A comprehensive review on dual approach to the vibration analysis: some dual techniques and application

 Vietnam Journal of Mechanics, VAST, Vol.42, No. 1 (2020), pp. 1 – 14
 DOI: https://doi.org/10.15625/0866-7136/14699
 A COMPREHENSIVE REVIEW ON DUAL APPROACH TO
 THE VIBRATION ANALYSIS: SOME DUAL TECHNIQUES
 AND APPLICATION
 N. D. Anh1,2
 1Institute of Mechanics, VAST, Hanoi, Vietnam
 2VNU University of Engineering and Technology, Hanoi, Vietnam
 E-mail: ndanh@imech.vast.vn
 Received: 16 October 2019 / Published online: 13 December 2019
 Abstract. This paper reviews key ideas of the researches on the dual approach to the
 vibration analysis. Three types of dual techniques, namely, forward - return dual tech-
 nique, global-local dual technique, weighted averaging dual technique, for the problem
 of equivalent replacement are summarized. Different implements and realizations of dual
 techniques to nonlinear vibration analysis and design of dynamic absorbers are reviewed.
 Finally, the challenging issues based on the dual techniques are discussed. A number of
 possibilities for developing analytical techniques related to dual techniques are proposed.
 Keywords: dual approach, dual technique, forward - return dual, global-local dual, weighted
 averaging dual.
 1. INTRODUCTION
 Oscillation plays an important role in our daily lives and nature. This reflects the fact
that life and nature are extremely diverse sets of motions. Fidlin wrote “Any motion is
deeply connected with one of the most fundamental properties of nature – its ability to
react with oscillations at any internal change or external influence” [1]. In all areas of hu-
man activities vibrational systems are increasingly abundant and varied and the study of
those systems is always required. Most phenomena in our world are essentially nonlinear
and described by nonlinear equations. We might simplify nonlinear phenomena as lin-
ear ones to make them easier to understand; however, for further investigation nonlinear
phenomena should be treated as nonlinear problems. Thus, the study of nonlinear prob-
lems is of crucial importance not only in all areas of physics but also in engineering and
in other disciplines. In particular, it appears that the analysis of vibration based on non-
linear mathematical models requires appropriate methods. Therefore, new methods for
analysis of nonlinear oscillations always cause concern of scientists and technicians. Re-
cently, a dual approach has been proposed to study the response of nonlinear systems [2]
 
c 2020 Vietnam Academy of Science and Technology
2 N. D. Anh
and some dual techniques have been developed, e.g. [3–27], based on the concept of du-
ality. Nature and life always contain dual inclinations. Those are perspectives that are
contradictory, or complementary to each other. Natural phenomena and human activ-
ities exhibit often dual characters which reflect two side processes or/and the relative
balance of two opposite sides. For illustration we may say attack – defense in a football
match, one way and return in an excursion, day and night. When a problem is consid-
ered it is quite often that one its side is given too much attention while another side is
almost or completely forgotten. This usual approach doesn’t reflect the real essence of
the problem in question and hence doesn’t yield an expected solution in many cases. The
main issue of a dual approach to a scientific problem is to always consider two different
(dual) aspects of the problem. This allows the study to become more harmonious and
reflects the essence of the problem. In the dual approach a dual technique is one that can
introduce two dual perspectives for the same problem in consideration. Let one needs
to investigate a problem. The use of the dual approach to this problem means that one
should introduce a dual technique that can reflect the essence of the problem.
 This paper reviews key ideas of the reported researches on the dual approaches to
the vibration analysis [3–27]. Various types of dual techniques, different implements and
realizations of dual approaches to vibration analysis and control are reviewed. Finally,
the challenging issues on the dual techniques are discussed.
 2. DUAL EQUIVALENT REPLACEMENT
 The first introduction of the dual approach to the nonlinear vibrations was suggested
in [2] for the problem of equivalent linearization. In science we usually needs to replace
approximately an object A described by the function A(x) with the object B described by
the function B(x). Thus, we will replace A(x) approximately with kB(x) where k is an
equivalent replacement coefficient to be found from an equivalent replacement criterion,
for example, by the mean-square error criterion
 D E
e2 = (A(x) − kB(x))2 → min, (1)
 k
where e = A(x) − kB(x) is the error of replacement, h·i is the deterministic averaging
operator or expectation operator for the case of random functions A(x), B(x). If B(x) is a
linear function of x, B(x) = x, the equivalent replacement is called equivale ... e some vibrating systems with nonlinear-
ities. The strongly nonlinear Duffing oscillator with third, fifth, and seventh powers of
the amplitude, the other strongly nonlinear oscillators and the cubic Duffing with discon-
tinuity are considered. The results obtained via this method are compared with the ones
achieved by the Min-Max Approach (MMA), the Modified Lindstedt – Poincare Method
(MLPM), the Parameter – Expansion Method (PEM), the Homotopy Perturbation Method
(HPM) and 4th order Runge–Kutta method.
10 N. D. Anh
 Further developments of this weighted averaging have been presented in [24,25] for
nonlinear systems appearing in practical engineering and physical problems. In [26, 27]
the equivalent linearization method with weighted averaging has been used to inves-
tigate the nonlinear vibration of microbeams based on the nonlinear elastic foundation
through the modified couple stress theory [26] and the nonlinear vibration of nanobeams
under electrostatic force through the nonlocal strain gradient theory [17]. The obtained
results demonstrate that this method is very convenient for solving nonlinear equations.
 Anh and Nguyen [6,7] suggested approximate analytical solutions of the optimal
tuning ratio of the DVA for the H∞ optimization by using the idea of local averaging
and dual replacement to the equivalent linearization method. Based on the idea of the
weighted dual mean square error criterion, the authors of [8] give an analytical approach
to the design of the three-element dynamic absorber for damped structures under ground
motion by replacing the original damped structure by an equivalent undamped structure.
Comparisons have been done to validate the accuracy of the obtained results. In [17] a
simple method to determine the approximate analytical solutions of the nontraditional
DVA when the damped primary structure is subjected to ground motion is proposed. The
main idea is based on the dual mean square error criterion of the equivalent linearization
method to replace approximately the original damped structure by an equivalent un-
damped one. Comparisons have been done to validate the effectiveness of the obtained
results. The global-local approach is used in [18] to give approximate analytical solutions
of H∞ optimization for all standard, three-element and non-traditional DVAs attached
to damped primary structures. The study is based on the global-local criterion of the
equivalent linearization method in order to replace approximately the original damped
structure by an equivalent undamped one.
 4. SOME PERSPECTIVE ASPECS OF DUAL APPROACH
 In this section we suggest some further perspective research aspects of the dual ap-
proach. First the weighted dual mean square error criterion has been investigated for
many stochastic nonlinear systems; however, it wasn’t used to study deterministic non-
linear vibrational systems. Thus, numerical simulations can be carried out to check the
accuracy of this criterion for free and forced vibrations in those systems. A new expres-
sion for the weighted parameter p different from (16) may be proposed to improve the
accuracy of the weighted dual mean square error criterion.
 Similarly, the global-local mean square error criterion can be modified in order to
be applicable to deterministic nonlinear vibrational systems. Second, comprehensive re-
searches about the local averaging operator should be investigated for periodic functions.
For ω-periodic function x(ωt) the conventional averaged value of x(ωt) over one period
is defined as
 2π/ω 2π
 ω Z 1 Z
 hx(ωt)i = x(ωt)dt = x(τ)dτ. (34)
 2π 2π
 0 0
 A comprehensive review on dual approach to the vibration analysis: Some dual techniques and application 11
 The local averaged value of x(ωt) is defined by replacing 2π in (34) by a positive
value r
 r
 1 Z
 hx(φ)i = x(φ)dφ, φ = ωt. (35)
 r r
 0
 If substituting the local averaging operator determined in (35) into the mean-square
error criterion (1) for the equivalent replacement of A(ωt) by kB(ωt) one gets the local
mean-square error criterion
 r
 1 Z
e2 = (A(φ) − kB(φ))2 dφ → min, (36)
 r r k
 0
which leads to
 r
 Z
 A(φ)B(φ)dφ
 0
 k(r) = r . (37)
 Z
 B2(φ)dφ
 0
 The behavior of the equivalent replacement coefficient k(r) depending on the local
value r needs to be investigated. The question of how r can be chosen is open and mini-
mal values of k(r) are ones of interest.
 5. CONCLUSIONS
 The study of nonlinear problems is of crucial importance not only in all areas of
physics but also in engineering and in other disciplines. In particular, it appears that
the analysis of vibration based on nonlinear mathematical models requires appropriate
methods. Therefore, new methods for analysis of nonlinear oscillations always cause
concern of scientists and technicians. Nature and life always contain dual inclinations.
Those are perspectives that are contradictory, or complementary to each other. There-
fore, scientific research also needs to reflect these properties. The main issue of a dual
approach to a scientific problem is to always consider the two different (dual) aspects of
the problem. This allows the study to become more harmonious and reflect the essence
of the problem. In the dual approach a dual technique is one that can introduce two dual
perspectives for the same problem in consideration.
 This paper reviews key ideas of the reported researches on the dual approaches to
the vibration analysis. Three types of dual techniques, namely, forward - return dual
technique, global-local dual technique, weighted averaging dual technique for the prob-
lem of equivalent replacement are summarized. Different implements and realizations
of dual techniques to nonlinear vibration analysis and design of dynamic absorbers are
reviewed. Finally, the challenging issues based on the dual techniques are discussed. A
number of possibilities for developing analytical techniques related to dual techniques
12 N. D. Anh
are proposed. The review of the above studies shows that the dual approach is an ap-
propriate one and the dual techniques are effective tools for studying random and deter-
ministic nonlinear vibrational systems. Meanwhile the extension of the dual approach to
other scientific problems is of high interest.
 ACKNOWLEDGEMENTS
 The paper is supported by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.04-2018.12.
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