Assessment of Structural Monitoring by Analyzing Some Modal Parameters: An Extended Inventory of Methods and Developments

The SHM (structural health monitoring) evaluation consists to determining the modes (resonances) of vibration characteristic of the structure and each of them is represented by its modal parameters which can be obtained experimentally and can be analyzed by different procedures. In the present paper (except subsections 2.1.1, 3.1.1) the coefficients (including the coefficient of displacement) are constant; in this regard, it is made an inventory of some methods of classical and non-classical mathematics with the specific computing scheme. All methods of classical mathematics that were considered, i.e., second-order linear nonhomogeneous differential equations, Laplace operational method, analytical conditional form, approximations with error evaluation (according to contraction principle) are inventoried and then developed—and the first two methods by comparison. As methods of the non-classical mathematics the dyadic wavelet method, the approximation (transform) fuzzy method and the grammatical evolution method are inventoried and then the first is developed. In addition, there are illustrated the calculus techniques by few examples and also computing wavelet coefficients for healthy and damaged structures. Notice that in the subsection 2.1.1 in which the coefficients (including the coefficient of displacement) are considered variable there are some transformations of (non)homogeneous equations in the scaled forms. In addition in the subsection 3.1.1 exact solutions, respectively approximations with error evaluation (according to contraction principle) relative to remarkable equations in literature (including the corresponding standard form) are obtained by using the results of 2.1.1.


Introduction
Health Management is defined as being the process of making optimal specific decisions, respectively recommendations about operation, mission and maintenance actions, based on the health evaluation data gathered by Health Monitoring Systems (HMS). As a part of Health Management procedure, the technology of Structural Health Monitoring (SHM) [1] is based on the process of implementing a damage detection and characterization strategy with applications in aerospace, civil and mechanical engineering infrastructure domain [1][2][3][4][5][6]. Each mode can be represented as a SDOF (single degree of freedom system) and characterized by its modal parameters: natural (modal or resonant) frequency, modal damping, and a mode shape-the change of structural properties causes their variation (it was experimentally proved the correlation between damage severity and modal parameters). The experimental modal parameters are obtained from a set of the frequency response function data (e.g., processing the SDOF or MDOF). In literature a large number of analytical and experimental studies have been conducted to establish correlations between damage severity and modal parameters. The wavelet analysis can make possible to evaluate the health of structures, see [1][2][3][4][5][6]. For example, in his research work Newland [7][8][9][10] applied the harmonic wavelet analysis to the study of vibration of the structures caused by underground trains and road traffic. The signal is analyzed with a flexible time-frequency window that narrows for high frequencies and extends for low frequencies, i.e. breaking up of the signal into shifted and scaled versions of a "mother wavelet" (scaling function) or "father wavelet" (basis function).
In the time domain, the mathematical model of SDOF is a 2nd-order differential equation of the form where f(t) represents the external excitation m is system mass, k stiffness, c viscous damping coefficient and x(t) the displacement response. In general, the displacement response x = x(t) is the solution of Cauchy problem where m, c ∈ ℝ * + , k, b ∈ C 0 I , I ⊆ ℝ + , k = k(t) is the variable displacement coefficient (in the Sects. 2.1.1, 3.1.1) and b = b(t) represents the disturbance. Next (except Sects. 2.1.1, 3.1.1) the displacement coefficient k is constant, i.e. k ∈ ℝ and the Eq. implies b m is positive definite) and that affects all the system parameters. Also, for the unforced damped SDOF system, the value of the damping ratio = c m 2 √ k m , k m ∈ ℝ * + determines the way the system oscillations go to zero, i.e., the cases underdamped ( 0 < < 1 ), overdamped ( > 1 ), respectively critically damped ( = 1 ) system. On the other hand the constant case allows "a calibration of dyadic wavelets" (see Sect. 4).

Second Order Linear Nonhomogeneous Differential Equations with Constant Coefficients
The solution of Cauchy problem (2), (3) is obtained as a particular solution of a 2nd-order linear nonhomogeneous differential equation with constant coefficients [12,13] is naturally as b m be quasi-polynomial consequently x p is quasi-polynomial with the association according to the method of non-determined coefficients. For some developments see Sect. 3.1.

Variable Coefficients
Consider a 2nd-order nonhomogeneous linear equation in the canonical form with temporally notation and variable coefficients I the disturbance and the associated homogenous equation Observation 2.1. 1.1 (solutions) The general solution of (9), (10) are respectively where x 1 , x 2 , respectively x p are particular solutions of (10) respectively of (9) (and the associated Wronskian of x 1 , x 2 , W(x 1 , x 2 ) is non-zero). In addition the solution of Cauchy problem is according to the initial conditions (see 2.1).
The scaled form (implicitly relative to a 0 ) of (9) is there are amibiguities in the context); the scaled form is native if a 0 = 1 . The scaled form of (10) is The absolute scaled form (implicitly relative to a 0 ) of (9) is ; the absolute scaled form of (10) is ii) (dual scaled forms) Particularly, the dual scaled form (implicitly relative to a 2 ≠ 0 ) of (9) is ; the dual scaled form of (10) is The absolute dual scaled form of (9) is ; the absolute dual scaled form of (10) is Observation 2.1.1.2 (Wronskian, see [14]) The definition expression of the Wronskian associated with x 1 , x 2 is (10) a 0ẍ + a 1ẋ + a 2 x = 0.
, ∈ ℝ * the equation (13) the equation (13) is transformed in dt, This form is formally mentioned as an equation with constant coefficients in [14]; relative to point ii) in the particular This form is mentioned as a canonical form relative to the equation (10) in [14]. Notice that for this form it is true (see ii) (reduction to the Riccati equation) By the change of function it is obtained the Riccati equation u = u 2 + u + , where = −1, = −a 1 , = −a 2 . Indeed, it is successively obtained ̇x = xu , ẍ =̇xu + xu , ẍ + a 1ẋ + a 2 x =̇xu+ xu + a 1 xu + a 2 x = x(u 2 +̇u + a 1 u + a 2 ) = 0 , ̇u = u 2 + u + = 0 with = −1, = −a 1 , = −a 2 . This is mentioned in another form in [14]. Notice that it is true For the forms (30), respectively (29) the "reduced" Riccati equation is respectively

(conditional reduction)
The "absolute" Bernoulli equation where ∈ ℝ * has the general integral Proof By the change of function (see Theorem 2.1. Finally, the general integral is given by (41). ◻ (ii) (the linear image of a pseudo-linear extension) In [RN] 1 are presented Notingher models as first and higher order pseudo-linear differential equations. In addition it is defined the linear image of a pseudo-linear equation, respectively a pseudo-linear extension of a linear equation; a pseudo-linear extension is convenient relative to the approximation of the Cauchy problem solution with the contraction principle.

Laplace Operational Method
According to the derivation theorem, the linearity of Laplace operator L and Sect. 2.1 the operational equation associated with Cauchy problem is (see [11,15]) The solution of (45) is

Analytical Conditional Form
If the disturbance b m has an analytical form, i.e. b m = ∑ n∈ℕ n t n , then the displacement x m ay have the form x(t) = ∑ n∈ℕ n t n , respectively on I b = [0, r b ], I x = [0, r x ] ⊂ ℝ + (relative to the positive parts of the convergence intervals [14]). The initial conditions involve 0 = x 0 , 1 = v 0 ; in addition are obtained recurrence relations between coefficients (according to the Eq. 3) and finally result as it is possible that x have no an analytical form on the interval I x . For some developments see Sect. 3.3.

Approximations with Error Evaluation
The Cauchy problem (3) becomes y (as vectorial differential equation with positively defined functions). In terms of existence and uniqueness theorem (see [11,15]) the approximations chain (in a Banach subspace F 0 in relation to the norm ‖ ⋅ ‖ u ) is given by or respectively analytically by (45) 1 [RN] M. Rebenciuc, P. Notingher, Notingher models as pseudolinear differential equations of first and higher order: An inventory of methods and developments, manuscript.

Dyadic Wavelet Method
It will be assumed that SDOF system (1) is dynamically forced with a harmonic function (i.e., under harmonic loading [3,16]). Formally, the more general wavelet solution of the Cauchy problem for the governing equation of motion of the system (1) is where x w p (t) is the particular wavelet solution of the problem, respectively the eighenfunction x w o (t) which represents the solution for the homogeneous equation associated to the Eq. (3) and the two coefficients c 1 , c 2 (see (5)) denoted with K 1 , K 2 are determined by applying the initial conditions X w (0) = x 0 and Ẋ w (0) = v 0 to solution (52), see Sect. 4.

Approximation (Transform) Fuzzy Method
In general, when a realistic problem is transformed into deterministic initial value problem of ordinary differential equations (e.g., 2) the following dilemma appears, namely if that model is optimal. For example, the initial value may not be known exactly and the function F t, x,̇x(t), f (t) from SDOF may contain uncertain parameters. Thus, if these parameters are estimated through certain measurements, then inevitably errors occur. The analysis of the effect of these errors leads to the study of the qualitative behavior of the solutions of Eq. (2) and in this case, it would be natural to use fuzzy initial value problems FIVPs, see [17]. Particularly, FIVP is a topic very important as much of the theoretical point of view as well as of their applications, for example, in the civil engineering domain [18]. In general, FIVPs do not always have solutions which we can obtain using analytical methods. In fact, many of real physical phenomena are almost impossible to solve by this technique. For this reason, in literature has proposed numerical methods to approximate the solutions of FIVPs: the predictor-corrector method PCM, the continuous genetic algorithm CGA, the artificial neural network approach ANNA (see [19]), etc. The homotopy analysis method HAM proposed by Liao [20][21][22] is effectively and easily used to solve some classes of linear and nonlinear problems without linearization, perturbation, or discretization. The HAM is based on a basic concept in topology, more precisely on the homotopy. It is introduced an auxiliary parameter to construct the so-called "zero-order deformation equation" and the HAM provides us with a family of solution expressions which depend of this auxiliary parameter-the convergence region and rate of solution series can be controlling by the HAM (by choosing a proper value of the auxiliary parameter). Thus, using HAM for the linear and nonlinear cases the results obtained are very effective.

Grammatical Evolution Method
An EA (evolutionary algorithm), particulary the canonical GA (genetic algorithm), can be characterised by where x(t) is the population of encodings at iteration t and r, v, s are "evolutionary" operators, respectively of replacement, of variation (crossover and mutation) and of selection (see [23]); Cauchy solution problem of point 1 belongs about "population of art". In this regard a first variant is an adaptation of the hybrid algorithm (for solving linear and partial differential equations) proposed by He et al. in [24]; hybridization consists of combining techniques of EA and some classical numerical methods, see [25] and [26] for a literature survey relative to EA in China.
The two, respectively three variant are adaptations of the GP (genetic programming) algorithms proposed by Burgess [27], by Cao et al. [28] and by Iba and Sakamoto [29], that induce the underlying differential equation from experimental data. GP as a tree-representations of computer programs was developed in Koza's four books-the first is [30] and was popularized in [31]; in addition see [32] for new developments in key applications of GP (and its variants) such as specialized applications, hybridized systems (which "marry" GP to other technologies) and a detailed look of some recent GP software release.
The four variant is an adaptation of the GE (grammatical evolution) algorithm proposed by Tsoulos and Lagaris [33]; GE as a grammar-based form of GP was developed in [34,35] (financial modelling) and [36] (dynamic environments).
GE uses a genotype-phenotype mapping where the linear genom selects production rules from BNF (Backus-Naur Form) grammar to map down to a syntactically correct program; as translation model the modulo rule defines a degenerate mapping from those l bits codon value to a choice ch for a production rule where s is the current (active) non-terminal and r s1 , … , r sn s is the production rules for s, j ∈ 1, … , n s (and in addition many codon value map the same choice of rules and usually l = 8).
The process of replacing non-terminals is of if: i. a phenotype (a complete computer program) is generated (all the non-terminals in the expression being mapped are transformed into terminals); ii. the end of genome is reached-in which case the wrapping operator is invoked (the return of the genom reading frame to the left hand side of the genome once again-until the maximum number of wrapping events has occurred); an incompletely mapped individual is assigned the lowest possible fitness value.
In [37] are analyzed the integer-based genotypic representation and the binary-based genotypic representation and associated mutation operator-with a statistically significant advantage of the integer-based genotypic representation. In [38] is stated a low locality of the GE representation as a non-correspondence neighboring genotype-neighboring phenotype. In this regard (for further research)-relative to the partial adapted grammar G C is analysed looking for an admissible chromosome for the distance x(t) = e t sin( t) (see point 1 and Sect. 2.7.1). GP and particularly GE have structural difficulty (it is unable to search effectively for some solutions); more specifically, GE outperforms standard GP if optimal solutions are composed of very narrow and deep structure and in contrast, where optimal solutions require more dense trees GP outperforms GE (see [39]). According to [33] it is considered "general form" of Cauchy problem (2) g(0, x(0),̇x(0)) = 0, g = g 1 , g 2 , corresponding model M c (t) expressed in the grammar G C (see Sect. 2.7.1). The fitness value of c is penalty function, ∈ ℝ * + is the scale parameter. Solutions are expressed in an analytical closed form-particularly exact solution is recovered if b m is quasi-polynomial. For derivatives it uses three different stacks-respectively for M c , Ṁ c , M c according differentiation rules adopted by the various automatic differentiation methods (see [30]); specifically constants creation are made using a meta-grammar (grammar of grammars, see [36]). As an update of the four variant control sequences of point 1 (developed in Sect. 3.1) can be obtained with control genes and with the behaviour switching BNF grammar definition (BNF-BS, see [40]).

Admissible Chromosome-Relative to G C
The partial adapted grammar G C = s 0 , N, T, R (which creates the constants of the C = { , } ⊂ T , see [37]):

Some Developments of Classical Mathematics
In this section are developed the methods which are presented in Sects.

Some Developments for Variable Coefficients
Free vibration of single-degree-of-freedom(SDOF) systems with periodic time-dependent coefficients (mass and stiffness) have been extensively investigated. So the equation with non-periodically time-varying coefficients are considered in particular.

(scaled "reduced" canonical form)
i) Let be the equation of type Konofin [41], respectively the scaled form (see Sect. 1) where m ∈ ℝ * + and the associated homogeneous equation, respectively a particular case with The particular form has the particular solution Another particular solution x 2 is given (according to Observations 2.1.1.2, 2.1.1.4. i)) by (64) (which is calculated according to the analytical expression of the exponential). ii) The equation of type Kolenef [42], respectively the scaled form is (61), where m = m(t) , k = k(t) are positive definite-particularly m(t) = m 1 t 1 , k(t) = k 1 t 2 , 1 , 2 ∈ ℝ (at the first step time interval). The associated homogeneous equation of the scaled particular form where 1 − 2 = 2 , m 1 > 4k 1 has the particular solution w h e r e r ∈ {r 1 , r 2 } , ⋅ 1 2 , i.e. relative to function variation in ; for solutions G k m has an exponential aspect for < 1 , a "linearizat" exponential aspect for = 1 and a periodical aspect for > 1.
4. (general case) "Any of the mass, damping or stiffness coefficients of a mechanical oscillator may depend on time" (see [44]). Consequently in the equation (3) (see section 1) c m , k m ∈ R * + become c m = c m (t) , k m = k m (t) positively defined and c m , k m ∈ C 0 I . In addition in the standard form (69) where k m = 2 0 , c m = 2 0 , 0 , 1 ∈ ℝ * become 0 = 0 (t) , 1 = 1 (t) , 0 , 1 ∈ C 0 I . Notice that the scaled and absolute dual scaled form of the associated homogeneous equation is where |s � | a 1 = + 2 , = 4̇0 + 1 2 0 and ( |s � | a 1 ) is con- In [45] a function for describing the variation of mass of a SDOF system with time is an arbitrary one positive definite and the variation of the stiffness is expressed as a functional relation with the mass function, i.e. c m of (3) (see Sect. 1) becomes c m =ṁ m + c m , where c is of Li [45]. Using appropriate change of variable the governing differential equation is reduced to a Bessel equation or other analytically solvable equation (see [45]). Relative to the "reduced" canonical form ( iv) c m ∈ [1,9] which involves t ∈ [t 1 , t 9 ] . For the standard form (see point 4) the case c m of the update ii) of (3.4) becomes ̄2 0 ≤ 2̄0 iff 2 0 ≤ 2 0 iff 0 ≤ 2 iff ̄0 ≤ 2 , 0 > 0 ; in addition for the update iv) it is true c m ∈ [1,9]  where v = v(x) is the potential energy of a non-linear system and in particular it is v(x) = k 2 x 2 − x 4 with corresponding particular differential equation (see [46]) respectively where (t)f (̇y) is a "separable" function with (t) a coefficient (see [47]) are 2nd-order pseudo-linear differential equations (see 2.1.1); these may be convenient relative to the approximation of the Cauchy problem solution with the contraction principle relative to 2nd-order linear differential equations.

Laplace Operational Method
Through the specific of problems occurs x c o = e t x 0 cos( t) + q 0 ( ) sin( t) , according to displacement theorem (see [11,15]).

Analytical Conditional Form
For as the positive parts of the convergence intervals for the associated MacLaurin series there are (absolute) punctual, respectively uniform (normal) convergence if I x =Ĩ x , respectively I b =Ĩ b , respectively if I x ⊂Ĩ x , respectively I b ⊂Ĩ b . A sufficient condition for analytical form (associated with the necessary condition is that uniformly bounded derivatives on I-with uniform (normal) convergence for bounded interval I (see [14]). The equation of (3) becomes

Approximations with Error Evaluation
Function h ∶ D h → ℝ 2 , D h ⊆ ℝ 3 , verifies the hypothesis of the existence and uniqueness theorem (see [11,15])locally, relative to the parallelepiped

Dyadic Wavelet Method
In some differential problems such as those corresponding to certain physical processes two main directions have been pursued with regard to the applicability of the wavelets [1-10, 16, 48-58] : determining the solution using wavelets with compact support and fulfillment of the conditions of the problem (initial or boundary) with some restrictions, e.g. [48]. It appeared a dilemma, namely, which wavelet family from a variety is the best choice to satisfy the axioms of multiresolution analysis [54,55]-a major disadvantage of the wavelet theory being the arbitrary character of their choice. For example, the Daubechies [53] wavelets generate a large number of calculations, even for the one simpler differential problem. In this respect, the dyadic wavelets package (defined analytic functions, infinitely derivable and band-limited) is the most appropriate tool for studying processes located in a Fourier domain. With this choice solving such problems is a trivial one the wavelet solution of the problem being accessible. Therefore, based on the properties of wavelets like good localization and resolution control the concept of orthonormal wavelet series [7][8][9][10] and representation of a function f ∈ L 2 (ℝ) by such an expression were introduced. To illustrate this, dyadic wavelet approximation represents a numerical method used to obtain numerical solutions of some physical problems. An orthonormal dyadic wavelet is a function ∈ L 2 (ℝ) , such that the family D n T s [ ] n, s∈ℤ is an orthonormal basis for   (89) 0 h|0 q 0 q (t) n k|i j i j (t) n k|i j i j (t) + k n k (t) n k|i j̄ i j (t) n k|i j̄ i j (t) + k n k (t) = f (t). (99) 0 l|0 h and s = 1, 2 represents the order of derivatives and the connection coefficients n r|n k , s = 1, 2 . As well, the connection coefficients of the harmonic wavelet functions [s] n k|n r are given by respectively (100) � Λ [2] 0 h�0 l + c Λ [1] 0 h�0 l + + k M+1 [2] 0 h�0 l = � Γ [2] n k�n r + c Γ [1] n k�n r + k 2M+1 � Γ [2] n k�n r + c Γ [1] n k�n r + k 2M+1  (52) is found. Also, are used so-called harmonic connection coefficients and their conjugates (for ∀ s ∈ ℕ the derivation order) which are provided by the derivatives of harmonic wavelet basis, for more technical details of computation see [16,49,52]. Coefficients K 1 , K 2 of the homogeneous wavelet solution are determined by applying the initial conditions to (52).
As a conclusion few wavelet-based methods to damage detection in a structure will be summarized further. Variation of wavelet coefficients represents a monitoring process normally used and its main scope is that identifying the existence and severity of damages in a machine or civil structure. In this regard, it has been established that the damages in a structure results in the variation of the wavelet coefficients-usually caused by the change of modal properties of a structure (after it experiences damage, see [1]). Another method is local perturbation of wavelet coefficients for damage localization in structures. It implies not only the existence, but also determine the location of damage, because the wavelet coefficients tend to show irregularity near the crack. Reflected wave method caused by local damage-this technique is used to localize damages, and measuring the severity of the damage is possible by studying the wave velocity.

Conclusions
Equation (3)-where the displacement coefficient is constant is a particular case of the original Eq. (2) -where the displacement coefficient is variable; the original purpose of the paper it was "a calibration of dyadic wavelets" (of a method of non-classical mathematics for obtaining Cauchy problem solution CPS of (2)) on the particular case of (3), see Sects. 2.5, 4. In this approach "of calibration" has been useful first (as method of classical mathematics) 2nd-order linear nonhomogeneous differential equations "method" by which has been established appearance structural of CPS (see Sects. 2.1, 3.1); Laplace operational method is "an interesting update" in which x c o (t) ("the homogeneous" part of CPS) is independent of disturbance (see Sects. 2.2, 3.2). Further according to analytical conditional form "method" result as it is possible that the displacement x(t) have no an analytical form on interval (see Sects. 2.3, 3.3), i.e., the product theorems relatively the series are not pointwise operational (according to Sect. 3.1). Finally approximations with error evaluation "method" (as "a string method" (106) [1] n k|n r = 2 n ⋅ 3 i, k = r 2 n (r − k) 2 , k ≠ r, of classical mathematics) gives a approximation z N where