Fixed Points on Active and Passive Dynamics of Active Hydraulic Mounts with Oscillating Coil Actuator

: Active hydraulic mounts with an inertia track, decoupler membrane, and oscillating coil actuator (AHM-IT-DM-OCAs) have been studied extensively due their compact structure and large damping in the low-frequency band. This paper focuses on a comprehensive analysis of the active and passive dynamics and their fixed points in mid-low-frequency bands, which will be helpful for parameter identification. A unified lumped parameter mechanical model with two degrees-of-free-dom is established. The inertia and damping forces of the decoupler/actuator mover may be neglected, and a nonlinear mathematical model can be obtained for mid-low-frequency bands. Theoretical analysis of active and passive dynamics for fluid-filled state reveals the amplitude dependence and a fixed point in passive dynamic stiffness in-phase or active real-frequency characteristics. The amplitude dependence of local loss at the fluid channel entrance and outlet induces the ampli-tude-dependent dynamics. The amplitude-dependent dynamics constitute a precondition for fixed points. A single fixed point in passive dynamics is experimentally validated, and a pair of fixed points in active dynamics for an AHM-IT-DM-OCA is newly revealed in an experiment, which pre-sents a new issue for further analysis.


Introduction
Fixed points are common phenomena in mechanical vibrations theory and practices. They often appear in frequency-response functions (FRFs) of damped systems and have rich physical meanings and wide applications. A pair of fixed points, P and Q in a damped dynamic absorber, has become a classic case in the research and application concerning fixed points, which have played a huge role in vibration engineering. Ormondroy and Den Hartog [1] proposed that the damping in a vibration absorber could make the absorber work better over a wider range of frequencies. They analyzed the FRFs of vibration absorbers with or without damping and pointed out the existence of optimum damping. Hahnkamm [2] referred to the intersections of P and Q on the FRFs of damped absorbers as fixed points and pointed out that the fixed points were independent of damping. The variation of fixed points with parameters was investigated and a method was proposed for obtaining the optimum natural frequency ratio by equalizing the heights of P and Q. Additionally, this work also proposed that the damping ratio was optimal when the fixed points were at the peaks, but no analytical solution was provided. Brock [3] obtained the optimum damping ratios for three cases, i.e., the optimum natural frequency ratio, a natural frequency ratio of 1, and a natural frequency of 0 (with only linear damping and no spring), by making the FRF curve of a damped absorber pass through the two fixed points horizontally, i.e., making the fixed points P and Q have peak values. Finally, Den Hartog [4] summarized research results concerning dynamic absorber fixed points in his book "Mechanical Vibrations", referring to the fixed point theorem. This theorem provides a classical optimization method, which is still widely used in dynamic absorber optimization. Regarding the literature, this research was the first whereby vibration theory paid any attention to the fixed-point phenomenon. It has not only led to fruitful theoretical results, but has also played an important role in the practice of vibrations.
As a typical damped system, the hydraulic mount also has fixed points in its dynamic characteristics. Fan et al. [5] have shown experimentally that the dynamic characteristics of a first-generation of hydraulic mount, i.e., a passive hydraulic mount with inertia track (PHM-IT), has fixed points in five different dynamics (dynamic stiffness modulus, loss angle, dynamic stiffness in-phase, dynamic stiffness out-of-phase, and viscous damping), as shown in the following expression: where k* is dynamic stiffness, a complex number, k is dynamic stiffness modulus, a positive real number, which is also called dynamic stiffness,  is loss angle, k′ is dynamic stiffness in-phase, an in-phase component of dynamic stiffness, k″ is dynamic stiffness out-ofphase, an orthogonal component of dynamic stiffness, and c is viscous damping. That study has also demonstrated an analytical solution for the fixed point in dynamic stiffness in-phase, and then a parameter identification method has been proposed based on the distinct features of a fixed point and a horizontal segment in dynamic stiffness in-phase. Based on the study of distinct features in the dynamics of PHM-IT [5], the active and passive dynamic characteristics of active hydraulic mounts with an inertia track, decoupler membrane, and oscillating coil actuator (AHM-IT-DM-OCAs) are studied in this paper. The distinct features are systematically explored for the dynamic characteristics in the fluid-filled state in terms of amplitude dependence, fixed point, resonance peak, and horizontal segment.
The discussion is divided into three sections. Section 1 introduces the typical structure and working principle, establishes a unified lumped parameter mechanical model, and explains the definition of active and passive dynamics and the meanings of high-, mid-, and low-frequency bands. In Sections 2 and 3, a nonlinear lumped parameter mathematical model with two degrees-of-freedom (DOF) is proposed. The mathematical model contains an inertia track and a mover, and the active and passive dynamics are focused on separately. Since the mover resonance frequency is far greater than the fluid channel resonance frequency, the model is simplified into a 1-DOF nonlinear model for mid-lowfrequency bands, which is used to study the amplitude dependence and fixed point of the dynamic properties, and then validated experimentally. Figure 1 shows the schematic diagram of (a) a passive hydraulic mount with inertia track and decoupler membrane (PHM-IT-DM) and (b) an AHM-IT-DM-OCA. The upper end is connected to engine, and the lower end is connected to body or chassis. In AHM-IT-DM-OCA, the coil skeleton is rigidly connected to the skeleton of the decoupler membrane to serve as an actuator mover. Alternating current (AC) power is applied to the coil. Then, the coil oscillates up and down under the alternating ampere force, which is induced in a constant magnetic field of a permanent magnet. The ampere force is the active force to be controlled. The active force is transmitted to the vehicle body through a secondary path to have the force transmitted from the engine to the vehicle body through the primary path canceled. The active force received by the actuator mover acts on the decoupler and simultaneously changes the dynamics of the hydraulic mount. The working mechanism of AHM-IT-DM-OCA can be understood either from the perspective of the actuator actively reducing the force transmitted to the vehicle body or from the perspective of the actuator actively adjusting the dynamic characteristics of the hydraulic mount.  Referring to the hydraulic mount mechanical models [5][6][7][8][9], a unified lumped parameter mechanical model for AHM-IT-DM and PHM-IT-DM is built, as shown in Figure 2. The displacement of the engine side is y1 and the corresponding reaction force, i.e., the force acting on the mount, is f1. The displacement of circularly flowing fluid in a horizontal plane relative to the wall of the fluid channel is y2, and the mover displacement is y3 (i.e., the decoupler for PHM-IT-DM). The displacement of the chassis side is y5, and the force transmitted to the chassis is f5. The pressure fluctuations in the upper and lower fluid chambers relative to the static state are p1 and p2, respectively. The active force of the actuator is fa. These are variables for the lumped parameter model. The vertical dynamic stiffness in-phase of main rubber spring is denoted as k1, the viscous damping is c1, and the equivalent mass on engine side is m1. The reciprocating motion of the main rubber spring squeezes out and pumps in the fluid for the upper fluid chamber. The equivalent pumping piston area is denoted as A1. Define bulk stiffness as a ratio of pressure change to corresponding volume change of a fluid chamber, i.e., dp/dV, in units of GN/m 5 ; denote the bulk stiffness of main rubber spring as K1 and that of rubber bellow as K2. Due to the wrinkled rubber bellow, K2 is several orders of magnitude smaller than K1. Thus, K2 is set to zero, and the pressure fluctuation in the lower fluid chamber is ignored and set p2 = 0 [5,7−9]. Denote the length of inertia track as l2, the cross-sectional area as A2, the wet perimeter of cross section as L2, and the hydraulic diameter as d2 = . Denote the mass of decoupler/mover as m3, the linear stiffness as k3, and the viscous damping as c3. Denote the equivalent piston area of the decoupler membrane as A3 and the bulk stiffness equivalent to linear stiffness k3 as K3. The equivalency relationship is as follows:

Mechanical Model of Active Hydraulic Mount (AHM)
These are parameters for the lumped parameter model. It should be noted that, for the active and passive hydraulic mounts shown in Figure  1, the basic structure is a hydraulic engine mount with inertia track and decoupler membrane, so the mechanical model can be unified. As to specific components, the model parameter values are different. Specifically, the mass m3 in the mechanical model refers to the mass of the decoupler for a PHM-IT-DM in Figure 1a and refers to the mass of the actuator mover for an AHM-IT-DM-OCA in Figure 1b. The active force fa is always zero for a PHM-IT-DM, while for an AHM-IT-DM-OCA, the fa is related to the excitation current i(t) or voltage u(t). When the actuator is powered off, an AHM-IT-DM-OCA will degenerate to a PHM-IT-DM.
The dynamic characteristics for an AHM can be divided into active and passive dynamics. The so-called passive dynamics refer to FRFs with input being the engine-side displacement and outputs being the reaction force of the engine side and the force transmitted to the chassis side when the chassis side is fixed and the actuator is powered off. These are drive-point and cross-point dynamic characteristics, respectively. The crosspoint dynamics are primary path transfer functions in the control of an AHM. The socalled active dynamics refer to FRFs with input being current or voltage applied to actuator and outputs being engine-side reaction force and the force transmitted to chassis side when the engine side and chassis side being fixed. The active dynamics from the excitation current or voltage to the force transmitted to the chassis side are secondary path transfer functions in the control of an AHM. The transfer functions of primary and secondary paths affect the performance, stability, and the operating bandwidth for an active control mount.
An AHM-IT-DM involves the resonance of two subsystems, namely the resonance of the fluid channel and the resonance of mover. Experimental studies have shown that the natural frequency fn3 of a fluid-filled mover is typically above 200 Hz, which is far greater than the natural frequency fn2 of the fluid channel (typically less than 20 Hz). Because the two natural frequencies are far apart and based on the desire to simplify the inertia and damping forces in different frequency bands based on mechanical vibrations theory, the frequency bands are divided into low-, mid-, and high-frequency bands, as shown in Figure 3. The frequency bands below the natural frequency fn3 of the mover shall also be referred to as mid-low-frequency bands and the corresponding dynamics as the mid-lowfrequency dynamics. The frequency bands above the natural frequency fn2 of the fluid channel shall be referred to as mid-high-frequency bands and the corresponding dynamic characteristics as the mid-high-frequency dynamics.

Analysis and Experimental Validation of Passive Dynamics
In this section, a nonlinear mathematical model for mid-low-frequency bands is established for passive dynamics of AHM-IT-DM-OCA, and the FRFs in fluid-filled state with the electromagnetic actuator off are studied. The engine-side displacement y1 is used as an excitation, while the engine-side reaction forces f1 and the force transmitted to chassis side f5 are used as outputs. These are the drive-point dynamic stiffness and cross-point dynamic stiffness of a mount.

Nonlinear Lumped Parameter Mathematical Model for Passive Dynamics
Based on the unified mechanical model as shown in Figure 2, set m1 = 0, y5 = 0, and fa = 0 to obtain the following nonlinear mathematical model for passive dynamics [8,9]: The equations are, in order, the Bernoulli equation for fluid flowing in inertia track, differential equation for mover motion, and the fluid continuity equation and equilibrium equation for engine-side reaction forces f1 and transmitted force to chassis side f5, respectively. For the fluid flowing in the inertia track, a nonlinear Bernoulli equation is used to account for the loss factor of fluid flowing along the fluid channel and local loss at the entrance and outlet [10].

Analysis of Mid-Low-Frequency Passive Dynamics and the Amplitude Dependence and Fixed Points (f << fn3)
As stated above, the natural frequency fn2 of the inertia track is much smaller than fn3 of the mover, that is, fn2 << fn3. Considering mechanical vibrations theory [11], for an excitation frequency f << fn3 in mid-low-frequency bands, the decoupler displacement y3 is essentially in-phase with excitation. The excitation force is mainly balanced by elastic restoring force. Under this condition, the inertia and damping forces of the decoupler [6−8,11] may be neglected. Simplification of Equation (3) leads to the following mathematical model:

High-
where f5 = f1 indicates that, in mid-low-frequency bands, drive-point passive dynamics are the same as cross-point ones, so there is no need to treat them differently. Eliminating the intermediate variables p1 and y3, it can be rewritten as follows: where Ku1 is bulk stiffness of upper fluid chamber surrounded by main rubber spring and decoupler membrane. It consists of K1 and K3 in-series [12], as follows: The in-series relationship makes the total bulk stiffness smaller than the smaller one of two bulk stiffnesses, that is, Ku1 < min (K1, K3). Equation (5) also shows that the various DOFs in AHM-IT-DM are decoupled from each other.
The multiple-scales perturbation analysis of second-order nonlinear damped vibration system has shown that [13], in terms of first-order approximation, the natural frequency is not affected by damping. In this case, the natural frequency of inertia track is as follows: Experiments and numerical simulations have shown that [5], under excitation of harmonic displacement y1, the responses of upper chamber pressure p1, inertia fluid displacement y2, decoupler/mover displacement y3, and force transmitted to the chassis f5 are all simple harmonics with the same frequency. Therefore, the following expressions can be assumed for the excitation and response:  (9) Substituting Equation (8) into Equation (9) and the second formula of Equation (5), a set of algebraic equations are obtained. Solving them, the FRF of H2 for fluid channel and the dynamic stiffness k * for a hydraulic mount can be obtained: Equation (10) shows the FRF, H2, contains response amplitude Y2 of inertia fluid, an unknown variable. Y2 and H2 may be calculated using a search method. For a given excitation amplitude Y1, H2 is calculated while Y2 gradually increases from 0. When is the response amplitude of inertia fluid. Then, H2 and k * can be determined. Numerical simulations have shown the searched solutions have good properties, and each excitation amplitude has a unique response [5].
Equation (11) shows the mid-low-frequency passive dynamics are dependent on the response Y2, which is uniquely determined by the excitation Y1. Therefore, the mid-lowfrequency passive dynamics of AHM-IT-DM are not only frequency-variant, but also amplitude-variant, i.e., there are different frequency-variant dynamic characteristics curves for different excitation amplitudes. The fundamental cause for amplitude-variant dynamics lies in the amplitude-dependent damping caused by local loss at the entrance and outlet. The amplitude-variant dynamic characteristics is a prerequisite for the existence of fixed points. In other words, for a lumped parameter model that only considers linear damping of the fluid, the dynamic characteristics are independent of the excitation amplitude, and it is impossible to express amplitude-variant dynamics. Thus, the linear model for AHM-IT-DM could not reveal the phenomenon concerning fixed points.
Based on Equation (11), the passive dynamics in mid-low-frequency bands can be analyzed, and a fixed point will be shown in passive dynamics. The following three situations are discussed: (1) Low-frequency band when where * 0 k ,  0 k , and  0 k are dynamic stiffness, dynamic stiffness in-phase, and dynamic stiffness out-of-phase when  → 0 , respectively. This result shows that, when the excitation frequency approaches to zero, the dynamics are that of the main rubber spring.
(2) Mid-frequency band when where  * k ,   k , and   k are dynamic stiffness, dynamic stiffness in-phase, and dynamic stiffness out-of-phase when  →, respectively. The result shows, when the fluid channel becomes almost turned off due to increasing fluid resistance with frequency, the passive dynamic stiffness in-phase in the mid-frequency band becomes constant,   (15) Note that Y1 and Y2 in Equation (14) are always positive, so H2,1 is a pure imaginary number, and the imaginary part is negative. The loss angle is therefore  = 2 2 , indicating that the inertia fluid has a phase resonance when  = 1 . By equating the real and imaginary parts on both sides, the following can be obtained: Substituting Y2 back into Equation (15), the dynamics can be obtained:  k k A K . This is the fixed point, referred to as R, on passive dynamic stiffness in-phase.

Experimental Validation of Amplitude Dependence and Fixed Point for Mid-Low-Frequency Passive Dynamics
The analysis shows that the distinct feature of the passive dynamics of AHM-IT-DM-OCAs is that a fixed point appears in mid-low-frequency bands due to amplitude dependence. To verify the amplitude dependence and fixed point, the following experiments are designed and carried out.
Two types of hydraulic mount, as shown in Figure 1, are tested. Figure 1a is PHM-IT-DM and Figure 1b Figure 1a. When excitation amplitude Y1 is 0.4, 0.6, 0.8, and 1.0 mm (curves 5-8, respectively), the dynamics, i.e., the dynamic stiffness k, loss angle  , dynamic stiffness in-phase k′, dynamic stiffness out-of-phase k", and viscous damping c, exhibit their fixed points P2, Q2, R2, S2, and T2, respectively. The fixed points R2 and the amplitude dependence in mid-low-frequency dynamics are clearly demonstrated as theoretically predicted in the analysis above. Curves 1-4 are the dynamics of a first-generation hydraulic mount, i.e., a PHM-IT, and the corresponding fixed points P1, Q1, R1, S1, and T1, have been reported in [5]. It is noted that, due to differences in the construction and rubber materials between the previously tested PHM-IT and newly tested PHM-IT-DM, the parameter values are different, as are the fixed points. Figure 5a−e show the experimental mid-low-frequency passive dynamics and fixed points of an AHM-IT-DM-OCA shown in Figure 1b. When the excitation amplitude Y1 is 0.4, 0.6, 0.8, and 1.0 mm (curves 1-4, respectively), the passive dynamics also exhibit fixed points P3, Q3, R3, S3, and T3, respectively, and amplitude dependence, as predicted in the analysis above.
Note that, in five different dynamics curves, as shown in Figures 4 and 5, there always exists a peak in each curve, which is the resonance peak induced by fluid channel resonance.    force f5 Experimental studies have shown that there exists a single fixed point on each of the five dynamics curves in mid-low-frequency bands. Equation (18) just gives the theoretical solution of the fixed point, R, in dynamic stiffness in-phase. The solutions of the other four fixed points involve a challenging mathematical problem, i.e., solvable in terms of radicals of fifth order and higher algebraic equations. Abel rigorously proved that general algebraic equations higher than fourth order could not have general-form radical solutions, named the Abel-Ruffini theorem [14,15]. To determine whether the solutions of the other four fixed points belong to special equations with solutions in terms of radicals, it is necessary to study their solvability criteria based on the group theory, and this remains a topic for further investigation.

Analysis and Experimental Validation of Active Dynamics
In this section, the FRFs of a fluid-filled AHM-IT-DM-OCA with both the engine side and chassis side fixed and with the electromagnetic actuator operating are studied. The current i serves as an input excitation, and the reaction force f1 and the force transmitted f5 to the chassis side are outputs.

Nonlinear Lumped Parameter Zodel for Active Dynamics
Based on the AHM-IT-DM mechanical model in Figure 2, setting y1 = 0 and y5 = 0, the relationship between the actuator active force and excitation current is considered. With the oscillating coil as actuator, the following relationship exists: where B is magnetic induction intensity, l is length of coil, and kM is the voice coil constant, or so-called electromechanical coupling coefficient. The nonlinear mathematical model for active dynamics of AHM-IT-DM-OCAs is as follows:

Analysis of Mid-Low-Frequency Active Dynamics and the Amplitude Dependence and Fixed Points (f << fn3)
Using the same simplification as in passive dynamics (Section 3.2), we neglect the inertia and damping forces of the actuator mover in mid-low-frequency bands, f << fn3. The following nonlinear mathematical model is obtained: where C, D, and λ are the same as those in Equations (10) and (11). Equations (23) and (24) show that the mid-low-frequency active dynamics F1/I and F5/I are also frequency-and amplitude-dependent. If the amplitude I of excitation current is different, the response amplitude Y2 will also be different. Then, the FRF curves are different. The amplitude dependence of the active dynamics may also imply the existence of a fixed point. We discuss the following three cases.
(1) Low-frequency band when  → 0 (f << fn2): (2) Mid-frequency band when  → (fn2 << f << fn3): This result means the following. First, f5 and fa have opposite directions in the midfrequency band, or f5 lags fa by a phase angle of π, and the two are synchronously reversed. Second, the active FRFs of the mid-frequency band have only a real part, while the imaginary part is zero, which indicates the amplitude-frequency characteristics are equal to the real-frequency characteristics. Third, the active dynamics in the mid-frequency band tend to a constant, Fi,,4, an identical horizontal segment in the curves.
(3) Resonance frequency band where  → 1 ( n2 → ff ): Using the same analysis method as in Section 3.2, we obtain the following: where aM = 2 3 3 E k A A K . This result shows, when λ = 1, i.e., the excitation frequency is equal to the fluid channel resonance frequency, the imaginary part still exhibits amplitude dependence, but the real part is a constant, which is independent of excitation amplitude. This means that, under different excitation amplitudes, the real-frequency characteristic curve family all pass through the point where the frequency is fn2 and the real part is M 1 u 1 3 3 A K k A K , which is another fixed point on active real-frequency characteristics.

Experimental Validation of Amplitude Dependence and Fixed Point for Mid-Low-Frequency Active Dynamics
The analysis shows that the distinct features of active dynamics of AHM-IT-DM-OCAs are amplitude dependence and a fixed point in mid-low-frequency bands. Figure 6 shows the test rig for active dynamics of an AHM-IT-DM-OCA. The upper and lower ends of the mount are fixed, and the actuator is excited with a burst random current i. The forces f1 and f5 at upper and lower ends are tracked by two force sensors of PCB 208C02 and the current is tracked by a current clamp of CHAUVIN ARNOUX E3N. The FRF curves of f1 and f5 with current i as input are obtained, as shown in Figure 7. Figure 7 shows the tested active dynamics in mid-low-frequency bands, which clearly reflect the amplitude dependence and fixed points. Figure 7a also shows that the mid-band amplitude-frequency characteristics and their real part tend to be the same constant, i.e., the curves converge to an identical horizontal segment. The experimental phenomena, such as amplitude dependence, resonance peak, and horizontal segment, agree well with theoretical predictions in mid-frequency band. However, Figure 7 also shows the paired fixed points on each characteristic curve of amplitude, phase, and real and imaginary parts, such as P5-P5-2, Q5-Q5-2, R5-R5-2, and S5-S5-2, respectively. This is the first time that the paired fixed points have been demonstrated in experimental results. The analytical results regarding passive dynamics of AHM-IT-DM-OCA agree well with experiment, while the active dynamics also agree with the experiment, except the fixed point. The fundamental difference in active and passive dynamics is that the electromagnetic actuator is powered on or powered off. The mechanism for paired fixed points in active dynamics should be investigated in the future, considering the actuator structure and dynamic characteristics of electromagnetic induction.

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The fundamental reason for amplitude-dependent dynamics is the amplitude dependence of local loss at the entrance and outlet. Amplitude-dependent dynamics represent a precondition for the existence of a fixed point.

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Since the inertia of the decoupler/mover may be neglected in mid-low-frequency bands, the drive point and cross point dynamics are identical.

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A single fixed point in passive dynamics for an AHM-IT-DM-OCA is revealed in the analysis and experiment. In the meantime, a pair of fixed points in active dynamics is newly revealed in the experiment. This paired appearance of fixed points is a new issue, and its mechanism should be investigated considering the dynamic characteristics of electromagnetic induction.

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