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  1. Introduction to acceleration sensors

    I am working on my final year project which involves the use of the Nitendo Wii Nunchuk. The Wii Nunchuk operates based on a 3-axis accelerometer, a joystick and two push buttons. I carried out a small theoretical research on the accelerometer, and in this wiki page I am going to briefly explain what an accelerometer is, how it works, types of accelerometers, how to selection an accelerometer for a project requirement, and shows some applications of accelerometers.

    Physically, acceleration is a vector quantity having both direction and magnitude that is defined as the rate at which an object changes its velocity with respect to time. It is a measure of how fast speed changes . An object is accelerating its velocity is changing. [1] In order to measure acceleration, an acceleration sensor called accelerometer is used. Accelerometer measures in units of g. A g is the acceleration measurement for gravity which is equal to 9.81 m/s². However, depending on altitude, this measurement can be 10 m/s²  in some place.


  2. Working principle of an accelerometer

    The design of an accelerometer is based on the application of physics phenomenon. In aviation, accelerometers are based on the properties of rotating masses. In the world of industry, however, the design is based on a combination of Newton's law of mass acceleration and Hooke's law of spring action. This is the most common design applied to the making of accelerometers, and therefore, in this wiki page I will focus on explaining the accelerometer's working principle  based on this combination of Newton's law and Hooke's law. Figure 1 shows a simplified spring-mass system. In figure 1a, the mass of mass m is attached to a spring at equilibrium position x0 which in turn is attached to the base. The mass can slide freely on the base. Suppose that the base friction is negligible. Figure 1b shows the mass is moving to the right by a displacement of Δx = x -  x0. Since the mass is slowing down, the direction of acceleration vector is to the left. In this case, the mass is subject to the force according Newton's second law and Hooke's law.


    Figure 1. A simplified spring-mass system accelerometer. [2]
    Reprinted from Process Control Instrumentation Technology book

    According to Newton's second law, if a mass, m, is undergoing an acceleration, a, then there must be a force, F, acting on the mass with a magnitude of F(N) = m(kg)a(m/s²) (1)
    Hooke's law states that if a spring of spring constant k is extended from its equilibrium position (x0) by a distance of Δx = x -  x0 (x is the current position with reference to x0), then there must be a force acting on the spring given by F(N) = k(N/m)Δx(m) (2) [2].

    Equating equations (1) and (2) yields
           ma = k Δx
    <=> a = (kΔx) / m (3)

    where k is spring constant in N/m
    Δx is the displacement in m
    m is the mass in kg
    a is acceleration in m/s²

    Equation (3) shows the measurement of acceleration in relation to mass, spring constant and spring extension. This is equivalent to the the linear equation y = λ x where y represents acceleration a, λ = k /m being a constant, and x being the displacement of the mass. This is how the design of an accelerometer is based on. The design and types of accelerometer based on spring-mass system differ in how this displacement measurement is made, and this comes to discussion on section 3 of this wiki page.

  3. Characteristics of spring-mass system based accelerometers

    The equation (3) as a result of the analysis from figure 1 and b holds true on the assumption that there is no friction applied to the mass m. In practice, other parameters such as natural frequency and damping coefficient need to be considered since they have a profound effect on the application of accelerometers. 

    The mass-spring system exhibits oscillations at some characteristic natural frequency. This natural frequency is given by
    f = 1/2π √(k /m)
    where f is natural frequency in Hz
    k is spring constant in N/m
    m is seismic mass in kg

    If there was no friction in the spring-mass system, the mass would oscillate forever. This is, however, not the case in reality. The friction that causes the system to rest is defined by a damping coefficient that has a unit of s^-1. The effect of oscillation is described by periodic damped signal which has the equation as follows. [2,2]

    XT(t) = Xo e^-μt sin(2πft)
    where XT(t) is transient mass position
    μ is damping coefficient
    f is natural frequency

    Now two parameters that affect the accelerometer have been described. If the spring-mass system is exposed to a vibration, then the resultant acceleration is given by
    a(t) = -w^2 x0 sinwt (4)
    Applying equation (4) to equation (3) yields
     Δx = -mx0/k w^2 sinwt


    Figure 2 a. Response of spring-mass system to vibration compared to w^2 prediction b. Effect of various peak motion [2,3]
    Reprinted from Process Control Instrumentation Technology book

    Define f as the applied frequency, and fN is the natural frequency of the accelerometer. When f < fN , the natural frequency has little effect on the operation of the accelerometer. when f > fN, the accelerometer output is independent of the applied frequency. An important point worth noting is that with the applied frequency much larger than the natural frequency, the accelerometer becomes a measurement of vibration displacement . At near the resonance of accelerometer's natural frequency, the output of the accelerometer becomes high non-linear.
    A rule of thumb is that with f < fN, the safe maximum applied frequency should be f < 2/5 fN.With f > fN, the minimum applied frequency should be f > 5/2 fN.[2,4] So to best avoid the severe effect of resonance, the accelerometer should not be used near the resonance of their frequency because the output will become non-linear. Furthermore, the accelerometer should not be used with the applied frequency larger than the natural frequency if the measurement system is meant to measure acceleration.This has an important implication in the selection of the accelerometer for an application.


  4. Types of accelerometers

    So far, I have discussed mainly spring-mass based accelerometers since it is the focus of this wiki page. In addition to that, there are many other accelerometers that are designed based on other physical phenomenon. In fact, what makes a difference between the types is the sensing element and their operating principles.

    4.1 Potentiometric accelerometer

    This is a type of an accelerometer which bases its working principles on the spring-mass system. The potentiometric accelerometer employs a mass (seismic mass), a spring, a dashpot, and a resistive element. The seismic mass is connected between a spring and a dashpot. The wiper of the potentiometer is connected to the mass.The following figure illustrates the structure of the potentiometric accelerometer.




    Figure 3. Structure of a potentiometric accelerometer [3]

    The way it works is simple. It measures the motion of the seismic mass by attaching the wiper arm to the spring-mass system. When the mass is moving, the position of the wiper changes according, thus changing the resistance of the resistive element. Since the natural frequency fN of the potentiometer accelerometer is generally less then 30Hz, this type of accelerometer should be used in low frequency vibration measurements. 

    4.2 Hall effect accelerometer

    Hall effect accelerometer is based the working principles of on spring-mass system. The output voltage varies according to a change in magnetic field from the magnet which is attached on a seismic mass. The mass deflects because of the forces due to acceleration. The output Hall voltage is calibrated in terms of acceleration.


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  5. Selection of an accelerometer

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  6. Applications of accelerometers

    Accelerometers can measure: vibrations, shocks, tilt, impacts and motion of an object.

    As a rule of thumb, in low-frequency applications (having a bandwidth on orders from 0 to 10 Hz), position and displacement measurements generally provide good accuracy. In the intermediate-frequency applications (less than 1 kHz), velocity measurement is usually favored. In measuring high-frequency motions with appreciable noise levels, acceleration measurement is preferred

     

     

     

     

     

     



 

 

 

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