A Multi Branch Torsional Vibration System

Published Date: 02 Nov 2017

or drive-train systems, such as mining machinery, petroleum and chemical

machinery, steel rolling machinery and automobiles[1-3]. For example, such

kinds of multi-branch torsional systems are found in coupled engine installations

using both turbines and reciprocating engines. The rapidly growing field of screw

compressors is a new addition to the similar multi-branch systems of marine

installations, auto differentials, and other geared installations. For instance,

recent advances in the shipbuilding industry have resulted in the construction of

large and powerful vessels with extremely complex propulsion systems. These

systems frequently are multi-branched, having two or more drive units.

Improper design parameters for machinery will cause serious torsional

vibration. Tortional vibration occurs when an excitation frequency gets close to

the natural frequency of the system. This will lead to noticeable sound

disturbances, severe shaking, and component fatigue problems. For example, it

is a commonly known fact that crankshaft failures can occur in internal

combustion engine driven installations when the operational speed range

contains significant torsional critical speeds. Because of the pulsating natural of

the gas pressure in the cylinder and the inertia of the reciprocating parts, severe

torsional stresses can develop in the main shafting. The result is either reduction

of shaft life or fatigue failure. In order to avoid fatigue failure of crank shafts in

such capital intensive machinery, it is essential that the following facts should be

carried out at the design stage: calculation of natural frequencies and modes,

harmonic analysis of excitation torques, selection of critical speeds and severe

orders, and calculation of maximum torsional stresses in operating speed range.

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Torsional vibration usually exists at one or more periods of the operating

range in torsional systems. It is very important to analyze and pre-estimate

critical speeds or torsional natural frequencies and mode shapes of the vibration

systems in the design stage. This way, future disastrous and costly repairs of the

machinery will be controlled. The study of analysis theory and methods of

multi-branch torsional vibrations is becoming an important subject as the

complexity of modern machines increases. The analysis of multi-branch torsional

vibration grows more and more important in industries moving toward large scale

systems, high speeds and automation. These include mining, rolling steel, oil and

chemical, machinery and shipbuilding industries.

Besides, a mechanical power transmission system is usually one part of a

machine, which is often subjected to static or periodic torsional loading that

necessitates the analysis of the torsional characteristics of the system. For

example, the drive train of a typical automobile is subjected to a periodically

varying torque[2]. This torque variation occurs due to the cyclical nature of the

internal combustion engine that supplies the power[3]. If the frequency of the

engine’s torque variation matches one of the resonance frequencies of the

engine/drive train system, large torsional deflections and internal shear stresses

can occur. Continued operation of the machinery under such conditions can lead

to early fatigue failure of system components.

Thus, an engineer designing such a system needs to be able to predict its

torsional natural frequencies and to easily determine what effects design

changes might have on those natural frequencies.

An efficient and accurate method and program for predicting torsional

natural frequencies of a piece of machinery should be capable of modeling the

important characteristics of the system in a timely manner. Accurately modeling a

system in the early stages of a design can reduce costs by decreasing the

number of changes needed at later stages in the design process. In the case of

modeling torsional system characteristics, it is common to find machinery with

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vibration dampers, tuned absorbers, and multiple shafts connected by gear trains

that can significantly affect the system’s dynamic performance. An accurate

model of the system must be flexible enough to account for such components.

However, a balance must be maintained between the accuracy of the model

being created and the amount of time and effort needed to create the model.

Therefore, a valuable design tool for torsional analysis would allow the engineer

to quickly create a model of the system that provides insight about the system

characteristics.

1.2 Literature Review

Many skilled researchers have conducted extensive investigations in this

field. However, the current studies on multi-branch vibration systems are

essentially an extension of traditional theory and methods for some particular

cases. This is especially true for Holzer’s method and the transfer matrix method.

The transfer matrix method for determining natural frequencies of torsional

systems is an extension and the matrix form of the Holzer method in which the

equations relating the displacements and internal forces of the system are written

in matrix form.

Wilson [4] gives a historical review of the early development of modern

torsional analysis. It is reported that failures in marine and aeronautical drive

trains were the original source of interest in the dynamic torsional behavior of

machinery.

Nestorides [5] describes methods for modeling the various elements of

torsional systems. These references include methods for determining equivalent

inertias and/or stiffnesses for a variety of machinery components including

crankshafts, flywheels, couplings, absorbers, etc. It is common for machinery

systems to consist of multiple shafts geared together in non-branched or

branched systems. Both references describe a method for modeling

non-branched, multi-shaft systems as an equivalent single-shaft system as well

as a procedure for performing Holzer method calculations for branched systems.

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Pestel and Leckie [6] describe the transfer matrix technique for analyzing a

branched-torsional system, which involves reducing the branched system to an

equivalent single-shaft system. This method requires lumping the characteristics

of the branch at the point on the main system where the branch is attached. That

technique requires the elimination of the branch’s state relations from the global

transfer matrix and can result in numerical difficulties when using a root finding

routine to determine natural frequencies. These numerical difficulties result from

infinity wraps that can be observed by plotting the characteristic determinant

curve for a branched system. The transfer matrix method can be used to a wide

variety of problems including the determination of natural frequencies and mode

shapes for undamped and damped torsional systems. In the process of

determining the eigenvalues of a torsional system or the system’s response to a

torsional excitation the boundary conditions of the model must be applied.

Pilkey and Chang [7] present a generalized method for applying the

boundary conditions to a torsional transfer matrix model that is useful in

developing an algorithm to accomplish the desired analysis. Pilkey and Chang

also present a number of useful torsional transfer matrices and describe a

computer program, TWIST, capable of performing torsional analysis for branched

systems.

Tavares and Prodonoff [8] presents a new modelling procedure for using in

analyses of torsional vibration of gear-branched propulsion systems, which has

evolved from considerations on the use of constrained finite element equilibrium

equations.

Shaikh [9, 10] developed a general and direct method for the analysis of

branched systems, in which transfer matrices were used in Holzer-type solutions.

He considered that the method should be applicable not only for torsional

vibration systems but also for other branched systems. In this method, no

matrix inversions (or equivalent operations) were required to account for

branches at a junction. A single determinant giving natural frequencies was

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reached irrespective of the number of branches and junctions. Thus, the method

is straightforward, compact, and economical for computer solutions.

Dawson and Davies [11] developed a globally convergent iteration technique

for application to residual function value vibration analysis methods as an

extension to the method proposed by Shaikh. This method yielded a fully

automatic, efficient method regardless of the natural frequency distribution or

frequency range of the problems. The iteration formula in the extended method

required the first and second derivatives of the residual determinant as well as

the determinant itself. The method of derivation of these derivatives via both a

matrix transfer and Holzer procedure was presented. Illustrative examples of the

application of the extended method to the solution of the torsional natural

frequencies of marine geared drive systems were presented which demonstrated

the power and efficiency of the extended method, irrespective of the natural

frequency distribution or the frequency range of the problem.

Eshleman [12] used the transfer matrix method to build up a refined

mathematical model of the engine and end item power shafts. He utilized the

model to determine their natural frequencies, mode shapes, torsional motions

and stresses. The mathematical model is composed of a finite number of

elements which simulate lengths of continuous, massive, elastic shaft with end

attached lumped mass and springs.

Sankar [13] presents one multi-shaft torsional transfer matrix approach that

maintains the state information for the entire model in the global transfer matrix.

This method involves building the transfer matrix for each branch separately,

applying compatibility relations at the junction where the branches join, and then

using the boundary conditions to find the characteristic determinant of the system.

However, that method is cumbersome for complicated systems with multiple

branches.

Sankar [14] developed a new method based on the extended transfer matrix

method to analyze free vibration of multi-branch torsional vibration systems. The

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method was radically different from the traditional methods in that an extended

transfer matrix relation was formulated for each branch. For this, the calculations

were propagated from the junction and proceed simultaneously in all branches

toward their respective ends. Then by substituting the compatibility and

equilibrium conditions, a frequency dependent characteristic matrix was

formulated. This procedure reduced the size of the matrix and automatically

eliminated the need of any additional operation such as matrix inversion and the

solution of a system of equations for the formulation of the characteristic matrix.

Finally, the boundary conditions were applied to the matrix relation and the

natural frequencies were determined from the roots of a frequency determinant

derived from the characteristic matrix.

Mitchell [15] has modified a multi-rotor transfer matrix approach for

geared-torsional systems which was originally developed by Hibner [16] for

shafts experiencing lateral vibrations. This multi-rotor transfer matrix approach is

a simple and effective method for modeling multi-shaft systems. The model

building procedure associated with this method can be readily generalized for

application in a computer program.

Abhary [17] advocates the use of a semi-graphical approach for modeling

lumped-parameter torsional systems. The graphical part of the technique is

simply a bookkeeping tool to aid the analyst in performing equivalence

calculations for systems with several branches. Once the equivalent model has

been created, the author suggests writing the equations of motion for the system

in matrix form and performing an eigenvalue analysis with the aid of a

commercial software package. However, for complicated systems the necessary

equivalence calculations can become time consuming and tedious. Therefore,

this technique is not optimal for use in a design tool for torsional analysis.

Edwards and Gray [18-19] developed a torsional vibration analysis program

suitable for handling multi-junction multi-branched systems with damping and

excitation torques (provided by engines, compressors, marine propellers, etc.)

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applied at many points in the individual branches. The method used a 2 by 2

matrix method for branched systems. This method was considered to be a major

advancement on other methods of vibration analysis, since it is faster and

cheaper for repeated use than either the current Holzer’s table method, the field

matrix method or the iteration methods. It was also more flexible, permitting the

user to include modifications and allowed the user to ask various questions

concerning the behavior of specific parts of the installation investigated, to which

specific answers were provided without having to evaluate all the conditions at all

points in the system. But, this method is not good for computer programming,

and the problems solving process needs interference by users.

Blanding [20] describes a transfer matrix computer program that implements

the Hibner/Mitchell multi-rotor transfer matrix approach for analyzing the

three-dimensional, harmonically forced response of multiple-shaft systems. This

three-dimensional response includes not only torsional response but also axial

and lateral responses. This model includes coupling terms between the different

degrees of freedom. The program has the capability to represent the time-varying

stiffness of a pair of meshing spur gears. In addition gear mesh errors can also

be included in the model to determine their effects on the response. These added

modeling capabilities increase the program’s ability to model a system accurately

and as such are significant contributions to the development of the transfer

matrix method for modeling rotors. However, including such advanced modeling

capabilities comes at the expense of increasing the complexity of the program.

Tsai and Kuang [21] also report of a computer program which implements the

multi-rotor transfer technique for coupled lateral-torsional vibration analysis of

geared rotors. Tsai and Kuang present an example uncoupled torsional analysis

of a three-shaft system. However, some of the parameters used for modeling the

system have inappropriate units. Therefore, the results they obtained cannot be

used as a test case for a new computer program.

Doughty and Vafaee [22] report on a transfer matrix computer program

capable of determining the damped natural frequencies and mode shapes of

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simple torsional systems. Two example problems with solutions are provided to

demonstrate the technique. However, analysis using the program is limited to

systems for which an Infinity Wraps equivalent single-shaft system can be

developed. The root search method used in this case is a Newton-Raphson

algorithm that has certain drawbacks. The Newton-Raphson technique requires

an initial root estimate that can affect the success of the routine in finding roots.

In addition this root-finding method requires the calculation of derivatives of the

function being considered. For the transfer matrix problem these derivatives must

be approximated in a somewhat arbitrary fashion which can also affect the

success of the root-finding efforts. Huang and Horng [23] also describe a transfer

matrix computer program that uses the Newton-Raphson technique for finding

the roots of damped torsional systems. This program implements the Pestel and

Leckie branching technique for a two-shaft system. Because this technique

keeps track of only the main system state values and not those of the branch,

calculating the complete eigenvectors for a two-shaft system using their program

requires two separate system models.

Many other researchers have conducted extensive investigations into

multi-branch torsional vibration, such as Mahalingam [24~26], Gilbert [27],

Hundal [28], Dawson and Davies [29], Wang [30], Rawtani [31], Lai [32], Robert

[33], Shigley and Mischke [34], Thomson and Dahleh [35], Whally and Ebrahimi [36]. Although there are some extensive investigations in this field, the current studies on multi-branch vibration systems are essentially an extension of traditional theory and methods for some particular cases. This is especially true for Holzer’s method and the transfer matrix method. The above studies are also limited in their applications and most of them are not systemized. The studies must also be adapted for computer solutions and make methods adaptable to all cases of multi-branch tortional vibration. Wang said that work is still needed to develop a more efficient and accurate method for analyzing multi-branch torsional vibration systems [37]. A few of the references gives the same suggestion, including Huang [38], Jaksic and Boltezar [39], and Mandal, Sivakumar and Kumar [40].