applied

sciences
Article
The Extended Galerkin Method for Approximate Solutions of
Nonlinear Vibration Equations
Ji Wang 1, * and Rongxing Wu 1,2

1 Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University,
818 Fenghua Road, Ningbo 315211, China; wurongxing98@163.com
2 Institute of Applied Mechanics, Ningbo Polytechnic, 1069 Xinda Road, Beilun District, Ningbo 315800, China
* Correspondence: wangji@nbu.edu.cn; Tel.: +86-574-8760 0467

Featured Application: This is a novel procedure for solving nonlinear equations of vibrations
with asymptotic solutions. It is an extension to the popular Galerkin method by adding an in-
tegration of time over one period of vibrations. The method is applicable to a broad class of
nonlinear equations as a systematic procedure for approximate solutions.

Abstract: An extension has been made to the popular Galerkin method by integrating the weighted
equation of motion over the time of one period of vibrations to eliminate the harmonics from thee
deformation function. A set of successive equations of coupled higher-order vibration amplitudes is
resulted, and a nonlinear eigenvalue problem is obtained for the frequency-amplitude dependence of
nonlinear vibrations with successive displacements. The subsequent solutions of vibration frequencies
and deformation are consistent with other successive approximate methods, such as the harmonics
balance method. This is an extension of the Galerkin method which has broad applications for
asymptotic solutions, particularly for problems in solid mechanics. This extended Galerkin method
can also be utilized for the analysis of free and forced nonlinear vibrations of structures as a new



technique with significant advantages in calculations.


Citation: Wang, J.; Wu, R. The
Keywords: vibration; nonlinear; frequency; approximation; Galerkin
Extended Galerkin Method for
Approximate Solutions of Nonlinear
Vibration Equations. Appl. Sci. 2022,
12, 2979. https://doi.org/10.3390/
app12062979
1. Introduction
The Galerkin method has been a popular choice for the approximate calculation of nat-
Academic Editor: Filippo Berto
ural frequencies of elastic components and structures, particularly if there are no analytical
Received: 28 December 2021 solutions or the equation of vibrations is hard to modify and solve [1–4]. Applications of
Accepted: 14 March 2022 the Galerkin method can be found in the literature with details of implementation and some
Published: 15 March 2022 novel techniques [5–7]. Engineers and students can use the method conveniently because
Publisher’s Note: MDPI stays neutral
it usually involves calculations of weighted integrations of functions from equations of
with regard to jurisdictional claims in equilibrium or motion over the physical domain of problem. Of course, it is also the basis
published maps and institutional affil- of the finite element method with omnipresent application nowadays. There are plenty of
iations. resources on the Galerkin method, including popular textbooks [8–10], but the essence is
the minimization of the error from an approximate solution through the diminishing of the
weighted integration of the error over the solution domain.
It is evident that most discussions and applications of the Galerkin method are on the
Copyright: © 2022 by the authors. linear analysis of problems of structural vibrations. If the vibrations are related to material
Licensee MDPI, Basel, Switzerland. nonlinearity or larger deformation, there are not many discussions on the utilization of the
This article is an open access article Galerkin method, at least in a systematic manner, because the method is known primarily
distributed under the terms and for linear problems as demonstrated. In a recent effort to study nonlinear vibrations of
conditions of the Creative Commons
elastic solids, it was found that such problems can be solved effectively with an extension
Attribution (CC BY) license (https://
of the Galerkin method through adding integration over time with the full expression
creativecommons.org/licenses/by/
of displacement. In fact, this is an extension to the standard procedure for effective and
4.0/).

Appl. Sci. 2022, 12, 2979. https://doi.org/10.3390/app12062979 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 2979 2 of 8

accurate solutions of nonlinear vibration problems with the resulting nonlinear eigenvalue
equations. Furthermore, it is inspiring that such an extension of the Galerkin method
for nonlinear vibration problems can be useful for problems with periodic properties.
Through this extension, the Galerkin method is now capable of analyzing both linear
and nonlinear vibrations as a unified technique and also adds another powerful tool for
nonlinear vibration problems as demonstrated in this paper and forced nonlinear vibrations
of a multiple-degree-of-freedom system [11]. The objective of this paper is to demonstrate
the effectiveness of the extended Galerkin method in solving nonlinear vibrations and
promote its adoption in the study and development of tools and techniques for more broad
nonlinear problems.

2. The Galerkin Formulation

2.1. State of the Art
The utilization of the Galerkin method in this study starts from a general nonlinear
differential equation which is frequently encountered in vibrations and solid mechanics in
the form of [8–10]  . .. 
N u, u, u, uk , ε, t = 0, (1)
. ..
where u, u u , uk , ε, and t are the displacement, derivatives with respect to time t, nonlinear

term with power of k, small parameter, and time, respectively. It should be emphasized
that the displacement function u is defined in the physical domain V which is not shown
but implied. Then the standard Galerkin method requires
Z  . .. 
N u, u, u, uk , ε, t δudV = 0, (2)
V

with δu as arbitrary variation of displacement. Equation (2) is usually referred to as the

weak form of Equation (1). It should be pointed out that in the standard Galerkin method,
there is no integration over time because the temporal variable is not considered in the
formulation and solution process. The effectiveness and validity of the standard Galerkin
method is based on the principle of the least square method, which implies an accurate
solution is achieved with the displacement solution in Equation (2).
Without losing generality, the approximate solution is assumed as

u= ∑ An cos nωt, (3)

where An , n, and ω are amplitudes, integers, and the angular frequency, respectively.
With the solution in Equation (3), it is obvious that

u = ∑ An cos nωt, uk = (∑ An cos nωt)k ,

. .. (4)
u = −ω ∑ nAn sin nωt, u = −ω 2 ∑ n2 An cos nωt.

Now by substituting Equation (4) into Equation (1), it yields

 
N An , Akn , ω m , cosm nωt , sinn nωt, ε = 0, (5)

where m is a combination of integers n and k.

As it is known, generally, there is no analytical solution to such a nonlinear equation,
and approximate methods such as the harmonic balance method (HBM) [12], the Krylov–
Bogoliubov–Mitropolsky method (KBM) [13], the Lindstedt–Poincaré method (L-P) [14],
the homotopy analysis method (HAM) [15,16], and variations have been utilized to find
solutions with different degrees of accuracy at conditions [8–10].
Appl. Sci. 2022, 12, 2979 3 of 8

2.2. The Extended Galerkin Formulation

As an alternative approach, treating time as an independent variable, the Galerkin
method in one period of vibrations is used to let [5–7]
Z T  . ..  2π
N u, u, u, uk , ε, t δudt = 0, T = , (6)
0 ω
Implying the best approximation of displacement amplitudes over the period of
vibrations. The full advantage of the Galerkin method is taken with a reasonable choice
of the displacement function, but the addition of integration over time should be noted
because the periodic property has been taken into consideration with the generalization of
the weighted integration. It has to be pointed out that Equation (6) is the Galerkin method
with the addition of temporal variable in one period of vibrations. This procedure has not
been used before for either linear or nonlinear vibrations. For the vibration analysis of a
continuous system, it is the extended Galerkin method because of the additional integration
over time domain of one period of vibrations.
Since the variation of solution in Equation (3) is

δu = ∑ δAn cos nωt, (7)

the arbitrary δAn will enforce the vanish of the weighted integration as the optimal approx-
imation to the equation through the known technique of the Galerkin method.
As a result of this operation, Equation (6) will be
Z T  
N An , Akn , ω m , cosm nωt , sinn nωt, ε cos nωtdt = 0, (8)
0

for amplitudes An and vibration frequency ω as a system of coupled nonlinear algebraic

equations.
Clearly, this approach is based on the known fact that the Galerkin method and the
weighted integration over a proper interval of time will provide a reasonable approximation
to the equation, which cannot be solved exactly. The Galerkin method has been widely used
with linear equations and spatial domains as the basis of many approximation procedures
of analytical and numerical nature but applying to the time domain for nonlinear vibrations
is not considered until recently with the Rayleigh–Ritz method by the first author [11,17,18].
Due to the interchangeability or equivalence of the Galerkin and Rayleigh-Ritz methods,
it is intuitive that the Galerkin method should also yield solutions to nonlinear vibration
equations similar to other methods like the harmonic balance method [12], as it has been
stated with the same formulation in Equation (8) by Liu and Chen [19]. The Galerkin
approach has also been implied by Nayfeh and Mook but not formulated in the same
manner [20]. A similar procedure with integration over a quarter of vibration period
is demonstrated by He [21] and Anderson et al. [22]. Since the procedure presented
in this study has not been used before, an extra exploration of these two methods for
asymptotic solutions to typical nonlinear vibration equations is recommended based on
current approaches. Naturally, this is an extension to the standard Galerkin method with
the interval of time as one period of the fundamental mode of vibrations with the underline
principle of the least square method.

3. Application Examples
3.1. The Duffing Equation
A similar approach in solving Duffing equation with good results has been reported
as the extended Rayleigh–Ritz method (ERRM) by Wang [17,18,23]. The current Galerkin
method is more flexible if the Lagrangian functional is not readily available, as it is known
in many cases of nonlinear vibration problems [24].
Appl. Sci. 2022, 12, 2979 4 of 8

Now, the popular Duffing equation is considered as [8–10]

..
u + u + εu3 = 0, (9)

where ε is a parameter which is usually assumed small as also in this study. Solution
techniques for larger ε are available for numerical analysis, but it is not the topic of this
paper. This is one of the standard nonlinear vibration equations and also is used as a
test problem for different solution techniques. Naturally, Duffing equation is taken as a
test problem for the simplicity and accuracy of the currently proposed extended Galerkin
method (EGM) with integration over time.
The asymptotic solution now is formulated as

M
u= ∑ A2j+1 cos(2j + 1)ωt, (10)
j =0

where M and A2j+1 (ε) are an integer and amplitudes, and a substitution of Equation (10)
into Equation (9) will give

N = A1 1 − ω 2 cos ωt + A3 1 − 9ω 2 cos 3ωt + εA31 cos3 ωt

(11)
+3εA21 A3 cos2 ωt cos 3ωt + 3εA1 A23 cos ωt cos2 3ωt + εA33 cos3 3ωt.

From Equation (6), the weighted integrations will be

Z T Z T
N cos ωtdt = 0, N cos 3ωtdt = 0, (12)
0 0

or
A1 (1− ω 2 ) 3εA3 3εA2 A 3εA A2
2 + 8 1 + 81 3 + 41 3 = 0,
A3 (1−9ω 2 )
(13)
εA3 3εA2 A 3εA3
2 + 8 1 + 41 3 + 8 3 = 0.
The typical initial conditions are
.
u(t = 0) = A, u(t = 0) = 0, (14)

which are related to the solutions through

A1 + A3 = A, A1 = A − A3 . (15)

Substituting Equation (15) into the first part of Equation (13), it gives

3εA2 3εAA3 3εA23

ω2 = 1 + − + . (16)
4 4 2
Substituting Equation (15) into the second part of Equation (13) and making necessary
simplifications by dropping higher-order terms of ε, it yields

εA3 εA3 3εA2

 
1
A3 = = 1 − . (17)
4 8 + 6εA2 32 4

Substituting Equation (17) into Equation (16), the approximate frequency solution is

3εA2 3ε2 A4
ω = 1+ − , (18)
8 256
Appl. Sci. 2022, 12, 2979 5 of 8

while other approximate solutions from the harmonic balance method (HBM) [12], the
Krylov–Bogoliubov–Mitropolsky method (KBM) [13], the Lindstedt–Poincaré method (L-
P) [14], and the homotopy analysis method (HAM) [15,16] are

3εA2 15ε2 A4
ω = 1+ 8 − 256 , (KBM),
3εA2 21ε2 A4 (19)
ω = 1+ 8 − 256 , (L − P),
3εA2 9ε2 A4
ω = 1+ 8 + 192 , (HAM).

It is clear that through a simple and elegant procedure, the proposed extended Galerkin
method presented an approximate solution for Duffing equation with the small parameter
ε. Although there are differences in the solution in Equation (18) in comparison with
Equation (19) with the coefficient for the ε2 A4 from other approximate techniques and
variations [25–27], they are in the same degree of accuracy, nonetheless. To illustrate the
accuracy of the extended Galerkin method shown in this study, a numerical analysis has
been completed to a multi-degree-of-freedom nonlinear vibration problem under excitation
with comparisons to results from other methods [11]. The same results exactly from the
harmonic balance method are obtained with the new method. Another demonstration of
the accuracy for the nonlinear surface wave analysis in comparison with the finite element
analysis is also presented recently [24]. The efficiency of the proposed solution procedure
can be seen through operations of integration, which is generally a one step process in
comparison with the split and combination of harmonic terms. Since the main objective of
this study is to validate the extended Galerkin method which can be used to solve nonlinear
vibration equations with approximate solutions, the numerical comparisons of specific
solutions and parameters can be done with details by following the procedure shown here
in separated studies in the future. The close results in Equation (18) show that a new and
convenient technique for solving such a nonlinear vibration equation has been presented
and validated with the fundament vibration frequency.

3.2. The Van Der Pol Equation

As a popular nonlinear vibration equation different from the Duffing equation, van
der Pol equation is also a typical test problem for solution techniques and accuracy [27].
The standard form of van der Pol equation is [9,28]
..
 .
u + u = ε 1 − u2 u, (20)

where ε is a small parameter in this study. The approximate displacement of the first two
frequencies is assumed to be

u = A1s sin ωt + A1c cos ωt + A3s sin 3ωt + A3c cos 3ωt , (21)

where A A1s , A1c , A3s , A3c and ω are amplitudes and frequency to be determined.

Now, the general form of the nonlinear vibration equation with the approximate
solution in Equation (21) is written as

N = A1s 1 − ω 2 sin ωt + A1c 1 − ω 2 cos ωt + A3s 1 − 9ω 2 sin 3ωt

+ A3c 1 − 9ω 2 cos 3ωt − ε 1 − A1s sin ωt + A1c cos ωt + A3s sin 3ωt


(22)

2 i
+ A3c cos 3ωt ω A1s cos ωt − A1c sin ωt + 3A3s cos 3ωt − 3A3c sin 3ωt .

Applying the general procedure of the extended Galerkin method, it requires

RT RT
0 N ( A, ω, ε, t) sin ωtdt = 0, 0 N ( A, ω, ε, t) cos ωtdt = 0,
RT RT (23)
0 N ( A, ω, ε, t) sin 3ωtdt = 0, 0 N ( A, ω, ε, t) cos 3ωtdt = 0.
Appl. Sci. 2022, 12, 2979 6 of 8

Substituting Equation (22) for Equation (23), it yields

2 n
2 h
2
2 io
A1s A1c − A3c εω + A1c A1c + A1c A3c + 2 −2 + A3c + A3s


εω
s c s 2


+2A1 −2 + A1 A3 εω + 2ω = 0,

2 s n
2 h
2
2 io
c A1 + A3 εω + A1s A1s − A1s A3s + 2 −2 + A3c + A3s
s


A1 εω
c s c 2


−2A1 −2 + A1 A3 εω + 2ω = 0,
n
3
2
2 h
2
2
2 io (24)
4A3 − A1 − 3A1 A1 + 6A3c A1c + 3A3c −4 + 2 A1s + A3c + A3s
s c c s εω
−36A3s ω 2 = 0,
n
3
2
2  s h
2
2 io
4A3c + − A1s + 6A3s A1s + 3 A1c A1 + 2A3s + 3A3s −4 + A3c + A3s

εω
−36A3c ω 2 = 0.

Since van der Pol equation is dominantly even by nature, the focus is on the even
function solution of time through the elimination of coefficients accordingly.
From the first one of Equation (24), it is clear that if higher-order coefficients vanish,
simple solutions are A1c = 2 and A1s = 0, which are known from approximate solutions of
other methods [9]. With these known coefficients, the frequency solution from the above
equations is
1 As
ω 2 = 1 + A3s εω = 1 + 3 ε, ω ≈ 1. (25)
2 2
A substitution of Equation (25) into Equation (24) will give

2εω 2ε ε 3A3s ε 3ε2

A3s = = = − , A c
3 = − = − . (26)
1 − 9ω 2 1 − 9ω 2 4 1 − 9ω 2 32
In summary, approximate solutions of amplitudes and frequency from the above
equations are

ε 3ε2 ε2
A1c = 2, A1s = 0, A3s = − , A3c = − , ω2 = 1 − . (27)
4 32 8
The solution from the KBM method is [9]

ε ε2
A1c = 2, A1s = 0, A3s = − , A3c = 0, ω 2 = 1 − . (28)
4 8
Clearly, using the solution technique and procedure of the extended Galerkin method,
a reasonably accurate solution to the van der Pol equation is also obtained. With the limited
assumption of approximation and a simple procedure, the results are quite reasonable
and accurate. Again, the approximate solutions present a good validation of the extended
Galerkin method for lower-order asymptotic solutions. Further examinations of accuracy
can be made with higher-order solutions and parameters by following the procedure in a
systematic manner.

3.3. Results and Discussion

From the two examples analyzed and solved by applying the extended Galerkin
method to popular nonlinear vibration equations, it is clear that the equivalent approximate
solutions are obtained with a new procedure. The fundamental solutions for both Duffing
and van der Pol equations shown in Equations (18) and (27) are close to other methods. By
making comparisons with existing solution methods, it shows that the extended Galerkin
method is consistent with the standard Galerkin method and more systematic in generating
the equations for undetermined amplitudes for approximate solutions. The extended
Galerkin method can be used for higher-order solutions of such nonlinear vibrations in a
systematic manner and iterative procedure.
Appl. Sci. 2022, 12, 2979 7 of 8

4. Conclusions
A procedure based on the popular Galerkin method has been proposed for solutions
of nonlinear vibrations with a good approximation of the natural frequency and asymptotic
deformation. It is done by representing the deformation with a harmonic series, and
the nonlinear equation is solved approximately in the fundamental period of vibrations
through adding integration with the weighting function and harmonic terms over time.
Then a nonlinear eigenvalue problem is obtained just like in the standard Galerkin method.
By solving the nonlinear eigenvalue problem, the natural frequency and mode shapes are
obtained approximately and asymptotically. The procedure is actually the extension of
the Galerkin method by adding an integration of the weighting function with time over
one period of the fundamental vibration mode. The same procedure can also be applied
to linear vibration equations to obtain identical results from the Galerkin method. The
effectiveness and accuracy are stemmed from the Galerkin method itself and the periodicity
of vibrations to guarantee excellent approximation with the weak form of the original
problem in differential equations, as demonstrated in approximate solutions in this study
and earlier numerical results [11,24]. This is a unique and systematic procedure for the
analysis of both linear and nonlinear vibrations of elastic structures and solids by the
Galerkin method with possible applications in other fields involving solutions of linear and
nonlinear differential equations, as was proposed and suggested in an earlier study [11].
The simplicity, elegance, effectiveness, and accuracy with advantages of operations of
integrations, particularly with symbolic mathematical tools, offer a favorable and powerful
method for nonlinear vibration analysis and approximate solutions for more general
nonlinear differential equations. Further applications and rigorous formulation of this
technique for the analysis of nonlinear vibrations of solids and structures will provide
a significant addition to available methods for some complications like the presence of
external excitations and bias fields in addition to free vibrations shown here.

Author Contributions: Conception, formulation, revision, and submission of the paper were done
by J.W., while the mathematical procedure, derivation, calculation, drafting, and check were made by
R.W. All authors have read and agreed to the published version of the manuscript.
Funding: This research is supported by the National Natural Science Foundation of China (Grant
11672142) with additional support through the Technology Innovation 2025 Program (Grant 2019B10122)
of the Municipality of Ningbo, Zhejiang Province, China.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data of this study is available from authors and author’s website.
Conflicts of Interest: The funders had no role in the design of the study; in the collection, analyses,
or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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