Determination of External Forces—Application to the Calibration of an Electromagnetic Actuator

[+] Author and Article Information
Izhak Bucher

 Faculty of Mechanical Engineering, Technion, Haifa 32000, Israelbucher@technion.ac.il

Moshe Rosenstein

 Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel

J. Vib. Acoust 128(5), 545-554 (Apr 10, 2005) (10 pages) doi:10.1115/1.2346699 History: Received May 19, 2003; Revised April 10, 2005

A new method to estimate time-varying external forces acting upon a vibrating structure is presented in this paper. The method is developed for a force-calibrating device which is designed to be rigid in the operating frequency band and, therefore, it was believed that simple force-gauges should recover the applied forces. It has been observed that the inertia of the vibrating calibration device distorts the estimated force, even when the excitation is only one-third of the first natural frequency. Unlike traditional methods, the frequency response need not be inverted, alternatively a smoothed, Lagrange multiplier based estimation method is formed. With the proposed method, an electromagnetic excitation device can be calibrated such that the electromagnetic forces can be compared with theoretical estimates. The unique features of the proposed method are: (a) Compensation for inertia forces; (b) incorporation of measured reaction forces in a mixed analytical and experimentally obtained model; (c) possible independence on elastic stiffness forces; and (d) closed form, integral equations. New equations relating the externally applied forces and various measurable physical parameters are established, for example: The dependency upon the air-gap, current, and magnetic flux can be found.

Copyright © 2006 by American Society of Mechanical Engineers
Topics: Force , Calibration
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Figure 13

Time histories of the measured displacements (top), the computed and fitted generalized coordinate (middle), and the computed acceleration according to Eq. 45 (bottom)

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Figure 14

Top: The estimated contribution of the inertia force. Middle: Measured reaction-forces. Bottom: Reconstructed magnetic forces, left, right, and mean.

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Figure 15

Reconstructed mean magnetic force as a function of the dc and ac current. Several experiments were conducted to demonstrate the repeatability of the estimation method.

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Figure 16

Directly reconstructed magnetic force compared with a magnetic flux based estimation

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Figure 17

Estimated phase shift of the current based force estimation relative to the directly estimated force

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Figure 18

Generalized model for the electromagnetic force (fitted surface) and directly estimated force—dots

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Figure 11

Time histories of the measured current and displacements versus the fitted Fourier series. Measured and fitted match exactly.

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Figure 12

Left: Measured distribution of the first three harmonics of beam’s deflection all proportional to φ1(x). Right: Normalized amplitude of the first five harmonics, showing nearly identical deformation shapes (all normalized for comparison). The harmonics were obtained at 20, 40, 60, 80, and 100Hz (measured with scanning laser sensor).

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Figure 1

Two-dimensional structure showing the reaction and the external forces

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Figure 2

Two-dimensional structure undergoing a virtual (rigid body) displacement due to the reaction force R3

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Figure 3

Two-dimensional structure showing the reactions and the external forces applied by an electromagnetic device

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Figure 4

Schematic overview of the experimental system showing the various components

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Figure 5

Left: A two mass system subject to an external force. Right: The same system where the force gauge has been released.

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Figure 6

Estimated force of Fig. 5. Shown are ∣fs∣-force gauge reading, ∣fm∣-inertia force, ∣fe∣-estimated force using Eq. 30. ∣H21(ω)∣,∣H22(ω)∣ are the relevant frequency response functions.

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Figure 7

Estimated force in Fig. 5 assuming 0.5% damping and 1% noise. Shown are, fs-force gauge reading, fm-inertia force, fi-displacement based force estimate, and fe-estimated force using Eq. 30.

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Figure 8

A statically indeterminate system undergoing a virtual displacement

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Figure 9

Locations of the force gauges, sensors (s1s2 at L1, L4), magnetic forces (F1F2 at L2, L3), and the reactions at force-gauges—R1r2 (acting at both ends of the beam)

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Figure 10

Photograph of the experimental rig




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