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

Authors: Nicolae Herişanu; Vasile Marinca
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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]:
(1.1)mu¨+k1(1-qsinω2t)u+k2u3=fsinω1t,
with the initial conditions
(1.2)u(0)=A,u˙(0)=0,
which can be written in the more convenient form
(1.3)u¨+ω2u-αusinω2t+βu3-γsinω1t=0,
where
(1.4)ω2=k1m,α=k1qm,β=k2m,γ=fm,
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
(1.5)τ=Ωt;u(t)=Ax(τ),
the original Equation (1.3) becomes
(1.6)Ω2x′′+ω2x-αxsinω2Ωτ+βA2x3-γAsinω1Ωτ=0,
with the initial conditions
(1.7)x(0)=1,x′(0)=0,
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:(2.1)F(τ,x,x′,x′′)=0.
The variational principle for (2.1) can be easily established if there exists a functional
(2.2)J=∫τ1τ2L(τ,x,x′)dτ,
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 functionL(τ,x,x′)  such that Euler’s equation of the functional (2.2) coincide with the system (2.1), that is,
(2.3)∂L∂x-ddτ(∂L∂x′)=F(τ,x,x′,x′′).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 solutionx¯ of (2.1) depends on several parameters C1,C2,…,Cs:
(2.4)x¯=x¯(τ,C1,C2,…,Cs),
such that the action functional (2.2) becomes
(2.5)J(C1,C2,…,Cs)=∫τ1τ2L(τ,x¯(τ,Ci),x¯′(τ,Ci)dτ),i=1,2,…,s.The parametersCi from (2.5), called convergence-control parameters, can be determined optimally applying the Ritz method [8]:
(2.6)∂J∂C1=∂J∂C2=⋯=∂J∂Cs.From (2.6) and from the initial condition (1.7)1 which becomes
(2.7)x¯(0,C1,C2,…,Cs)=1,
we can obtain optimally the parameters Ci,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
(2.8)L(τ,x,x′)=-12Ω2x′2+12ω2x2-12αx2sinω2Ωτ+14βA2x4-γAxsinω1Ωτ.If we considers=3 in (2.4), then the approximate solution of (1.6) can be written as
(2.9)x¯(τ)=C1cosτ+C2cosω1Ωτ+C3cosω2Ωτ.Also, we can choose this approximate solution in the form(2.10)x¯(τ)=C1cosτ+C2cosω1+ω2Ωτ+C3cosω1-ω2Ωτ+C4cosΩ+ω1Ωτ,
or
(2.11)x¯(τ)=C1cosτ+C2cosΩ-ω1Ω+C3cosΩ+ω1Ωτ+C4cosΩ-ω2Ωτ+C5cosΩ+ω2Ωτ,
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:
(2.12)J(C1,C2,C3)=K0+K1cos2πω1Ω+K2cos2πω2Ω+K3cos4πω1Ω+K4cos4πω2Ω+K5cos6πω2Ω+K6cos2π(ω1-ω2)Ω+K7cos2π(ω1+ω2)Ω+K8cos2π(2ω1+ω2)Ω+K9cos2π(2ω1-ω2)Ω+K10cos2π(ω1+2ω2)Ω+K11cos2π(ω1-2ω2)Ω+K12sin2πω1Ω+K13sin2πω2Ω+K14sin4πω1Ω+K15sin4πω2Ω+K16sin6πω1Ω+K17sin6πω2Ω+K18sin8πω1Ω+K19sin8πω2Ω+K20sin2π(ω1+ω2)Ω+K21sin2π(ω1-ω2)Ω+K22sin4π(ω1+ω2)Ω+K23sin4π(ω1-ω2)Ω+K24sin2π(2ω1+ω2)Ω+K25sin2π(2ω1-ω2)Ω+K26sin2π(3ω1+ω2)Ω+K27sin2π(3ω1-ω2)Ω+K28sin2π(ω1+2ω2)Ω+K29sin2π(ω1-2ω2)Ω+K30sin2π(ω1+3ω2)Ω+K31sin2π(ω1-3ω2)Ω,
where
(2.13)K0=π2(ω2-Ω2)(C12+C22+C32)+αΩ12ω2(3C12+3C22+2C32)-αΩω2C124(4Ω2-ω22)+αΩω2C228(4ω12-ω22)+αΩ(ω1-ω2)C1C22[Ω2-(ω1-ω2)2]+αΩ(ω1+ω2)C1C22[Ω2-(ω1+ω2)2]-αΩω2C1C3Ω2-4ω22+αΩC2C34(2ω2-ω1)+αΩC2C34(2ω2+ω1)-γΩω1C1A(Ω2-ω12)+γΩC24Aω1+γΩC32A(ω1-ω2)+γΩC32A(ω1+ω2)+3πβA216(C14+C24+C34+4C12C22+4C12C32+4C22C32),K1=γΩω1C1A(Ω2-ω12),K2=αΩω2C124(4Ω2-ω22)-αΩ(C12+C22)4ω2,K3=-λΩC24Aω1,K4=αΩω2C1C3Ω2-4ω22,K5=-αΩC3224ω2,K6=αΩ(ω2-ω1)C1C22[Ω2-(ω2-ω1)2]-γΩC32A(ω1-ω2),K7=-αΩ(ω1+ω2)C1C22[Ω2-(ω1+ω2)2]-γΩC32A(ω1+ω2),K8=-αΩC228(2ω1+ω2),K9=αΩC228(2ω1-ω2),K10=-αΩC2C34(ω1+2ω2),K11=-αΩC2C34(2ω2-ω1),K12=Ωω1(Ωω1+ω2)C1C2Ω2-ω12+βA28[Ω2C13C2+ΩC1C2ω1(2C12+3C22+10C32)Ω2-ω12+3Ω2C13C29Ω2-ω12],K13=Ωω2(Ωω2+ω2)C1C3Ω2-ω22+βA28[Ω2C13C3+ΩC1C3ω2(2C12+10C22+3C32)Ω2-ω22+3Ω2C13C39Ω2-ω22],K14=-(ω2+ω12)ΩC228ω1+βA232[5Ωω1C12C22Ω2-ω12-2ΩC22(3C12+C22+3C32)ω1],K15=-(ω2+ω22)ΩC328ω2+βA232[5Ωω2C12C32Ω2-ω22-2ΩC32(3C12+3C22+C32)ω2],K16=3βA2Ωω1C1C238(Ω2-9ω12),K17=3βA2Ωω2C1C338(Ω2-9ω22),K18=βA2ΩC24128ω1,K19=βA2ΩC34128ω2,K20=-(ω2+ω1ω2)ΩC2C32(ω1+ω2)+βA24[3Ω(ω1+ω2)C12C2C34Ω2-(ω1+ω2)2-3ΩC2C3(2C12+C22+C32)2(ω1+ω2)],K21=-(ω2+ω1ω2)ΩC2C32(ω1-ω2)+βA24[3Ω(ω1-ω2)C12C2C34Ω2-(ω1-ω2)2-3ΩC2C3(2C12+C22+C32)2(ω1-ω2)],K22=-3βA2ΩC22C3232(ω1+ω2),K23=-3βA2ΩC22C3232(ω1-ω2),K24=3βA2Ω(2ω1+ω2)C1C22C38[Ω2-(2ω1+ω2)2],K25=3βA2Ω(2ω1-ω2)C1C22C38[Ω2-(2ω1-ω2)2],K26=-βA2ΩC23C38(3ω1+ω2),K27=-βA2ΩC23C38(3ω1-ω2),K28=3βA2Ω(ω1+2ω2)C1C2C328(Ω2-(ω1+2ω2)2],K29=3βA2Ω(ω1-2ω2)C1C2C328(Ω2-(ω1-2ω2)2],K30=-βA2ΩC2C338(ω1+3ω2),K31=-βA2ΩC2C338(3ω2-ω1).The values of the parametersC1,C2,C3 and the frequency Ω are obtained from (2.6) which become:
(2.14)∂J∂C1=∂J∂C2=∂J∂C3=0,
and from the initial condition (2.7) which can be written as
(2.15)C1+C2+C3=1.


## 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:(3.1)A=1,m=2,k1=200,k2=1000,q=0.1,ω1=2.1,ω2=1.1,f=6,
or for the corresponding coefficients of (1.6):
(3.2)ω=10,α=10,β=500,γ=3.
The parametersC1,C2,C3 and the frequency Ω are obtained from (2.14) and (2.15):
(3.3)C1=0.981045,C2=0.00376699,C3=0.0151884,Ω=21.5167.
The approximate solution of (1.1) in this case becomes
(3.4)u(t)=0.981045cosΩt+0.00376699cosω1t+0.0151884cosω2t.
Figure1 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:(3.5)A=0.8,ω1=2.7,ω2=1.0.
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:
(3.6)C1=0.966202,C2=0.00106843,C3=0.0327299,Ω=18.0654,
and consequently, the approximate solution of (1.1) becomes in this case:
(3.7)u(t)=0.772961cosΩt+0.000854743cosω1t+0.0261839cosω2t.
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:(3.8)A=2,ω1=2.5,ω2=1.5.
In this case, following the procedure described above, we obtain
(3.9)C1=0.989719,C2=0.00281954,C3=0.00746148,Ω=39.6388,
and the approximate solution of (1.1) becomes
(3.10)u(t)=1.97944cosΩt+0.00563909cosω1t+0.014923cosω2t.
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 Figures1–3 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 parametersCi 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.



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