Download Project 1 Groups A - E

The University of Jordan
Department of Mechatronics Engineering
0908341 Measurements and Instrumentation
Dr. Mohammad Kilani
Group Project 1 (Due 26 June 2012)
Problem 1:
A liquid-in-glass thermometer is to be used for measuring a dynamically changing temperature signal. The liquid inside
the thermometer’s bulb has a mass 𝑚 and a specific heat 𝑐. The bulb has a surface area 𝐴, and a convection heat transfer
coefficient ℎ. The temperature in the thermometer at time 𝑇(0) = 𝑇𝑜
1.
Using an energy balance on the thermometer bulb to derive the differential equation relating the temperature of the
liquid in the bulb, 𝑇(𝑡), to the temperature of the medium to be measured 𝑇𝑚 (𝑡), and write the differential equation
in the following standard form:
𝑑𝑇
𝜏
+ 𝑇 = 𝑇𝑚 (𝑡)
𝑑𝑡
where 𝜏 is the time constant. Express 𝜏 in terms of 𝑚, 𝑐, ℎ, and 𝐴.
2.
Find an analytical expressions for 𝑇(𝑡) for 𝑡 ≥ 0 for (i) step input, 𝑇𝑚 (𝑡) = 𝑇∞ , (ii) ramp input, 𝑇𝑚 (𝑡) = 𝑘𝑡, and (iii)
periodic input, 𝑇𝑚 (𝑡) = 𝐴𝑐𝑜𝑠(Ω𝑡).
3.
Make a plot for 𝑇(𝑡) vs. 𝑡 for 0 ≤ 𝑡 ≤ 10𝜏, for (i) step input, 𝑇𝑚 (𝑡) = 100, (ii) ramp input, 𝑇𝑚 (𝑡) = 10𝑡, and (iii)
periodic input, 𝑇𝑚 (𝑡) = 100𝑐𝑜𝑠(2π𝑡). Let 𝑇𝑜 = 0 in the plots and generate four different curves on the same plot
for 𝜏 = 0.1 s., 𝜏 = 1 s., 𝜏 = 5 s. and 𝜏 = 10 s.
4.
If the liquid in the thermometer is ethanol (density ρ = 800 kg/m3 , and specific heat capacity c = 2.4 × 103 J/kg℃ )
and the bulb of the thermometer has a cylindrical shape with an outer diameter of 3 × 10−3 m, find the minimum
exposed length 𝑙 of the cylindrical shape of the bulb needed to give (i) a time constant τ = 0.1, (ii) steady state error
in case of ramp input of less than 2/𝑘 ℃, and (iii) amplitude ratio in case of periodic input of less than 50%.
5.
The thermometer in part (4) is to be modeled as a simple RC circuit consisting of a capacitor, a resistor, and voltage
source connected in series. Design the circuit and select suitable values for R and C.
Problem 2:
A spring scale is to be used for measuring a dynamically changing force signal. The scale may be analyzed as a simple
single degree of freedom spring-mass-damper system with mass 𝑚, spring stiffness 𝑘, and damping coefficient, 𝑐.
Assume an initial deflection of the scale 𝑥(0) = 0.
1.
Use Newton’s 2nd law of motion to derive the differential equation relating the deflection of the spring, 𝑥(𝑡), to the
excitation force 𝑓(𝑡), and write the equation in the following standard form:
𝑑2𝑥
𝑑𝑥
1
+ 2𝜁𝜔𝑛
+ 𝜔𝑛 2 𝑥 = 𝑓(𝑡)
2
𝑑𝑡
𝑑𝑡
𝑚
where 𝜔𝑛 is the natural frequency and 𝜁 is the damping ratio. Express 𝜔𝑛 and 𝜁 in terms of 𝑚, 𝑐 and 𝑘.
2.
For a step excitation force 𝑓(𝑡) = 𝐹,
a. Find an analytical expressions for 𝑥(𝑡) for 𝑡 ≥ 0.
b. Define the response ratio 𝑅𝑟 = 𝑥 ⁄𝑥𝑠 where 𝑥𝑠 = 𝐹/𝑘 and plot 𝑅𝑟 vs. 𝜔𝑛 𝑡 for 0 ≤ 𝜔𝑛 𝑡 ≤ 4𝜋 . Generate five
different plots on the same graph corresponding to 𝜁 = 0, 𝜁 = 0.25, 𝜁 = 0.5, 𝜁 = 1 and 𝜁 = 2.
3.
For a periodic excitation force 𝑓(𝑡) = 𝐹𝑜 cos(𝜔𝑡)
a. Find an analytical expressions for 𝑥(𝑡) for 𝑡 ≥ 0.
b. Define the amplitude ratio 𝐴𝑟 = 𝑥𝑜 ⁄𝑥𝑠 where 𝑥𝑜 is the amplitude of the steady response and 𝑥𝑠 = 𝐹𝑜 /𝑘, and
define the frequency ratio 𝐹𝑟 = 𝜔⁄𝜔𝑛 and plot 𝐴𝑟 vs. 𝐹𝑟 on a logarithmic scale for 0 ≤ 𝐹𝑟 ≤ 4. Generate six
different plots on the same graph corresponding to 𝜁 = 5, 𝜁 = 2, 𝜁 = 1, 𝜁 = 0.7, 𝜁 = 0.3 and 𝜁 → 0.
c. Plot the phase shift of the sinusoidal steady response vs. 𝐹𝑟 for 0 ≤ 𝐹𝑟 ≤ 4.
d. What value of ζ would result in the broadest frequency range over which 𝐴𝑟 ≅ 1