Electromagnetic Field and Variable Inertia Analysis of a Dual Mass Flywheel Based on Electromagnetic Control

Abstract

The moment of inertia of the primary flywheel and the secondary flywheel in a dual mass flywheel (DMF) directly affects the vibration damping performance in an automotive driveline. To enable better minimization of vibration and noise by changing the moment of inertia of the DMF to adjust the frequency characteristics of the automotive driveline, a new variable inertia DMF structure is proposed by introducing electromagnetic devices. The finite element simulation model of the electromagnetic field of an electromagnetic device is established, the electromagnetic field characteristics in the structure are analyzed, and the variation in the electromagnetic force under different air gaps and current conditions is obtained. The electromagnetic force test system of the electromagnetic device is constructed, and the validity of the finite element simulation analysis of the electromagnetic field of the electromagnetic device is verified. A mechanical model of the electromagnetic device is established to analyze the characteristics of the displacement of the moving mass in the structure as well as the variation in the moment of inertia of the DMF at different rotational speeds and currents. The maximum adjustable proportion of its moment of inertia can reach 15.07%. A torsional model of the automotive driveline is established to analyze the effect of variable inertia DMF on the resonance frequency of the system under different currents. The results show that the electromagnetic device introduced in the DMF can realize the active adjustment of the moment of inertia and enable the resonance frequency to decrease with increasing rotational speed, which expands the idea of optimizing the vibration damping performance of the DMF and provides a reference for better control of the torsional vibration of the automobile or other mechanical transmission systems.

1. Introduction

A dual mass flywheel (DMF) is a kind of torsional vibration damping device for automotive drivetrains first developed by the LuK company in Germany which has better vibration damping performance compared to the traditional follower disk torsional vibration damper [1,2,3]. With the continued research and resulting improvement in technology, there are now more and more high-performance DMF products, and more and more automobiles are equipped with DMF [4,5]. The research on DMF has mainly focused on the areas of vibration damping performance analysis, innovative structure design with multiple levels of stiffness, nonlinear vibration and other nonperimeter spring structures—for example, Schaper [6] developed an analytical model of the torsional vibration of a revolving long coil spring DMS and studied the different torque characteristics observed when increasing or decreasing the torsion angle. Gharehbolagh [7] investigated the nonlinear vibration of three-cylinder engines equipped with DMF. A nonlinear dynamic model of a DMF is presented in [8] and verified by test bench data. Quattromani [9] established a nonlinear three-dimensional dynamics analytical model of a circumferential long arc spring DMF considering centrifugal force, contact, friction, etc., and analyzed the effect of its main parameters on the vibration damping performance of the transmission system. Zu [10] verified that the semi-active control of the torsional vibration of an automotive driveline equipped with magnetorheological fluid DMF can effectively mitigate the torsional vibration of the system, especially the vibration response in start–stop conditions, through joint simulation and experimental analysis. A dynamic parameter adjustment and design method for a multi-stage torsion-stiffened DMF is proposed by Wang [11] and analyzes the vibration damping performance of a three-stage torsion-stiffened DMF. Chen [12] designed a magnetorheological fluid DMF with adjustable damping and optimized the damping effect by providing appropriate input currents according to the damping requirements under different operating conditions.

The vibration damping performance of a DMF in the automotive driveline is not only influenced by the stiffness and damping of the elastic structure, but also directly related to the moment of inertia of the primary flywheel and the secondary flywheel [13], and the vibration state of the automotive driveline is not the same at different excitation frequencies corresponding to different engine speeds. In general, the main parameters of DMF products designed for an automotive driveline cannot be adjusted, and it is difficult to adjust the stiffness and damping parameters to better suppress torsional vibration, and it is also difficult to adapt them to other drivetrain systems. If variable inertia technology can be introduced into a DMF according to the working conditions of the engine, so that the inertia moment of the DMF can be adjusted and the theoretical research on its vibration suppression design can be carried out in depth, it will have the possibility of suppressing torsional vibration in the further expand automotive transmission system. Variable inertia technology is widely used in the mechanical field, mainly involving flywheel structures. Thus, Xu [14] developed a two-terminal mass-based vibration absorber. Yang [15] proposed a flywheel capable of adaptively generating the variable moment of inertia and tested its performance by mounting the variable inertia flywheel on a double-ended hydraulic device. Dong [16] investigated the semi-active regulation of the moment of inertia of a magnetorheological variable inertia flywheel. A new variable inertia flywheel that can slow down the speed decay while having the advantages of high efficiency and low starting torque is proposed by Aneke [17]. Kondoh [18] applied an active variable inertia structure to a fixed speed flywheel energy storage system. Kushwaha [19] proposed a variable inertia flywheel for stabilizing the speed of a hydraulic motor. A flywheel with symmetrically placed mass spring dampers is presented by Li [20]. Zhang [21] designed a variable inertia flywheel that can provide a diesel generator with better resistance to load disturbances. Zu et al. [22] validated the torsional vibration of automotive transmission systems equipped with magnetorheological fluid DMF through joint simulation and experimental analysis. Semi-active control can effectively alleviate system torsional vibration, especially the vibration response during start–stop conditions.

In order to enable the adjustment of the natural frequency characteristics of the system by actively changing the moment of inertia of the DMF to reduce the system vibrations and noise when the vehicle is in different operating conditions, a new variable inertia DMF structure based on electromagnetic control is proposed. The electromagnetic field and variable inertial characteristics of the electromagnetic device in DMF are studied through theoretical analysis, finite element simulation and experiments. A torsional model of the automotive driveline is constructed and the effect of variable inertia on the resonant frequency of the system at different currents is analyzed.

2. Electromagnetic Analysis of the Devices

2.1. The Structure of the Variable Inertia DMF

As shown in Figure 1a, the DMF mainly consists of a primary flywheel, an arc spring and a secondary flywheel. To improve this structure, an electromagnetic device is incorporated into the primary flywheel (region W in Figure 1a), the main structure of which consists of two permanent magnets, springs and coils, as shown in Figure 1b. The coil is wound on permanent magnet 1, which is fixed in the radial groove of the primary flywheel, and permanent magnet 2 (moving mass) is connected to the other end of the radial groove of the primary flywheel by a spring. The radial position of permanent magnet 2 in the primary flywheel under the action of the centrifugal force, the spring force and the electromagnetic force when the DMF is operated at different speeds is adjusted by changing the current in the coil to change the moment of inertia.

2.2. Analysis of Electromagnetic Forces

The square of the magnetic induction in the air gap and the total area of the magnetic lines of force that pass through the poles determine the value of the electromagnetic force. Assuming that the magnetic induction is uniformly distributed along the surface of the poles and that the medium between the two permanent magnets is air, the magnitude of the electromagnetic force is [22]:

$$F_{\mathrm{e}}={\displaystyle \frac{B^{2}S}{2{\mu }_0}}$$

(1)

where B is the magnetic induction, S is the cross-sectional area of the pole, and µ₀ is the air permeability.

The simplified structure of the electromagnetic device is shown in Figure 2. The electromagnetic field generated by the magnetic field and current of the permanent magnet exist in the device, and the electromagnetic force is analyzed using the magnetic circuit theory, because the input current to the coil is direct current, the eddy current loss and the core itself are small, the leakage flux of the coil and the permanent magnet is neglected and it is assumed that the magnetic potential lands evenly on the air gap and the permanent magnet. The coil dissipates the heat well.

The turns and current of coil are N and I, respectively. The air gap is δ, the cross-sectional areas of the permanent magnets 1 and 2 are S₁ and S₂, the heights of the permanent magnets 1 and 2 are h₁ and h₂, respectively, the magnetic induction of the permanent magnets is B_r, the coercive force is H_c, and the relative permeability of the permanent magnets is µ_r.

According to Kirchhoff’s second law, the algebraic sum of the magnetic pressures of the segments in any closed magnetic circuit is equal to the algebraic sum of the magnetic potentials. The algebraic sum of the magnetic pressures of the segments is constantly equal to zero when returning to a point that is a full circle around the circle, and one can obtain the following equation [23]:

$$\{\begin{array}{c}{\displaystyle \frac{B_1}{{\mu }_r{\mu }_0}}{h}_1+2{\displaystyle \frac{B_1}{{\mu }_0}}{\displaystyle \frac{\delta }{2}}+2{\displaystyle \frac{B_1}{{\mu }_0}}{\displaystyle \frac{d_1}{2}}+{\displaystyle \frac{B_1}{{\mu }_0}}\left(2\delta +h_1\right)=IN+F_{\Phi 1}\\ {\displaystyle \frac{B_2}{{\mu }_{\mathrm{r}}{\mu }_0}}{h}_2+2{\displaystyle \frac{B_2}{{\mu }_0}}{\displaystyle \frac{\delta }{2}}+2{\displaystyle \frac{B_2}{{\mu }_0}}{\displaystyle \frac{d_2}{2}}+{\displaystyle \frac{B_2}{{\mu }_0}}\left(2\delta +h_2\right)=F_{\Phi 2}\end{array}$$

(2)

where F_Φ1 and F_Φ2 are the magnetic kinetic potentials of permanent magnets 1 and 2, respectively, and B₁ and B₂ are the magnetic field induction strengths of permanent magnets 1 and 2, respectively.

The equations for B₁ and B₂ are obtained from Equation (3):

$$\{\begin{array}{l}{B}_1={\mu }_0{\displaystyle \frac{NI+F_{\Phi 1}}{\frac{h_1}{{\mu }_r}+3\delta +d_1+h_1}}\hfill \\ B_2={\mu }_0{\displaystyle \frac{F_{\Phi 2}}{\frac{h_2}{{\mu }_r}+3\delta +d_2+h_2}}\hfill \end{array}$$

(3)

The magnetic potentials F_Φ1 and F_Φ2 are calculated according to the following equation:

$$\{\begin{array}{l}{F}_{\Phi 1}=H_c{h}_1\\ F_{\Phi 2}=H_c{h}_2\end{array}$$

(4)

The relative permeability µ_r of a permanent magnet is calculated as

$${\mu }_r={\displaystyle \frac{B_r}{H_c{\mu }_0}}$$

(5)

From the magnetic circuit model and according to Equation (1), the electromagnetic force Fe between the two permanent magnets is obtained as:

$$F_{\mathrm{e}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{B_1}^2{S}_1}{{\mu }_0}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{B_2}^2{S}_2}{{\mu }_0}}$$

(6)

2.3. Electromagnetic Field Simulation Analysis

The electromagnetic field and the electromagnetic force of the electromagnetic device are analyzed using finite element simulation software COMSOL5.5, and a two-dimensional axisymmetric model for magnetic field analysis of the electromagnetic device is established [24]. As shown in Figure 3, the rectangular blocks on the left side above and below in the figure represent two permanent magnets, respectively, while the rectangular block on the right side of permanent magnet 1 is the coil area, and the two L-shaped areas are the sleeves, and the permanent magnets are all N48 powerful neodymium-iron-boron magnets, and the material of the coil is copper, the material of the sleeve is 304 stainless steel, and the other parts are air domains. In addition, regions I–VI and the direction of the arrow in the figure are the set paths for extracting the results of the magnetic induction strength simulation.

The permanent magnet selects the remanent magnetism model under the Ampere law option and sets the parameters such as coil current and number of turns. Triangular cells are used to divide the mesh, as shown in Figure 4, and the mesh is encrypted using the demarcation layer option for the structural connection between the coil and permanent magnet 1, with a maximum grid cell of 5 mm and a minimum grid cell of 0.02 mm.

Permanent magnet 2 is a movable part with a maximum displacement of 20 mm and its air gap δ₀ = 20 mm relative to permanent magnet 1 in the starting position. The dimensional parameters in the simulation model are listed in

Table 1. The dimensions in the model.

Parameter	Value	Parameter	Value
Permanent magnet 1 diameter d1 (mm)	30	Permanent magnet 2 diameter d2 (mm)	40
Permanent magnet 1 heights h1 (mm)	40	Permanent magnet 1 heights h2 (mm)	20
Sleeve1 outside diameter d3 (mm)	24	Sleeve2 outside diameter d4 (mm)	50
Sleeve1 heights h3 (mm)	24	Sleeve2 heights h4 (mm)	44

. The characteristic parameters of the simulation model are listed in

Table 2. The characteristic parameters of the model.

Parameter	Value	Parameter	Value
Air permeability µ0 (H/m)	1.25 × 10–5	Permanent magnet remanence Br (T)	1.43
Permeability of stainless steel	1.09	Coercivity of Permanent magnets Hc (kA/m)	10,430

.

Solving two-dimensional finite element simulations of steady-state electromagnetic fields of electromagnetic devices at different currents. The distribution of magnetic lines of force on the structure of the electromagnetic device with different air gap δ and the cloud diagram of magnetic induction intensity at a coil current of 4 A are shown in Figure 5. The lines of magnetic force are in a closed loop from the N-pole of the permanent magnet, through the air and back to the S-pole of the permanent magnet.

As shown in Figure 6, the magnetic induction intensity varies with the paths I-VI marked in Figure 3. It can be seen that the magnetic induction intensity reaches its peak at the location where the structural shape changes, and gradually increases in the direction of the arrow in regions I and IV, and gradually decreases in regions III and VI. In the area of II and V, that is, on the side of the permanent magnets 1 and 2, the magnetic induction shows the properties of the two ends of the large one in the middle of the small one, since the magnetic lines of force of the permanent magnets are converged from the two poles, which leads to this that the magnetic induction intensity increases at both ends of the permanent magnets. Since the magnetic field of the permanent magnet at the intersection of paths IV and V and the magnetic field generated by the coil overlap most strongly, the magnetic induction strength of permanent magnet 2 is also greatest at this position. The trend in the magnetic induction intensity of the two permanent magnets with the direction of the path arrows under different air gaps δ is close to the trend, but there is a numerical difference, and the maximum value of the magnetic induction intensity of the permanent magnets increases as the air gap δ decreases from 20 mm to 1 mm from 1.67 T to 1.78 T. The magnetic induction intensity of the two permanent magnets with different air gap δ is close to the trend in the path arrow direction, but there is a numerical difference.

Figure 7 shows the variation in the electromagnetic force with the air gap δ for different currents. As the input current increases, the electromagnetic force becomes larger, and the curve trend is more pronounced in the air gap δ is located in the range of 1–10 mm, this is due to the air has a large magnetoresistance, when the magnetic line of force in the air through the attenuation of the large, so the larger the air gap, the electromagnetic force due to the large magnetoresistance, the large loss and become smaller.

2.4. The Electromagnetic Force Test

Figure 8 shows the constructed electromagnetic force test device. Two permanent magnets are fixed to the base by means of brackets. The air gap δ is varied by adjusting the distance between the brackets. Coils are wrapped around the permanent magnets 1, the input current to the coils can be varied by an adjustable DC power, and the electromagnetic force between the two permanent magnets can be fed to a computer using a force sensor, a digital transmitter, and a USB-485 serial cable.

Figure 9 shows the comparison between the test results and the simulation results of the electromagnetic force acting on permanent magnet 2 with different air gaps at 0 A and 4 A coil current. As can be seen in the figure, the variation in the electromagnetic force with air gap obtained from the test is close to the simulation results. The maximum relative error between the two is 3.57%. The validity of the results of the analysis of electromagnetic fields is verified.

3. Analysis of the Moment of Inertia

3.1. Analysis of Forces

When the DMF works, permanent magnet 2 and the sleeve can slide along the radial slot of the primary flywheel. The mechanical analysis of permanent magnets 2 and sleeves in a single electromagnetic device is investigated. As shown in Figure 10, point O is the rotational center. In the mechanical analysis, permanent magnet 2 and the sleeve are considered as a whole (a moving part), and the mass is m. The stiffness and initial length of the spring are k and l₀. The distance from the spring attached to the primary flywheel to point O is r₀. The distance from point O to the center of mass of the moving part is r. ω is the DMF rotational angular velocity.

The centrifugal force F_c of the moving mass is:

$$F_{\mathrm{c}}=m{\omega }^{2}r$$

(7)

When the spring deformation is Δl, the force Fs on the moving mass is:

$$F_{\mathrm{s}}=k\Delta l$$

(8)

The air gap δ can be expressed by the following equation:

$$\delta ={\delta }_0-\Delta l$$

(9)

The length l of the spring after deflecton under forces is:

$$l=l_0+\Delta l$$

(10)

The distance r is:

$$r=r_0+l+{\displaystyle \frac{h_2}{2}}$$

(11)

F_m and F_s are simplified in the computational model, the details of the simplification process can be found in the literature [25].

When the variable inertia dual mass flywheel works stably, the device is in equilibrium and the following equation can be obtained:

$$F_{\mathrm{c}}-F_{\mathrm{s}}-F_{\mathrm{e}}=0$$

(12)

When the current in the coil is stable, the relationship between the electromagnetic force and the air gap conforms to the following equation [26]:

$$F_{\mathrm{e} }=C_1{\left({\displaystyle \frac{1}{C_2+\delta }}\right)}^2+C_3$$

(13)

where C₁, C₂ and C₃ are constants, the air gap δ is expressed in meters.

From Equations (7)–(13), the following equation can be obtained:

$${\displaystyle \frac{m{\omega }^2}{C_1}}\left(r_0+l_0+{\displaystyle \frac{h_2}{2}}\right)+{\displaystyle \frac{m{\omega }^2-k}{C_1}}\Delta l-{\displaystyle \frac{C_3}{C_1}}={\left({\displaystyle \frac{1}{C_2+{\delta }_0-\Delta l}}\right)}^2$$

(14)

The simulation results at I = 4 A are nonlinearly fitted and the fitting results are shown in Figure 11. In this condition, C₁ = 0.1319, C₂ = 0.01529, C₃ = 0. The Δl can be calculated by substituting the parameters into the above equation for different rotational speeds, and other variables such as the distance r from the center of mass of permanent magnet 2 to the point O of the rotary center and the size of the air gap δ can also be calculated.

In the electromagnetic device, the mass of the moving mass is m = 0.3 kg, the stiffness of the spring is k = 30 N/mm, r₀ = 90 mm, l₀ = 36 mm. The variation curves of the spring deflecton Δl, the distance r from the center of mass of permanent magnet 2 to the point O of the slewing center, and the size of the air gap δ at different rotational speeds for the input current I = 4 A are obtained and shown in Figure 12 and Figure 13.

It can be seen that when the rotational speed is 0 r/min, the spring is in compression due to the electromagnetic force, and as the rotational speed rises to reach 500 r/min, the spring returns to its initial length. When the rotational speed continues to rise, the centrifugal force of permanent magnet 2 is greater than the electromagnetic force, the spring begins to enter the stretching state, when the rotational speed reaches 1487 r/min, the spring deformation reaches 20 mm, at this time, the displacement of permanent magnet 2 is maximum, and the air gap reaches the minimum value, even if the rotational speed continues to rise, permanent magnet 2 does not continue to move.

Figure 14 shows the curves of the spring force F_s on permanent magnet 2 and the centrifugal force F_c of permanent magnet 2 as a function of rotational speed. Due to the electromagnetic force, the centrifugal force is 0 N and the spring force is −73 N at a speed of 0 r/min, when the spring is in compression. As the rotational speed continues to rise, the centrifugal force of permanent magnet 2 and the spring force by which it is acted upon increase gradually, when the rotational speed is increased to 500 r/min, the spring force is 0 N. When the rotational speed is 1487 r/min, the spring force reaches a maximum value of 600 N. Thereafter, as the rotational speed is increased, the spring force is no longer changed and the centrifugal force continues to increase.

3.2. Calculation of the Moment of Inertia

The moment of inertia of the moving part around the rotational center can be expressed as [27]:

$$J_m={\displaystyle \frac{1}{12}}m\left({\displaystyle \frac{3}{4}}{d_3}^2+{h_3}^2\right)+m{r}^2$$

(15)

The wire diameter of the spring in an electromagnetic device is small relative to the mean diameter of the spring, and the mass of the spring can be viewed as concentrated in the mean diameter (the helix). The mass of the spring is m_s, the center diameter D_s2, the pitch P_s, the helical rise angle is α, and the number of working turns of the spring is n_s. The total spring length L_s = πD_s2n_s/cosα and the wire density is m_s/L_s. Take half of the total height of the spring structure to be analyzed, and take the length of ds tiny section of the spring at the point M of height h, as shown in Figure 15, over the point M to do the plumb line of the plane Oxz, the point of intersection is N, this section of the spring center of mass of the point M to the x-axis distance l_PM is:

$$l_{PM}=\sqrt{h^2+{\displaystyle \frac{D_{\mathrm{s}2}^2}{4}}{\sin }^2\theta }$$

(16)

where θ is the angle between the line ON and the x-axis.

The spring of height l/2 corresponds to n_s/2 turns of the helix, so the value of θ corresponding to height h is:

$$\theta =2\pi {\displaystyle \frac{h}{\frac{l}{2}}}{\displaystyle \frac{n_{\mathrm{s}}}{2}}=2\pi n_{\mathrm{s}}{\displaystyle \frac{h}{l}}$$

(17)

The moment of inertia of this tiny segment of spring is:

$$d{J}_{\mathrm{s}}={\displaystyle \frac{l_{\mathrm{PM}}^2{m}_{\mathrm{s}}}{L_{\mathrm{s}}}}ds$$

(18)

The following equation can be obtained from the geometric relationship:

$$ds={\displaystyle \frac{1}{\sin \alpha }}dh$$

(19)

$$\tan \alpha ={\displaystyle \frac{P_{\mathrm{s}}}{\pi D_{\mathrm{s}2}}}$$

(20)

$$P_{\mathrm{s}}={\displaystyle \frac{l}{n_{\mathrm{s}}}}$$

(21)

Based on the above equation, the spring rotational moment of inertia around the x-axis in Figure 15 is obtained:

$$J_{\mathrm{s}}=2{\displaystyle \frac{m_{\mathrm{s}}}{l}}{\displaystyle {\int }_0^{\frac{l}{2}}\left(h^2+\frac{D_{\mathrm{s}2}^2}{4}{\sin }^2\left(2\pi n_{\mathrm{s}}\frac{h}{l}\right)\right)}dh=m_{\mathrm{s}}\left({\displaystyle \frac{l^2}{12}}+{\displaystyle \frac{D_{\mathrm{s}2}^2}{8}}\right)$$

(22)

The distance r_s from the center of mass of the spring to the center of rotation of the DMF is:

$$r_{\mathrm{s}}=r_0+{\displaystyle \frac{l}{2}}$$

(23)

The moment of inertia of the spring about the center of rotation of the DMF is:

$$J_{\mathrm{s}}^{\prime }=J_{\mathrm{s}}+m_{\mathrm{s}}{r_{\mathrm{s}}}^2$$

(24)

There are eight sets of electromagnetic devices in the variable inertia DMF, then the moment of inertia of the variable inertia DMF is:

$$J=J_{\mathrm{u}}+8\left(J_{\mathrm{m}}+J_{\mathrm{s}}^{\prime }\right)$$

(25)

where J_u is the inertia expression of the immutable part in the structure.

In the DMF, J_u = 0.1 kg·m², the mass of the spring is m_s = 6.7 × 10⁻² kg and the center diameter is D_s2 = 20 mm. Figure 16 shows the rotational inertia of the DMF versus rotational speed for different currents.

Before the moving mass reaches the limit position, the moment of inertia of the variable inertia DMF increases with the rotational speed, and when the moving mass reaches the limit position, the moment of inertia of the variable inertia DMF does not change even if the rotational speed continues to increase. Since different currents produce different electromagnetic forces on the moving mass, the rotational speed at which the variable inertia DMF reaches the limit position is different for different currents. The effect of the current on the moment of inertia becomes larger as the speed increases because the centrifugal force is small at low speeds, the air gap δ is large, and the electromagnetic forces are also less sensitive to changes in the value of the current. In addition, the variable inertia DMF has a minimum moment of inertia of 0.01552 kg·m² and a maximum moment of inertia of 0.01786 kg·m². The maximum adjustable proportion of its moment of inertia can reach 15.07%.

4. Analysis of the Intrinsic Frequency of Variable Inertia DMF

Since the noise and vibration of the automotive driveline mainly occur when the engine starts and operates at high speeds, the driveline model equipped with a variable inertia DMF is simplified to analyze the low-order and high-order intrinsic frequencies of the automotive driveline. An equivalent model of the driveline dynamics is shown in Figure 17, where J₁ is the sum of the moment of inertia of the engine crankshaft and the primary flywheel, J₂ is the moment of inertia of the secondary flywheel of the DMF and the clutch disc, J₃ is the equivalent moment of inertia of the driveline, K₁ is the average torsional stiffness of the DMF, and K₂ is the equivalent torsional stiffness of the driveline.

From the literature [28], the first- and second-order resonant frequencies f₁ and f₂ of the system are:

$$f_{1,2}={\displaystyle \frac{1}{2\pi }}\times \sqrt{{\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}}$$

(26)

where a = J₁J₂J₃, b = −K₁J₃(J₁ + J₂) − K₂J₁(J₂ + J₃), c = K₁K₂(J₁ + J₂ + J₃).

Each moment of inertia parameter of the original equivalent model is: J₁ = 0.15 kg·m², J₂ = 0.055 kg·m², J₃ = 2.34 kg·m², K₂ = 10 N·m/°, K₂ = 574 N·m/°. The first-order resonant frequency f₁ of the system is lower than the excitation frequency (10–13.3 Hz) corresponding to the engine idle speed (600–800 r/min). The engine idle resonance is avoided, and the current can be adjusted to change f₁ during the starting process to prevent the resonance phenomenon from occurring during the starting process. The second-order resonance frequency f₂ of the system is larger than the excitation frequency corresponding to the normal driving speed of the engine, which can isolate the resonance rotational speed outside the operating speed of the engine, effectively eliminating the effects of resonance on the engine and driveline.

Since the intrinsic frequency of the system is directly related to the moment of inertia of each degree of freedom, the moment of inertia of J₁ increases after the introduction of the electromagnetic device, and the value of the increased moment of inertia, J, varies at different rotational speeds, as shown in Figure 16. Therefore, the first- and second-order resonant frequencies f₁ and f₂ of the system change with the rotational speed, as shown in Figure 18. As the rotational speed increases, the first- and second-order resonant frequencies f₁ and f₂ of the system are decreasing. This allows the engine to run at lower idle speeds, which can reduce energy consumption. It can also be noticed that the current has less effect on the frequency characteristics at low speeds, while the frequency sensitivity to the current magnitude increases as the speed increases.

(1)

A variable inertia DMF based on an electromagnetic device is proposed. The electromagnetic device is analyzed by finite element simulation of electromagnetic field, and the relationship between the electromagnetic force with air gap and current is obtained. The results show that the electromagnetic force becomes larger as the current I in the coil increases, while the electromagnetic force decreases gradually with the increase in the air gap δ. The electromagnetic force of the device was tested and the test results were compared with the analysis results, which showed that the maximum error value of the two was 3.57%, verifying the accuracy of the finite element analysis

(2)

A mechanical model of the electromagnetic device was developed to analyze the forces on the electromagnetic device during rotation. The results show that the moment of inertia of the variable inertia DMF is directly related to the rotational speed and current, and its moment of inertia can be adjusted by changing the current. The moment of inertia of variable inertia DMF varies over a range of 15.07% of the maximum rotational inertia, and this work provides a theoretical reference for the control and attenuation of rotor system vibration.

(3)

An analytical model of the intrinsic frequency characteristics of an automotive driveline has been established, and the effects of the variable inertia of the electromagnetic device on the first-order and second-order resonant frequencies of the system have been analyzed at different currents. The results show that the first- and second-order resonant frequencies f₁ and f₂ of the system decrease as the speed increases. As a result, it allows the engine to run at a lower idling speed, which can reduce energy consumption.