A Study of a Magnetic Damper Using Rare-Earth Magnets and a Pinned Displacement Magnifying Mechanism Taichi Matsuoka and Kenichiro Ohmata Department of Mechanical Engineering Informatics, Meiji University -- Higashimita, Tama-ku, 4-857 Kawasaki, Japan Abstract. A new type of magnetic damper using rare-earth magnets and a pinned displacement magnifying mechanism has been developed. The magnifying ratio and the resisting force characteristics of the damper and its effects of vibration suppression for a piping system were discussed experimentally, theoretically and numerically.. Introduction Magnetic damping has the advantages of a linear characteristic, noncontact mechanism and being good at heat, and has been studied by many researchers [ ]. The authors proposed two types of magnetic dampers in the previous papers [][3]: one is a passive damper consisting of a ball screw, an aluminum disk and rare-earth magnets and the other is a semiactive damper consisting of a ball screw, a brake disk, a brake shoe and a magnetostrictive actuator. These dampers are a little expensive because of the ball screw. In this paper, a new type of magnetic damper using rare-earth magnets has been developed in order to obtain an economical passive magnetic damper which gives damping within a relative displacement in the region of. ~ mm. The magnetic damper consists of a pinned displacement magnifying mechanism [4], rare-earth magnets, a copper plate and coil springs. The trial magnetic damper was made and the resisting force characteristics were measured. The experimental results are compared with the theoretical results. The frequency responses and the seismic responses of a three-dimensional piping system supported by the damper were measured using an electrohydraulic type shaking table. The experimental results are compared with the calculated results obtained by a finite element method (ANSYS), and the effects of vibration suppression of the damper are discussed.. Construction and resisting force characteristics of the magnetic damper Figure shows the conceptual sketch of the magnetic damper. The magnetic damper consists of a displacement magnifying mechanism, rare-earth magnets, a copper plate, linear bearings, a push rod, a load column, rod ends and coil springs. The displacement magnifying mechanism is composed of two levers, three connecting rods and eight pins (ball bearings). It is possible to add the friction force to the magnetic damper. The coil springs in the magnetic damper are removable. For small oscillations, the magnifying ratio λ is approximately given by ( a + b)( c + d) λ = () ac The exact solution for λ can be obtained numerically by using Fig. and following equations.
x x ( a + b)( a + b + c) x θ =, θ =, θ 3 = a ae f {a c + ( a + b) x } () a + b f θ 4 = sin sinθ ( cosθ 3 ), xr = ( c + d) sinθ 4 c c Thus the input displacement x, velocity v and acceleration α are magnified by λ times, and the displacement x r, velocity v r and acceleration α r of the copper plate become x (3) r = λ x, vr = λv, α r = λα Since the copper plate moves across the magnetic fluxes due to the rare-earth magnets, the eddy-current damping force proportional to the velocity v r is generated in the copper plate. This force F M is given by the following equation [3] [7] B hlwc FM = v r (4) ρ where B is the magnetic flux density, h the thickness of the copper plate, l, w the length and the width of the rare-earth magnets respectively, ρ the resistivity of the copper plate and C the dimensionless parameter decided by the shape of both the magnetic flux and the copper plate. The total resisting forces F of the magnetic damper is given by B hlwc F v m k x (5) = λ + rλ α + λ ρ where m r is the equivalent mass of the magnifying mechanism and the copper plate. If the friction force f is added to the magnetic damper, the total resisting force F is given by F = F + f λ sign( v) (6) where sign(v) is the sign function which takes - or corresponding to a minus or plus sign of v. 3. Resisting force characteristics of the damper The trial damper whose magnifying ratio λ is 8.7 was made using rectangular rare-earth magnets of 5 5 mm and copper plates of thickness 6 mm. The construction and the experimental condition of the damper are shown in Fig. 3 and Table, respectively. In order to obtain a larger value of C, the rare-earth magnets were attached to the top of the push rod instead of the copper plate and the copper plates were fixed to the inner sides of the damper case in the trial damper. The damper was attached between a shaking table and a rigid wall through a load cell as shown in Fig. 4, and the resisting force characteristics of the damper when it was subject to harmonic excitations having amplitude of mm and frequencies of,, 3, 4 and were measured. The experiments were carried out for three different numbers of magnets, i.e. pair, pairs and 4 pairs. The springs were removed in these experiments. Figure 5(a), (b) shows the experimental results, together with the calculated results, in the cases of pairs and 4 pairs of magnets, respectively. It will be seen from Fig. 5 that the damper has about the same resisting force as Eq. (5). 4. Frequency responses and seismic responses of a piping system supported by the damper The trial damper was attached at a corner of a three-dimensional piping system rigidly supported at both ends as shown in Fig. 6, and the frequency responses and the seismic responses of the piping were measured using an electrohydraulic type shaking table. The equations of motion of the piping system in matrix form are given by [ M ]{} u&& + [ A] {} u + γ [ A] {} u& + { F} = [ M ]{} && z T {} u = { u L, u }, {} y = { y, L, y } T, {} && z = {&& z, L, && z} T, n n
A A m [ A] = M M, [ M ] = m + m, {} F n L L A A n nn O M = f i M where [M] is the mass matrix, [A] the influence coefficient matrix, {F} the resisting force vector, {u}, {z} the relative displacement and input displacement vectors respectively and γ the internal damping ratio of the pipe. The analytical model of the three-dimensional piping was divided into 5 beam elements and Eqs. (7) were calculated using the time history analysis in the finite element analysis software ANSYS/Structural. The solution time was about 3 minutes under a Windows PC with 8 MB memories. The experimental condition of the piping system is given in Table. The length of each straight part of the piping system is.5 m and the outside diameter of the copper pipe is 4 mm. Figure 7 shows the frequency responses of the piping system when it was subject to a vertical sinusoidal acceleration of amplitude 3 m/s. It is apparent from the Fig. 7 that the vertical displacement at the corner of the piping decreases to about /5 to that of the experiments without the damper in the case of pairs of magnets and the magnetic damper has enough damping for suppressing the vibration of the piping system. Next, the El Centro (94) NS component and Akita (983) NS component normalized to be 3 m/s and m/s at the maximum acceleration respectively were inputted to the vertical direction of the shaking table and the vertical acceleration and the relative displacement (deflection) at the corner of the piping were measured by an accelerometer and an inductance-type displacement transducer. The experiments were also carried out in the case of without the damper, and the experimental results were compared with the calculated results. The maximum response acceleration and deflection at the corner of the piping in the vertical direction are given in Table 3(a), (b), and the response waves at the corner of the piping are shown in Fig. 8(a), (b). It can be seen from the Table 3(a), (b), Fig. 8(a), (b) that the maximum displacement at the corner of the piping decreases to about / to that of the experiments without the damper in the case of pairs of magnets. It is also apparent from these table and figure that the experimental results agree with the calculated results to some degree and the validity of the calculations was confirmed. i d O m n (7) 5. Conclusion In this paper, a magnetic damper using rare-earth magnets and a pinned displacement magnifying mechanism was made, and its resisting force characteristics and the effects of vibration suppression of the damper applied to a three-dimensional piping system were discussed experimentally, theoretically and numerically. The results may be summarized as follows: () The resisting force of the magnetic damper is given by the sum of the magnetic damping force, the inertia force and the spring force. The displacement magnifying ratio of the trial magnetic damper is about 8.7. () The maximum displacement at the corner of the piping decreases to about /5 to that of the experiments without the damper when the piping is subject to a sinusoidal displacement. (3) The maximum deflection at the corner of the piping decreases to about / to that of the experiments without the damper when the piping is subject to a seismic acceleration. (4) The experimental results agree with the calculated results to some degree and the validity of the calculations was confirmed.
6. Acknowledgement The authors wish to thank Mr. Y. Okano of Meiji University for his assistance in carrying out the experiments. This research was supported by the Grant-in-Aid for Scientific Research (B) of the Ministry of Education, Science and Culture of Japan. This support is gratefully appreciated. References [] Schieber, D., Optimal dimensions of rectangular electromagnet for braking purposes, IEEE Trans. on magnetics, Vol. MAG-, No. 3, 975, pp.948-95. [] Weinberger, M. R., Drag force of an eddy current damper, IEEE Trans. on aerospace and electronic systems, Vol. AES-3, No., 977, pp.97-. [3] Nagaya, K., Kojima, H., Shape characteristics of the magnetic damper consisting of a rectangular magnetic flux and a rectangular conductor, Bull. JSME, Vol. 5, 98, pp.36-3. [4] Nagaya, K., Kojima, H., On a magnetic damper consisting of a circular magnetic flux and a conductor of arbitrary shape. Part I : Derivation of the damping coefficients, Trans. ASME, Vol. 6, 984, pp.46-5. [5] Nagaya, K., Kojima, H., On a magnetic damper consisting of a circular magnetic flux and a conductor of arbitrary shape. Part II : Applications and numerical results, Trans. ASME, Vol. 6, 984, pp.5-55. [6] Kanamori, M., Ishihara, Y., Shape optimization of conductor slab on an electromagnetic damper by boundary element method combined with finite element method, Trans. JSME (in Japanese), Vol. 56, No. 57, C, 99, pp.698-73. [7] Asami, T., Hosokawa, Y., A practical expression for design of a magnetic damper (Improvement of the convergence in the Nagaya-Kojima expression), Trans. JSME (in Japanese), Vol. 6, No. 58, C, 995, pp.587-59. [8] Seto, K., Vibration control method using magnetic damping, Trans. JSME (in Japanese), Vol. 56, No. 55, C, 99, pp.79-86. [9] Kobayashi, H., Aida, S., Development of a houde damper using magnetic damping, Proc. Vibration Isolation, Acoustics and Damping in Mechanical Systems, ASME DE-Vol. 6, 993, pp.5-9. [] Aida, Y., et al., Dynamic vibration absorber using magnetic spring and damper, Seismic Engineering, ASME PVP-Vol. 3, 995, pp.439-445. [] Matsuhisa, H., Nishihara, O., Dynamic vibration absorber for a ropeway carrier, Proc. 997 ASME Design Engineering Technical Conference, DECT97 / VIB-3944, 997, pp.85-94. [] Ohmata, k., Yamakawa, I., Ball screw type damper using rare-earth magnets, Proc. th International Workshop on Rare-Earth Magnets an Their Applications, Vol. II, 989, pp.65-73. [3] Ohmata, K., Nakahara, Y., Noguchi, O., Hybrid damper using a magnetostrictive actuator and rare-earth magnets, Proc. st International Conf. on Motion and Vibration Control, Vol., 99, pp.645-65. [4] Matsuoka, T., Ohmata, K., A study of magnetic dampers using a pinned displacement enlargement mechanism, Proc. th Symposium on Electromagnetics and Dynamics (in Japanese),, pp.65-68. N c d b x, v S x r v r a (Brake shoe) Rare-earth magnet Copper disk Linear bearing Spring Push rod Lever Pin Load column Rod end Fig. Conceptual sketch of the magnetic damper Fig. Analytical model
Table Experimental condition of the trial damper Length a, b, 4 mm Lever Length c, d, 4 mm Length e, f 4, 3 mm Material Nd-Fe-B Length l mm Magnet Width w 5 mm Thickness 5 mm Open flux B.4 T Mass 5 g Material CP Length mm Copper plate Width mm Thickness h 6 mm Resistivity ρ.7-8 Ωm (Brake shoe) Rare-earth magnet Copper disk Linear bearing Spring Push rod Lever Pin Load column Rod end Resisting force [N] 3 - - Experiment Hz Hz 3 Hz 4 Hz Calculation Hz Fig. 3 Construction of the trial damper -3 -.5 - -.5.5.5 Displacement [mm] (a) In the case of pairs of magnets 3 4 5 Resisting force [N] 5-5 Experiment Hz Hz 3 Hz 4 Hz Calculation Hz Amplifier A/D converter Personal computer. Load cell. Damper 3. Displacement transducer 4. Shaking table 5. Induction motor - -.5 - -.5.5.5 Displacement [mm] (b) In the case of 4 pairs of magnets Fig. 4 Experimental apparatus Fig. 5 Resisting force characteristics
Y Z 5 X 5 & z& Fig. 6 Analytical model Anchor Added mass 5kg Damper 5 Table Experimental condition of the three-dimensional piping Material CBE Length of straight parts L.5 m Outside diameter d 4.3 mm Wall thickness t.5 mm Young s modulus E GPa Density µ 85 kg/m 3 Internal damping ratio γ.3 Displacement ratio 9 8 7 6 5 4 3 Experiment (pair) (pairs) Calculation (pair) (pairs) 4 6 8 Frequency [Hz] Fig. 7 Frequency responses at the corner of the piping system Table 3 Maxima of the response at the corner of the piping (a) El Centro NS (3 m/s ) Experiment Calculation Accel. Disp. Accel. Disp. & y& [m/s ] & y& [m/s ] u m m m m[mm] 4.98 5.97 4.56 6.55 ( pairs of magnets) 3.3 3.9 3.4 3.5 ( pair of magnets) 5.76 3.4 3.35 4.7 (b) Akita NS ( m/s ) Experiment Calculation Accel. Disp. Accel. Disp. & y& [m/s ] u m [mm] [m/s m & y& ] u m m[mm] 4.54 5.5 4.49 4.9 ( pairs of magnets) 3..77.8.5 ( pair of magnets) 3.54 3.7.76 3.65
z [m/s ] y [m/s ] y [m/s ] z [m/s ] Input acceleration Input acceleration y [m/s ] y [m/s ] (pairs) (pairs) y [m/s ] y [m/s ] (pair) (a) In the case of El Centro NS component (pair) (b) In the case of Akita NS component Fig.8 Response waves at the corner of the piping