1.8mm and δ=0.05mm, there are also two regions of SN bifurcation, which are 150rad/s<ω<220rad/s and ω>550rad/s, and the Hopf bifurcation region is ω>250rad/s.Figure 5
Dynamic model of the rotor-NES system when gyroscopic effect is ignored.Figure 6
Rotor system used for numerical simulation.Figure 7
Frequency response diagrams of the system in different cases.
(a)
h
=
5
m
m and δ=0.025mm
(b)
h
=
1.8
m
m and δ=0.025mm
(c)
h
=
1.8
m
m and δ=0.05mmComparing the three cases, it can be concluded that, with the increase of nonlinearity and magnitude of unbalance eccentricity, the regions of local bifurcations become broader and the possibility of SMR behavior occurring becomes larger.


## 4.3. SMR Analysis

### 4.3.1.h=5mm and δ=0.025mm

Whenh is 5 mm, the nonlinear stiffness is 0.54 N/mm3; the critical forces under different rotating speeds are shown in Figure 8. It can be seen that Fcritical1 is greater than 0.25 mm in most frequencies, so SMR behavior cannot happen.Figure 8
Critical forces under different exciting frequency whenh=5mm.The phase portrait of the system whenω=314rad/s is shown in Figure 9, from which it can be seen that the phase trajectory flows into a fixed point, which means the system can only reach stable state. This can also be proved by the amplitude-frequency response curves of the system shown in Figure 10, which are numerically obtained by applying Runge-Kutta method. It can be seen from Figure 10 that the amplitudes of the main system and the NES are all very large at resonance region and no SMR behavior happens.Figure 9
Phase portrait of the slow motion of the system whenδ=0.025mm and h=5mm.Figure 10
Amplitude-frequency response curves of the system whenδ=0.025mm and h=5mm.


### 4.3.2.h=1.8mm and δ=0.025mm

Whenh is 1.8 mm, the nonlinear stiffness is 4.77 N/mm3; the critical forces under different rotating speeds are shown in Figure 11. It can be seen that now A is greater than Fcritical1, so SMR behavior has the possibility of occurring. The phase portrait of the system when ω=314rad/s is shown in Figure 12. It can be seen from the phase portrait that there exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2. This provides the possibility of SMR behavior occurring.Figure 11
Critical forces under different exciting frequency whenh=1.8mm.Figure 12
Phase portrait of the slow motion of the system whenδ=0.025mm and h=1.8mm.Figure13(a) shows the slow motion return map of ω=314rad/s which starts from θ=0rad, and Figure 13(b) shows the one-dimensional map when ω=314rad/s. It can be judged from the one-dimensional map that SMR behavior can happen, as all the points starting from θ1,θ2 are mapped into the same interval.Figure 13
Slow motion return map and one-dimensional map of the system whenω=314rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapBut the SMR behavior can only be sustained in the small range near the resonance region. For example, whenω=200rad/s, the slow motion return map and one-dimensional map are shown in Figure 14. It can be seen from Figure 14(a) that the phase orbit flows to a fixed point and cannot jump to upper folder line. From Figure 14(b), it can be seen that all the points are mapped into a single point which is outside of the interval θ1,θ2; this means no SMR behavior can happen.Figure 14
Slow motion return map and one-dimensional map of the system whenω=200rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure15, from which it can be seen that the SMR behavior can only happen in a small interval near the resonance region.Figure 15
Amplitude-frequency response curves of the system whenδ=0.025mm and h=1.8mm.Figure16 shows the responses of the rotor and the NES when ω=314rad/s. It can be seen from Figures 16(a) and 16(b) that the responses are not in steady state, the amplitude of the response of the rotor system is low, and that of the NES is high, which means the energy is absorbed by the NES.Figure 16
Responses of the rotor and the NES whenω=314rad/s.
(a)
Response of the rotor system
(b)
Response of the NES


### 4.3.3.h=1.8mm and δ=0.05mm

The phase portrait of the system whenω=314rad/s is shown in Figure 17. It can be seen that the phase portrait is similar to the former case; there also exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2.Figure 17
Phase portrait of the slow motion of the system whenr=0.5mm and h=1.8mm.Figures18(a) and 18(b) are the one-dimensional maps of the system when ω=200rad/s and ω=314rad/s, respectively. It can be seen from the maps that SMR behavior happens at both the two frequencies, which proves that the SMR region is larger than the former case.Figure 18
One-dimensional map of the system whenω=200rad/s and ω=314rad/s.
(a)
ω
=
20
0
r
a
d
/
s
(b)
ω
=
31
4
r
a
d
/
sThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure19, from which it can be seen that the SMR occurrence range is much larger than the former case.Figure 19
Amplitude-frequency response curves of the system whenδ=0.05mm and h=1.8mm.


## 4.3.1.h=5mm and δ=0.025mm

Whenh is 5 mm, the nonlinear stiffness is 0.54 N/mm3; the critical forces under different rotating speeds are shown in Figure 8. It can be seen that Fcritical1 is greater than 0.25 mm in most frequencies, so SMR behavior cannot happen.Figure 8
Critical forces under different exciting frequency whenh=5mm.The phase portrait of the system whenω=314rad/s is shown in Figure 9, from which it can be seen that the phase trajectory flows into a fixed point, which means the system can only reach stable state. This can also be proved by the amplitude-frequency response curves of the system shown in Figure 10, which are numerically obtained by applying Runge-Kutta method. It can be seen from Figure 10 that the amplitudes of the main system and the NES are all very large at resonance region and no SMR behavior happens.Figure 9
Phase portrait of the slow motion of the system whenδ=0.025mm and h=5mm.Figure 10
Amplitude-frequency response curves of the system whenδ=0.025mm and h=5mm.


## 4.3.2.h=1.8mm and δ=0.025mm

Whenh is 1.8 mm, the nonlinear stiffness is 4.77 N/mm3; the critical forces under different rotating speeds are shown in Figure 11. It can be seen that now A is greater than Fcritical1, so SMR behavior has the possibility of occurring. The phase portrait of the system when ω=314rad/s is shown in Figure 12. It can be seen from the phase portrait that there exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2. This provides the possibility of SMR behavior occurring.Figure 11
Critical forces under different exciting frequency whenh=1.8mm.Figure 12
Phase portrait of the slow motion of the system whenδ=0.025mm and h=1.8mm.Figure13(a) shows the slow motion return map of ω=314rad/s which starts from θ=0rad, and Figure 13(b) shows the one-dimensional map when ω=314rad/s. It can be judged from the one-dimensional map that SMR behavior can happen, as all the points starting from θ1,θ2 are mapped into the same interval.Figure 13
Slow motion return map and one-dimensional map of the system whenω=314rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapBut the SMR behavior can only be sustained in the small range near the resonance region. For example, whenω=200rad/s, the slow motion return map and one-dimensional map are shown in Figure 14. It can be seen from Figure 14(a) that the phase orbit flows to a fixed point and cannot jump to upper folder line. From Figure 14(b), it can be seen that all the points are mapped into a single point which is outside of the interval θ1,θ2; this means no SMR behavior can happen.Figure 14
Slow motion return map and one-dimensional map of the system whenω=200rad/s.
(a)
Slow motion return map
(b)
One-dimensional mapThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure15, from which it can be seen that the SMR behavior can only happen in a small interval near the resonance region.Figure 15
Amplitude-frequency response curves of the system whenδ=0.025mm and h=1.8mm.Figure16 shows the responses of the rotor and the NES when ω=314rad/s. It can be seen from Figures 16(a) and 16(b) that the responses are not in steady state, the amplitude of the response of the rotor system is low, and that of the NES is high, which means the energy is absorbed by the NES.Figure 16
Responses of the rotor and the NES whenω=314rad/s.
(a)
Response of the rotor system
(b)
Response of the NES


## 4.3.3.h=1.8mm and δ=0.05mm

The phase portrait of the system whenω=314rad/s is shown in Figure 17. It can be seen that the phase portrait is similar to the former case; there also exists a region between θ1 and θ2, in which all orbits come to lower folder line N1 and have the possibility of jumping to upper folder line N2.Figure 17
Phase portrait of the slow motion of the system whenr=0.5mm and h=1.8mm.Figures18(a) and 18(b) are the one-dimensional maps of the system when ω=200rad/s and ω=314rad/s, respectively. It can be seen from the maps that SMR behavior happens at both the two frequencies, which proves that the SMR region is larger than the former case.Figure 18
One-dimensional map of the system whenω=200rad/s and ω=314rad/s.
(a)
ω
=
20
0
r
a
d
/
s
(b)
ω
=
31
4
r
a
d
/
sThe amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure19, from which it can be seen that the SMR occurrence range is much larger than the former case.Figure 19
Amplitude-frequency response curves of the system whenδ=0.05mm and h=1.8mm.


## 4.4. Effect of Linear Stiffness

The linear stiffnessk-2 can affect the dynamics of the system, so here the effect of k-2 is analyzed. Figures 20(a) and 20(b) are the amplitude-frequency response curves of the rotor and the NES with different k-2, respectively. It can be seen from Figure 20 that the responses of the rotor become lower with the increase of linear stiffness. Although the linear stiffness becomes larger, the NES keeps its effectiveness.Figure 20
Amplitude-frequency response curves of the system with different linear stiffness.
(a)
Response of the rotor system
(b)
Response of the NES


## 4.5. Comparison with Linear Absorber

The comparison of the NES performance with linear absorber is carried out. The parameters of the linear NES are chosen to bem2=0.1kg, k-2=987N/m, and c-2=3Ns/m. The parameter of the NES is h=1.8mm. At δ=0.05mm, the results are shown in Figure 21, in which the amplitudes of the rotor system with NES, with linear absorber, and without NES or absorber are all shown. It can be seen from Figure 21 that both the NES and linear absorber can reduce the amplitude of the rotor system in the resonance region, but the system with linear absorber has two large amplitude areas besides the resonance region, while the system with NES only has one large amplitude area which is in the resonance region. In a very large frequency range, the system with NES has lower vibration amplitude than the system with absorber.Figure 21
Amplitude-frequency response curves of the system with different linear stiffness.


## 4.6. Gyroscopic Effect

Whenl0=l/3, the gyroscopic effect is large and it can affect the dynamics of the system greatly. The numerical simulation can be carried out based on (10). In this case, the stiffness parameters of the rotor system can be obtained ask-1=729/8EI/l3=1.85×105 N/m,k-rθ=k-θr=-81/4EI/l2=-1.52×104 Nm/rad,k-θθ=81/6EI/l=3.76×103 Nm2/rad.Whenh=1.8mm and δ=0.05mm, the responses of the rotor and the NES are analyzed and shown in Figure 22. It can be seen that although the gyroscopic effect changes the natural frequency of the rotor system, the NES can still reduce the amplitude of the rotor system in the resonance region.Figure 22
Amplitude-frequency response curves of the system when considering gyroscopic effect.
(a)
Response of the rotor system inx direction
(b)
Response of the rotor system iny direction


## 5. Conclusions

A NES consisting of negative stiffness magnetic spring and positive coil springs is proposed in this paper. The principles and fundamental characteristics of the NES are studied analytically and numerically, and some conclusions are as follows:(
1
) The proposed NES is effective, and the vibration amplitude of the rotor system can be largely declined by the NES.(
2
) The larger nonlinear stiffness and larger unbalance eccentricity broaden the SMR occurrence range in the frequency domain.(
3
) Compared with the linear absorber, the NES only has one large amplitude area which is in the resonance region and has better performance over a wider frequency range.(
4
) The NES is still effective when the gyroscopic effect is large.Further work needs to experimentally investigate the effectiveness of the NES on the rotor test rig.



---

*Source: 4132607-2017-12-14.xml*