=Paper=
{{Paper
|id=Vol-3896/paper21
|storemode=property
|title=Mathematical and computer modelling to assess accuracy and adequacy in torque measurement
|pdfUrl=https://ceur-ws.org/Vol-3896/paper21.pdf
|volume=Vol-3896
|authors=Andrii Sverstiuk,Taras Dubynyak,Nadia Shostakivska,Myroslava Yavorska,Mykola Poshyvak
|dblpUrl=https://dblp.org/rec/conf/ittap/SverstiukDSYP24
}}
==Mathematical and computer modelling to assess accuracy and adequacy in torque measurement==
https://ceur-ws.org/Vol-3896/paper21.pdf
Mathematical and computer modelling to
assess accuracy and adequacy in torque
measurement
Andrii Sverstiuka,b,∗,†, Taras Dubynyakb,†, Nadia Shostakivskab,†, Myroslava Yavorskab,†
and Mykola Poshyvakb,†
a
I. Horbachevsky Ternopil National Medical University, Maidan Voli, 1, Ternopil, 46002, Ukraine
b
Ternopil National Ivan Puluj Technical University, Rus'ka str. 56, Ternopil, 46001, Ukraine
Abstract
In this paper, a mathematical and computer modelling approach is considered to assess the accuracy
and adequacy of torque measurement by strain gauge dynamometer. Computer modelling allows
predicting the performance of a strain gauge dynamometer and simulating it under various
mechanical and physical data. Strain gauging is based on the phenomenon of the strain effect, which
is a change in the active resistance of the conductor of the primary transducer (strain gauge) under the
influence of mechanical stresses and strains. The use of strain gauges in scientific and technical
research allows monitoring deformations and stresses under static and dynamic loads. The main task
of this work is to develop and control the torque on the motor shaft, and to assess the adequacy of the
mathematical model and the degree of accuracy of the results obtained using the developed model of
the experimental data or test problem
Keywords ⋆1
strain gauge, shaft torque, engine, mathematical model, computer modelling
1. Introduction
Tensometry, as a set of methods and tools for determining the stress-strain states of objects and
structures, is widely used not only to measure the degree of deformation, but also to determine
the weight in the control of belt conveyors, the weight of vehicles (cars, railroad cars), to
substantiate the reliability and safety of nuclear power structures, etc [1].
⋆
ITTAP’2024: 4th International Workshop on Information Technologies: Theoretical and Applied Problems,
October 23-25, 2024, Ternopil, Ukraine, Opole, Poland
1∗
Corresponding author.
†
These authors contributed equally.
sverstyuk@tdmu.edu.ua (A. Sverstiuk); d_taras@ukr.net (T. Dubyniak); shostakivska@ukr.net (N. Shostakivska);
myavorska@gmail.com (M. Yavorska); claruspuer01@gmail.com (M. Poshyvak)
0000-0001-8644-0776 (A. Sverstiuk); 0000-0003-1529-6951 (T. Dubyniak); 0000-0002-7732-6186 (N.
Shostakivska); 0000-0001-8033-7348 (M. Yavorska); 0009-0009-9655-4123 (M. Poshyvak)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
Research in the field of torque measurement using strain gauges has been actively
developing for several decades. One of the pioneers in this field are scientists who studied the
accuracy and reliability of strain gauge measurements.
Special attention should be paid to the works of American and European researchers, in
particular, R. Williams and K. Martin, who studied the influence of temperature and vibration on
the accuracy of measurements [2].
In addition, the German scientist Hans-Peter Schmidt is known for his work on increasing
the sensitivity of tension dynamometers through the optimization of design solutions. In the
context of the adequacy of mathematical models, a significant contribution was made by
Japanese scientists such as Y. Takehara, who developed effective methods for correcting errors
caused by non-stationary operating conditions. Their research laid the foundation for modern
approaches to modeling and accuracy analysis of measuring systems using strain gauges [3].
The more we know about the object of study and the better and more accurate its
mathematical model we can build, the more accurately we can predict its behavior and manage
it more effectively. Mathematical modeling of the behavior of processes of various origins and
their optimal control are of key importance for conducting systematic research in many areas of
activity [4].
The main objective of this work is to develop a torque control unit that can be built into
existing standard equipment during its modernization or repair.
Testing of various types of engines and power generation equipment to improve design,
increase operational reliability, reduce thermal, mechanical, electrical and other losses, and
increase efficiency requires the creation of more advanced measuring equipment to determine
the torques transmitted by the rotating shaft.
The use of torque measurement equipment, which is caused by the need to determine the
power of power equipment, as well as to conduct operational and resource tests of mobile power
units (engines of tractors, agricultural machinery and other vehicles) to determine the load
resistance forces in mechanisms with a rotating output shaft and similar tasks and problems [5].
2. Overview of torque measurement methods and
tools
The applications for torque measurement devices are quite diverse, as are the requirements for
them. Therefore, there are many torque measurement systems available.
The classification of devices for measuring torque is based on various features: purpose,
design features, operating principle, operating conditions, accuracy of the information obtained,
etc [6].
The most convenient for use is the classification of torque measurement by the principle of
operation, according to which all instruments and devices for measuring torque can be divided
into:
Mechanical: mechanical brakes, electric brakes, aerodynamic brakes with differential
mechanism, inductor brakes, drive motors.
Hydraulic: chamber, volumetric, disk with oil film.
Optical: photoelectric, photoelastic, stroboscopic, optoelectric, optomechanical, holographic.
Electric, which are divided into:
1 with a sensor on a rotating shaft, capacitive, string, isotope;
2 with the sensor not on the rotating shaft - nonius, time phases;
3 strain gauges, which are divided into two types:
- with a sensor on the rotating shaft - non-contact, ohmic;
- with the sensor not on the rotating shaft - magnetoelastic, using eddy currents.
Most of them are used to measure torque in stationary and laboratory conditions.
Their advantages include simplicity of design and ease of use. However, they have a number
of disadvantages, including the inability to use them in the presence of vibrations, shaft runout,
large fluctuations in humidity and ambient temperature.
Based on the tests, it was found that only vibration-frequency, time, phase, ohmic, and non-
contact torque measurement methods are suitable for long-term and stable use in the field.
Vibration frequency sensors are manufactured with shafts made of special high-quality steels
[7]. Due to the variety of shaft sizes of tractors and agricultural machines, these sensors are not
widely used for torque measurement.
Time and phase methods of torque measurement are used on shafts with a large base (shaft
length) or a large torsion angle.
The disadvantages of these methods include the dependence of the measurement error on
the shaft speed and its runout. In this regard, time and phase methods of measuring torque can
only be used exclusively under local operating conditions. The simplest and most reliable way to
record torque is to directly measure the strain on the surface of the shaft under test using strain
gauges. Such measurements of torque are called ohmic, and in most cases, wire sensors are used.
As a rule, a bridge scheme with two to four load cells is used to measure the torque value,
which are glued at an angle of 45 to the axis.
The use of a bridge circuit increases sensitivity, improves linearity of the characteristics,
reduces the sensitivity of the transducer to bending deformations, as well as to stresses arising
from compression or tension of the shaft, and reduces the influence of temperature on the
measurement process [8].
The advantages of strain gauges are their small size and simple mechanical design, while the
disadvantages are the readings taken from a rotating shaft to a fixed measuring device and the
fact that the bridge circuit does not fully compensate for the effects of bending deformations.
3. Search for analogs
The prototype dynamometer for torque measurement is GD1L3/04 "Strain gauge dynamometer
for torque measurement". The invention relates to measuring technology and can be used to
measure the torque on the shafts of various machines.
The purpose of the invention is to increase the measurement accuracy of the strain gauge
and simplify its mounting on shafts.
The strain gauge shown in Figure 1 consists of a housing 1 with a drive part (sprocket,
pulley), in which ball bearings 2 are installed, which are seated with inner rings on the outer part
of the hub 3 and rest against the shoulder through a set of adjustable shims 4. The elastic
element 5 is made in the form of a hollow cylinder with flanges, with strain gauges 6 glued to its
outer surface, centered relative to the common axis of the tensiodynamometer by fitting a
special bore machined into the compressive surface of the rear flange onto the protruding outer
ring of the bearing, and connected by the front flange to the hub 3, and from the rear - to the
drive part on the body 1 [9].
Figure 1: Prototype of the torque measuring unit
Placing the housing 1 with the drive part on the bearings 2 relieves the elastic element 5
from the radial forces perceived by the drive part and also makes it possible to transmit torque to
the elastic element 5 from the hub 3 with little resistance. To ensure the rigidity of the
connection of the elastic element 5 with the hub and the body 1 with the drive part, each flange
connection has two reamer bolts 7, which are placed diametrically opposite, the remaining bolts
8 are of the usual design.
The connecting rod 9, in addition to its direct purpose - connecting the machine shaft with
the rotor of the current collector and the output conductors of the strain gauge circuit laid on its
surface, also serves for axial fixation of the strain gauge on the machine shaft with the help of a
washer 10. The strain gauge works as follows. The torque from the machine drive is transmitted
by means of a V-belt or chain to the drive part (pulley, sprocket) on the body 1. From it to the
elastic element 5, and then to the hub 3, and then through the keyway of the shaft on which the
strain gauge is mounted. The deformation of the elastic element is perceived by strain gauges 6,
the electric current of the unbalance of the strain gauge circuit is transmitted through the
conductors laid along the rod 9 to the measuring equipment through the current collector.
The use of an elastic element in the strain gauge in the form of a hollow cylinder with a wall
thickness selected from the ratio [10]
(1)
where dc is the diameter of the annular cross-section of the elastic element along the center
line, allows to create on its surface by varying the geometric dimensions of the annular cross-
section of the elastic element the maximum level of tangential stresses equivalent to the
measured torque and at the same time a significant value of the torsional stiffness of the elastic
element, which provides a high natural frequency of the strain gauge to increase the
measurement accuracy.
In addition, the design of the strain gauge with a hollow elastic element reduces the number
of additional parts and connections between the drive part and the machine shaft to the lowest
possible level, which greatly simplifies its mounting on the shaft and removal without any
disassembly, reducing assembly and manufacturing costs and metal consumption [11].
4. Comparative analysis and selection of a solution to
the task
Strain gauges are divided into two types according to their design features:
- with the placement of strain gauges on the moving elements of the drive;
- with the placement of strain gauges on non-moving drive elements;
The forces in the latter are transmitted to the strain gauges, which are fixed stationary, from
the drive, which can be rotated under the influence of torque due to the installation on bearings,
auxiliary transmissions (gearing on an elastic torsion shaft, a cable through a pulley on the axis
of rotation of the drive, installation of the drive on a lever, etc.) [12].
The advantages of these designs are the absence of current collectors from moving
elements, which reduces measurement error and simplifies the electrical part of the device. The
disadvantages are the bulkiness and complexity of the mechanical design, and the presence of
additional errors due to mechanical transmissions, which in turn creates impossible conditions
for the use of the device in difficult field conditions.
Strain gauges with strain gauges placed on moving elements, when optimally designed,
have minimal dimensions and a maximum simple structure that compensates for errors caused
by the information signal taken from rotating elements.
To solve this problem, many designs of both contact and non-contact types have been
created.
So, to solve this issue, we focus on the design of a device for measuring torque with strain
gauges mounted on the moving elements of the drive based on the copyright certificate
GD1L3/04 "Strain gauge dynamometer for measuring torque" [13].
5. Mathematical representation and calculation of
model adequacy
5.1 Estimates of strain gauge resistance measurement errors
Selecting a modeling object and building its mathematical model.
A mathematical model is used to describe certain properties of a projected object that
are significant at the stage of a particular design procedure [14]. It can reflect:
- a set and interconnection of the constituent elements of an object when solving
problems of linking structural elements to certain spatial positions (for example, PCB
tracing) or relative moments of time (for example, the sequence of technological
operations).
- geometric properties of the object in terms of spatial forms and the relative position
of its elements.
- quantitative and qualitative relationships between external conditions, process
parameters, and parameters of the system in which the processes under study occur.
Atthe same time, itis necessaryto indicate the area of adequacy of the adopted description (the
limits of the system parameters within which the adopted description reflects the properties of the
designed object with an accuracy not less than the specified one).
When modeling the behavior of a technical object in conditions close to its operating
conditions, it should be borne in mind that the technical object operates under the
influence of three main factors [15]:
1. energy source that causes the desired processes to occur in the object;
2. external actions from the environment; among these actions, one can distinguish
useful actions that also cause the necessary (useful) processes in the technical object, and
harmful actions that cause deviations in the functioning of the technical object from the
planned one;
3. a receiver (load), which is another real (material) object that perceives movement,
energy, or an information signal from the technical object in question; the receiver can
also influence the behavior of the technical object in both a positive and negative sense.
In view of the above, a model that simulates a technical object under conditions close
to its operating conditions should include the relevant components [16].
5.2 Mathematical formulation and method of solving the problem
At the initial stage of formulating a mathematical model, we turn to the conceptual
statement of the problem - a list of the main issues to be solved by means of mathematical
modeling, as well as a set of hypotheses regarding the properties and behavior of the modeling
object [17].
A conceptual model is built as an idealized model of an object.
According to the accepted hypotheses, a set of parameters describing the state of the object
and a list of laws describing the behavior of the object and the relationship between the object's
parameters and the environment are determined. A mathematical problem statement
(mathematical description of an object) is a set of mathematical relations that describe the
behavior of a modeling object. Let's turn to the analytical representation of the dependence of
the measured resistance on the supply voltage and the reference resistances in the measuring
bridge.
In this case, the working formulas are the ratios that define the effect of a change in
resistance in the bridge arm on the unbalance voltage:
(2)
where U0 is the supply voltage, r0 is the constant resistance, U is the unbalance voltage
and the effect of temperature on the change in resistances in the bridge arms:
. (3)
5.3 Implementation of the model in the form of a program
To model these dependencies, we used the MATLAB environment, in particular, work with
symbolic variables [18].
clear all
r0=1000;
U0=100;
% effect of changes in resistance in the bridge arm on the unbalance voltage
%(inverse characteristic)
syms U
dr=4*U*r0/U0/(1-2*U/U0)
subplot(3,1,1)
ezplot(dr,[0,5]),grid
% effect of temperature change on the change in resistance in the bridge arm
%(inverse characteristic)
T0=20;
t=[T0:10:T0+10*T0];
k1=.001;
k2=.000005;
R=r0*(1+k1*(t-T0)+k2*(t-T0).^2);
subplot(3,1,2)
plot(R-r0,t,'o'),grid
n=length(t);
Q(1:n,1)=R(1:n)-r0;
Q(1:n,2)=t(1:n);
T=1.5e-006*dr^3-.0015*dr^2+.84*dr+22;
% dependence of the measured resistance on the unbalance voltage
% (calibration characteristic)
subplot(3,1,3)
ezplot(T,[0,5]),grid
% dependence of the measured resistance on the unbalance voltage and
resistance
% in the arms of the balanced bridge (full calibration characteristic)
syms U r0
dr=4*U*r0/U0/(1-2*U/U0);
T=1.5e-006*dr^3-.0015*dr^2+.84*dr+22;
figure
ezsurf(T,[0,5,100,1000])
box
% sensitivity of the measuring unit to changes in the measured voltage and
resistance in
% arms of the balanced bridge
Su=diff(T,U)
Sr=diff(T,r0)
figure
subplot(2,1,1)
ezsurf(Su,[0,5,100,1000])
box
subplot(2,1,2)
ezsurf(Sr,[0,5,100,1000])
box
% measurement error
dU=1;
dr0=5;
dT=abs(Su)*dU+abs(Sr)*dr0
figure
ezsurf(dT,[0,5,100,1000])
5.4 Modeling results
-(40 U)/(U/50 - 1)
200
100
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
U
200
100
0
0 50 100 150 200 250 300 350 400
22 -...- (168 U)/(5 (U/50 - 1))
150
100
50
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
U
Figure 2: Inverse characteristic for assessing the effect of resistance change in the bridge arm on
the unbalance voltage (top graph); Inverse characteristic for assessing the effect of temperature
change on resistance change in the bridge arm (middle graph); Dependence of measured
resistance on unbalance voltage (bottom graph)
Figure 3: Dependence of the measured resistance on the unbalance voltage and resistance in the
arms of the balanced bridge (full calibration characteristic)
Figure 4: Sensitivity of the measuring unit to changes in the measured voltage and resistance in
the arms of a balanced bridge
Figure 5: Measurement error as a function of unbalance voltage and balancing resistance in the
bridge arm
Figure 6: Sensitivity of the measuring unit to changes in the measured voltage and resistance in
% of the balanced bridge arms
5.5 Checking the adequacy of the model
The adequacy of the model is understood as the degree to which the results obtained using
the developed model correspond to the data of the experiment or test problem.
The purpose of the adequacy test is to:
- ensure that the accepted set of hypotheses is valid at the stage of the conceptual and
mathematical model;
- establish that the accuracy of the results obtained corresponds to the accuracy specified in
the terms of reference.
In models for performing estimation calculations, an error of 10...15% is considered
satisfactory. In models used in control and monitoring systems, an error of 1...2% or less is
acceptable.
The reasons for the inadequacy of the mathematical model may be as follows:
– the values of the specified model parameters do not correspond to the permissible range
of these parameters;
– the accepted system of hypotheses is correct, but the constants and parameters in the
defining relations are not set accurately enough;
– the system of hypotheses used is incorrect.
If the results are inadequate, the model should be adjusted, considering the reasons in the
above sequence.
Errors in measuring the strain gauge resistance are a component of the torque measurement
error. The obtained estimates allow us to track the influence of an external factor (such as
temperature) and the accuracy of the system parameters (such as reference resistances in the
arms of the measuring bridge) on the final result - the value of the measured torque.
Conclusion
Errors in measuring the strain gauge resistance are a component of the torque measurement
error. Computer modelling allows to predict the performance of the strain gauge and simulate it
under various mechanical and physical conditions. The obtained estimates allow us to track the
influence of an external factor (such as temperature) and the accuracy of the system parameters
(such as reference resistances in the arms of the measuring bridge) on the final result - the value
of the measured torque.
References
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measurements. Journal of Measurement Science, 45(3), 120-135.
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environments. Mechanical Systems and Signal Processing, 67(1), 42-54.
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