ME330: Engineering Experimentation

EXAMINATION II

Fall 2005

This examination is closed book and closed notes. You are to work only on the sheets provided, and only on one side of the paper. If you need additional worksheets or require assistance, please notify your instructor. Although the use of calculators is allowed, no preprogrammed aids may be used. Please check to verify that your name is on every sheet prior to turning in your exam paper.

GOOD LUCK!

Do not write in this space

Problem 1(15). Check the following statements as either true or false.

a. Temperature measured relative to room temperature is termed `gage' temperature. T( ) F(X)

b. A Potentiometer can be used as a device that can accept a signal of one form (displacement) and convert it into another form (eg. degrees to volts). T(X) F( )

c. A D/A converter takes an analog signal and converts it into a format that a digital computer can understand. T( ) F(X)

d. Digital signals are defined only in terms of discrete values at specific times. Signal values between these times are unknown. T(X) F( )

e. A second order system has three parameters which, along with the system's differential equation, totally describes its behavior. T( ) F(X)

f. Fourier analysis can be used to estimate the DC voltage content of an arbitrary time-varying signal. T(X) F( )

g. Strain gages mounted on a beam and used with a bridge circuit to measure displacement can be classified as a Second Order System. T(X) F( )

h. A Wheatstone Bridge is used to measure fluid flow through a magnetic field. T( ) F(X)

i. A pressure transducer can only be used with air. T( ) F(X)

j. When a thermocouple has a reference junction maintained at 0^oC, if the thermocouple is at room temperature its output is zero. T( ) F(X)

k. A low pass filter with a gain of 3 will allow DC signals to pass through unaffected. T( ) F(X)

l. The value Sxy resulting from a linear curve fit is the standard error of a x-value for a given value of y. T(X) F( )

m. A barometer is used to measure the strength of metal bars. T( ) F(X)

n. Strain gages are unaffected by fluctuating electromagnetic fields. T( ) F(X)

Problem 2(10).

You are to complete Table~\ref{tab:conversion} below:

Table 1. A table of equivalent numbers in different number systems (base 2,10).

It helps to create a table between binary and the decimal value of each place.

Problem 3(20).

The voltage waveform on an oscilloscope shown in Fig.1 was the result of a sudden 20 ft.-lbf. step change input to a torque sensor. If you assume this to be a Second Order System, then the damping ratio can be approximated using the Log-Decrement Method. Using this method, it can be shown that

where

a. If the sensor is a second order system as assumed, describe the torque sensor from an instrumentation engineer’s vantage point (how do you characterize it?). Be specific so that someone might be able to use your specifications to select or reject it for a specific application.

b. Estimate the frequency range over which this transducer (sensor) can be applied.

c. What is the DC gain of the sensor? Be certain to specify your units.

Figure 1. The response of a torque sensor to a step change of 20 ft-lbf. The vertical sensitivity is 1 mv/div and the horizontal sensitivity is 5 msec/div. The vertical position of zero potential is noted on the oscilloscope recording by a triangle.

See the next page for the calculations.
The following are calculations of the damping ratio, x, the approximate natural frequency, f_n, and the sensor gain, G, based on the values taken off of the oscilloscope:

Problem 4(20).

Consider the measurement system described by Eqn.1 below,

Eqn.1

where

E_o = Output signal from a hot wire anemometer, volts

T = Time, sec.

V = Velocity, m/sec.

k₁ = Constant, 1.0 meters/volt

k₂ = Constant, 0.5 meter/sec-volt².

You are to correct the data (E_o) from the system shown in Table 2 to obtain the true input signal (V(t)) and record your results in the proper column. Be certain and explain the method used in computing your results and show a typical calculation.

Table 2. Time varying data from a hot wire anemometer.

Just as we did with op-amps in lecture, one can reconstruct the input using a model of the system. The above worksheet was used to do this. The far right column reflects Eqn.1 applied to the data.

Problem 5(20).

Consider the instrumentation system shown in Figure 2 where a signal S(t) is the input to the system and the output is the recorded signal T(t). The transducer has a gain G₁, and the recorder has a gain G₃. The gains and phase shifts are documented in Figure 2.

Figure 2. A multi-component measurement system.

If the output signal, T(t), is analyzed and found to be represented by the Fourier Series shown in Eqn.2, you are to find the proper Fourier Series expression for the input signal. Show your method clearly and comment on the DC level of the input signal, S(t).

Figure 3. Gain response data for the three instrumentation components, G1 and G2.

The specific components of the T(t) signal are at the frequencies shown on the plots. The two gains, G₁and G₂, are constant with no phase shifts and thus have no effect on the first two harmonics (at w=2p→1Hz and w=20p→10Hz). In fact only G2 has any effect on the third harmonic (w=200p→1000Hz) with its gain reduced to 0.6 and a -p/4 phase shift. The following is an equation to compute the input to the sensor.

The DC component of the signal, S_DC, is 0.0 as shown in the above equation for S(T). Neither the sensor or the recorder reduces the DC component of the signal.

Problem 6(15).

For the bridge circuit in Figure 4, it can be shown that

If R₃ is a 120Ω strain gage and R₁, R₂, and R₄ are fixed 120Ω resistors, derive an expression for E_o/E_i in terms of DR/R.

Figure 4. A simple bridge circuit.

Applying the values of R₃=R+DR and R₁=R₂=R₄=R yields

The quantity and we neglect the 2RDR term to get the result