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.
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.
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 point in plane 2 (as shown in Figure 1) can be calculated aswhere is the residual flux density.
The point charge of a point in plane 3 is
So the interaction force between points and iswhere is the relative permeability of the permanent magnets.
When the central magnet moves in the radial direction for a distance , the interaction force between planes 2 and 3 can be calculated as
The interaction force between plane 2 and plane can be written aswhere , , , and .
And also, the interaction force between plane 1 and plane can be written as
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
Equations (4)–(7) can be solved numerically and the relationship between and different can be obtained. Then the radial stiffness of the negative stiffness magnetic spring can be obtained by
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.
As shown in [6], the nonlinearity of the magnet spring can be ignored when distance is large. But when 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
(a) mm
(b) mm
The approximated stiffness coefficients of different magnet distances are shown in Table 2. It can be seen from Table 2 that the polynomial fitting effect is rather good and is rather large when is small.
2.3. Structure of the NES
Assembling the magnet spring and coil springs together, the NES is formed, whose structure is shown in Figure 3. 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.
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, when and , when choosing the coil spring with total stiffness , the above linear stiffness of the NES is , while its nonlinear stiffness is . 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 Figure 4, and the dynamic equations can be written aswhere “” denotes , , , and are mass, diameter moment of inertia, and polar moment of inertia of the rotor, respectively, is the damping of the rotor, , , , and are stiffness of bending and rotating of the rotor, respectively, , , and are mass, damping, and stiffness of the NES, respectively, is the unbalance eccentricity of the rotor, is the rotating speed of the rotor system, and is the whirling speed of the rotor 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 Figure 4 and the dynamic equations can be written aswhere .
We assume that
Equations (11) can be simplified towhere “′” denotes and , , , , , , , , and .
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
Substitute (14) into (13) and extract only the components; one obtainswhere .
To obtain the steady state responses of the system, equate the time derivatives of (15) to zero () and substitute the first equation of (15) into the second equation; one obtains
The second equation of (16) can be changed to
The steady state response can be obtained from (17), and then the response 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:
Introduce multiple scale expansion as
Substituting (19) into (18) and setting the coefficients of powers of equal to zero, the following equations are obtained:
Integrate the first equation of (20) and obtainwhere is arbitrary function of higher-order time scales.
When , parameters in order stay invariant and reaches an asymptotic equilibrium , which is
Assuming , it can be obtained thator equivalentlywhere .
The magnitude can be obtained from (24) and the angle can be obtained as
If the parameters of the system satisfy (14), the system may give rise to strongly modulated responses. Taking the limit in the second equation of (19), one can obtain
It can be written in a more convenient form:where , , and .
By taking complex conjugate of (27) and assuming , one can obtain
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
We assume that the first pair of folded singularities are and and the second pair of folded singularities are and . and can be obtained from the first equation of (29) that
, , , and can be obtained from the second equation of (30):
From (31), it can be seen that the critical amplitude of for occurrence of the saddle-node bifurcation is
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 point in the interval and computing the fast jump from to , in which can be obtained byand can be obtained as
(2) The second step is computing the slow branch from to , which can be solved by (28).
(3) The third step is computing the fast jump from to , in which can be obtained byand can be obtained as
(4) The fourth step is computing the slow branch from to , which can be solved by (28).
If can be mapped into interval after the 4 steps, the SMR behavior is sure to happen. If 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 Table 1. The rotor system is shown in Figure 6. We assume that the length of the shaft is 370 mm and the diameter is 10 mm and the modulus of elasticity is . When , 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 . The other parameters of the rotor system are shown in Table 3.
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 are and , and , and and .
The frequency responses of 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 and , 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 and , and there is also a region that can show Hopf bifurcation, which is . In the case of and , there are also two regions of SN bifurcation, which are and , and the Hopf bifurcation region is .
(a) and
(b) and
(c) and
Comparing 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. and
When 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 is greater than 0.25 mm in most frequencies, so SMR behavior cannot happen.
The phase portrait of the system when 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.
4.3.2. and
When 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 is greater than , so SMR behavior has the possibility of occurring. The phase portrait of the system when is shown in Figure 12. It can be seen from the phase portrait that there exists a region between and , in which all orbits come to lower folder line and have the possibility of jumping to upper folder line . This provides the possibility of SMR behavior occurring.
Figure 13(a) shows the slow motion return map of which starts from , and Figure 13(b) shows the one-dimensional map when . It can be judged from the one-dimensional map that SMR behavior can happen, as all the points starting from are mapped into the same interval.
(a) Slow motion return map
(b) One-dimensional map
But the SMR behavior can only be sustained in the small range near the resonance region. For example, when , 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 ; this means no SMR behavior can happen.
(a) Slow motion return map
(b) One-dimensional map
The amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure 15, from which it can be seen that the SMR behavior can only happen in a small interval near the resonance region.
Figure 16 shows the responses of the rotor and the NES when . 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.
(a) Response of the rotor system
(b) Response of the NES
4.3.3. and
The phase portrait of the system when 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 and , in which all orbits come to lower folder line and have the possibility of jumping to upper folder line .
Figures 18(a) and 18(b) are the one-dimensional maps of the system when and , 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.
(a)
(b)
The amplitude-frequency response curves numerically obtained by applying Runge-Kutta method are shown in Figure 19, from which it can be seen that the SMR occurrence range is much larger than the former case.
4.4. Effect of Linear Stiffness
The linear stiffness can affect the dynamics of the system, so here the effect of is analyzed. Figures 20(a) and 20(b) are the amplitude-frequency response curves of the rotor and the NES with different , 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.
(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 be , , and . The parameter of the NES is . At , 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.
4.6. Gyroscopic Effect
When , 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 as N/m, Nm/rad, Nm2/rad.
When and , 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.
(a) Response of the rotor system in direction
(b) Response of the rotor system in 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:
The proposed NES is effective, and the vibration amplitude of the rotor system can be largely declined by the NES.
The larger nonlinear stiffness and larger unbalance eccentricity broaden the SMR occurrence range in the frequency domain.
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.
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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to gratefully acknowledge the National Natural Science Foundation of China (Grant no. 51475085) for the financial support for this study.