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Mechanical Accelerometer System Analysis Revised 5/16/2016 12:25:00 a)
Domingo A. López Saldaña Psysics Engineering Student, Universidad de Guanajuato, León,Gto, Mx. Loma del Bosque #103, Lomas del Bosque, Lomas del Campestre, 37150 León, Gto. b)
Isaac E. Castro Estrada Physics Engineering Student, Universidad de Guanajuato, León,Gto, Mx. Loma del Bosque #103, Lomas del Bosque, Lomas del Campestre, 37150 León, Gto.
In this article we present the Mechanical Accelerometer System analysis first finding the mathematical model that represents that system with the corresponding transfer function method.
I.
INTRODUCTION
Mathematical models of physical systems are key elements in the design and analysis of control systems. The dynamic behavior is generally described by ordinary differential equations. There exists a wide range of systems, including mechanical, hydraulic, and electrical. Since most physical systems are no linear, we must to use approximations through Laplace Transform methods. Then, we will proceed to obtain the input-output relationship for components and subsystems in the form of transfer functions.
acceleration due to gravity is at a minimum) and orientation (because tilting something changes the way gravity acts on it and the force it feels). Accelerometers are also widely used in inertial navigation and guidance systems in such things as airplane and ship autopilots. Another very common use in transportation is in automobile airbags: when an accelerometer detects a sudden change in a car's speed, signaling an imminent collision, it triggers an electrical circuit that makes the airbags inflate.
IV. WHAT IS ACCELERATION? II. ACCELEROMETERS Speed is a handy measurement that tells you how quickly you can get from one place to other. A car´s top speed is generally a good indication of how powerfull an engine it has. Acceleration is much more interesting than speed and more useful: it`s how quickly something can speed up or peed down Measuring acceleration is a bit more tricky than measuring speed because it involves figuring out how speed changes over a period of time. How do you measure acceleration? Not surprisingly, with a device called an accelerometer.
III. WHAT ARE ACCELEROMETERS USED FOR? Accelerometers are the stuff of rocket science. Mounted in spacecraft, they're a handy way to measure not just changes in rocket speed but also apogee (when a craft is at its maximum distance from Earth or another mass, so its
If you have a certain force and you apply it to a mass, you'll make the mass accelerate. Newton's second law of motion relates force, mass, and acceleration through this very simple equation: Force=Mass(Acceleration) or Acceleration=Force/Mass In other words, acceleration is the amount of force we need to move each unit of mass.
V. HOW DO ACCELEROMETERS WORK? This equation is the theory behind accelerometers: they measure acceleration not by calculating how speed changes over time but by measuring force. How do they do that? Generally speaking, by sensing how much a mass presses on something when a force acts on it. There are many different types of accelerometers. The mechanical ones are a bit like scaled-down versions of
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passengers sitting in cars shifting back and forth as forces act on them. They have something like a mass attached to a spring suspended inside an outer casing. When they accelerate, the casing moves off immediately but the mass lags behind and the spring stretches with a force that corresponds to the acceleration. The distance the spring stretches (which is proportional to the stretching force) can be used to measure the force and the acceleration in a variety of different ways. Seismometers (used to measure earthquakes) work in broadly this way, using pens on heavy masses attached to springs to register earthquake forces. When an earthquake strikes, it shakes the seismometer cabinet but the pen (attached to a mass) takes longer to move, so it leaves a jerky trace on a paper chart.
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Where: F1= Is the force toward to the right due to rocket´s movement. . Force due to the mass displacement with respect to the accelerometer case. . Spring Restoring Force Force due to the friction
Next we will analyze an example of a mechanical accelerometer using its transfer function and the root locus method.
VI. MECHANICAL ACCELEROMETER A mechanical is used to measure the acceleration of a levitated test sled, as shown in Figure 1. The test sled is magnetically levitated above a guide rail a small distance δ. The accelerometer provides a measurement of the acceleration a(t) of the sled, since the position y of the mass M, with respect to the accelerometer case, is proportional to the acceleration of the case (and the sled). The goal is design an accelerometer with an appropriate dynamic responsiveness. We wish to design an accelerometer with an acceptable time for the desired measurement characteristic, y=qa(t), to be attained (q is a constant).
Figure 2. “M” mass Free-body diagram
Therefore, from Figure 1 and (2) we have:
Since ( ) is the engine force, we have ̈ Figure 1. An accelerometer mounted on a jet-engine test sled
To solve this problem first sketch the free-body diagram for the mass M in which are showed all forces on the body (Figure 2.). Applying equilibrium conditions in x-axis we have:
∑
̇
( )
or ̈
̇
( )
We selected the coefficients where b/M=3, k/M =2, F(t)/ =Q(t), and we consider the initial conditions y(0)= -1 and ̇ ( ) We the obtain the Laplace transform equation, when the force, and thus Q(t), is a step function, as follows:
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(
( )
( )
̇ ( ))
(
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( )
( ))
( )
( )
Since Q(s)=P/s, where P is the magnitude of the step function, we obtain (
( )
)
(
( )
)
( )
or (
(
) ( )
)
Thus the output transform is
( )
( (
) )
( (
)(
) ) Figure 3. Accelerometer response
Expanding in partial fraction form yields ( )
For this case, using the previous parameters, the Transfer Function will be:
We the have ( (
) | )
)(
. Thus,
Similarly, k2=+P and ( )
(
[
(
)
) )
and its Inverse Transform
)
Therefore, the output measurement is
( )
( (
( )
( )
*
(
)
(
)+
0
Plotting y(t) in Figure 4, we now see the proportional response is reached at t=1. ]
0
The plot of y(t) is shown in Figure 3 for P=3. We can see that y(t)is proportional to the magnitude of the force after 5 seconds. Thus in steady state, after 5 seconds, the respose y(t) is proportional to the acceleration, as desired. If this period its excessively long, we must increase the spring
constant, k, and the friction b, while reducing the mass, M.
If we are able to select the components so that b/M=12 and k/M =32, the accelerometer will attain the proportional response in 1 second.
Figure 4. Accelerometer proportional response reached in t=1
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VII. CONCLUSIONS Our accelerometer model was an open loop system with a 3 poles and 2 zeros transfer function. Its input was the Force F(t) and the output the mass M position y(t). Using the partial fraction method we found the response to a step function in t-domain. From the plot we see after 3 seconds the system turns stable and reach a proportional response of the acceleration. Additionally using the parameters where b/M=12 and k/M =32, we obtain this proportional response in t=1 s.
VIII. APPENDIX
1. 2.
b)
3.
http://www.explainthatstuff.com/accelerometers.html , retrieved 15/05/2016. [1] C. Dorf, Richard, H. Bishop, Robert. Modern Control Systems, 11th Edition (2008).
c)
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