Article
citation information:
Zoul, V., P. Kováč. A brief overview about
the development of torsional vibration calculation and education of methods for
their calculation. Scientific Journal of
Silesian University of Technology. Series Transport. 2018, 99, 205-211. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2018.99.19.
Václav ZOUL[1],
Petar KOVÁČ[2]
A BRIEF OVERVIEW ABOUT THE DEVELOPMENT OF
TORSIONAL VIBRATION CALCULATION AND EDUCATION OF METHODS FOR THEIR CALCULATION
Summary. Torsional vibration calculation (TVC) is among the
basic calculations that are required to support propulsion projects for various
vehicles, ships and construction machines. A number of powerful software
packages for these calculations already exists. However, the need to respect
the physical properties of propulsion systems that are more complex than before
has arisen. At the same time, knowledge about system properties and the
capabilities of computing systems are also growing, resulting in an increase in
new software systems, which could be used for these calculations. In this
paper, these new trends are briefly described, with attention paid not only to
practical use, but above all to how and to what extent these themes should be
presented to students.
Keywords: propulsion systems;
torsional vibration calculation; trends in development of torsional vibration
calculation; simulation methods; GT-Power
1. INTRODUCTION
TVC is among the basic calculations
that are required to support propulsion projects for vehicles, ships and
construction machines. These calculations have been known and applied for more
than 100 years initially in response to ship shaft snapping, which started with
use of piston steam machines in about 1870.
More calculation methods have been
developed since the 19th century and used for TVC. These methods are described
in detail in numerous publications, as well as currently used in practical
settings.
The basic principles of torsional
vibration are naturally presented on basic educational courses on machinery
dynamics. In the case of more advanced courses for students studying fields
related to the construction of piston engines and vehicles, torsional vibration
problems are explained in much more detail, with an emphasis on the connection
between basic mechanical vibrations and the practical application of TVC to
real propulsion systems. Students find this interesting, as they directly see
how knowledge from pure theory is reflected and used in praxis.
For the calculation of torsional
vibrations, methods are employed, which are the basis for commercial software.
There are even methods described in “bibles” in this field, such as
[1,2,3]. Typical torsional vibration programs, which are equipped not only with
proven computing procedures but also with subroutines enabling easy and fast
input and the economic output of results (user-friendly), are used by ship
classification societies, for example, “Det Norske Veritas” [4].
Let us not forget the last standard
[5], which gives a detailed overview of the whole issue.
At present, simulation methods are
increasingly being applied for TVC purposes, which must also be introduced to
students.
The purpose of this paper is to
introduce to involved experts the scope of the content presented to students on
courses covering TVC, as well as the latest developments in this field.
2. BASIC PRINCIPLE OF
THE COURSES
The lectures related to
TVC are based on the fact that, if professional computations are made, it is
also necessary to use one of the professional software packages. In the manuals
for such packages, there is usually enough information on how to use them.
However, there is often a lack of information on the theoretical foundation of
the software, which often means that less experienced users view the software
as a “black box”. For this reason, the theoretical basis of TVC and
the basic methods for its applications should be explained.
It should be also taken
in account that not every workplace will have access to highly sophisticated
software, bought at considerable expensive. As such, it is important to
recognize that it is possible to perform a number of less demanding
calculations using simpler means, such Excel.
3. NATURAL FREQUENCY
CALCULATION
Basic information about
an oscillating system facilitates the calculation of natural frequencies and
vibratory mode shapes (normal elastic curves), which correspond to them.
For this purpose, the
Holzer method [6] is presented, which, despite being first published in 1907,
is easy to apply due to simple algorithmization. In addition, this method,
based on the search for equality between kinetic and potential energy, gives a
good explanation of the physical principle of resonant phenomena.
The second method for
the calculation of natural frequencies, as presented in lectures, is based on
seeking out frequencies that correspond to the state, when the determinant of
the equation of motion reaches the value of 0. The calculation is again easy to
carry out using matrix algebra and macroprogramming, both of which are parts of
Excel.
For users with MATLAB at
their disposal, it is naturally easy to use the subroutine “eig”.
4. FORCED VIBRATION
CALCULATION
4.1. Steady states,
linear systems
TVC, until recently, was
based on the frequency domain approach and focused primarily on steady-state
solutions.
In lectures, the
so-called “direct method” is explained, in which the system of
differential equations is transferred, using the assumed form of the solution
(sinus), in a system of algebraic equations, which is then easily solved using
the MATLAB equation solver.
The method is, in
principle, suitable for solving linear differential equations. If some weak
non-linearities are present in the system, which can be replaced by their
equivalent linear values, such as equivalent linear stiffness or equivalent
linear damping, and when these non-linearities are dependent on the oscillation
frequency or the instantaneous mean load in the connections between masses,
these non-linearities can also be respected.
In lectures, the
transfer matrix method is also explained [7]. The method resolves physical
vibration problems in a similar way to the “direct method”; the
same applies to the steady-state vibration of linear systems. However, the
programming is more transparent, uses smaller matrixes and is therefore more
suitable for systems with fewer degrees of freedom and also for smaller
computers equipped with simple software. In addition, the principle can be
applied not only to torsional vibration but also to a combination of different
types of oscillation, such as linear or bending oscillation. Therefore, an
acquaintance with this method is useful.
Another common method is
the “modal method”, which transfers the original equation of motion
to a new system in which every equation only corresponds to one mode of
vibration. This method, as described, for example, by Ker Wilson [1] or in [2],
is common applied, even if it is not quite exact for systems with stronger
damping. In fact, it uses the results of the natural frequency calculations of
an undamped system to calculate forced vibration. However, this method has a
number of advantages, particularly in terms of low demand for computing
technique. There is also plenty of experience with this method. Nevertheless,
the method is especially included in lectures because it significant helps us
to understand what happens in the system when it oscillates and the effect of
excitation, damping, mass and stiffness, when they are located in certain
positions in the oscillating system.
4.1. Transition states,
non-linear systems
To solve transient states in
systems with non-linear equations of motion or a combination of both, it is
necessary to default to the solution in the time domain. This means it is
necessary to use a numerical integration method [8].
Naturally and generally
in these cases, it is the most common and most accessible to use the simulation
and model-based system Simulink, which is integrated with MATLAB.
For the application of
simulation calculations, particular in the case of dynamics in drive lines,
Latchet’s book [9] could be very helpful.
In recent years, this
approach has been increasingly successful in the simulation of various
processes including the dynamics of drive lines using the SimulationX software
produced by the ESI ITI company from Dresden (see Conference Proceedings [10]).
A recent application of
SimulationX, in connection with the model description language Modelica, was
described by D. Grimm from Daimler AG in his paper in [10]. Similar articles
e.g., [11] have been produced by MAN B&W, while [12,13] describes the
selection of calculating software for the Croatian Register of Shipping.
The aforementioned
packages are already far more adaptable to professional use. Efforts have also
focused on the rapid calculation of alternatives, offering an opportunity to
optimize design parameters quickly and effective [14].
5. SIMULATION, MODELING
IN GT-POWER
Rather than SimulationX,
another powerful simulation software package called GT-Power is at the disposal
of the Czech Technical University in Prague. The software is managed and
distributed by the US company, Gamma Technologies, from Westmont, IL.
GT-Power
is the industry-standard engine performance simulation package, which is
used to predict engine performance quantities, such as power, torque,
volumetric efficiency, fuel consumption and turbocharger performance. For
problems connected with torsional vibration, GT-Power can
facilitate the following calculations:
- Crankshaft
torsional vibrations (time and frequency domain)
- Viscous and
rubber dampers
- Engine
unbalanced force moments, balance shafts
- Three-dimensional
block vibrations and mounting system analysis, vibration isolators
It is
clear that the calculation possibilities are considerably wider in this regard
than for previous non-simulation methods. Calculations, as can be seen, also
include such effects that were not previously considered but are nevertheless
related to alternating torques and to torsional vibrations, which is evident,
for example, in the case of oscillations of the entire engine.
Another
undeniable advantage is that, if the excitation effects of working cycles in
individual cylinders (tangential pressures) are not known in advance, then it
is possible to calculate them using GT-Power. The introduction of more
sophisticated models for the stiffness and damping of elastic members, as well
as for vibration damping, involving vibration dampers and pendulum eliminators,
is one of the advantages of the system.
Last
but not least, GT-Power allows us to respect the influence of engine
regulation, which in turn facilitates a solution to problems such as diesel
engine “hunting” when idle.
These
are all reasons why GT-Power calculations are included in the learning process.
6. COMPARISON OF DIFFERENT METHODS
OF TORSIONAL VIBRATION CALCULATION
Although it would appear
that TVC and its results, using the different methods listed above, must be the
same; in practice, however, the matter is not so simple.
The main reason is that
each method works in some cases with slightly different models of
mass-stiffness system components, such as models of dynamic stiffness or
damping. This is especially true of computation in the frequency domain.
On the other hand,
numerical integration allows us to more accurately respect the dynamic
properties of system elements compared with simple equivalent linearization. In
turn, we can use more sophisticated models of elastic and damping elements,
such as those described in [15].
Another problem concerns
the phase shifts of each harmonic response component, which may be affected by
the damping present in the system. Inaccurate phase shifts could influence the
synthesis values of the torsional angle and load ([16], Chapter 4.19).
Therefore, only scalar sums should be considered in these cases.
To make the process of
entering input values into the program and clarify the way in which to
understand the calculation results, TVC was carried out using the
above-described methods. Each calculation was performed on the same diesel
generator assembly with an elastic coupling. Subsequently, the way in which
each of these methods respects the dynamical model of the drive line was
assessed, along with the difficulty in preparing the calculation and
determining time necessary to perform it.
The
first calculation methods, which will be mentioned further, employ the
“direct” method, while the second employ “modal
analysis”, both of which are programmed in MATLAB.
Both
these two calculations were compared with the GT-Power calculation. For
transient states, in this case the engine starting period, Simulink and
GT-Power were used.
Programming
with the use of the “modal method” seems to be easy and quick to
perform. It looks sufficient when the calculation of an elastic coupling is
necessary and when only one mode of vibration is decisive. For more complicated
systems, it is necessary to take into account the fact that the phase shift
between different vibratory modes is not quite correct. The calculation is
based on a simplified assumption about the normal elastic curves determined for
the undamped system. Therefore, the synthesis of vibration caused under
different vibratory modes and by different harmonic components of excitation
could involve scalar summation only, as mentioned above.
Programming
using the “direct method” is slightly complicated, mainly because
the extent of the resolved systems and thus the calculation are usually
greater. The method is widespread and used above all as a basis for
professional programs. The limitation is that it only solves stationary states
and only respects certain types of weaker nonlinearities.
The use
of Simulink is useful when no other software packages, specializing in
non-linear and transient torsional vibration, are available. Simulink
calculations, for the same drive line, produce the same results compared to
GT-Power calculations, but the time necessary for the latter calculations is
disproportionately higher.
7. SUMMARY AND
CONCLUSIONS
TVC is
one of the basic calculations needed for the design of drive lines. These
calculations have a long tradition and therefore developed to a very good
level.
When
teaching students, the basic methods are firstly described to help them
understand the nature and solution of the problem and to apply them using
simple, but generally accessible, computer software. Afterwards, the
application of existing professional software packages is shown.
However,
methods are developing all the time, while a recent trend in using simulation
programs is clearly visible.
They
are mainly two software packages used in this field: SimulationX and GT-Power.
These packages are able to solve similar problems in a similar range, to a much
large extent than in the case of classic older programs.
GT-Power
is available at the Czech Technical University, meaning that its students are
familiar with the software, at least in terms of the basic forms of
calculation.
In the
course of conducting the activities mentioned in the paper, selected examples
were presented, which could be solved by students in seminars.
This
theme will continue with investigations into other more complicated
applications of GT-Power for advanced systems, such as in the form of a
solution to a system with a two-mass flywheel or with a pendulum torsional
vibration eliminator.
Nevertheless,
improvements in the calculation methodology should continue to be made, but
only in direct connection with the calculation of real drive lines and by
comparing the calculated values with the measurement results.
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Received 05.03.2018; accepted in revised form 26.05.2018
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