Chapter VII Experimental Uncertainty Analysis
Chapter VII – Experimental Uncertainty Analysis
CHAPTER VIII
Measurement of Solid-Mechanical Quantities
1. Measuring strain with electrical resistance strain gages
An electrical resistance strain gage is used to measure strain a structure and is an essential components in several sensors: pressure sensors, force sensors, …
Furthermore, electrical resistance strain gages are inexpensive and quite reliable.
Principle
Measuring the strain is essential to determine the stress on a structure (knowing the material properties).
For elastic materials stressed in a single direction (uniaxial stress):
Or: Stress = Module of Elasticity Strain
Now, how this is related to electrical resistance? The resistance of a wire is given by:
With: R is the resistance; is the resistivity; L is the wire length and A is the cross sectional area of the wire.
Then, we can write:
However;
We will call dD/D the transverse strain t; and the relation between the axial and the transverse strain is:
Where v is the Poisson’s ratio.
Finally, the equation for the resistance can be written as:
From here the strain gage factor can be defined as:
Typical strain gage
To define the state of strain on a surface, we have to define two orthogonal linear strains x and y and a third strain called the shear strain xy. To do this, we will use three strain gages in an arrangement called a strain rosette. Two configurations are possible: a rectangular rosette and equiangular rosette.
(a)rectangular rosette; (b) equiangular rosette.
For the rectangular rosette:
For the equiangular rosette
2. Measuring displacement with a potentiometer
A potentiometer is a variable voltage divider. It is mainly composed of a resistance that varies with the position of a slider. The most commonly used potentiometers are linear or angular. Linear potentiometers can be used to measure displacements as small as 0.1 to 0.2 in and as high as 1 ft. For angular potentiometers they can measure up to 3500 (multiple rotations)
3. Measuring linear velocity with a linear velocity transducer
The principle is to use a permanent magnet core past a fixed coil. As the north pole of the magnet approaches the coil, the magnetic lines will cut across the coils and generate a voltage. This voltage is proportional to the velocity. If now we impose that the number of coils varies with position, we can get the velocity and the position.
However, a problem occurs when the magnet is at the center of the coil, since the voltage induced by both poles will cancel each others, leading to a zero net voltage. This problem is avoided by using two coils.
4. Measuring acceleration and vibration
4.1. Piezoelectric accelerometers
Certain materials, example: quartz crystal, have the ability to generate an electrical charge when they are deformed. This ability or characteristic is essential to several sensors.
Practically speaking, the piezoelectric material is coated with a conducting material such as silver. When a certain load is applied, electrons move to one of the conducting faces and away from the other. The result is a charge being stored in the inherent capacitance of the piezoelectric material. Two possible configuration are possible: longitudinal effect and transverse effect.
- Longitudinal effect
In this case, the charge is given by:
Where F is the force applied and d is the piezoelectric coefficient of the material.
- Transverse effect
In this case, the charge is given by:
For both cases, the charge will result in a voltage. To be applied in real life experiments, this charge must be sensed with minimal dissipation. This is achieved using a charge amplifier.
Theoretically speaking, piezoelectric sensors should be able to measure quasi-static forces; however, this is limited by charge decay with time. Therefore, the sensors based on piezoelectric materials are more suitable for high frequency measurements.
Piezoelectric materials are used to measure acceleration. A typical setup includes: housing, a mass called the seismic mass, and a piezoelectric sensing element. On this setup, an initial force in applied is order to create a preload. Piezoelectric accelerometers are available in many ranges up to 1000 g.
4.2. Vibrometers
A vibrometers is usually used to measure ground motion in earthquakes and sometimes the vibration of machines. Their mode of measurement is quite different, when compared to piezoelectric accelerometers.
This sensors measures usually the displacement of the base not the acceleration (although their linked mathematically speaking). It is most sensitive to vibrations with moderate frequencies and fairly large displacement amplitudes.
5. Measuring force with a strain gage load cell
The force can be related to the deflection () and to the resulting strain () using linear functions:
C1 and C2 are constants.
Practically speaking, two arrangements are frequently used: the cantilever beam and the hollow cylinder. In both cases, four strain gages are used: two as tensile gages and two as compressive gages.
(a) Cantilever beam; (b) hollow cylinder
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Instrumentation and Measurements \ LK\ 2009