Types of Motion

Introduction

Motion sensors are designed to measure the rate of change of position, location, or displacement of an object that is occurring. If the position of an object as a function of time is x(t), then the first derivative gives the speed of the object, u(t}, which is called the velocity if a direction is also specified. If the speed of the object is also changing, then the first derivative of the speed gives the acceleration. This is also the second derivative of the position.
The primary form of motion sensor is the accelerometer. This device measures the acceleration, a(t), of an object. By integration of Equations (5.16) and (5.17) it is easy to show that the accelerometer can be used to determine both the speed and position of the object as well. Thus, in the accelerometer we have a sensor that can provide acceleration, speed (or velocity), and position information.

The design of a sensor to measure motion is often tailored to the type of motion that is to be measured. It will help you understand these sensors if you have a clear understanding of the types of motion.

The proper unit of acceleration is meters per second squared (m/s²). Then speed will be in meters per second (m/s) and position of course in meters (m). Often, acceleration is expressed by comparison with the acceleration due to gravity at the Earth's surface. This amount of acceleration, which is approximately 9.8 m/s², is called a "gee," which is given as a bold g in this text.

EXAMPLE 5.11
An automobile is accelerating away from a stop sign at 26.4 ft/s². What is the acceleration in m/s² and in gs?

Solution
To find the acceleration in m/s², we simply convert the feet to meters according to 2.54 cm/in and 12 in/ft:

In terms of gs, we then have

where the subscript on a indicates the units are gs. Thus, this acceleration away from the stop sign provides an acceleration of about 80% of that caused by gravity at Earth's surface.

Rectilinear

This type of motion is characterized by velocity and acceleration which is composed of straight-line segments. Thus, objects may accelerate forward to a certain velocity, deaccelerate to a stop, reverse, and so on. There are many types of sensors designed to handle this type of motion. Typically, maximum accelerations are less than a few
gs, and little angular motion (in a curved line) is allowed. If there is angular motion, then several rectilinear motion sensors must be used, each sensitive to only one line of motion. Thus, if vehicle motion is to be measured, two transducers may be used, one to measure motion in the forward direction of vehicle motion and the other perpendicular to the forward axis of the vehicle.
Some sensors are designed to measure only rotations about some axis, such as the angular motion of the shaft of a motor. Such devices cannot be used to measure the physical displacement of the whole shaft, but only its rotation.

Vibration

In the normal experiences of daily living, a person rarely experiences accelerations that vary from 1 g by more than a few percent. Even the severe environments of a rocket launching involve accelerations of only 1 g to 10 g. On the other hand, if an object is placed in periodic motion about some equilibrium position as in Figure 5.21, very large peak accelerations may result that reach to 100 g or more. This motion is called vibration. Clearly, the measurement of acceleration of this magnitude is very important to industrial environments, where vibrations are often encountered from machinery operations. Often, vibrations are somewhat random in both the frequency of periodic motion and the magnitude of displacements from equilibrium. For analytical treatments, vibration is defined in terms of a regular periodic motion where the position of an object in time is given by

where x(t) = object position in m
x_o = peak displacement from equilibrium in m
w = angular frequency in rad/s

The definition of w as angular frequency is consistent with the reference to w as angular speed. If an object rotates, we define the time to complete one rotation as a period T that corresponds to a frequency f = 1/T. The frequency represents the number of revolutions per second, and is measured in hertz (Hz), where 1 Hz = 1 revolution per second. An angular rate of one revolution per second corresponds to an angular velocity of 2p rad/s, because one revolution sweeps out 2p radians. From this argument, we see that f and w are related by
Because f and w are related by a constant, we refer to w as both angular frequency and angular velocity.

Now we can find the vibration velocity as a derivative of Equation (5.20):

and we can get the vibration acceleration from a derivative of (5.22):

Vibration position, velocity, and acceleration are all periodic functions having the same frequency. Of particular interest is the peak acceleration:

We see that the peak acceleration is dependent on w², the angular frequency squared. This may result in very large acceleration values, even with modest peak displacements, as Example 5.12 shows.

EXAMPLE 5.12
A water pipe vibrates at a frequency of 10 Hz with a displacement of 0.5 cm. Find (a) the peak acceleration in m/s2, and (b) g acceleration.

Solution
The peak acceleration will be given by

where w = 2pf = 20 p rad/s and x_o = 0.5 cm = 0.005 m
a_peak = (20 p)² (0.005)
a_peak = 19.7 m/s²

b. Noting that 1 g = 9.8 m/s², we get
A 2-g vibrating excitation of any mechanical element can be destructive, yet this is generated under the modest conditions of Example 5.12. A special class of sensors has been developed for measuring vibration acceleration.

Shock

A special type of acceleration occurs when an object that may be in uniform motion or modestly accelerating is suddenly brought to rest, as in a collision. Such phenomena are the result of very large accelerations, or actually decelerations, as when an object is dropped from some height onto a hard surface. The name shock is given to decelerations that are characterized by very short times, typically in the order of milliseconds, with peak accelerations over 500 g. In Figure 5.22 we have a typical acceleration graph as a function of time for a shock experiment. This graph is characterized by a maximum or peak deceleration a_peak, a shock duration T_d, and bouncing. We can find an average shock by knowing the velocity of the object and the shock duration, as considered in Example 5.13.


EXAMPLE 5.13
A TV set is dropped from a 2-m height. If the shock duration is 5 ms, find the average shock in g.

Solution
The TV accelerates at 9.8 m/s2 for 2 m. We find the velocity as

If the duration is 5 ms, we have or 122 g. No wonder that the TV breaks apart when it hits ground!

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Related Links:
Accelerometer Principles
Types of Accelerometers
Motion Sensor Applications

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