An Optimal Approach to Study the Nonlinear Behaviour of a Rotating Electrical Machine

Abstract

We introduce a novel analytical approximate technique, called Optimal Variational Method (OVM), to investigate the nonlinear behaviour of a rotating electrical machine modelled as an oscillator with cubic elastic restoring force and time variable coefficients. The proposed procedure involves the presence of some initially unknown convergence-control parameters whose values are later optimally determined. Comparisons between the obtained results and exact ones reveal that OVM is very effective and convenient providing highly accurate results.

1. Introduction

Mathematical modelling of many physical and engineering systems often leads to nonlinear ordinary or partial differential equations. In order to analyze such mathematical models, an effective method is required to provide solutions conforming to physical reality. In some cases, inherent difficulties are overcome by replacing a nonlinear differential equation with a corresponding linear differential equation that approximates the original one close enough to give useful results, but this is only a harsh approximation.

There are some known approaches intended to obtain approximate solutions to nonlinear dynamical systems. The most common and widely studied methods for solving nonlinear differential equations are the perturbation methods [1–3], but almost all perturbation methods are based on such an assumption that a small parameter must exist in the equation under investigation. The approximate solutions obtained through perturbation methods are valid, in most cases, only for small values of parameters, and moreover, there is no criterion on how small the parameters should be.

In order to overcome these limitations related to small parameters, several effective methods were developed, such as the method of Lie group [4], the Adomian decomposition method [5], the weighted-linearization method [6], and the method of harmonic balance [7].

The aim of this paper is to propose a novel variational approach to study the nonlinear behaviour of a rotating electrical machine modelled as an oscillator with cubic elastic force and time variable coefficients.

Rotating electrical machines are complex dynamical systems exhibiting typical problems from rotor dynamics, especially for high speeds or heavy duty rotor shafts with high inertial loads [8, 9]. An important engineering challenge in this field is to develop such mathematical models and methods, which allows quite accurate predictions of the dynamical phenomena at the stage of design, when compared against experimental measurements obtained after the stage of machine building. These models and methods should be made available in order to provide an easy-to-use tool to operate design modifications when these may be required to suppress undesired dynamic phenomena.

The most common sources of vibration in rotating machines are related to the unbalanced forces of the rotor, shaft misalignment and nonlinearity of the bearing stiffness, variable elasticity, bad bearings and mechanical looseness, and other electrical and mechanical faults which generate nonlinear vibration in the system.

These phenomena should be controlled in order to make the machine run smoothly and reliably. It is well known that rotating parts cannot be perfectly balanced. From engineering point of view, it is impossible to make any rotor perfectly mass balanced. Hence, residual unbalance is always present to some extent, even though the rotating structure is well constructed, but if it deteriorates, then damaging vibration occurs. Supplementary problems could arise in case of some horizontal rotating machines, when the gravity effect is not negligible for certain stiffness conditions. In case of gravity deflection, the shaft center leaves the bearing centerline, which leads to vibration occurrence. Misalignment could also occur in the electrical rotating machine after some amount of running. All these could be highly detrimental, affecting the integrity of the system [8, 9].

In this paper, a new analytical procedure, namely, the Optimal Variational Method (OVM) is employed to study the problem of nonlinear vibrations of an electric machine supported by nonlinear bearings characterized by nonlinear stiffness of Duffing type while the entire system is subject to a parametric excitation caused by an axial thrust and a forcing excitation caused by an unbalanced force of the rotor. In these conditions, the dynamical behaviour of the investigated electrical machine will be governed by the following second-order strongly nonlinear differential equation [8]: 𝑚𝑢̈+𝑘₁1−𝑞sin𝜔₂𝑡𝑢+𝑘₂𝑢³=𝑓sin𝜔₁𝑡, (1.1) with the initial conditions 𝑢(0)=𝐴,𝑢̇(0)=0, (1.2) which can be written in the more convenient form 𝑢̈+𝜔²𝑢−𝛼𝑢sin𝜔₂𝑡+𝛽𝑢³−𝛾sin𝜔₁𝑡=0, (1.3) where 𝜔²=𝑘₁𝑚𝑘,𝛼=¹𝑞𝑚𝑘,𝛽=²𝑚𝑓,𝛾=𝑚, (1.4) and the dot denotes derivative with respect to time and 𝐴 is the initial amplitude of the oscillations. Note that it is unnecessary to assume the existence of any small or large parameters in (1.3).

Equation (1.3) describes a system oscillating with an unknown period 𝑇. We switch to a scalar time 𝜏=Ω𝑡. Under the transformation 𝜏=Ω𝑡;𝑢(t)=𝐴𝑥(𝜏), (1.5) the original Equation (1.3) becomes Ω²𝑥^′′+𝜔²𝜔𝑥−𝛼𝑥sin²Ω𝜏+𝛽𝐴²𝑥³−𝛾𝐴𝜔sin¹Ω𝜏=0, (1.6) with the initial conditions 𝑥(0)=1,𝑥^′(0)=0, (1.7) where the prime denotes derivative with respect to 𝜏.

2. A Novel Variational Method and Solutions

In order to show the basics of OVM, we consider the following differential equation: 𝐹𝜏,𝑥,𝑥^′,𝑥^′′=0. (2.1) The variational principle for (2.1) can be easily established if there exists a functional 𝐽=^𝜏2_𝜏1𝐿𝜏,𝑥,𝑥^′𝑑𝜏, (2.2) which admits as extremals the solutions of (2.1), where 𝐿 is the Lagrangian of the system (2.1) and [𝜏₁,𝜏₂] is the domain of interest.

This problem is based on the study of the conditions under which there exists a function 𝐿(𝜏,𝑥,𝑥^′)  such that Euler’s equation of the functional (2.2) coincide with the system (2.1), that is, 𝜕𝐿−𝑑𝜕𝑥𝑑𝜏𝜕𝐿𝜕𝑥^′=𝐹𝜏,𝑥,𝑥^′,𝑥^′′. (2.3)

On physical grounds, the primary significance of this problem, called “the inverse problem in Newtonian mechanics” [10] rests on the fact that the acting forces in Newtonian system (2.1) need not necessarily be derivable from a potential. Equation (2.2) is called action functional or action for short.

In our procedure, we assume that the approximate solution 𝑥 of (2.1) depends on several parameters 𝐶₁,𝐶₂,…,𝐶_𝑠: 𝑥=𝑥𝜏,𝐶₁,𝐶₂,…,𝐶_𝑠, (2.4) such that the action functional (2.2) becomes 𝐽𝐶₁,𝐶₂,…,𝐶_𝑠=^𝜏2_𝜏1𝐿𝜏,𝑥𝜏,𝐶_𝑖,𝑥^′𝜏,𝐶_𝑖𝑑𝜏,𝑖=1,2,…,𝑠. (2.5)

The parameters 𝐶_𝑖 from (2.5), called convergence-control parameters, can be determined optimally applying the Ritz method [8]: 𝜕𝐽𝜕𝐶₁=𝜕𝐽𝜕𝐶₂=⋯=𝜕𝐽𝜕𝐶_𝑠. (2.6)

From (2.6) and from the initial condition (1.7)₁ which becomes 𝑥0,𝐶₁,𝐶₂,…,𝐶_𝑠=1, (2.7) we can obtain optimally the parameters 𝐶_𝑖,𝑖=1,2,…,𝑠 and the frequency Ω of the system (2.1).

We remark that the condition (1.7)₂ is identically verified by the solution (2.4). On the other hand, the expression of the solution (2.4) is not unique.

The validity of the proposed approach is illustrated on (1.6). In this case, the Lagrangian of (1.6) can be written as 𝐿𝜏,𝑥,𝑥^′1=−2Ω²𝑥^′2+12𝜔²𝑥²−12𝛼𝑥²𝜔sin²Ω1𝜏+4𝛽𝐴²𝑥⁴−𝛾𝐴𝜔𝑥sin¹Ω𝜏. (2.8)

If we consider 𝑠=3 in (2.4), then the approximate solution of (1.6) can be written as 𝑥(𝜏)=𝐶₁cos𝜏+𝐶₂𝜔cos¹Ω𝜏+𝐶₃𝜔cos²Ω𝜏. (2.9)

Also, we can choose this approximate solution in the form 𝑥(𝜏)=𝐶₁cos𝜏+𝐶₂𝜔cos¹+𝜔₂Ω𝜏+𝐶₃𝜔cos¹−𝜔₂Ω𝜏+𝐶₄cosΩ+𝜔₁Ω𝜏, (2.10) or 𝑥(𝜏)=𝐶₁cos𝜏+𝐶₂cosΩ−𝜔₁Ω+𝐶₃cosΩ+𝜔₁Ω𝜏+𝐶₄cosΩ−𝜔₂Ω𝜏+𝐶₅cosΩ+𝜔₂Ω𝜏, (2.11) and so on.

Substituting (2.9) into (2.8) and this into (2.10) we have the following results for 𝜏₁=0 and 𝜏₂=2𝜋: 𝐽𝐶₁,𝐶₂,𝐶₃=𝐾₀+𝐾₁cos2𝜋𝜔₁Ω+𝐾₂cos2𝜋𝜔₂Ω+𝐾₃cos4𝜋𝜔₁Ω+𝐾₄cos4𝜋𝜔₂Ω+𝐾₅cos6𝜋𝜔₂Ω+𝐾₆𝜔cos2𝜋₁−𝜔₂Ω+𝐾₇𝜔cos2𝜋₁+𝜔₂Ω+𝐾₈cos2𝜋2𝜔₁+𝜔₂Ω+𝐾₉cos2𝜋2𝜔₁−𝜔₂Ω+𝐾₁₀𝜔cos2𝜋₁+2𝜔₂Ω+𝐾₁₁𝜔cos2𝜋₁−2𝜔₂Ω+𝐾₁₂sin2𝜋𝜔₁Ω+𝐾₁₃sin2𝜋𝜔₂Ω+𝐾₁₄sin4𝜋𝜔₁Ω+𝐾₁₅sin4𝜋𝜔₂Ω+𝐾₁₆sin6𝜋𝜔₁Ω+𝐾₁₇sin6𝜋𝜔₂Ω+𝐾₁₈sin8𝜋𝜔₁Ω+𝐾₁₉sin8𝜋𝜔₂Ω+𝐾₂₀𝜔sin2𝜋₁+𝜔₂Ω+𝐾₂₁𝜔sin2𝜋₁−𝜔₂Ω+𝐾₂₂𝜔sin4𝜋₁+𝜔₂Ω+𝐾₂₃𝜔sin4𝜋₁−𝜔₂Ω+𝐾₂₄sin2𝜋2𝜔₁+𝜔₂Ω+𝐾₂₅sin2𝜋2𝜔₁−𝜔₂Ω+𝐾₂₆sin2𝜋3𝜔₁+𝜔₂Ω+𝐾₂₇sin2𝜋3𝜔₁−𝜔₂Ω+𝐾₂₈𝜔sin2𝜋₁+2𝜔₂Ω+𝐾₂₉𝜔sin2𝜋₁−2𝜔₂Ω+𝐾₃₀𝜔sin2𝜋₁+3𝜔₂Ω+𝐾₃₁𝜔sin2𝜋₁−3𝜔₂Ω, (2.12) where 𝐾₀=𝜋2𝜔²−Ω²𝐶²₁+𝐶²₂+𝐶²₃+𝛼Ω12𝜔₂3𝐶²₁+3𝐶²₂+2𝐶²₃−𝛼Ω𝜔₂𝐶²₁44Ω²−𝜔²₂+𝛼Ω𝜔₂𝐶²₂84𝜔²₁−𝜔²₂+𝛼Ω𝜔₁−𝜔₂𝐶₁𝐶₂2Ω²−𝜔₁−𝜔₂²+𝛼Ω𝜔₁+𝜔₂𝐶₁𝐶₂2Ω²−𝜔₁+𝜔₂²−𝛼Ω𝜔₂𝐶₁𝐶₃Ω²−4𝜔²₂+𝛼Ω𝐶₂𝐶₃42𝜔₂−𝜔₁+𝛼Ω𝐶₂𝐶₃42𝜔₂+𝜔₁−𝛾Ω𝜔₁𝐶₁𝐴Ω²−𝜔²₁+𝛾Ω𝐶₂4𝐴𝜔₁+𝛾Ω𝐶₃2𝐴𝜔₁−𝜔₂+𝛾Ω𝐶₃2𝐴𝜔₁+𝜔₂+3𝜋𝛽𝐴²𝐶16⁴¹+𝐶⁴₂+𝐶⁴₃+4𝐶²₁𝐶²₂+4𝐶²₁𝐶²₃+4𝐶²₂𝐶²₃,𝐾₁=𝛾Ω𝜔₁𝐶₁𝐴Ω²−𝜔²₁,𝐾₂=𝛼Ω𝜔₂𝐶²₁44Ω²−𝜔²₂−𝛼Ω𝐶²₁+𝐶²₂4𝜔₂,𝐾₃=−𝜆Ω𝐶₂4𝐴𝜔₁,𝐾₄=𝛼Ω𝜔₂𝐶₁𝐶₃Ω²−4𝜔²₂,𝐾₅=−𝛼Ω𝐶²₃24𝜔₂,𝐾₆=𝛼Ω𝜔₂−𝜔₁𝐶₁𝐶₂2Ω²−𝜔₂−𝜔₁²−𝛾Ω𝐶₃2𝐴𝜔₁−𝜔₂,𝐾₇𝜔=−𝛼Ω₁+𝜔₂𝐶₁𝐶₂2Ω²−𝜔₁+𝜔₂²−𝛾Ω𝐶₃2𝐴𝜔₁+𝜔₂,𝐾₈=−𝛼Ω𝐶²₂82𝜔₁+𝜔₂,𝐾₉=𝛼Ω𝐶²₂82𝜔₁−𝜔₂,𝐾₁₀=−𝛼Ω𝐶₂𝐶₃4𝜔₁+2𝜔₂,𝐾₁₁=−𝛼Ω𝐶₂𝐶₃42𝜔₂−𝜔₁,𝐾₁₂=Ω𝜔₁Ω𝜔₁+𝜔²𝐶₁𝐶₂Ω²−𝜔²₁+𝛽𝐴²8Ω²𝐶³₁𝐶₂+Ω𝐶₁𝐶₂𝜔₁2𝐶²₁+3𝐶²₂+10𝐶²₃Ω²−𝜔²₁+3Ω²𝐶³₁𝐶₂9Ω²−𝜔²₁,𝐾₁₃=Ω𝜔₂Ω𝜔₂+𝜔²𝐶₁𝐶₃Ω²−𝜔²₂+𝛽𝐴²8Ω²𝐶³₁𝐶₃+Ω𝐶₁𝐶₃𝜔₂2𝐶²₁+10𝐶²₂+3𝐶²₃Ω²−𝜔²₂+3Ω²𝐶³₁𝐶₃9Ω²−𝜔²₂,𝐾₁₄𝜔=−²+𝜔²₁Ω𝐶²₂8𝜔₁+𝛽𝐴²325Ω𝜔₁𝐶²₁𝐶²₂Ω²−𝜔²₁−2Ω𝐶²₂3𝐶²₁+𝐶²₂+3𝐶²₃𝜔₁,𝐾₁₅𝜔=−²+𝜔²₂Ω𝐶²₃8𝜔₂+𝛽𝐴²325Ω𝜔₂𝐶²₁𝐶²₃Ω²−𝜔²₂−2Ω𝐶²₃3𝐶²₁+3𝐶²₂+𝐶²₃𝜔₂,𝐾₁₆=3𝛽𝐴²Ω𝜔₁𝐶₁𝐶³₂8Ω²−9𝜔²₁,𝐾₁₇=3𝛽𝐴²Ω𝜔₂𝐶₁𝐶³₃8Ω²−9𝜔²₂,𝐾₁₈=𝛽𝐴²Ω𝐶⁴₂128𝜔₁,𝐾₁₉=𝛽𝐴²Ω𝐶⁴₃128𝜔₂,𝐾₂₀𝜔=−²+𝜔₁𝜔₂Ω𝐶₂𝐶₃2𝜔₁+𝜔₂+𝛽𝐴²43Ω𝜔₁+𝜔₂𝐶²₁𝐶₂𝐶₃4Ω²−𝜔₁+𝜔₂²−3Ω𝐶₂𝐶₃2𝐶²₁+𝐶²₂+𝐶²₃2𝜔₁+𝜔₂,𝐾₂₁𝜔=−²+𝜔₁𝜔₂Ω𝐶₂𝐶₃2𝜔₁−𝜔₂+𝛽𝐴²43Ω𝜔₁−𝜔₂𝐶²₁𝐶₂𝐶₃4Ω²−𝜔₁−𝜔₂²−3Ω𝐶₂𝐶₃2𝐶²₁+𝐶²₂+𝐶²₃2𝜔₁−𝜔₂,𝐾₂₂=−3𝛽𝐴²Ω𝐶²₂𝐶²₃32𝜔₁+𝜔₂,𝐾₂₃=−3𝛽𝐴²Ω𝐶²₂𝐶²₃32𝜔₁−𝜔₂,𝐾₂₄=3𝛽𝐴²Ω2𝜔₁+𝜔₂𝐶₁𝐶²₂𝐶₃8Ω²−2𝜔₁+𝜔₂²,𝐾₂₅=3𝛽𝐴²Ω2𝜔₁−𝜔₂𝐶₁𝐶²₂𝐶₃8Ω²−2𝜔₁−𝜔₂²,𝐾₂₆=−𝛽𝐴²Ω𝐶³₂𝐶₃83𝜔₁+𝜔₂,𝐾₂₇=−𝛽𝐴²Ω𝐶³₂𝐶₃83𝜔₁−𝜔₂,𝐾₂₈=3𝛽𝐴²Ω𝜔₁+2𝜔₂𝐶₁𝐶₂𝐶²₃8Ω²−𝜔₁+2𝜔₂²,𝐾₂₉=3𝛽𝐴²Ω𝜔₁−2𝜔₂𝐶₁𝐶₂𝐶²₃8Ω²−𝜔₁−2𝜔₂²,𝐾₃₀=−𝛽𝐴²Ω𝐶₂𝐶³₃8𝜔₁+3𝜔₂,𝐾₃₁=−𝛽𝐴²Ω𝐶₂𝐶³₃83𝜔₂−𝜔₁. (2.13)

The values of the parameters 𝐶₁,𝐶₂,𝐶₃ and the frequency Ω are obtained from (2.6) which become: 𝜕𝐽𝜕𝐶₁=𝜕𝐽𝜕𝐶₂=𝜕𝐽𝜕𝐶₃=0, (2.14) and from the initial condition (2.7) which can be written as 𝐶₁+𝐶₂+𝐶₃=1. (2.15)

3. Test Examples

The validity of the proposed procedure for solving the investigated problem is illustrated on three examples, considering different parameters and initial amplitudes.

Case a. As a first example, we consider the following set of parameters: 𝐴=1,𝑚=2,𝑘₁=200,𝑘₂=1000,𝑞=0.1,𝜔₁=2.1,𝜔₂=1.1,𝑓=6, (3.1) or for the corresponding coefficients of (1.6): 𝜔=10,𝛼=10,𝛽=500,𝛾=3. (3.2)
The parameters 𝐶₁,𝐶₂,𝐶₃ and the frequency Ω are obtained from (2.14) and (2.15): 𝐶₁=0.981045,𝐶₂=0.00376699,𝐶₃=0.0151884,Ω=21.5167. (3.3)
The approximate solution of (1.1) in this case becomes 𝑢(𝑡)=0.981045cosΩ𝑡+0.00376699cos𝜔₁𝑡+0.0151884cos𝜔₂𝑡. (3.4)
Figure 1 shows the comparison between the present solution (3.4) and numerical integration results obtained using a fourth-order Runge-Kutta scheme.

Figure 1 

Comparison between the numerical solution of (1.1) and the approximate solution (3.4) for 𝐴=1, 𝑚=2, 𝑘1=200, 𝑘2=1000, 𝑞=0.1, 𝜔1=2.1, 𝜔2=1.1, 𝑓=6.

Case b. As a second example, we consider the following parameters: 𝐴=0.8,𝜔₁=2.7,𝜔₂=1.0. (3.5) Keeping the rest of the coefficients unchanged and following the same procedure, we obtain the optimal values of the convergence-control parameters 𝐶₁,  𝐶₂, 𝐶₃ and the frequency Ω: 𝐶₁=0.966202,𝐶₂=0.00106843,𝐶₃=0.0327299,Ω=18.0654, (3.6) and consequently, the approximate solution of (1.1) becomes in this case: 𝑢(𝑡)=0.772961cosΩ𝑡+0.000854743cos𝜔₁𝑡+0.0261839cos𝜔₂𝑡. (3.7)
The comparison between the present solution (3.7) and numerical integration results in the second case is presented in Figure 2.

Figure 2 

Comparison between the numerical solution of (1.1) and the approximate solution (3.7) for 𝐴=0.8, 𝑚=2, 𝑘1=200, 𝑘2=1000, 𝑞=0.1, 𝜔1=2.7, 𝜔2=1.0, 𝑓=6.

Case c. Finally, for the last example we consider the following set of parameters: 𝐴=2,𝜔₁=2.5,𝜔₂=1.5. (3.8) In this case, following the procedure described above, we obtain 𝐶₁=0.989719,𝐶₂=0.00281954,𝐶₃=0.00746148,Ω=39.6388, (3.9) and the approximate solution of (1.1) becomes 𝑢(𝑡)=1.97944cosΩ𝑡+0.00563909cos𝜔₁𝑡+0.014923cos𝜔₂𝑡. (3.10)
The comparison between the obtained approximate analytical solution (3.10) and numerical integration results in the third case is presented in Figure 3.

Figure 3 

Comparison between the numerical solution of (1.1) and the approximate solution (3.10) for 𝐴=2, 𝑚=2, 𝑘1=200, 𝑘2=1000, 𝑞=0.1, 𝜔1=2.5, 𝜔2=1.5, 𝑓=6.

It can be observed from Figures 1–3 that the approximate analytical results obtained using the proposed procedures are in good agreement with the numerical ones.

4. Conclusions

In this work, an optimal variational approach is employed to propose a new analytic approximate solution for nonlinear conservative oscillations of an electrical machine. The construction of our variational approach is different from traditional approach especially concerning the involvement of some initially unknown parameters 𝐶_𝑖 called convergence-control parameters, whose optimal values ensure a fast convergence of the approximate analytical solutions. This is in fact the main advantage of the OVM, which provides us with a simple and rigorous way to control and adjust the convergence of the solutions. The proposed procedure is valid even if the nonlinear equation does not contain any small or large parameters. Three test examples illustrate that the proposed analytical approach is very effective and yields accurate results comparing to those obtained via numerical integration using a fourth-order Runge-Kutta method.