STRAIN GAGE

Force, Stress, and Strain

If an object receives an external force from the top, it internally generates a repelling force to maintain the original shape. The repelling force is called internal force and the internal force divided by the cross-sectional area of the object (a column in this example) is called stress, which is expressed as a unit of Pa (Pascal) or N/m². Stress is the force an object generates inside by responding to an applied external force. Suppose that the crosssectional area of the column is A (m²) and the external force is P (N, Newton). Since external force = internal force, stress, σ (sigma), is:

       σ = P/A   (Pa or N/m²)     (1)

 When a bar is pulled, it elongates by ΔL, and thus it lengthens to L (original length) + ΔL (change in length). The ratio of this elongation (or contraction), ΔL, to the original length, L, is called strain, which is expressed in ε (epsilon):

        ε₁ = ΔL (change in length) / L (original length)

Strain in the same tensile (or compressive) direction as the external force is called longitudinal strain. Since strain is an elongation (or contraction) ratio, it is an absolute number having no unit. Usually, the ratio is an extremely small value, and thus a strain value is expressed by suffixing “x10^-6 (parts per million) strain,” “μm/m” or “με.”

The pulled bar becomes thinner while lengthening. Suppose that the original diameter, d₀, is made thinner by Δd. Then, the strain in the diametrical direction is:

       ε₂ = –Δd / d₀

Strain in the orthogonal direction to the external force is called lateral strain. Each material has a certain ratio of lateral strain to longitudinal strain, with most materials showing a value around 0.3. This ratio is called Poisson’s ratio, which is expressed in ν (nu):

       ν = |ε₂ / ε₁| = 0.3

With various materials, the relation between strain and stress has already been obtained experimentally. Figure below graphs a typical relation between stress and strain on common steel (mild steel). The region where stress and strain have a linear relation is called the proportional limit, which satisfies the Hooke’s law.

       σ = E.ε       or       σ/ε = E

The proportional constant, E, between stress and strain in the equation above is called the modulus of longitudinal elasticity or Young’s modulus, the value of which depends on the materials. As described above, stress can be known through measurement of the strain initiated by external force, even though it cannot be measured directly.

Polarity of Strain

There exist tensile strain (elongation) and compressive strain (contraction). To distinguish between them, a sign is prefixed as follows:

• Plus (+) to tensile strain (elongation)
• Minus (–) to compressive strain (contraction)

Strain Gage

A strain gage is a sensor whose resistance varies with applied force; It converts force, pressure, tension, weight, etc., into a change in electrical resistance which can then be measured.

Structure of Strain Gages

There are many types of strain gages. Among them, a universal strain gage has a structure such that a grid-shaped sensing element of thin metallic resistive foil (3 to 6μm thick) is put on a base of thin plastic film (15 to 16μm thick) and is laminated with a thin film.

Principle of Strain Gages

Metal wires can be used as strain gages. Stretching of the wire changes its geometry in a way that acts to increase the resistance. For a metal wire, we can calculate the gage factor  as follow:

        
        R = ρL/A = ρL/πr² = 4ρL/πD²
       dR = (4L/πD²) dρ + (ρ/πD²) dL− (8ρL/πD³) dD
       dR/R = dρ/ρ + dL/ L − 2dD/D 

Then

         K = (dR/R)/(dL/L) = (dρ/ρ)/(dL/L) + 1 − (2dD/D)/(dL/L)

Since

       −(dD/D)/(dL/L)

is defined as Poisson’s ratio, v, we have the Gage Factor:

       K = 1 + 2υ + (dρ/ρ)/(dL/L)

For different metals, this quantity depends on the material properties, and on the details of the conduction mechanism. In general, metals have gage factors between 2 and 4.

Now, since the stress times the area is equal to the force, and the fractional change in resistance is equal to the gage factor times the fractional change in length (the strain), and stress is Young’s modulus times the strain, we have 

       F = σA = EA(dL/L) = (EA/K)(dR/R)

or

       dR/R=FK/EA

So the fractional change in resistance of a strain gage is proportional to the applied force and is proportional to the gage factor divided by Young’s modulus for the material. Clearly, we would prefer to have a large change in resistance to simplify the design of the rest of a sensing instrument, so we generally try to choose small diameters, small Young’s modulus, and large gage factors when possible. The elastic limits of most materials are below 1%, so we are generally talking about resistance changes in the 1%–0.001% range. Clearly, the measurement of such resistances is not trivial, and we often see resistance bridges designed to produce voltages that can be fed into amplification circuits.

Examples of strain gage applications

There is an uncountable number of different applications for strain gages. Only a few are listed here:

Experimental stress analysis. Diagnosis on machines and failure analysis. 

• multi axial stress fatigue testing, proof testing
• residual stress
• vibration measurement
• torque measurement
• bending and deflection measurement
• compression and tension measurement
• strain measurement

Sensors for machines, automotive, research etc.

• force measurement in machine tools
• aerospace
• impact sensors
• dental sensors
• medical sensors
• automotive, motor sport
• Biometrics
• tension sensors
• web tension
• force on hydraulic or pneumatic press

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ACCELEROMETERS

Acceleration is a physical characteristic of a system. The measurement of acceleration is used as an input into some types of control systems. The control systems use the measured acceleration to correct for changing dynamic conditions.

What is Acceleration? 

Acceleration: the time rate of change of velocity

or               : the time rate of change of the time rate of change of  distance

What are the units?

Acceleration is measured in (ft/s²) or (m/s²)

What is a “g”?

A “g” is a unit of acceleration equal to Earth’s gravity at sea level
   g = 32.2 ft/s² or g = 9.81 m/s²

What is the time rate of change of velocity?

• When plotted on a graph, velocity is the slope of distance versus time
• Acceleration is the slope of velocity versus time 

What is an accelerometer?

An accelerometer is an electromechanical device that will measure acceleration forces. These forces may be static, like the constant force of gravity pulling at your feet, or they could be dynamic - caused by moving or vibrating the accelerometer.

What are accelerometers useful for?

By measuring the amount of static acceleration due to gravity, you can find out the angle the device is tilted at with respect to the earth. By sensing the amount of dynamic acceleration, you can analyze the way the device is moving.
At first, measuring tilt and acceleration doesn't seem all that exciting. However, engineers have come up with many ways to make really useful products using them.

An accelerometer can help your project understand its surroundings better. Is it driving uphill? Is it going to fall over when it takes another step? Is it flying horizontally or is it dive bombing your professor? A good programmer can write code to answer all of these questions using the data provided by an accelerometer. An accelerometer can help analyze problems in a car engine using vibration testing, or you could even use one to make a musical instrument.

In the computing world, IBM and Apple have recently started using accelerometers in their laptops to protect hard drives from damage. If you accidentally drop the laptop, the accelerometer detects the sudden freefall, and switches the hard drive off so the heads don't crash on the platters. In a similar fashion, high g accelerometers are the industry standard way of detecting car crashes and deploying airbags at just the right time.

Pocket Accelerometer

Typical  Accelerometer  Applications

• Tilt / Roll 
• Vibration / “Rough-road” detection 
• Can be used to isolate vibration of mechanical system from outside sources 
• Vehicle skid detection 
• Often used with systems that deploy “smart” braking to regain control of  vehicle 
• Impact detection 
• To determine the severity of impact, or to log when an impact has  occurred 
• Input / feedback for active suspension control systems 
• Keeps vehicle level

How do accelerometers work?

There are many different ways to make an accelerometer.

• Some accelerometers use the piezoelectric effect - they contain microscopic crystal structures that get stressed by accelerative forces, which causes a voltage to be generated. 

• Another way to do it is by sensing changes in capacitance. If you have two microstructures next to each other, they have a certain capacitance between them. If an accelerative force moves one of the structures, then the capacitance will change. Add some circuitry to convert from capacitance to voltage, and you will get an accelerometer. 

• There are even more methods, including use of the piezoresistive effect, hot air bubbles, and light. 

What things should you consider when buying an accelerometer?

Analog vs digital - First and foremost, you must choose between an accelerometer with analog outputs or digital outputs. This will be determined by the hardware that you are interfacing the accelerometer with. Analog style accelerometers output a continuous voltage that is proportional to acceleration. E.g. 2.5V for 0g, 2.6V for 0.5g, 2.7V for 1g. Digital accelerometers usually use pulse width modulation (PWM) for their output. This means there will be a square wave of a certain frequency, and the amount of time the voltage is high will be proportional to the amount of acceleration.

If you are using a BASIC Stamp, or any other microcontroller with purely digital inputs, you will most likely need to go for a digital output accelerometer. The disadvantage here is that it requires you to use the timing resources of the microcontroller to measure the duty cycle, as well as performing a computationally intensive division operation.

If you are using a PIC/AVR/OOPIC/Javelin with analog inputs, or a completely analog based circuit, analog is almost always the best way to go. Depending on the compiler, measuring analog acceleration can be as simple as acceleration=read_adc(); and can be done in a few microseconds.

Number of axis - For most projects, two is enough. However, if you want to attempt 3d positioning, you will need a 3 axis accelerometer, or two 2 axis ones mounted at right angles.

Maximum swing - If you only care about measuring tilt using earth's gravity, a ±1.5g accelerometer will be more than enough. If you are going to use the accelerometer to measure the motion of a car, plane or robot, ±2g should give you enough headroom to work with. For a project that experiences very sudden starts or stops, you will need one that can handle ±5g or more.

Sensitivity - Generally speaking, the more sensitivity the better. This means that for a given change in acceleration, there will be a larger change in signal. Since larger signal changes are easier to measure, you will get more accurate readings.

Bandwidth - This means the amount of times per second you can take a reliable acceleration reading. For slow moving tilt sensing applications, a bandwidth of 50Hz will probably suffice. If you intend to do vibration measurement, or control a fast moving machine, you will want a bandwidth of several hundred Hz.

Impedance/buffering issues - This is by far the single most common source of problems in projects involving analog accelerometers, because so few people thoroughly read the required documentation. Both PIC and AVR datasheets specify that for A-D conversion to work properly, the connected device must have an output impedance under 10kΩ. Unfortunately, Analog Devices' analog accelerometers have an output impedance of 32kΩ. The solution to this is to use a low input offset rail to rail op amp as a buffer to lower the output impedance.
 

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