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4 – FUNDAMENTALS

4 – FUNDAMENTALS shutting. Furthermore, controlling a mechanical system is much simpler in a frequency range considerably below its first resonant peak. The system’s resonant frequency fres and its eigenfrequency nearly coincide for weakly damped mechanical systems. They √ 1 − D2 . are coupled with the damping ratio D to fres = f (1) eig Analog to (4.10), the first eigenfrequency can be written in terms of f (1) eig k = . (4.11) 4πDm Dynamical step response measurements of the mechanical system allow to determine the damping ratio. Weakly damped systems show approximately PT2 behaviour in words of the control theory. Thus, several decaying oscillations occur before reaching a stationary state after a step or Dirac delta excitation respectively. Neighboring amplitudes ∆ of these oscillations peaks define the logarithmic decrement 6 δ to Hereafter, D can be calculated with δ = ln ∆i D = ∆i+1 . (4.12) δ √ . (4.13) π2 + δ2 In the verification phase of the MERTIS project, the MSTS shall persist several shaking tests simulating the launch phase. It is appropriate to maximize the damping ratio for avoiding high resonant peaks, which can destroy the MSTS. However, a high first eigenfrequency is required for fast shutting. These parameters obviously interact in an reciprocal manner when analyzing (4.11). The force generated by the VCA is equal to the dynamic force F in (4.9) and must counteract the totalized acceleration force, attenuation force and spring force. The goal for the MSTS FH design is therefore to minimize these forces by finding adequate values for the parameters m, k and c. 6 Consider the associatied graphic and the mathematical derivation of the logarithmic decrement for unknown stationary amplitudes presented in appendix A. 20

4.3 Electromagnetics 4.3.1 Introduction 4 – FUNDAMENTALS As applicable actuation principle, a voice coil actuator was disclosed in the shutter study. This electromagnetic actuator is capable to perform linear or rotary movements comparable to electric motors. The configuration of a VCA is in general a cylindric coil, which plunges into a setup of a centered magnet surrounded by a ferrite ring, like it can be found in conventional loudspeakers. The application of rotary VCAs is well established in hard disk drive heads. 4.3.2 Theory To introduce the relevant electromechanic parameters, a loudspeaker VCA shall be considered as shown in fig. 4.5 (left). The stroke of the coil will be generated by the Lorentz force FV CA induced by the coil current IC and the magnet field B of the centered permanent magnet. The Lorentz force can be written as FV CA = ICd LC × B, (4.14) whereas d LC indicates the differential of the coil wire cross section normal. The right hand grip rule defines the direction of FV CA. Due to the axisymmetric setup of the considered VCA, B is always perpendicular to d LC. Therefore (4.14) can be simplified to FV CA = ICLCB. (4.15) When assuming a homogeneous magnetic field within the coil region, a force factor KF can be determined written as KF = FV CA IC = LCB, (4.16) which characterizes every VCA 7 . Loudspeaker VCAs usually have a heavy magnet setup compared to the coil’s mass. So, it is obvious that normally the coil is moved for optaining high accelations. However, moving the magnet has significant advantages when its mass is in the same range as the mass of the coil. Here, the coil’s power leads do not move and possibly break after a high number of performed cycles. Thus, two VCA concepts can be deduced and will be named as 7 www.beikimco.com accessed on June 4, 2008. 21

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