Simulation of the misalignment process of an overhead crane in Matlab / Simulink
Komarov Alexandr1, Grachev Alexey2, Gabriel Anton3
1тел: +79117123468, komarov.ae@edu.spbstu.ru
2тел: +79052638554, grachev_aa@spbstu.ru
3тел: +79992020689, gabriel_as@spbstu.ru
Abstract
The object of the study is the angle of deviation of the crane from the perpendicular to the rails in a
horizontal plane. The aim of the work is the mathematical description of the crane’s bridge beam misalignment,
creation of the model in Matlab Simulink. Based on the resulting model, there was implemented a control system that
compensates for emerging misalignment by speeding up or slowing down crane drives.
Key words
Overhead crane, overhead crane’s misalignment, crane model in Matlab/Simulink, laser ranging device,
registering of misalignment, crane control system.
Introduction
The problem of the frame’s misalignment on the horizontal plane is relevant for all types of overhead cranes.
There are different reasons for the misalignment: uneven abrasion of wheels, the roughness of the crane track (crane
rails). The main parameter for the misalignment emerging is the mass center of system crane-load shift relative to the
crane’s center of symmetry. That results in the crane’s drives uneven loading which leads to running of one support
relative to another – misalignment.
Usually, this problem is not fixed by the manufacturer. It is known as a common issue with such types of
cranes. However, the misalignment causes the occurrence of additional stresses in a crane’s construction and the
installation structure. Uneven parts production, a decrease in the operation period, and even crane collapse can be
the result of the misalignment of the crane’s moving.
In the article [1] the usage of the additional adjusting control system was proposed for receiving the crane
operator’s control signals, adjustment of them, and generating a new control action on the drives. From the review of
the existing sensors and their installation for detecting the misalignment, it was determined that a laser ranging
device is best used as a sensor.
In the following article, it is considered a mathematical description of the crane’s misalignment process and
the development of a simple control system for fixing the misalignment.
1. Mathematical description
It is needed to create a crane’s mathematical model which will meet the required parameters to implement a
model of the control system in Matlab. To select a modeling object the overhead crane market was reviewed to find a
manufacturer which provides detailed technical specification of the crane in open access. Stahl Crane Systems
company was selected because of the review. A crane with the following technical parameters was selected as an
example for the modeling [2].
Table 1. Main technical characteristics of the crane
Load capacity
Crane weight
Crane span length
Max speed
Wheels’ diameter
16 000 kg
4 300 kg
15 000 mm
40 m/min
315 mm
More detailed specifications are described in the company’s catalog. Based on the technical specifications
provided by the manufacturer and overhead cranes’ design features [3][4] there was defined and selected all
parameters for the creation of the required crane mathematical model such as the wheels’ width and bearing rib,
crane rails type and dimensions. The block diagram of the created model is shown in Figure 1.
Figure 1. Scheme of the mathematical description problem
A search for scientific articles on this topic has revealed that most publications are about descriptions and
suggestions for solution to load swinging during transportation. But there are not so much researches about the topic
of the current work. However, the authors [5][6] consider the problem of the misalignment of metal structures of
overhead cranes from the point of view of stresses arising in the structure. Also, a method of mathematical
description of load swinging can be used for the current problem [7][8][9]. A mathematical model is described by
Euler-Lagrange formulas of the second kind, in which the Lagrangian operator is a difference between total kinetic
energy and total potential energy of the system. Crane’s x coordinate and angle of deviation of the crane from the
perpendicular to the motion vector are used as generalized coordinates. The scheme of the considered problem is
shown in Figure 2.
Figure 2. Scheme of the mathematical description problem
The potential energy of the system can be neglected in the current problem. Then the Lagrangian operator is
equal to the total kinetic energy of the system that consists of the kinetic energy of the rectilinear motion of the crane
and the load, the kinetic energy of crane’s rotation, and kinetic energy of the payload:
𝑚𝑥̇ 2
𝑇=
2
+
𝐽𝑐𝑟 𝛼̇ 2
2
+
𝐽𝑝𝑙 𝛼̇ 2
(1)
2
𝑚 – the sum of the crane mass and the payload mass;
𝐽𝑐𝑟 – moment of inertia of the crane rotation;
𝐽𝑝𝑙 – moment of inertia of the payload rotation.
To find moments of inertia for the crane and the payload, it is needed to know the coordinates of the mass
center, which depends on the position of the payload mass on the crane:
Where
𝐶𝑀 =
𝐿𝑐𝑟
2
+
𝐿𝑐𝑟
2
𝑚𝑐𝑟
𝑚𝑝𝑙
𝑋−
1+
(2)
𝐿𝑐𝑟 – length of the crane span;
𝑋 – the position of the payload relative to the left edge of the crane;
𝑚𝑐𝑟 – crane mass;
𝑚𝑝𝑙 – payload mass.
The moment of inertia of the crane rotation relative to the mass center can be described as a sum of two
moments of inertia of two rods rotating around the axis of rotation. The payload moment of inertia is described as a
rotation of a material point around the axis of rotation. As a result, the formula of the total kinetic energy has the
following form:
Where
𝑇=
(𝑚𝑐𝑟 +𝑚𝑝𝑙 )𝑥̇ 2
2
+
𝐶𝑀2
+𝐿𝑐𝑟2 +𝐶𝑀2 −3𝐿𝑐𝑟𝐶𝑀)𝛼̇ 2
𝐿𝑐𝑟
𝑚𝑐𝑟 (2
6
+
𝑚𝑝𝑙 (𝐶𝑀−𝑋)2 𝛼̇ 2
(3)
2
The traction force of the drives acting on the crane can be found by newton’s second law. Thereby, the
generalized force for the x coordinate is presented in the formula (4).
𝑄1 = 𝑅𝑤 cos 𝛼 ((
𝑚𝑐𝑟
2
+ 𝑚𝑝𝑙
𝐿𝑐𝑟 −𝐶𝑀 𝑑𝜔1
𝐿𝑐𝑟
)
𝑑𝑡
+(
𝑚𝑐𝑟
2
+ 𝑚𝑝𝑙
𝐶𝑀 𝑑𝜔2
𝐿𝑐𝑟
)
𝑑𝑡
)
(4)
Unequal values of the torques of the drives directed towards each other affect the crane rotation. The
generalized formula for the 𝛼 coordinate is presented in the formula (5).
𝑄2 = 𝑅𝑤 (𝐶𝑀 (
Where
𝑚𝑐𝑟
2
+ 𝑚𝑝𝑙
𝑅𝑤 – wheel radius.
𝐿𝑐𝑟 −𝐶𝑀 𝑑𝜔1
𝐿𝑐𝑟
)
𝑑𝑡
− (𝐿𝑐𝑟 − 𝐶𝑀) (
𝑚𝑐𝑟
2
+ 𝑚𝑝𝑙
𝐶𝑀 𝑑𝜔2
𝐿𝑐𝑟
)
𝑑𝑡
)
(5)
In the current work, the friction force of the wheel edges on the rails is not considered as if the control
system operates correctly and there is done a proper maintaining of the crane rails and the crane those forces will be
random and short. So, there is no need to model them. Substituting the obtained equations 3, 4, 5 into the EulerLagrange formula of the second kind, we get:
𝑅𝑤 𝑐𝑜𝑠 𝛼((
𝑥̈ =
{
𝛼̈ =
𝑚𝑐𝑟
𝐿 −𝐶𝑀 𝑑𝜔1
𝑚
𝐶𝑀 𝑑𝜔
+𝑚𝑝𝑙 𝑐𝑟
)
+( 𝑐𝑟 +𝑚𝑝𝑙 ) 2)
2
𝐿𝑐𝑟
𝑑𝑡
2
𝐿𝑐𝑟 𝑑𝑡
(𝑚𝑐𝑟 +𝑚𝑝𝑙 )
𝑚𝑐𝑟
𝐿 −𝐶𝑀 𝑑𝜔1
𝑚
𝐶𝑀 𝑑𝜔
+𝑚𝑝𝑙 𝑐𝑟
)
+(𝐿𝑐𝑟−𝐶𝑀)( 𝑐𝑟 +𝑚𝑝𝑙 ) 2)
2
𝐿𝑐𝑟
𝑑𝑡
2
𝐿𝑐𝑟 𝑑𝑡
3
𝐶𝑀
(𝑚𝑐𝑟 (2
+𝐿𝑐𝑟 2 +𝐶𝑀2 −3𝐿𝑐𝑟 𝐶𝑀)+3𝑚𝑝𝑙 (𝐶𝑀−𝑋)2 )
𝐿𝑐𝑟
(6)
3𝑅𝑤 (𝐶𝑀(
One of the main reasons for the crane misalignment occurrence is the dependence of the drive rotation
speed on the applied load torque. Maximum driver speed must be limited depending on the coordinate of the payload
position and its mass while modeling in Matlab. In the work [10] it is described a detailed model of the crane drives
which is redundant in this paper. In further work it is planned to make a more realistic model. As drives in the crane
model under study are used with frequency converter, such drive can be considered as a direct current motor with a
rigid mechanical characteristic from the point of view of the control system and mechanical characteristic.
Direct current motor is described using second-order aperiodic link consisting of two consecutive first-order
aperiodic links which describe electromagnetic and mechanical characteristics, respectively. Drive electromagnetic
characteristics with vector control might not be considered as it is extremely difficult to find electromagnetic
characteristics for the crane’s drive. Thus, the engine model is reduced to a first-order aperiodic link with PID control
and negative inverse connection on speed.
In the Stahl Crane Systems catalog [11] for crane drives, not enough parameters are provided to determine
drive mechanical characteristics. An analog of the Russian production with similar basic parameters was selected but
its parameters were also insufficient to determine the mechanical characteristic. Cage asynchronous motor АИР63В4
was selected. It is necessary to design a mechanical characteristic for the selected motor to implement in the model
the dependence of the maximum drive speed and the load torque generated by the motor. АИР63В4 motor
parameters are shown in table 2. The section of mechanical characteristics of interest is constructed for the selected
motor, as shown in Figure 3.
Table 2. АИР63В4 motor parameters
Parameter
Power, kW
Power factor
The ratio of the maximum moment to the
minimum
Synchronous speed, RPM
Speed at nominal torque, RPM
Marking
P
𝐾𝑝
ℎ
Value
0,33
0,76
2,2
𝑛0
𝑛𝑛
1500
1325
Figure 3. Section of the mechanical characteristic
Speed range from 1500 rpm to 1000 rpm can be considered linear and can be described by the equation of
a straight line with a slope coefficient 𝑘𝑛 = 77,6. Thus, the maximum speed of the motor is described by the formula
(7).
𝜔𝑚𝑎𝑥 = 𝑛0 −
Where
𝑍 – gear reduction rate;
𝑀𝑛 𝑍
𝑀𝑟 𝑘𝑛
(7)
𝑀𝑛 – nominal torque;
𝑀𝑟 – moment of resistance (load) acting on the drive.
The authors [12] have obtained a formula (8) for calculating the moment of resistance acting on the drives.
𝑀𝑐 = (𝑄 + 𝐺) (𝑘 +
𝑓𝑑
2
) 𝑘𝑝
(8)
𝑄, 𝐺 – crane and load masses, respectively, measured in tons;
𝑘 – rolling friction coefficient of the steel wheel on the rail;
𝑓 – reduced coefficient of friction in bearings;
𝑑 – axle diameter;
𝑘𝑝 – coefficient of friction of the rib.
The coefficient 𝑘𝑝 is assumed to be equal to one as the friction force of the wheels on the edges is not taken
into consideration.
As the control system is working with the step of the discreteness of ranging devices used to measure the
covered distance and drives’ speed is needed to be controlled, it is necessary to implement the numerical
differentiation block. The system will get the speed of the movement from the movement function due to the block. As
a method of numerical differentiation, the finite difference method on a three-point scheme is used.
Where
2. Matlab/Simulink implemetation
Figure 4 shows the final crane drive model developed in Matlab Simulink. For ease of use of the drive
model, the linear speed of the movement is used as input and output signals, and not the speed of rotation of the
motor itself. The maximum speeds depending on the load torque calculated in the previous part are specified as the
parameters of the restrictions in the saturation block.
Figure 4. Drive model in Matlab Simulink
The final model of the crane developed in Matlab Simulink is shown in Figure 5. The linear speeds of the left
and right drives are fed to the input of the system, respectively. Then these values are differentiated to obtain the
values of the accelerations of the left and right parts of the crane. Parts of the equations from formula (6) are written
in the amplification blocks. After the amplification blocks, the values of linear acceleration (in the upper part of the
diagram) and angular acceleration of the crane rotation (in the lower part of the diagram) are displayed. Using
integration blocks, the final output parameters of the model are:
− linear acceleration of the center of mass;
− linear speed of the center of mass;
− coordinate of the center of mass position;
− angular acceleration of crane rotation around the center of mass;
− the angular speed of the crane rotation around the center of mass;
− the angle of deviation of the crane from the motion vector in radians;
− the angle of deviation of the crane from the motion vector in degrees.
Figure 5. Crane model in Matlab Simulink
Also, Figure 6 shows a block that calculates the coordinates of the left and right parts of the crane, which will
be the output coordinates of the entire crane model. The final crane model is shown in Figure 7.
Figure 6. Block for calculating the coordinates of the right and left parts of the crane
Figure 7. Final crane model in Matlab Simulink
First of all, the control system must contain sensors that will ensure the availability of up-to-date information
about the position of the crane in space. Within the framework of work [1], a pair of laser rangefinders were selected
as sensors. Also, studies have shown that laser range finders are the least susceptible to changing environmental
conditions and it is more expedient to use them as sensors of the system [13] [14] [15].
The crane model, created in Matlab Simulink, outputs the positions of the right and left parts of the crane as
output signals. This already performs the functionality of rangefinders, but it is important to take into account that the
rangefinder signal is not constant, and the data on the position of the crane is discrete. It is also important to note that
rangefinders have some measurement error, and sometimes noise and interference are superimposed on the
measurements. In this paper, the noise power applied to the rangefinder signal is assumed to be zero, since noise
filtering requires a separate review and selection of filters.
Thus, the rangefinder model is reduced to a signal sampling block and adding white noise to this signal. At
the input, such a module takes the position of one of the sides of the crane, obtained from a previously made model,
and at the output, the distance measured by rangefinders, which is close to the real one. The resulting diagram of the
rangefinder block is shown in Figure 8.
Figure 8. Model of a laser rangefinder in Matlab Simulink
A PI regulator is used as a regulator of the control system. The difference between the readings of the laser
rangefinders is fed to the input of the regulator and the regulator reduces this difference to zero. The output of the
regulator is connected to the previously created numerical differentiation block to obtain the value of the correction
speeds for the drives.
The model of the control system in Matlab Simulink is shown in Figure 9. Figure 10 shows a model of the
entire system. It is a block of the crane model described earlier, two laser rangefinder modules and a control system
block.
Figure 9. Control system model in Matlab Simulink
Figure 10. Complete system model
3. Experiment in Matlab/Simulink
Figure 11 shows a graph of the drive transition process, the model of which is shown in Figure 4. The
desired speed is set to 0.6 m/s. This value is the maximum speed for the crane model under study. There is no load
on the drive, and it reaches the set speed of rotation.
Figure 11. Graph of the transition process in the drive model
To confirm the adequacy of the model and its compliance with the specified requirements, a number of
experiments were conducted. Figures 12 and 13 show the readings of the graphs of the model with the position of the
load in the center of the crane bridge and shifted to the right side of the direction of movement, respectively.
Figure 12. Graphs of model parameters with a load in the center of the crane
Figure 13. Graphs of model parameters with shifted load
According to the graphs, it can be concluded that the model describes the movement and misalignment of
the crane. There is no need to limit the misalignment angle, because the control system must minimize the angle and
it must not reach values that are unacceptable from the point of view of the geometry of the structure.
Figures 14 and 15 show the graphs of transition processes with the misalignment compensation system
turned off and on. The tests were carried out on the model shown in Figure 10. In this case, a load of 16,000
kilograms is located in the middle of the right half of the crane bridge. The second graph is scaled to demonstrate the
transition process in the amount of the crane misalignment.
Figure 14. Graph of rangefinder readings with the misalignment compensation system turned off
Figure 14. Graph of rangefinder readings with the misalignment compensation system turned on
According to the graphs showing one of the many experiments performed, it can be judged that such a
model is adequate. The amount of misalignment reaches 45 mm, which can be considered acceptable, because with
such geometrical characteristics of the crane and wheels, the misalignment value at which the wheel flanges will not
touch the rails can reach 100 mm.
Conclusion
As a result of this work, a mathematical model was obtained that describes the process of misalignment of
the load-bearing beam of an overhead crane. Based on the obtained mathematical model, a model was implemented
in Matlab Simulink, which reproduced the misalignment of the crane depending on the specified parameters. Then a
control system was implemented and simulated, which included two laser rangefinders as sensors that measure the
amount of misalignment, and a crane control block that corrects the speeds of the drives and thereby minimizes the
amount of misalignment of the crane.
Through a series of experiments, it was revealed that the developed control system copes with the task of
countering the crane misalignment. In the further development of the work, it is planned to improve the crane model
and control system in order to bring the processes closer to real ones.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Komarov A.E., Grachev A.A., Gabriel A.S. Analysis of the possibility of installation of the underway crane
crane beam control. 2020. No. 20. P. 17–26.
Official catalog of models of overhead cranes of the Stahl Crane Systems company [Electronic resource].
URL: https://stahlnw.ru/wp-content/uploads/docs/brochures/kranovye-sistemy-STAHL.pdf (date accessed:
04/26/2021).
Zhegulsky V.P., Lukashuk O.A. Design, construction and calculation of bridge crane mechanisms: a tutorial.
2016.184 p.
Sitthipong S. и др. Design analysis of overhead crane for maintenance workshop // MATEC Web Conf. 2018.
Т. 207. С. 1–5.
Lebedev V.A. and others. Research of malfunctions of metal structures of bridge-type cranes. P. 171-177
Galdin N.S., Kurbatskaya S.V., Kurbatskaya O.V. Mathematical modeling of the resistance force to the
movement of the bridge crane. S. 5-8.
Zhang Z., Chen D., Feng M. Dynamics model and dynamic simulation of overhead crane load swing systems
based on the ADAMS // 9th Int. Conf. Comput. Ind. Des. Concept. Des. Multicult. Creat. Des. - CAIDCD
2008. 2008. С. 484–487.
Kuznetsov A.P., Markov A.V., Shmarlovsky A.S. Mathematical models of gantry cranes. 2009.Vol. 8, No. 46.
P. 93–100.
Ismail R.M.T.R. и др. Nonlinear dynamic modelling and analysis of a 3-D overhead gantry crane system with
payload variation // EMS 2009 - UKSim 3rd Eur. Model. Symp. Comput. Model. Simul. 2009. С. 350–354.
Bukhlakov A.M., Gilev S.E., Zyuzev A.M. Imitation model of the electric drive of the bridge crane carrier.
2017.S. 326–329.
The official catalog of models of wheel blocks of the Stahl Crane Systems company [Electronic resource].
URL:
https://d1dv5w06e8cxfl.cloudfront.net/fileadmin/user_upload/Dokumente/Drucke/Produktinfos/Pi_Radblock_2
019-01.pdf (date accessed: 04/26/2021).
Karzhavin V.V., Kamenskikh S.F., Dushanin I.V. Calculation of crane mechanisms. Course project. 2013.115
s
Ryde J., Hillier N. Performance of laser and radar ranging devices in adverse environmental conditions // J. F.
Robot. 2009. Т. 26, № 9. С. 712–727.
Formsma O. и др. Realistic simulation of laser range finder behavior in a smoky environment // Lect. Notes
15.
Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics). 2011. Т. 6556 LNAI. С.
336–349.
Li J.F. и др. A novel positioning system of overhead crane // Proc. - Int. Conf. Electr. Control Eng. ICECE
2010. 2010. С. 209–212.
Отзывы:
Авторизуйтесь, чтобы оставить отзыв