Development of a torsional dynamic absorber using a magnetorheological elastomer for vibration reduction of a powertrain test rig

Abstract

This article presents the development of a torsional adaptive tunable vibration absorber using a magnetorheological elastomer for vibration reduction of a powertrain test rig. The magnetorheological elastomer used to develop the adaptive tunable vibration absorber consists of silicone polymer, silicone oil and magnetic particles with the weight percentages of 60%, 20% and 20%, respectively. Experimental testing is conducted to obtain the magnetorheological elastomer’s properties, such as Young’s modulus and the damping ratio, and effective formulas are derived to facilitate the design of the adaptive tunable vibration absorber. With the derived formulas, a magnetorheological elastomer–based adaptive tunable vibration absorber is designed and manufactured, and experimental testing is also conducted to validate the design. The results of experiments show that the magnetorheological elastomer–based adaptive tunable vibration absorber can work in a frequency range from 10.75 to 16.5 Hz (53% relative change). Both the designed and experimental results of the adaptive tunable vibration absorber’s frequencies are in good agreement. A powertrain model is used to validate the magnetorheological elastomer–based adaptive tunable vibration absorber’s effectiveness, and the numerical simulations show that the powertrain frequencies are shifted away from the resonant frequency; thus, the powertrain’s steady-state vibration can be significantly reduced. This magnetorheological elastomer–based adaptive tunable vibration absorber will be a promising new device for vibration reduction of vehicle powertrains.

Keywords

Adaptive tunable dynamic absorber vibration reduction powertrain vibration magnetorheological elastomer

Introduction

A magnetorheological elastomer (MRE) is a smart material whose mechanical properties can be magnetically controlled. As such, it is potentially a suitable material for developing structures of varying stiffness (Carlson and Jolly, 2000; Kallio, 2005).

One of the most popular applications of MRE material is that this material is used as an adaptive element for developing adaptive tuned vibration absorbers (ATVAs). Because of the increase in elastic modulus under the application of an external magnetic field, MRE-based ATVAs can work in a wide frequency range instead of a specific frequency or narrow bandwidth, as a traditional vibration absorber does. For instance, Ginder et al. (2001) used a MRE as a variable-spring-rate element to develop an ATVA, which was then experimentally investigated. The experimental results show that the frequency range of the ATVA is from 500 to 610 Hz under a magnetic field of 0.56 T. Deng et al. (2006) developed an MRE-based ATVA for vibration reduction of a beam with two supported ends. The authors reported that this ATVA works effectively in a frequency range from 55 to 81.25 Hz (relative frequency change 49%) in a magnetic field produced by an electromagnetic coil with a direct current (DC) current of 1.5 A. Albanese and Cunefare (2003) presented a state-switched absorber (SSA) using a MRE material and found that with iron particles 35% by volume, the MRE material has the largest magnetorheological effect and the natural frequency of the SSA can be tuned from 45 to 183 Hz (360% increase in frequency shift). In addition, Lerner and Cunefare (2008) tested the SSA operating in different working modes of MREs. These authors reported that the increase in the frequency range of the SSA is 183%, 473% and 510% for shear, longitudinal and squeeze modes, respectively.

Even though a great number of MRE-based ATVAs have been studied, their use has only been applied to single-degree-of-freedom (SDOF) systems. The application of ATVAs for multi-degrees-of-freedom (MDOF) systems is an area that has not been fully addressed so far. In addition, no ATVA using MREs has been applied to torsional vibration reduction of mechanical systems, which constitute a major proportion of engineering vibration problems, of which, vehicle powertrains are a typical example.

The powertrain is a crucial subsystem of vehicles and is also a source of vibration (Couderc et al., 1998; Crowther, 2004; Zhang et al., 2003). Because of the wide range of operating frequencies of the powertrain, the likelihood of the engine working speed being in the resonance area is very high. Moreover, the resonance can be unavoidable when the engine speed passes through one or more powertrain frequencies, for example, when the engine accelerates from idle to top working speeds. Consequently, powertrains may experience a high level of vibration if the acceleration time is not rapid enough, and this vibration reduces the comfort performance of vehicle (Hoang et al., 2009, 2011). Clearly, this vibration needs to be minimized.

Hoang et al. (2009) proposed a concept design of a MRE-based ATVA for steady-state vibration control of powertrains. The authors reported that by using a soft MRE developed by Abramchuk et al. (2006), the ATVA can work in a frequency from 7 to 70 Hz. In this study, the influence of the ATVA parameters and its location in the powertrain system was also examined. Although the soft MRE shows a significant increase in shear modulus, the damping of this MRE was not reported, and the damping model of another MRE reported by Zhou (2003) was used instead. As a result, the damping model may not accurately represent the damping properties of the soft MRE.

Hoang et al. (2011) used a new MRE reported by Chertovich et al. (2010) to develop a MRE-based ATVA for transient vibration control of a powertrain, in which the new MRE has significant increase in shear modulus and low damping. With the new MRE, the ATVA can work in a frequency between 10 and 55 Hz; thus, the ATVA can be tuned to deal with the resonance occurring in a powertrain transient state with excitation frequency of 20–40 Hz. Although the capacity of the MRE-based ATVA was numerically validated, two shortcomings are apparent. First, it is difficult and impractical to supply DC to the magnetic circuit, which includes three electromagnetic coils and a mild steel core, when the circuit rotates with the powertrain shaft. Second, with the rotating magnetic circuit, the inertia of the MRE-based ATVA will be high; worse, the three electromagnetic coils and mild steel core may generate the imbalanced forces on the powertrain system. As a result, the powertrain may experience a higher level of vibration.

This study presents the development of a torsional ATVA using a MRE for vibration control of powertrain systems in which a torsional MRE-based ATVA is experimentally validated. The first section is the introduction. The second section presents a MRE and experimental testing for determining its properties for the design of the ATVA. The third section proposes the design of a MRE-based ATVA and conducts an experiment to validate the ATVA design. Numerical simulations are conducted to validate the ATVA’s effectiveness for vibration control of a powertrain in a steady state in section ‘Numerical simulations’, and the last section presents the conclusion of this work.

A MRE and its characteristics

MRE preparation and experimental set-up

The MRE consists of a rubbery silicone polymer matrix; silicone oil, which serves as a plasticizer; and magnetic particles having weight fractions of 60%, 20% and 20%, respectively. The size of the magnetic particles is 6–7 µm. To enhance the MR effect, the mixed material was placed in a magnetic field before it was cut into rectangular prism samples 16 mm × 16 mm × 45 mm for testing. This MRE material was fabricated at the University of Wollongong, Australia. The test was conducted at the Dynamic and Solid Mechanics Laboratory, University of Technology, Sydney (UTS). The experimental set-up for measuring MRE Young’s modulus is shown in Figure 1.

The MRE sample was placed in the middle of two permanent cylindrical magnets (D-D50H12.5-N45-disc 50 mm diameter × 12.5 mm high). The fixture is a steel frame support, which is used to locate the handle. The fixture ensures that the distance between the two magnets can be adjusted easily by turning the handle. Consequently, the magnetic field applied to the MRE sample can be varied. The MRE magnetic flux density is measured by BELL 610 Gauss-meter. The force F and the length change of the MR specimen are provided by Device Instron. The MRE Young’s modulus is measured by using the following equation

E = \frac{σ}{ε} = \frac{F / A_{0}}{Δ l / l_{0}} = \frac{F l_{0}}{A_{0} Δ l}

(1)

where E is the Young’s modulus, F is the applied force, A₀ is the original cross-sectional area through which the force is applied, Δl is the length change of the MR specimen, and l₀ is the original length of the MRE specimen.

To measure the MRE damping ratio, a weight is attached to the MRE specimen. By measuring the vibration attenuation of the weight, the damping ratio can be calculated as

ς = \frac{δ}{\sqrt{{(2 π)}^{2} + δ^{2}}}

(2)

The logarithmic decrement δ = ln(x₁/x₂), in which x₁ and x₂ are vibration amplitudes measured from one cycle for the vibration of the weight. In other words, by measuring the free vibration of the weight, vibration amplitudes x₁ and x₂ can be determined; the damping ratio is then calculated by equation (2).

Experimental results and the proposed model

Young’s modulus and the damping ratio of the MRE are measured using the experimental set-up as presented in section ‘MRE preparation and experimental set-up’. To facilitate the ATVA design, models for both Young’s modulus and damping ratio are derived. The procedure for proposing these models is similar to those introduced by Hoang et al. (2011).

Young’s modulus of the MRE is approximated by the following equation

E = E (B) = {\begin{matrix} E_{0} & B \leq B_{0} \\ E_{0} + (E_{max} - E_{0}) \frac{B^{2}}{B_{S}^{2}} (3 - 2 \frac{B}{B_{S}}) & 0 < B < B_{S} \\ E_{max} & B \geq B_{S} \end{matrix}

(3)

Here, B₀ = 0.05 T is the value of magnetic field density, from which the MRE material is initially effected, B_S = 0.225 T is the saturated point, E₀ = 114.2 kPa and E_max = 270.9 kPa.

The experimental data and the proposed model of MRE Young’s modulus are shown in Figure 2.

Also, the damping ratio is proposed as equation (4)

ς = {\begin{matrix} 0.126 + 4.9794 B^{2} - 33.1958 B^{3} 0 \leq B < B_{C} \\ 0.1412 + 0.0256 B - 0.1281 B^{2} B_{C} \leq B \leq B_{max} \end{matrix}

(4)

with B_C = 0.1 T, B_max = 0.35 T, and the proposed damping ratio and experimental data are shown in Figure 3.

It can be seen that the experimental data and the proposed models in Figures 2 and 3 are in good agreement. These models will be used to design the MRE-based ATVA in the following section.

A proposed design of MRE-based ATVA and experimental validation

In this section, an MRE-based ATVA is designed for a powertrain test rig at the UTS. For more details of the UTS powertrain test rig, see Crowther (2004). In this application, the MRE-based ATVA will be mounted on the propeller shaft of the test rig; the shaft has a diameter of 86 mm.

Proposed design of MRE-based ATVA

The proposed ATVA design is shown in Figure 4, and the cross section of the ATVA is shown in Figure 5.

In Figure 4, the rotating part, which is the main part of ATVA, consists of an outer ring, inner ring and eight MRE specimens located in the gap between the rings, and two mild steel overlap sheets, which are used to direct the magnetic flux to the MRE specimens. The MRE specimens act as springs to ensure that the rotating part is a torsional SDOF system, and it is located on the powertrain propeller shaft. The ATVA is fixed on the frame support, and this support is used to locate the ATVA to the powertrain test bed.

It is noted that the eight MRE specimens operate as eight translational springs in tangent direction. These specimens create elastic forces between the outer and inner rings. The magnetic flux path and equivalent magnetic circuit of the ATVA are shown in Figure 6(a) and (b), respectively. In this design, to enhance the flux from the magnetic coil to the MRE specimens rather than the brass outer ring, two mild steel overlap sheets are bolted to both sides of the outer ring as shown in Figure 6(a).

Using Ohm’s law for the magnetic circuit, the magnetomotive force NI can be designed. Here $ϕ_{S}$ , $ϕ_{B}$ and $ϕ_{MRE}$ are magnetic flux through steel, brass and MRE, respectively; R_Air, R_Brass and R_MRE are magnetic reluctance of air, brass and MRE, respectively; N is the number of turns of the coil and I is the DC input current supplied to the electric coil. In this design, N = 1000, and input current varies from 0 to 5.75 A. Main parameters of the ATVA are shown in Table 1.

Table 1.

ATVA’s mechanical parameters.

Items	Material	Mass estimation	Dimensions (mm)
Outer ring	Brass	1.6 kg	R_i = 70, R_o = 90, L = 18
Inner ring	Brass	0.8 kg	R_i = 48, R_o = 57, L = 16
Overlap sheet	Mild steel	0.8 kg	R_i = 58, R_o = 90, L = 7
Mild steel core	Mild steel	7 kg	R_i = 74, R_o = 139, L = 7
Electric coil	Copper	5.3 kg	R_i = 92, R_o = 132, L = 34

ATVA: adaptive tuned vibration absorber; R_o: outer radius, R_i: inner radius, L: length.

Because the inner ring is fixed in the propeller shaft of powertrain, the inertia of ATVA can be calculated by

J_{A} = J_{Outer ring} + 2 J_{Overlap sheet}

(5)

Here, the outer ring and each of the overlap sheets are treated approximately as a hollow cylinder as shown in Figures 7 and 8.

By using the mass element of a cylinder $dm = ρ dV = ρ Lrdrd θ$ , as shown in Figure 8, the inertia of the holed cylinder can be calculated

\begin{matrix} J_{O} = \int r^{2} dm = ρ L \int_{0}^{2 π} \int_{R_{i}}^{R_{o}} r^{3} drd θ = ρ L \int_{0}^{2 π} d θ \int_{R_{i}}^{R_{o}} r^{3} dr \\ = 2 π ρ L \int_{R_{i}}^{R_{o}} r^{3} dr = \frac{π ρ L}{2} (R_{o}^{4} - R_{i}^{4}) \end{matrix}

It is noted that $m = π ρ L (R_{o}^{2} - R_{i}^{2})$ is the mass of the cylinder, thus, the moment of inertia

J_{O} = \frac{1}{2} m (R_{o}^{2} + R_{i}^{2})

(6)

With the parameters of the outer ring and overlap sheet as shown in Table 1, the inertia moment of the inner ring and each overlap sheet is calculated by equation (6), then, the inertia moment of ATVA J_A = 0.021 kg m² is calculated by equation (5).

It is assumed that each MRE specimen works as a translational spring, and the stiffness of a specimen can be calculated as follows (Lerner and Cunefare, 2008)

k = \frac{EA}{L}

(7)

Because there are eight MRE specimens, the torsional stiffness of the MRE-based ATVA can be expressed by

k_{A} = 8 \frac{EA}{L} d

(8)

Here d is the distance from the centre of a MRE specimen to the centre of the propeller shaft. In this design, d = 65 mm, and L = 35 mm and A = 7 mm × 8 mm are the length and the cross section, respectively, of a MRE specimen, and E is the MRE Young’s modulus, which was shown in Figure 2. The ATVA’s natural and damped frequencies can be calculated by equations (9) and (10)

f_{n} = \frac{1}{2 π} \sqrt{\frac{k_{A}}{J_{A}}}

(9)

f_{d} = f_{n} \sqrt{1 - ς_{A}^{2}}

(10)

and the damping coefficient can be calculated

c_{A} = 2 ω_{n} J_{A} ς_{A} = 4 π f_{n} J_{A} ς_{A}

(11)

With J_A = 0.021 kg m², the stiffness $k_{A}$ calculated in equation (8) and the damping ratio $ς_{A}$ as shown in equation (4), and the frequency of the designed ATVA can be calculated by equation (10).

Experimental validation

The experimental set-up for measuring the ATVA frequency is shown in Figure 9.

In this experiment, the device BELL 610 Gauss-meter is used to measure the magnetic field on the surface of each MRE sample. Two accelerometers Crossbow CXL01LF1 are also used to measure the free vibration of the ATVA. These sensors are connected to the Dynamic Signal Analyser 35665A (Hewlett Packard), which has two channels, numbered 1 and 2, to connect to the accelerometers.

It can be seen in Figure 9 that the accelerometers are used to pick up the signal of acceleration in tangent directions, and the signals are transferred to the channels of the analyser. The data from channels 1 and 2 are averaged in time domain, and fast Fourier transformation (FFT) is used to obtain the frequency domain of the free vibration of the ATVA. As a result, the free vibration of the MRE-based ATVA can be measured, and its natural frequency can be obtained.

Experimental results

By varying the input current from 0–5.75 A, the dependence between the measured magnetic flux density and the input current is shown in Figure 10.

Clearly, the magnetic flux density B of the magnetic circuit is proportional to the input current I. A comparison of the ATVA designed frequency, which is calculated by equation (10), and the experimental results are illustrated in Figure 11.

It is obvious that the ATVA frequencies for both experimental data and the design are in good agreement. The ATVA frequency range is from 10.75 to 16.5 Hz (the relative change in the frequency is 53%). It is noted that the maximum frequency of 16.5 Hz is measured at magnetic flux density B = 0.21 T (at maximum input current I = 5.75 A).

Numerical simulations

The MRE-based ATVA can work effectively in a tunable frequency range from 10.75 to 16.5 Hz. To show the capacity of the MRE-based ATVA for vibration control of powertrains, a simplified powertrain model consisting of inertias, stiffness and damping is shown in Figure 12.

The powertrain is modelled as a 4-degree-of-freedom system in which the engine is modelled by the first inertia. The second and third inertias represent the clutch or the torque converter (TC) and the transmission gear box, respectively. The drive line components of the powertrain are modelled by the fourth inertia.

The powertrain has a number of gear ratios, which are used to set the optimal engine speed according to the vehicle speed. These gears are characterized by varying the stiffness k₂. In this study, it is assumed that only the first transmission gear is used. When the powertrain is in a resonant range, the ATVA is considered to work effectively if the powertrain’s natural frequencies can be shifted away from the resonance; hence, powertrain’s steady-state response is reduced significantly.

By using Lagrange’s equation, the equation of motion of the system before adding the ATVA can be expressed as the equation as below

J \overset{\cdot\cdot}{θ} + C \overset{\cdot}{θ} + K θ = T

(12)

where $θ = {[θ_{1} θ_{2} θ_{3} θ_{4}]}^{T}$ and $T = {[T (t) 0 0 0]}^{T}$ are vectors of generalized coordinates and external torque.

The inertial matrix J and the stiffness and damping matrices, K and C, have the following forms

J = [\begin{matrix} J_{1} & 0 & 0 & 0 \\ 0 & J_{2} & 0 & 0 \\ 0 & 0 & J_{3} & 0 \\ 0 & 0 & 0 & J_{4} \end{matrix}]

(13)

\begin{matrix} K & = [\begin{matrix} k_{1} & - k_{1} & 0 & 0 \\ - k_{1} & k_{1} + k_{2} & - k_{2} & 0 \\ 0 & - k_{2} & k_{2} + k_{3} & - k_{3} \\ 0 & 0 & - k_{3} & k_{3} \end{matrix}]; \\ C & = [\begin{matrix} c_{1} & - c_{1} & 0 & 0 \\ - c_{1} & c_{1} + c_{2} & - c_{2} & 0 \\ 0 & - c_{2} & c_{2} + c_{3} & - c_{3} \\ 0 & 0 & - c_{3} & c_{3} \end{matrix}] \end{matrix}

(14)

By solving equation (12), both free and forced vibrations of powertrain can be obtained. As a result, powertrain’s vibration characteristics such as natural frequencies and frequency response can be obtained.

To investigate the effectiveness of the ATVA, the powertrain vibration parameters are set as J₁ = 0.4 kg m², J₂ = 0.05 kg m², J₃ = 0.1 kg m², and J₄ = 8 kg m²; c₁ = 6 N m s/rad, c₂ = 4 N m s/rad and c₃ = 4 N m s/rad and k₁ = 20,000 N m/rad, k₂ = 18,000 N m/rad and k₃ = 5350 N m/rad.

With these parameters, the powertrain has three natural frequencies f₁ = 13.8891 Hz, f₂ = 62.0286 Hz and f₃ =148.4882 Hz.

It is assumed that excitation frequency Ω is equal to the frequency of the engine speed. If the engine speed is 825 r/min, it gives Ω= 2π× 13.75 rad/s. Clearly, a resonance occurs in powertrain because f₁ = 13.8891 Hz is close to the excitation frequency. To deal with the resonance, the MRE-based ATVA is added to the powertrain as in Figure 13. Here, the input current I = 4 A is tuned, at which magnetic flux density = 0.14 T, as shown in Figure 10; thus, the ATVA’s frequency f = 13.75 Hz, as shown in Figure 11.

With the ATVA, the system has 5 degrees of freedom, and the equation of motion has the same form as equation (12) with $θ = {[θ_{1} θ_{2} θ_{3} θ_{4} θ_{A}]}^{T}$ , and the inertia matrix is expressed in the following form

J = [\begin{matrix} J_{1} & 0 & 0 & 0 & 0 \\ 0 & J_{2} & 0 & 0 & 0 \\ 0 & 0 & J_{3} & 0 & 0 \\ 0 & 0 & 0 & J_{4} & 0 \\ 0 & 0 & 0 & 0 & J_{A} \end{matrix}]

(15)

the stiffness and damping matrix have forms of

K = [\begin{matrix} k_{1} & - k_{1} & 0 & 0 & 0 \\ - k_{1} & k_{1} + k_{2} & - k_{2} & 0 & 0 \\ 0 & - k_{2} & k_{2} + k_{3} + k_{A} & - k_{3} & - k_{A} \\ 0 & 0 & - k_{3} & k_{3} & 0 \\ 0 & 0 & - k_{A} & 0 & k_{A} \end{matrix}]

(16)

C = [\begin{matrix} c_{1} & - c_{1} & 0 & 0 & 0 \\ - c_{1} & c_{1} + c_{2} & - c_{2} & 0 & 0 \\ 0 & - c_{2} & c_{2} + c_{3} + c_{A} & - c_{3} & - c_{A} \\ 0 & 0 & - c_{3} & c_{3} & 0 \\ 0 & 0 & - c_{A} & 0 & c_{A} \end{matrix}]

(17)

With the inertial matrix J and stiffness and damping matrices K and C, as shown in equation (15), (16) and (17), the vibration characteristics of the powertrain after adding the ATVA can be solved. For example, let T₀ = 3 N m, the powertrain’s steady responses for the fundamental frequency of powertrain f₁ = 13.5 Hz before and after adding the ATVA are shown in Figure 14.

It can be seen in Figure 14 that the powertrain’s vibration frequency response is reduced significantly after adding the ATVA. To be specific, the resonance f = 13.75 Hz was shifted away from the excitation frequency, and two new frequencies of 13 and 14.6 Hz are introduced. It confirms that the ATVA works effectively.

Conclusion

This work has presented a MRE-based ATVA for vibration control of powertrains. A MRE material was fabricated, and the MRE properties were measured. An ATVA was subsequently designed for the powertrain test rig in UTS and was experimentally investigated. The experimental results show that the ATVA can work in a frequency range from 10.75 to 16.5 Hz (53% relative change). Both the designed and experimental results of ATVA’s frequency are in good agreement. This MRE-based ATVA can be applicable not only to the UTS powertrain test rig but also to some other vehicle powertrains. However, a potential drawback is that with the number of wire turns N = 1000, the magnetic flux density B of 0.21 T at current of 5.75 A is small. In other words, the magnetic circuit of the ATVA should be optimized. This limitation will be improved in future work.

To show the capacity of the ATVA’s effectiveness for vibration control of powertrain systems, a powertrain fitted with the ATVA was numerically investigated. It was found that by adding the MRE-based ATVA, the powertrain frequencies could be shifted away from the resonant frequency. As a result, the forced vibration of the powertrain was reduced significantly. This finding confirms that the ATVA works effectively in a tunable range of frequency. Although the frequency of the ATVA was measured, the vibration reduction of the powertrain was not conducted in real time. This is also a shortcoming of this study. The experimental validation of real time control of the ATVA for the UTS powertrain test rig will be addressed in our future studies.

Footnotes

Acknowledgements

The support from Professor Peter Watterson, Mr Christopher Chapman, Mr Michael Tran and Mr Lifu Wang, from the University of Technology, Sydney, and Mr Tongfei Tian, from University of Wollongong for experimental testing is acknowledged.

Funding

This study was financially supported by the Australian Research Council (ARC DP1096847) and the University of Wollongong URC Small Grant.

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