# Magnetically Tunable Vibration Transmissibility for Polyurethane Magnetic Elastomers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{1/2}was observed (G: storage modulus, m: mass), indicating that the observed vibration is basically described by a simple harmonic oscillation.

## 1. Introduction

_{0}is the fractional change in elastic modulus and Δf/f

_{0}is the corresponding fractional change in natural frequency. This means that magnetic elastomers are useful for application in active dampers, because the natural frequency can be continuously changed by magnetic field strength. Komatsuzaki et al. have fabricated a dynamic vibration absorber using magnetorheological elastomers that demonstrates the ability of broadband vibration control [11,12]. They reported that the natural frequency shifted from 60 to 250 Hz by applying an electric current of 3.5 A [11]. Nguyen et al. developed a vibration isolator with a frequency shift of ~8 Hz by applying a magnetic field of 218 mT [13]. Fu et al. have also developed a semi-active/fully-active hybrid isolator using magnetorheological elastomer and piezoelectric material. They found that the natural frequency can be shifted by approximately 20 Hz by applying an exciting current of 1 A to the electromagnet [14].

## 2. Materials and Methods

#### 2.1. Synthesis of Magnetic Elastomers

_{w}= 2000, 3000), prepolymer cross-linked by tolyrene diisocyanate (Wako Pure Chemical Industries. Ltd., Osaka, Japan), a plasticizer (dioctyl phthalate, DOP, Wako Pure Chemical Industries. Ltd., Osaka, Japan), and carbonyl iron (CI, CS grade, BASF SE., Ludwigshafen am Rhein, Germany) particles were mixed using a mechanical mixer for several minutes. The median diameter of CI particles was 7.0 ± 0.2 μm, as determined by a particle size analyzer (SALD-2200, Shimadzu Co. Ltd., Kyoto, Japan). The saturation magnetization for CI particles was evaluated to be 245 emu/g by SQUID magnetometer (MPMS, Quantum Design Inc., San Diego, CA, USA). The mixed liquid was poured in a silicon mold and cured in an oven for 20 min at 100 °C. The weight concentration of DOP was defined by the ratio of DOP to the matrix without magnetic particles and it was fixed at 50 wt %; DOP/(DOP + matrix). The weight fraction of magnetic particles (CI-CS) was varied up to 70 wt %; CI/(CI + matrix), which corresponds to a volume fraction of 0.23.

#### 2.2. Vibration Experiments

_{1}and α

_{2}are the acceleration measured by sensor 1 (vibrating stage of exciter) and sensor 2 (after attenuation by magnetic elastomer), respectively. f and f

_{0}are the frequency and natural frequency, respectively. tanδ is the loss factor of elastomers. A cylindrical electromagnet (FSGP-40, Fujita Co. Ltd., Kuwana, Japan) with a weight of 300 g was used for generating magnetic fields. The electric potential and electric current were 22.3 V and 0.23 A, respectively, when a magnetic field of 60 mT was applied. The magnetic field strength was measured on the top and the center of the electromagnet by a Hall sensor (TM-601, Kanetec Co. Ltd., Ueda, Japan). An enlarged photograph of the electromagnet is also shown in Figure 1. There is no yoke on the electromagnet for a closed circuit of magnetic flux; therefore, the magnetic field strength is extremely weak. If an electromagnet generating uniform magnetic fields were used, the magnetic effect on the vibration property would be enhanced. Two kinds of weight with masses of 69 and 224 g were used to clear the effect of mass on the natural frequency or transmittance. The strain was calculated to be 1.4 × 10

^{−2}–5.1 × 10

^{−2}for a loading weight of 69 g and 4.7 × 10

^{−2}–1.7 × 10

^{−1}for a loading weight of 224 g using the values of storage modulus (=3G’). The strain dependence for magnetic elastomers was similar to that previously reported [9]. Most of magnetic elastomers studied here exhibited the nonlinear viscoelasticity; however, only samples without magnetic particles and φ= 0.03 demonstrated the linear viscoelasticity at whole strains.

#### 2.3. Dynamic Viscoelastic Measurements

^{−4}, and the frequency was constant at 1 Hz. Although the frequency is different from that in the vibration experiment, the viscoelastic data are listed in Table 1, as a reference. An electric current with 0.33 A was used for the rheological measurement which corresponds to a magnetic field of 60 mT. The sample was a disk 20 mm in diameter and 1.5 mm thick.

## 3. Results and Discussion

^{1/2}for magnetic elastomers with various volume fractions of magnetic particles. If the vibration observed here is a simple harmonic motion, the natural frequency relates to the elastic modulus G’ and mass m of the vibration object as the following relation [15,16],

^{1/2}·m

^{−1}·kg

^{−1/2}. This means that magnetic elastomers show the linear viscoelastic response at high load. In general, the nonlinear viscoelasticity is significant at high strains (i.e., at high load). This contradiction is not yet clear; however, the natural frequency might be strongly influenced by structures such as particle network at 0 mT or chain structure at 60 mT. This is because that their structures are collapsed under high strains. To fully elucidate the relation between natural frequency and nonlinear viscoelasticity, viscoelastic measurements at high frequency at around 100 Hz are necessary, and this will be reported in a subsequent paper.

## 4. Conclusions

^{1/2}revealed that the observed vibration can be basically explained by a simple harmonic motion. However, the material deviates from this simple harmonic model at low loads, warranting further study. The total weight of the device containing the electromagnet is only 320 g. We firmly believe that magnetic elastomers are useful for active dampers with compact size.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Photographs of the experimental set-up for the vibration measurements for polyurethane and magnetic elastomers under magnetic fields.

**Figure 2.**Frequency spectra of transmissibility for magnetic elastomers with various volume fractions of magnetic particles at (

**a**) 0 mT and (

**b**) 60 mT. Normalized frequency spectra for magnetic elastomers at (

**c**) 0 mT and (

**d**) 60 mT (loading weight 69 g).

**Figure 3.**Relationship between (

**a**) natural frequency or (

**c**) transmissibility at 0 and 60 mT and volume fraction of magnetic particles. Change in (

**b**) natural frequency or (

**d**) transmissibility and volume fraction of magnetic particles.

**Figure 4.**Relationship between natural frequency and (G’/m)

^{1/2}in Equation (3) for all magnetic elastomers under two different loads.

**Figure 5.**Relationship between the change in natural frequency and the value calculated from Equation (1) for all magnetic elastomers under two different loads.

**Table 1.**Loss factor tanδ determined from transmissibility spectra (loading weight 69 g), storage modulus G’, loss modulus G’’, and loss factor tanδ determined from rheological measurement for polyurethane and magnetic elastomers.

φ ^{1} | Fitting | Rheometer | ||||||
---|---|---|---|---|---|---|---|---|

tanδ ^{2} | G’ ^{3} (kPa) | G” ^{4} (kPa) | tanδ ^{5} | |||||

0 mT | 60 mT | 0 mT | 60 mT | 0 mT | 60 mT | 0 mT | 60 mT | |

0.00 | 0.32 | 0.29 | 17 | 17 | 2.3 | 2.3 | 0.13 | 0.13 |

0.03 | 0.28 | 0.28 | 20 | 20 | 1.9 | 1.9 | 0.10 | 0.10 |

0.08 | 0.26 | 0.26 | 29 | 33 | 2.8 | 3.1 | 0.09 | 0.09 |

0.16 | 0.33 | 0.32 | 36 | 48 | 3.5 | 4.3 | 0.10 | 0.09 |

0.23 | 0.36 | 0.33 | 44 | 61 | 4.0 | 5.9 | 0.09 | 0.09 |

^{1}Volume fraction of magnetic particles;

^{2}Loss factor;

^{3}Storage modulus at 1 Hz;

^{4}Loss modulus at 1 Hz;

^{5}Loss factor at 1 Hz.

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**MDPI and ACS Style**

Endo, H.; Kato, S.; Watanebe, M.; Kikuchi, T.; Kawai, M.; Mitsumata, T.
Magnetically Tunable Vibration Transmissibility for Polyurethane Magnetic Elastomers. *Polymers* **2018**, *10*, 104.
https://doi.org/10.3390/polym10010104

**AMA Style**

Endo H, Kato S, Watanebe M, Kikuchi T, Kawai M, Mitsumata T.
Magnetically Tunable Vibration Transmissibility for Polyurethane Magnetic Elastomers. *Polymers*. 2018; 10(1):104.
https://doi.org/10.3390/polym10010104

**Chicago/Turabian Style**

Endo, Hiroyuki, Shunsuke Kato, Mayuko Watanebe, Takehito Kikuchi, Mika Kawai, and Tetsu Mitsumata.
2018. "Magnetically Tunable Vibration Transmissibility for Polyurethane Magnetic Elastomers" *Polymers* 10, no. 1: 104.
https://doi.org/10.3390/polym10010104