FIGURE 17.4 Unit step time responses of a second-order system with various damping ratios. The maximum overshoot, delay, rise, settling times, and frequency of oscillations depend on the damping ratio. Smaller damping ratios give faster response but larger overshoot. In many applications, a damping ratio of 0.707 is preferred.
magnetic force is generated to oppose the motion of the mass displaced from the neutral, thus restoring neutral position — just as a mechanical spring in a conventional accelerometer would do. The advantages of this approach are the better linearity and elimination of hysteresis effects as compared to mechanical springs. Also, in some cases, electric damping can be provided, which is much less sensitive to temperature variations.
One very important feature of null-balance type instruments is the capability of testing the static and dynamic performances of the devices by introducing electrically excited test forces into the system. This remote self-checking feature can be quite convenient in complex and expensive tests where it is extremely critical that the system operates correctly before the test commences. They are also useful in acceleration control systems, since the reference value of acceleration can be introduced by means of a proportional current from an external source. They are usually used for general-purpose motion measurements and monitoring low-frequency vibrations. They are specifically applied in measurements requiring better accuracy than achieved by those accelerometers based on mechanical springs as the force-to-displacement transducer.
There are a number of different types of electromechanical accelerometers: coil-and-magnetic types, induction types, etc.
Coil-and-Magnetic Type Accelerometers
These accelerometers are based on Ampere’s law; that is: “a current carrying conductor disposed within a magnetic field experiences a force proportional to the current, the length of the conductor within the field, the magnetic field density, and the sine of the angle between the conductor and the field.”
Figure 17.6 illustrates one form of accelerometer making use of the above principle. The coil is located within the cylindrical gap defined by a permanent magnet and a cylindrical soft iron flux return path. It is mounted by means of an arm situated on a minimum friction bearing so as to constitute an acceleration-sensitive seismic mass. A pick-off mechanism senses the displacement of the coil under
© 1999 by CRC Press LLC
FIGURE 17.5 Bode plots of gains and phase angles against frequency of a second-order system. Curves are functions of frequencies as well as damping ratios. These plots can be obtained theoretically or by practical tests conducted in the frequency range.
acceleration and causes the coil to be supplied with a direct current via a suitable servo-controller to restore or maintain a null condition.
Assuming a downward acceleration with the field being radial (90°) and using Ampere’s law, the force experienced by the coil may be written as:
F = ma = ilB |
(17.9) |
or the current |
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(17.10) |
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Current in the restoring circuit is linearly proportional to acceleration, provided: (1) armature reaction effects are negligible and fully neutralized by the compensating coil in opposition to the moving coil, and (2) the gain of the servo system is large enough to prevent displacement of the coil from the region in which the magnetic field is constant.
© 1999 by CRC Press LLC
FIGURE 17.6 A basic coil and permanent magnet accelerometer. The coil is supported by an arm with minimum friction bearings to form a proof mass in a magnetic field. Displacement of the coil due to acceleration induces an electric potential in the coil to be sensed and processed. A servo system maintains the coil in a null position.
In these accelerometers, the magnetic structure must be shielded adequately to make the system insensitive to external disturbances or Earth’s magnetic field. Also, in the presence of acceleration, there will be a temperature rise due to i2R losses. The effect of these i2R losses on the performance is determined by the thermal design and heat transfer properties of the accelerometer. In many applications, special care must be exercised in choosing the appropriate accelerometer such that the temperature rises caused by unexpected accelerations cannot affect excessively the scale factors or the bias conditions.
A simplified version of another type of servo-accelerometer is given in Figure 17.7. The acceleration a of the instrument case causes an inertial force F on the sensitive mass m, tending to make it pivot in its bearings or flexure mount. The rotation θ from neutral is sensed by an inductive pickup and amplified, demodulated, and filtered to produce a current ia directly proportional to the motion from the null. This current is passed through a precision stable resistor R to produce the output voltage signal and is applied to a coil suspended in a magnetic field. The current through the coil produces magnetic torque on the coil, which takes action to return the mass to neutral. The current required to produce magnetic torque that just balances the inertial torque due to acceleration is directly proportional to acceleration a. Therefore, the output voltage e0 becomes a measure of acceleration a. Since a nonzero displacement θ is necessary to produce the current ia, the mass is not exactly returned to null, but becomes very close to zero because of the high gain amplifier. Analysis of the block diagram reveals that:
e0 R = (mra − e0Kc R) × (K pKa Ks ) (s2 ω2nl + 2ζ1 s ωnl + 1) |
(17.11) |
Rearranging this expression gives:
mrR K pKaa Ks = (s2 ω2nl + 2ζ1 s ωnl + 1+ KcKpKaa Ks ) e0 |
(17.12) |
© 1999 by CRC Press LLC
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FIGURE 17.7 A simplified version of a rotational type servo-accelerometer. Acceleration of the instrument case causes an inertial force on the sensitive mass, tending to make it pivot in its bearings or flexure mount. The rotation from neutral is sensed by inductive sensing and amplified, demodulated, and filtered to produce a current directly proportional to the motion from the null. The block diagram representation is useful in analysis.
By designing the amplifier gain, Ka is made large enough so that KcKpKaa/Ks >> 1.0; then:
e0 a (s) = K |
(s2 ω2nl + 2ζ1s ωnl + 1+ KcK pKaa Ks ) e0 |
(17.13) |
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where |
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(17.14) |
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(17.15) |
© 1999 by CRC Press LLC