Problems EXC 4 1920

Fundamentals of measurement and automation technology 1 (WS 2019/20)
4th Exercise: 50. und 51. KW

Error calculation. Dynamic characterization of zero order transmission elements.
Error calculation
Note: To solve the following tasks, also use the "Error Calculation Guide" available in the
network.
Problem 1L: water bath for chocolate coating

With the help of a water bath, chocolate coating (for decorating Christmas cookies) is to be kept at
a temperature of about 40°C. The temperature of the water bath is measured with a type K ther-
mocouple, which has an error of 2.5 K. The thermocouple is directly connected to a reference junc-
tion with a temperature of 0°C. This operating mode causes an error of 0.4% of the measuring
range, which is 100 K. The following measured values have been recorded.
number of measurement 1 2 3 4 5 6 7 8
ϑTE, i in °C 43,4 41,9 45,2 42,7 43,7 44,6 42,3 43,6
a) Determine the mean temperature ϑTE for the measurement with the thermocouple and the
associated absolute maximum error. Specify the result in the form ϑ = ϑTE + ∆ϑ TE,abs,max .
b) To reduce the uncertainty of measurement for ϑ an additional measurement is performed
with a mercury thermometer (measuring range from 20°C to 70°C) of error class 1. The
measured value is ϑHg = 43,1 °C . Which weighted mean value ϑres and which associated er-
ror ∆ϑres result for the temperature ϑ ? Specify the result in the form ϑ = ϑres + ∆ϑres .
Problem 2: relative probable error

Calculate the relative probable error of the following quantities
x 12
a) y 1 =
2x 2
b) y 2 = x 1 + x 2
3
x3
x + 2x 2
c) y 3 = 1
4x 2
n 2
 ∂y ∆xi 
first using the general expression ∆yrel,prob = ∑ 
i =1 
⋅ 
∂xi y 
from the Gaussian law of error
propagation and then using the rules for special cases, if possible.
Problem 3B: best fit line

During the calibration of a measuring system, the following measured values of the output variable
y were determined as a function of the input variable x:
x 1 2 3 4 5 6 7 8 9 10
y 0,67 2,23 3,46 5,37 4,35 5,93 6,00 9,22 8,89 9,32
1
The functional relationship is to be described with a compensation straight line y = ax + b. The

coefficients a and b, their errors and the correlation coefficient r are to be determined!
Problem 4L: radiation power measurement

When measuring the radiant power of a test specimen, the following measured values for the ra-
diant power P were determined as a function of the temperature T:
T in K 275 300 325 350 375 400 425 450 475 500
P in W 0,1028 0,0865 0,1373 0,31 0,3316 0,289 0,5644 0,5701 0,7676 0,8975
The functional connection between T and P is given by P(t) = p1T 4 + p0 .

a) Transform the measured values appropriately so that you can subsequently work with a linear
regression.
b) For the linear regression line, determine the slope, the axis intersection, and their errors.
c) Use the results from b) to determine the coefficients p0 and p1 of the functional relationship.
Dynamic characterization of zero order transmission elements
Note: To solve the following tasks, also use the "Static and dynamic characterization of
measuring elements" script available in the network.
Problem 5B: rotor turning mass

A drive torque MA and a braking torque MB act on a rotor turning mass. The output signal is the
change in the speed of the shaft. The input signal represents the difference between the two tor-
ques. Determine the type and characteristic values of the transmission element.
2
Problem 6: piezoelectric force transducer

A piece of quartz cut out of a single crystal is used as a piezoelectric force transducer. The upper
and lower cut surface is metallized and serves as an electrode. If a force acts on the quartz disc,
charges are induced, whereby the amount of charge increases linearly with the applied force with
the proportionality factor k. The quartz can be understood as a capacitor that stores charges,
which makes it possible to tap a voltage, see picture. The quartz block has the base area A, the
thickness d and the relative dielectric constant εr . Determine the type of transmission element
when the input signal is the change of the applied force ∆F and the output signal is the resulting
voltage change ∆U .
Problem 7: running rope

The speed of a pulley is determined by the speed of a running rope. The input variable is the dis-
placement of the rope, the output variable is the resulting change in the angular speed of the
pulley ∆ω. The search is for ∆ω = f(∆s). Which transmission element with which characteristic values
does this correspond to?
3
Problem 8: spring-damper-lever-system
Given is the shown mechanical system of a spring-damper-rod-system. The displacement change
of the suspension point ∆s is the input signal xe, which via a lever (with lever arms a and b) should
lead to the change of the force F, so that the lever remains in equilibrium. This is the output signal:
x a = ∆F . Assume a force balance to determine the type and characteristic quantities of the trans-
mission element. Consider the relationships for the spring force Fc and the damping force FD. The
spring constant c and the damping constant k are given.
____________________________________________________________________________________
Note: Tasks with B are example tasks, tasks with H are to be solved as homework
and tasks with L are tasks for the learning group letter.

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Fundamentals of measurement and automation technology 1 (WS 2019/20)

4th Exercise: 50. und 51. KW

Problem 1L: water bath for chocolate coating

Problem 2: relative probable error

Problem 3B: best fit line

The functional relationship is to be described with a compensation straight line y = ax + b. The

Problem 4L: radiation power measurement

The functional connection between T and P is given by P(t) = p1T 4 + p0 .

Dynamic characterization of zero order transmission elements

Problem 5B: rotor turning mass

Problem 6: piezoelectric force transducer

Problem 7: running rope

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