Magnetic Damping For Maglev

Magnetic damping is one of the important parameters that control the response and stability of maglev systems. An experimental study to measure magnetic damping directly is presented. A plate attached to a permanent magnet levitated on a rotating drum was tested to investigate the effect of various parameters, such as conductivity, gap, excitation frequency, and oscillation amplitude, on magnetic damping. The experimental technique is capable of measuring all of the magnetic damping coefficients, some of which cannot be measured indirectly.


INTRODUCTION
One of the key elements in controlling the dynamic characteristics of maglev systems is magnetic damping. Two types of magnetic damping can be introduced, active and passive magnetic damping. Several aspects of magnetic damping have been studied: (1) magnetic damping as a hnction of speed (Moon 1977, Iwamoto et al. 1974, Yamada et al. 1974); (2) damping constant as a function of frequency (Saitoh et al. 1992); (3) passive damping (Iwamoto et al. 1974(Iwamoto et al. , F'ujiwara 1980; and (4) active secondary suspension (Nagai and Tanaka 1992). It is fair to say that damping characteristics under various conditions are still not well characterized.
Different methods can be used to analyze or measure magnetic damping: direct method and indirect method. It appears that most of the past studies are based on the indirect method. A direct method was introduced recently to maglev system by Chen et al. (1993). The direct method is capable of measuring the self-induced and mutual magnetic damping. In the study of maglev response, all magnetic damping must be quantified; without those damping values, it is difficult to predict the response of maglev systems. This paper presents two series of tests t o quantify the passive magnetic damping. The purpose is to determine the effect of various parameters on magnetic damping such as conductivity, gap, excitation amplitude, and oscillation frequency.
Once magnetic damping and stiffness are know, it can be applied to maglev systems.

EXPERIMENTAL SETUP
The general experimental setup is shown in Fig. 1. It includes a rotating drum, shaker, force transducer, and displacement transducer. The drum is covered with aluminum sheet with 26.99 cm diameter, 14.61 cm wide, and 0.635 cm thick.
The rotating speed can vary from 0 to 3500 rpm with its speed from 1 m/s to 50 d s .
The shaker provides proper excitation force at given fkequencies and the impedance transducer measures the displacement of the supporting bar.
The force transducer to measure the magnetic force is shown in Fig. 2a. The magnet, 2.54 x 5.08 x 0.318 cm, is connected to aluminum plates, 5.0 x 7.6 x 0.8 cm, with copper brackets, and then supported by an aluminum bar, 2.64 cm wide, 22.86 cm long, and 1.27 cm thick (Fig. 2b). The aluminum bar is attached to the shaker at one end and to the magnet and aluminum plate at the other. One set of strain gauges is placed on the smaller section of the aluminum bar to measure the force due to the excitations at other end.
Without magnetic field, the force transducer is calibrated by dynamic method.
The supporting bar is fairly rigid with a natural frequency of larger than 100 Hz.
For a given excitation frequency and amplitude given to the supporting bar by the shaker, the inertia force of the magnet, aluminum plate, and support structure can be calculated. Typical results are shown in Fig. 3. Theoretically, the inertia force should be proportional to the square of excitation frequency. The curve given in Fig.   3 shows that the power is very close to 2. From the inertia force, displacement, and strain gauge, the calibration constant of the force transducer can be calculated. The force transducer measures the lift force with a sensitivity of about 1 volt for 120 g of force acting on the middle of the magnet and damping plate.

TEST CASES AND DATA ANALYSIS
The detailed arrangement of the magnet and aluminum plate is shown in Fig. 2b. Two cases are tested. In each test, the shaker provides an excitation at a given frequency with a specific amplitude. The displacement of the aluminum support measured by the displacement transducer and the forces consisting of the inertia force of the active element (magnet, aluminum plate, copper bracket, and aluminum support below the strain gauges) and magnetic force measured by the strain gauges are measured simultaneously. The RMS magnitude of the displacement and force as well as the phase angle between them at the excitation frequency are obtained. The magnetic stifhess and damping can be calculated from those data. Each test was performed at several excitation frequencies for the whole range of rotating speed.
The measured dynamic force is given as follows: where u is the displacement, m is the mass of the active element, and C and K are damping coefficient and stiffness to be determined, respectively. Note that m should also include magnetic mass. However, because the excitation frequency of maglev is fairly low, the magnetic mass will be very small and is negligible (Iwamoto et al. 1974).
Let the RMS values of the displacement and force be do and fo, respectively, and the phase angle between the two be 4. C and K are given by  Tests at three frequencies, 2,4, and 6 Hz, are conducted for the whole range of rotating speed for all test cases. Typical results are shown in Fig. 6 for Test A.4.
This figure contains the same information as those in Fig. 4 except that three excitation frequencies are included. The inertia force is proportional to the square of the excitation frequency. At 2 and 4 Hz, the inertia force is smaller, the measured force will be mainly attributed to magnetic force. For different excitation frequencies, the drastic change of phase angle between displacement and measured force also changes with excitation frequencies, but all of them are close to the characteristic speed. However, when the gap is 3 mm for Tests A.l, A.3, B.l, and B.3, magnetic damping values for the series A are larger than those of the series B. This means that the aluminum plate with higher purity will provide higher damping.

Gap
In the tests A.3, A.4, and A.5, the gap between the magnet and drum is kept at a constant, while the aluminum is placed at different gaps with respect to the drum, 3, 5.5, and 8 mm. Magnetic stiffnesses for different gaps are approximately the same. However, magnetic damping values depend on the gap.
As the aluminum plate moves closer to the drum, magnetic damping increases. The magnetic damping values for 3 and 5.5 mm are larger than those for 8 mm.
Tests A.l and A.3, or Tests A.2 and A.4, the gap between the aluminum and drum is fixed while the gap between the magnetic and aluminum is varied. We can also compare the magnetic damping values for the two sets of tests. When the magnet is further away from the drum, magnetic stiffness decrease and magnetic damping increases.

Frequency
The effect of excitation frequency is presented in Figs. 9 and 10 for four speeds, 0, 4.8, 8.3, and 37.6 d s . The magnetic stiffness is practically independent of the frequency while magnetic damping increase slightly with frequency. Figure 11 shows the magnetic stiffness and damping for test B.3 as a function of excitation amplitude at 33.8 m/s. Similar tests have also been performed at other speeds. Regardless of the speed, it is noted that excitation amplitude does not affect both magnetic stiffness and damping in the parameter range tested. This means the linear theory can be used if the displacement is small.

APPLICATIONS TO MAGLEV
Once the maglev damping and stiflhess are know, it can be applied to vehicle dynamics. As an example, consider a maglev vehicle, with the mass m, moving on a guideway. The equation of motion is where m is the total mass of the vehicle; C, is structural damping; C is magnetic damping including aerodynamic damping; K is magnetic stiffness; and q(t) is external excitation.
The natural frequency and modal damping ratio are For a given vehicle, C and K depend on clearance and speed. Using the experimental data measured in this experiment, we can analyze the system characteristics. For example, magnetic damping and stiffness obtained in Tests A. 1, From Eqs. (4) and (5) as well as static magnetic force, the natural frequencies and modal damping ratio can be calculated as a function of b. Furthermore, the response of a maglev system can be predicted from Eq. (3) for a given excitation q(t).

CLOSINGREMARKS
A direct method is used to measure magnetic damping and stiffness for two series of tests. The effect of various parameters are investigated: conductivity, gap, excitation amplitude and excitation frequency. The direct method of measurement is useful in determining magnetic damping and stiffness.
Once magnetic damping and stiffness are know, the dynamic response of maglev systems can be predicted. In addition, passive damping plate can be introduced to improve ride quality and to control stability. Other control techniques using the characteristics of magnetic forces can be applied t o maglev. This technique will be very useful to measure magnetic damping and stiffness coefficients for prototype.
prediction of maglev response as well as the control of maglev systems.
This will provide the necessary elements for the