Valery F Timkov
email: valerytimkov@gmail.com

A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron. Improving the Accuracy of Some Physical Constants

Keywords: Planck's length, mass, and time, the accuracy of physical constants, Euler number, Planck's Universal Proportions, gravitational quantization step

    A Method for Empirical Estimation of Planck's Length, Mass, and Time through the Characteristics of an Electron. Improving the Accuracy of Some Physical Constants

Introduction

The structure of knowledge about the surrounding physical world consists of laws, constants, initial conditions, limits, restrictions, and thresholds. Physical constants play a special role in this structure. The laws of nature can be conditionally represented as a composition of their dynamic and static parts. The dynamic part of the laws is represented by the variables of these laws, and the static part is represented by physical constants. The static part of the laws of nature reflects the This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172 P r e p r i n t n o t p e e r r e v i e w e d structure of matter and the dynamic part of its movement. Consequently, the nature of physical constants consists in the numerical display of individual elements of the structure of matter at its structural level, at which the scenario of the action of a physical law unfolds. The totality of all existing physical constants forms a system of knowledge about the numerical values of the basic elements of the structure of matter. Physical constants have a set of specific characteristics. These characteristics of physical constants are discussed and described, for example, in [START_REF] Weinberg | Overview of Theoretical Prospects for Understanding the Values of Fundamental Constants [and Discussion[END_REF][START_REF] Barrow | The Constants of Nature: From Alpha to Omega -The Numbers that Encode the Deepest Secrets of the Universe[END_REF][START_REF] Uzan | The Natural Laws of the Universe: Understanding Fundamental Constants[END_REF][START_REF] Uzan | Varying Constants, Gravitation and Cosmology[END_REF][START_REF] Uzan | The fundamental constants and their variation: observationaland theoretical status[END_REF][START_REF] Bjorken | Standard Model Parameters and the Cosmological Constant[END_REF][START_REF] Fritzsch | The Fundamental Constants in Physics[END_REF][START_REF] Wilczek | Fundamental constant[END_REF][START_REF] Duff | Trialogue on the number of fundamental constants[END_REF]. It follows from these publications that the main characteristics of physical constants are: universality, invariance, stability, experimental verification and experimental confirmation of numerical values, and accuracy. When using physical constants in practice, the characteristics of stability, experimental measurements of their values, and accuracy are of the greatest interest.

Stability and non-variability in time of the values of physical constants are directly related to their nature. A physical constant is stable to the extent that the structure of matter represented by this constant is stable. The problem of stability of physical constants is studied in publications [START_REF] Uzan | Varying Constants, Gravitation and Cosmology[END_REF][START_REF] Duff | Trialogue on the number of fundamental constants[END_REF][START_REF] Duff | How fundamental are fundamental constants?[END_REF][START_REF] Duff | Comment on time-variation of fundamental constant[END_REF][START_REF] Leite | ESPRESSO's Early Commissioning Results and Performance Related to Tests of Fundamental Constant Stability[END_REF][START_REF] Colaço | Constraining a possible time-variation of the speed of light along with the fine-structure constant using strong gravitational lensing and Type Ia supernovae observations[END_REF][START_REF] Landau | Variation of fundamental constants and white dwarfs[END_REF][START_REF] Lopez-Honorez | Variations in fundamental constants at the cosmic dawn[END_REF]. It follows from the publications that the greatest attention is paid to the study of the stability of constants: the speed of light in a vacuum, the fine structure constant, the ratio of the proton and electron masses, the elementary electric charge, and the gravitational constant. It is noted that there is a possibility of a certain instability of these constants in time. For a confident assessment of the instability of the constants, it is necessary to take into account the observation conditions in the experiments and increase the accuracy and stability of the measuring equipment.

The measurement of physical constants, their dimension, and increasing the accuracy of values are studied in publications [START_REF] Smith | The Measurement of the Fundamental Constants[END_REF][START_REF] Kawasaki | Measurement of the Newtonian Constant of Gravitation G by Precision Displacement Sensors[END_REF][START_REF] Mohr | Data and analysis for the CODATA 2017 special fundamental constants adjustment[END_REF][START_REF] Tegmark | Dimensionless constants, cosmology, and other dark matters[END_REF][START_REF] Petley | Towards the Next Evaluation of the Fundamental Physical Constants[END_REF][START_REF] Timkov | Improving the Accuracy of the Numerical Values of the Estimates Some Fundamental Physical Constants[END_REF][START_REF] Timkov | The Method for Increasing the Accuracy of Some Fundamental Physical Constants[END_REF][START_REF] Anderson | Measurements of Newton's gravitational constant and the length of day[END_REF][START_REF] Taylor | Determination of e/h, Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants[END_REF]. Almost all constants that are included in the CODATA listing [START_REF]Fundamental Physical Constants[END_REF] are measured. The greatest attention is paid to the experimental evaluation of numerical values: Newtonian constant of gravitation fine-structure constant elementary charge speed of light in vacuum Planck constant Boltzmann constant electron mass proton mass Compton wavelength To improve the accuracy of physical constants, two directions are used. The main direction is realized through the experiment, with the help of improving its methodology, carefully taking into account the conditions of observation, and improving the measuring equipment. An auxiliary direction is analytical when the analytical formula of a physical constant with low accuracy is known, which is expressed in terms of constants with higher accuracy. For example, let's increase the accuracy of the value of the electron mass constant from 10 to 11 decimal places. The following formula is presented in [START_REF] Timkov | Improving the Accuracy of the Numerical Values of the Estimates Some Fundamental Physical Constants[END_REF]: the moments of mass of a hypothetical Planck particle, as well as leptons and baryons, expressed in terms of their Compton wavelength (in particular, over 2pi), are equal to each other:

(1) , G ,  , q , c , h , k , e m , pr m . C  e m , P P e C C C pr Cpr n Cn m l m m m m m                    
where respectively: this is the Planck mass, as well as the masses:

electron, muon, tau, proton, neutron; this is the Planck length, as well as the Compton wavelength over 2pi: electron, muon, tau, proton, neutron.

In formula [START_REF] Weinberg | Overview of Theoretical Prospects for Understanding the Values of Fundamental Constants [and Discussion[END_REF], the proton has the highest accuracy and the smallest value of Uncertainty, therefore, for the condition of the task, formula (1) has the form:

(2)

From:

Then, according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF] and taking into account the NIST Uncertainty Machine [START_REF]NIST Uncertainty Machine[END_REF]:

(

The accuracy value of the Newtonian constant of gravitation is one of the lowest among all fundamental constants. The Newtonian constant of gravitation in 1969 [START_REF] Taylor | Determination of e/h, Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants[END_REF] was:

Now [START_REF]Fundamental Physical Constants[END_REF] Newtonian constant of gravitation has the value:

In the process of measurements from 1969 to the present, the value of the Newtonian constant of gravitation has changed significantly, and the accuracy of its value has increased by only one order of magnitude. This is the basis for this constant to change the priority of methods for estimating its values, that is, the main method for estimating the Newtonian constant of gravitation should be considered analytical, which makes it possible to increase the accuracy of measuring Newtonian constant of gravitation from 5 to 11 decimal places.

Physical constants can be divided into dimensionless and having dimensions. Constants having dimensions are mainly represented in units of electromagnetism or units of gravity. Since the electric charge is a function of the moment of mass, and the Coulomb law can be represented in the gravitational form [START_REF] Timkov | Electric Charge as a Function of the Moment of Mass. Gravitational Form of Coulomb`s Law[END_REF], the electromagnetic field is a special case of the gravitational field. This means that all the basic units of measurement of electromagnetism can be represented in units of gravity: meter, kilogram, and second. Based on the concepts: 1) electric charge is a function of the moment of mass, 2) the moments of mass of leptons and baryons, defined through their Compton wavelength over 2pi, are equal to each other and equal to the moment of mass of a hypothetical Planck particle, it is possible to improve the accuracy of some physical constants, for example, constant fine structure, elementary electric charge, electron mass, Bohr magneton.

This accuracy value corresponds to the accuracy of the characteristics of the electron and proton, that is, 11 decimal places. Improving the accuracy of measurements of the characteristics of the electron and proton will increase the accuracy of the above physical constants.

, , , , , ,

P e pr n m m m m m m    , , , , , , P C C C Cpr Cn l         .
         G 11 3 1 2 6.6732(31) 10 . G m kg s      11 3 1 2 6.67430(31) 10 . G m kg s     
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172

P r e p r i n t n o t p e e r r e v i e w e d

The idea of a connection between the characteristics of a hypothetical Planck particle and a real elementary particle is presented in [START_REF] Timkov | Fractal Structure of the Fundamental Constants. Numerical Evaluation of the Values of Fundamental Constants with Use of the Major Characteristics of Muon[END_REF]. The article proposes to connect the characteristics of the hypothetical Planck particle with the characteristics of the muon through the golden algebraic fractal.

A Method for Empirical Estimation of Planck's Length, Mass, and Time through the

Characteristics of an Electron

Let us estimate the characteristics of one mole of the hypothetical Planck particles, and then compare them with the characteristics of an electron. The physical meaning of the proposed method will not change if in its algorithm instead of the Avogadro constant: we will use the dimensionless Avogadro number: , meaning by this number the number of structural elements of matter in one mole. Planck length of one mole of hypothetical Planck particles with [START_REF]Fundamental Physical Constants[END_REF]:

where: is the Planck length [START_REF]Fundamental Physical Constants[END_REF].

Let us estimate how many times the Bohr radius [START_REF]Fundamental Physical Constants[END_REF] is greater than the Planck length of one mole of hypothetical Planck particles:

(

) 5
The number is close to twice Euler's number. The relative error of non-coincidence of their values is: Let the number be equal to twice the value of the Euler number, then:

Whence the Planck length, taking into account [START_REF]NIST Uncertainty Machine[END_REF], is equal to:

Let's estimate the mass of one mole of electrons :

where is the electron mass [START_REF]Fundamental Physical Constants[END_REF]. Let us estimate how many times the mass is greater than the Planck mass [START_REF]Fundamental Physical Constants[END_REF]:

.0221407 6 6 10 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172

A l N mo   

P r e p r i n t n o t p e e r r e v i e w e d

The number is close to the value of the number:

The relative error of non-coincidence of their values is: Let:

Then, taking into account formula ( 8):

(10)

Let's increase the accuracy of the fine-structure constant from 10 to 11 decimal places. The value of the fine-structure constant [START_REF] Timkov | Improving the Accuracy of the Numerical Values of the Estimates Some Fundamental Physical Constants[END_REF] in terms of the Bohr radius and the Rydberg constant [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF]: [START_REF] Duff | Comment on time-variation of fundamental constant[END_REF] Taking into account formulas ( 3) and ( 11), as well as [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF], the Planck mass has the value: [START_REF] Leite | ESPRESSO's Early Commissioning Results and Performance Related to Tests of Fundamental Constant Stability[END_REF] Let us estimate the quantum of the atomic unit of time [START_REF]Fundamental Physical Constants[END_REF] of one mole of electrons:

The value of quantum is approximately in times the Planck time [START_REF]Fundamental Physical Constants[END_REF]. The relative error of non-coincidence of their values is: Let's replace the attribute approximately with the attribute exactly, then, taking into account the formula [START_REF] Duff | Comment on time-variation of fundamental constant[END_REF] and [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF]: [START_REF] Landau | Variation of fundamental constants and white dwarfs[END_REF] We denote the dimensionless constant as and call it the Avogadro-Euler constant:

Then formulas [START_REF] Fritzsch | The Fundamental Constants in Physics[END_REF][START_REF] Leite | ESPRESSO's Early Commissioning Results and Performance Related to Tests of Fundamental Constant Stability[END_REF][START_REF] Landau | Variation of fundamental constants and white dwarfs[END_REF] can be represented as:

(16) (17)
Partial verification of formulas [START_REF] Smith | The Measurement of the Fundamental Constants[END_REF][START_REF] Kawasaki | Measurement of the Newtonian Constant of Gravitation G by Precision Displacement Sensors[END_REF][START_REF] Mohr | Data and analysis for the CODATA 2017 special fundamental constants adjustment[END_REF]. In accordance with formula (1) and taking into account the most accurate characteristics of a proton, it is true:

According to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]:

1 A 1 . 2 e    0.0037%. 1 1 . 2 A e     2 . d P A e m e N m        0 a R  0 3 12 7.29735256929 10 (1.1 10 ). 4 a R             8
         44 54
5.39144339732 10 (8. 13 10 ) .

au P AE t t s N         . P P pr Cpr m l m     (20) (21)
The relative error of non-coincidence of the result of formulas ( 20) and ( 21) is:

The moment of mass of a hypothetical Planck particle according to formulas [START_REF] Smith | The Measurement of the Fundamental Constants[END_REF][START_REF] Kawasaki | Measurement of the Newtonian Constant of Gravitation G by Precision Displacement Sensors[END_REF]:

(22)
The relative error of non-coincidence of the result of formulas ( 21) and ( 22) is:

Speed of light in a vacuum according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]:

Speed of light in a vacuum in terms of Planck length and Planck time according to CODATA 2018:

The relative error of non-coincidence of the result is:

The speed of light in a vacuum according to formulas ( 16) and ( 18):

The match of the result is 100%.

Tests have shown that the proposed estimates for Planck's length, mass, and time are more efficient than the estimates provided in CODATA 2018.

Improving the Accuracy of Some Physical Constants

It is known [START_REF] Timkov | Electric Charge as a Function of the Moment of Mass. Gravitational Form of Coulomb`s Law[END_REF] that the electric charge is a function of the moment of mass:

Then, taking into account formulas [START_REF] Duff | Comment on time-variation of fundamental constant[END_REF] and [START_REF] Tegmark | Dimensionless constants, cosmology, and other dark matters[END_REF], as well as [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF]: [START_REF]NIST Uncertainty Machine[END_REF] Elementary electric charge according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]:

The accuracy of the value of the elementary electric charge has been increased by 2 orders of magnitude, and the exact attribute should be removed. Planck constant taking into account formula [START_REF] Tegmark | Dimensionless constants, cosmology, and other dark matters[END_REF], as well as [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF]:

Planck constant according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]: 3.51767294176 10 .

P P AE e e C AE a m l k N m m g N m                   0.0000000002%.
                  q 2 7
10 .

P P q m l     
h J m s m l c c                   34 1 1 6.62607015 10 ( ). h J s exact   
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P r e p r i n t n o t p e e r r e v i e w e d

The accuracy of the value of Planck's constant has been increased by 3 orders of magnitude, and the exact attribute must be removed. Let us estimate the auxiliary constants. Estimation of the linear density of the Planck mass taking into account formulas ( 16) and ( 17):

(28)

Estimation of the linear rarefaction of the Planck mass taking into account formulas ( 16) and ( 17):

(29)

Estimation of the Planck charge taking into account formula ( 25):

(30)

Estimation of the Planck acceleration taking into account formulas ( 16) and ( 18):

(31)

Estimation of the static moment of the Planck mass taking into account formula ( 19):

Estimation of the dynamic moment of the Planck mass, that is, the moment of motion of the Planck mass taking into account formula ( 17):

(33)

Estimation of the Planck force (34)

Estimation of the Planck energy taking into account formulas [START_REF] Smith | The Measurement of the Fundamental Constants[END_REF][START_REF] Kawasaki | Measurement of the Newtonian Constant of Gravitation G by Precision Displacement Sensors[END_REF][START_REF] Mohr | Data and analysis for the CODATA 2017 special fundamental constants adjustment[END_REF][START_REF]NIST Uncertainty Machine[END_REF]:

(35) (36) (37)
where is the reduced Planck constant, that is, is the Dirac constant.

Taking into account formulas (35-37), as well as [START_REF]Fundamental Physical Constants[END_REF][START_REF]NIST Uncertainty Machine[END_REF], we estimate the value of the Planck temperature :

,

Pl  27 17 1 1
1.34649254157 10 (6. 14 10 ) .

P Pl P g m l k m       1 , Pl   8 1 2 37 1 1
7.42670285296 10 (1.12 10 ) . 1 1956010181.19(0.679) , 10

P Pl Pl P g l m k m            , P q 1 2 1 18
P P P J q c E t      P T
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172 P r e p r i n t n o t p e e r r e v i e w e d (38) where is the Boltzmann constant.

Planck temperature according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]:

The relative error of non-coincidence of values with formula (38) is: Let's estimate the value of the Newtonian constant of gravitation

(39) (40) (41)
The Newtonian constant of gravity according to CODATA 2018 [START_REF]Fundamental Physical Constants[END_REF]:

The accuracy of the Newtonian constant of gravitation increased by 6 orders of magnitude.

Estimation of the Newtonian Constant of Gravitation Based on Experimental Data

Let's check the Newtonian constant of gravitation estimate based on experimental data. It was shown in [START_REF] Timkov | Quantization of the Gravitational Field. Theoretical and Experimental Substantiation of the Gravitational-Electromagnetic Resonance. The Physical Nature of the Quantum of the Gravitational Field. Why the Speed of Light in Vacuum is Constant[END_REF] that the gravitational field has a quantum wave character. Any gravitating object of the observable Universe, like the Hubble spheres, including the Hubble sphere itself, creates a standing gravitational wave in space-time. If we represent this gravitating object as a material point, then the structure of the gravitational field around this point can be represented as a set of nested spherical equipotential surfaces centered at this material point and separated from each other by the length of a standing gravitational wave, which is the quantization step. The length of the quantization step is proportional to the mass of the gravitating object, and the coefficient of proportionality is equal to the linear rarefaction of the Planck mass. The quantum of the gravitational field is the square of the speed of light in a vacuum, and the physical nature of the quantum of the gravitational field is the kinematic gravitational viscosity of the gravitational field of the observable Universe. The gravitational field has two fundamental properties: 1) it bends spacetime around any gravitating object, 2) it has a kinematic gravitational viscosity, due to which there is a finite rate of transmission of physical interaction in nature -the speed of light in vacuum. The main proof of the quantum-wave nature of the gravitational field is the presence in nature of a gravitational-electromagnetic resonance at the frequency of a standing gravitational wave, which is shown in the experiments presented in [START_REF] Timkov | Planck`s Universal Proportions. Gravitational-Electromagnetic Resonance[END_REF][START_REF] Timkov | Gravitational-Electromagnetic Resonance of the Sun as One of the Possible Sources of Auroral Radio Emission of Planets in Kilometric Range[END_REF][START_REF] Timkov | Gravitational-Electromagnetic Resonance of the Sun in the Low-Frequency of Radio Spectrum of the Jupiter[END_REF]. In accordance with the concept of the quantumwave nature of the gravitational field, any gravitating object of the observable Universe, like the , where is the Hubble constant, then, in accordance with [START_REF] Timkov | Planck`s Universal Proportions. Gravitational-Electromagnetic Resonance[END_REF],

the following formulas are true:

(42)

where is the energy of any gravitating object, is the energy of the baryonic matter of the observable Universe.

Based on the concept of Planck's Universal Proportions [START_REF] Timkov | Planck`s Universal Proportions. Gravitational-Electromagnetic Resonance[END_REF], the following formulas are true:

(43)

It follows from formulas (43) that by knowing the values of the proportional ratios of Planck's length, mass, and time and experimentally determining one of the three characteristics of a gravitating object, one can estimate the other two unknown characteristics.

Taking into account formula (43), the formulas are true:

(44) (45)
In [START_REF] Timkov | Planck`s Universal Proportions. Gravitational-Electromagnetic Resonance[END_REF], the results of an experiment on estimating the parameters of the Earth's gravitationalelectromagnetic resonance, namely, the quantization step of the Earth's gravitational field are given:

Using the estimation of the value of the Earth quantization step, let's estimate its effective radius , corresponding to the standard acceleration of gravity on the Earth .

It is known that the mass of the Earth can be estimated through the known values of the standard acceleration of gravity and the effective radius:

From:

(47) , m , q S d t U M U R U T 0 1 U T H  0 H 2 2
; ; ; ; ; ,

P P P q d U U U P U U U P P E E E m c E F S E t E M c E F R E T t t             E U E 0 ; ; . q U U U P P P Pl U P q U P d U P d U S M R M m l m m m c M H l S R t t T t t T            2
S R l l l F m m M m c t t T t l                   3 3 3 1 2 3 2 2 . q U P Pl P P d U U S R l G c m t m t M T           0.00443474 . q S m   e R  g  2 2
2

.

e e q g R g R M M G S c              2 . q e S c R g     
It follows from formula (47) that the effective radius of the Earth and the standard acceleration of gravity do not directly depend on the mass of the Earth, but depend on the quantization step of its gravitational field. According to CODATA 2018:

Then:

(48)

The value of the effective radius of the Earth was obtained by calculation on the basis of experimental data on the standard acceleration of gravity and the quantization step of the Earth's gravitational field. In [33], the value of the Earth's mass is given:

Then the estimate of the value of the Newtonian constant of gravitation is: To estimate the mass of the Earth using formula (50), estimate the value of the Newtonian constant of gravitation:

(51)

The value of the Newtonian constant of gravitation according to formula (51) is close to its value according to formulas (39 -41).

Conclusions

A method for empirical estimation of the Planck length, mass, and time based on the characteristics of an electron is proposed. This method relates the characteristics of a hypothetical Planck particle with the characteristics of leptons and baryons, which makes it possible to improve the accuracy of the following physical constants: Planck's length, mass, and time, Newtonian constant of gravitation, Planck's constant, fine structure constant, elementary electric charge, electron mass, temperature Planck, as well as all physical constants that can be expressed analytically in terms of Planck's length, mass, and time, for example, the Hartree energy, the Bohr magneton, the von   This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172 P r e p r i n t n o t p e e r r e v i e w e d

Klitzing constant. With an increase in the accuracy of the characteristics of the electron and proton, the accuracy of all the above physical constants will be increased accordingly.

  the standard acceleration of gravity and the effective radius of the Earth were established through the experiment, and the mass of the Earth was determined by calculation through the Newtonian constant of gravitation, which means that in the formula (49) there should be a result in accordance with CODATA 2018: But this result does not exist. Therefore, both the mass of the Earth according to [33] and the value of the Newtonian constant of gravitation according to CODATA 2018 have erroneous estimates. Estimating the mass of the Earth through Planck's Universal Proportions: (50)

  Hubble spheres, including the Universe itself, has three self-sufficient and interrelated characteristics: mass quantization step delay time for the propagation of a light signal over a distance equal to the quantization step . Then, if this is the mass of the baryonic matter of the observable Universe (the Hubble sphere), this is the radius of the Hubble sphere, this is the Hubble time, that is:

	P T		P k E		32 1.41673240714 10 (4.31 10 ) , 22 K  
	k
	32 1.416784 10 . K  0.0036%. : G 20 1 3 2 1 10 (2.87 0 ) P T  11 . kg m s      11 20 1 3 2 6.67478765001 10 (3.51 10 ) 6,674787 6 5001  , kg m s        2 P c m    2 3 P l c     11 20 1 3 2 2 5 10 (3.5 10 ) , 6,67478765001 P kg m t s c           1 2 . m kg s G G G 3   P r e p r i n t n o t p e e r r e v i e w e d 11 6.67430 10 G   
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This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.

This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4468172 P r e p r i n t n o t p e e r r e v i e w e d

CRediT authorship contribution statement

The author confirms that all stages of preparation of this paper: Idea, Conceptualization, Methodology, Supervision, Writing & Review & Editing Original Draft, Data Curation, Software, Investigation, Formal Analysis, Validation, Project Administration are performed by the author alone.

Data Availability

Data will be made available on request.

Declaration of competing interest

The author declares that he has no conflict of interest that could affect the work presented in this paper.


Joseph Lardies
email: joseph.lardies@univ-fcomte.fr

Ephane Gouttebroze
email: stephane.gouttebroze@univ-fcomte.fr

Identication of modal parameters using the wavelet transform

Keywords: Wavelet transform, Modal parameters, Ambient vibration, Random decrement technique

The wavelet transform is used as a time-frequency representation for the determination of modal parameters such as natural frequencies, damping ratios and mode shapes of a vibrating system. It is shown that using a particular form of the son wavelet function, results are improved compared to those obtained with the traditionally Morlet wavelet function. The accuracy of this new technique is conrmed by applying it to a numerical example and to ambient vibration measurements of a tower excited by wind.

Introduction

In recent years, researchers in applied mathematics, theoretical physics and signal processing have developed powerful wavelet techniques, called wavelet transform, for the multiscale representation and analysis of signals [START_REF] Chui | An introduction to wavelets[END_REF][START_REF] Meyer | Wavelets, algorithms and applications[END_REF][START_REF] Torresani | Analyse continue par ondelettes[END_REF]. In contrast to Fourier-transform-based approaches where a window is used uniformly for spreaded frequencies, the wavelet transform localises the information in the time-frequency plane, using short windows at high frequencies and long windows at low frequencies. By this way, the characteristics of nonstationary signals can be more closely monitored and the transient behaviour and discontinuities in the signals can be better investigated. For example, if there is an instantaneous impulse disturbance at a certain time interval, it may contribute to the Fourier transform but its location on the time axis will be lost. However, by the wavelet transform, both time and frequency information can be obtained. The wavelet transform is much more local: instead of transforming a pure time description into a pure frequency description, the wavelet transform nds a good compromise, a time-frequency description. An important area of application where these properties are relevant is the analysis of mechanical structures under ambient excitation.

Studies have been conducted on identication of modal parameters including natural frequencies, damping coecients and mode shapes of structural systems using ambient vibration records without input measurements (see Ref. [START_REF] Wang | Element-level system identication with unknown input[END_REF] for an important literature review, Refs. [START_REF] Feng | Identication of a dynamic system using ambient vibration measurements[END_REF][START_REF] Ibrahim | Random decrement technique for modal identication structures[END_REF]). Solutions are usually based on smooth spectral curves obtained from the random decrement method associated with signal processing techniques. Recently, stochastic sub-space methods [START_REF] Lardies | Modal analysis of random vibrating systems from multi-output data[END_REF][START_REF] Peeters | Reference based stochastic subspace identication for output-only modal analysis[END_REF][START_REF] Kullaa | System identication of the tower of a wind turbine[END_REF] have been developed in modal analysis of mechanical systems excited randomly and without knowledge of the excitation. These methods use an important number of matrices and the computation time is important.

In this paper, a method applying the wavelet transform for the determination of modal parameters of a structure under ambient excitation is presented. The structure is a 310 m tall TV tower constructed in the city of Nanjing in China. Acceleration responses of this structural system are used in the wavelet transform to identify its dynamic characteristics. The wavelet transform is a time-frequency representation and decouples automatically the modal components. Thus, in the analysis of a multidegrees of freedom system, the wavelet transform does not need any band-pass ltering procedure. In practice, damping coecients in vibrating systems are the most critical to estimate since they are the most sensitive to noise and measurement errors. The paper shows an ecient method to estimate the damping coecients using the wavelet transform, with the modied Morlet wavelet function. These damping coecients are also estimated using autocorrelation functions of signals ltered around natural frequencies. Finally, the wavelet transform is applied to determine the mode shapes of the structure.

The contribution of the paper to other techniques presented in Refs. [START_REF] Fasana | A road bridge dynamic response analysis by wavelet and other estimation techniques[END_REF][START_REF] Staszewski | Identication of damping in MDOF systems using time-scale decomposition[END_REF] consists essentially in the use of the random decrement method to transform random signals in free vibration responses and the use of a modied Morlet wavelet function to improve the resolution: close modes can be identied.

Review of the wavelet transform

Mathematically, wavelet transforms are inner products of the signal x(t) and a family of wavelets [START_REF] Chui | An introduction to wavelets[END_REF][START_REF] Meyer | Wavelets, algorithms and applications[END_REF][START_REF] Torresani | Analyse continue par ondelettes[END_REF]. Let (t) be the analysing wavelet called also the mother wavelet of the analysis or the wavelet 'prototype'. The corresponding family of wavelets consists of a series of son wavelets, which are generated by dilatation and translation from the mother wavelet (t) shown as follows:

a; b (t)= 1 √ a t -b a ;a ¿ 0;b ∈ R; ( 1
)
where a is the dilatation or scale parameter dening the support width of the son wavelet and b the translation parameter localising the son wavelet function in the time domain. The factor 1 √ a is used to ensure energy preservation in the wavelet transform. The dierence between these wavelets is mainly due to the dierent lengths of lters that dene the wavelet and scaling functions. Wavelets must be oscillatory, must decay quickly to zero (can only be nonzero for a short period) and must integrate to zero [START_REF] Chui | An introduction to wavelets[END_REF].

The idea of the wavelet transform is to decompose a signal x(t) into wavelet coecients W (a; b) using the basis of son wavelets a; b (t). Under the hypothesis that x(t) satisfy the condition:

+∞ -∞ |x(t)| 2 dt¡∞;
(2) which implies that x(t) decays to zero as t →± ∞ , the wavelet transform of x(t) is expressed by the following inner product in the Hilbert space:

W (a; b)= x(t); a; b (t) = +∞ -∞ x(t) * a; b (t)dt; (3)
where the asterisk stand for complex conjugate. This shows that the wavelet transform is a linear scalar product normalised by the factor 1= √ a and this scalar product is a measure of the uctuation of the signal x(t) around the point b at the scale a. The scaling operation is nothing more than performing stretching and compressing operations on the son wavelet, which in turn can be used to obtain the dierent frequency information of the signal to be analysed. The compressed version is used to satisfy the high-frequency needs, and the dilated version is used to meet low-frequency requirements. Then, the translated version is used to obtain the time information of the signal to be analysed. In this way, a family of scaled and translated wavelets is created and serves as the base, the base for representing the signal to be analysed. In other words, the wavelet transform W (a; b) can be considered as functions of translation b with each xed scale a. It gives the information of x(t) at dierent levels of resolution and also measures the similarity between the signal x(t) and each son wavelet a; b (t). Note that the wavelet transform represents also the convolution between the signal x(t) and the wavelet function. This implies that a wavelet can be used for feature discovery if the wavelet used is similar to the feature components (eventually eigenfrequencies and damping coecients) hidden in the analysed signal. For the function (t) to qualify as an analysing wavelet, it must satisfy the admissibility condition [START_REF] Chui | An introduction to wavelets[END_REF][START_REF] Meyer | Wavelets, algorithms and applications[END_REF][START_REF] Torresani | Analyse continue par ondelettes[END_REF]:

0 ¡c = +∞ -∞ | (!)| 2 |!| d!¡∞; (4)
where (!) is the Fourier transform of (t). Then the wavelet transform can be inverted and the signal x(t) recovered:

x(t)= 1 c +∞ -∞ +∞ -∞ W (a; b) a; b (t) da db a 2 : (5)
Note that since | (!)| tends to 0 when ! tends to ±∞, the Fourier transform of the wavelet can be considered as a band-pass lter.

For practical purposes, the possibility of time-frequency localisation arises if the wavelet g(t)i s a window function, which means that (t) decays to zero as t →±∞:

+∞ -∞ | (t)| dt¡∞ (6)
and the wavelet transform analyses a signal x(t) only at windows dened by the wavelet function (t). If one assumes a fast decay of (t): the values of (t) are negligible outside a given time domain interval, the transform becomes local in time domain, in this interval.

The frequency localisation can be explained when the wavelet transform is expressed in terms of the Fourier transform. Note X (!) the Fourier transform of the signal x(t) and a * (a!)e j!b the Fourier transform of the son wavelet * (tb)=a. Using the Parseval's theorem [START_REF] Meyer | Wavelets, algorithms and applications[END_REF], we obtain

W (a; b)= √ a 2 +∞ -∞ X (!) * (a!)e j!b d! (7)
and the frequency localisation depends on the scale parameter a. Note that this operation is equivalent to a particular lter band analysis in which the relative frequency bandwidth !=! are constant and related to the parameters a; b and to the frequency properties of the wavelet. The local resolution of the wavelet transform in time and frequency is determined by the duration and bandwidth of analysing functions given by t = a t and f =f =a, where t and f are the duration and bandwidth of the wavelet function. The resolution of the analysis is therefore good for high dilatation in the frequency domain and for low dilatation in time domain.

Before the calculation, the relevant parameters must be discretised for being computed by a computer. Instead of continuous dilatation and translation, the wavelet may be dilated and translated discretely by selecting a=a m 0 and b=nb 0 a m 0 where a 0 and b 0 are xed values with a 0 ¿ 1;b 0 ¿ 0;m;n∈ Z and Z is the set of positive integers. We obtain then a discretised son wavelet and a corresponding discrete wavelet transform which provides a decomposition of a signal into sub-bands with a bandwidth that increases linearly with frequency. In the case of dyadic discretisation, the most popular method, we have a 0 = 2 and b 0 = 1 and each spectral band is approximately one octave wide. In this form, the wavelet transform can be viewed as a special kind of spectral analyser.

Application of the wavelet transform to modulated signals

Consider the case of a signal x(t) modulated in amplitude:

x(t)=A(t) cos(!t) (8)
and x a (t) its analytic signal dened as

x a (t)=x(t)+jH [x(t)]; (9)
where H [x(t)] is the Hilbert transform of x(t):

H [x(t)] = 1 +∞ -∞ x() 1 t - d: (10)
If x(t) is assumed asymptotic, that is if the phase of the signal varies much faster than the amplitude, it is shown [START_REF] Torresani | Analyse continue par ondelettes[END_REF] that the complex signal:

z(t)=A(t)e j!t (11)
constitutes a good approximation of the analytic signal and is used for the determination of the wavelet transform. Following Torresani [START_REF] Torresani | Analyse continue par ondelettes[END_REF] the wavelet transform of the signal x(t)i s

W (a; b)= x(t); a; b (t) = 1 2 x a (t); a; b (t) = 1 2 √ a +∞ -∞ A(t)e j!t * t -b a dt: (12)
Now, the basic idea is to develop the amplitude A(t) around t = b (point where the son wavelet a; b (t) is maximum) using Taylor's formula. Neglecting terms of order superior to one, we have

W (a; b)= 1 2 √ a +∞ -∞ (A(b)+o(A ′ (b))) e j!t * t -b a dt (13)
the primes indicating a derivative. Using the Fourier transform of * (tb)=a and neglecting terms of order one (innitely small), we obtain

W (a; b)= √ a 2 A(b) * (a!)e j!b : (14)
We generalise to signals modulated in amplitude and frequency:

x(t)=A(t) cos('(t)): (15)
The wavelet transform of these signals can be approximated by means of asymptotic techniques as

W (a; b)= √ a 2 A(b) * (a' ′ (b)) e j'(b) : (16)
The linearity of the wavelet transform can be applied to P signals modulated in amplitude and frequency:

x(t)= P k=1 A k (t) cos(' k (t)) (17)
and we obtain

W (a; b)= √ a 2 P k=1 A k (b) * (a' ′ k (b)) e j' k (b) : (18)
As a consequence, assuming that the wavelet amplitude is localised at a certain value ! = ! 0 in the frequency domain (central frequency of the wavelet), the wavelet transform modulus |W (a; b)| is localised near the P curves with equations

a = k (b)=! 0 =' ′ k (b): (19)
The dilatation parameter a has been calculated in order to maximise the amplitude of * (a' ′ k (b)).

Modal parameters determination

The free response of a viscously damped single degree of freedom system is

x(t)=B e -!nt cos(! d t + 0 ) ( 20
)
with ! n the undamped natural frequency, ! d = ! n 1 -2 the damped natural frequency and the damping ratio. If the system is underdamped, that is if the damping ratio is smaller than 1, the signal x(t) can be considered asymptotic, and therefore the results obtained previously can be used considering:

A(t)=B e -!nt ; (21)
'(t)=! d t + 0 ⇒ ' ′ (t)=! d : (22)
The wavelet transform of the damped sinusoid is

W (a; b)= √ a 2 B e -!nb * (a! d )e j(! d b+ 0) (23)
and its modulus is localised from Eq. ( 19) at a constant value of the dilatation parameter noted a 0 :

a = a 0 = ! 0 =! d : (24)
The dilatation parameter a 0 , which maximises the amplitude of the wavelet transform, corresponds and is related to the analysed modal frequency of the system. For a xed value of the dilatation parameter (a = a 0 ), the wavelet transform modulus is

|W (a 0 ;b)| = √ a 0 2 B e -!nb | * (a 0 ! d )| (25)
and applying the logarithm to this function we obtain

ln|W (a 0 ;b)| = -! n b +ln √ a 0 2 B| * (a 0 ! d )| : (26)
Thus the damping ratio of the system can be estimated from the slope of the straight line of the logarithm of the wavelet transform modulus, assuming that the natural frequency ! n has been previously estimated (for example, by FFT plot X (!) of the signal x(t)).

Assuming the Fourier transform of the dilated wavelet real, condition which is fullled currently, the wavelet transform phase is given by

Arg(W (a 0 ;b)) = ! d b + 0 ⇒ d db Arg(W (a 0 ;b)) = ! d (27)
and the plot of (d=db) Arg(W (a 0 ;b)) should be constant in time and equal to the damped natural frequency ! d .

The damping and frequency estimation procedures, based on the wavelet transform presented above, can be extended to multidegrees of freedom systems. Consider now the free response of a, P degrees of freedom system

x(t)= P k=1 B k e -k ! nk t cos(! d k t + 0k ); ( 28
)
where k is the damping ratio, ! nk the undamped natural frequency and ! d k the damped natural frequency associated to the kth mode. From Eq. ( 18), the wavelet transform of the multidegrees of freedom system is

W (a; b)= √ a 2 P k=1 B k e -k ! nk b * (a! d k )e j(! d k b+ 0k ) : (29)
The wavelet transform is a signal decomposition procedure working as a lter in the time-frequency domain: it analyses a signal only locally at windows dened by the son wavelet. Thus, multidegrees of freedom system can be decoupled into single degrees of freedom. For a xed value of the dilatation parameter (a = a i ), which maximises * (a! d ), only the mode associated with a i gives a relevant contribution in the wavelet transform, while all the other terms are negligible. Thus the wavelet transform of each separated mode i =1; 2;:::;P becomes

W (a i ;b)= √ a i 2 B i e -i !nib * (a i ! di )e j(! di b+ oi ) : (30)
Clearly, the wavelet transform oers a decoupling of multidegrees of freedom systems into single modes. Using Eq. (30) associated with Eqs. ( 26) and ( 27), it is possible to follow the amplitude and the phase variations in the time domain of each modal component and to estimate the corresponding damping ratio and natural damped frequency associated to the isolated mode. We obtain for each ith mode:

ln|W (a i ;b)| = -i ! ni b +ln √ a i 2 B| * (a i ! di )| ; (31) Arg(W (a i ;b)) = ! di b + 0i ⇒ d db Arg(W (a i ;b)) = ! di : (32)
This technique requires a previous choice of the value of the dilatation parameter a i corresponding to the analysed mode and the resolution of the wavelet transform depends on the value of this scale parameter. For a given wavelet function, the resolution of the wavelet transform is determined by the dilatation parameter a i , thus the choice of the analysing wavelet is important.

Choice of the analysing wavelet

There are a number of dierent complex and real-valued functions used as analysing wavelets and satisfying the admissibility condition. One of the most known and widely used is the Morlet wavelet dened by (t)=e j!0t e -t 2 =2 ;

(33

)
where ! 0 is the central wavelet frequency. The dilated version of the Fourier transform of (t)i s real and is given by

(a!)= √ 2 e -1=2(a!-!0) 2 : (34)
In practice, to verify the admissibility condition, the value of ! 0 is chosen superior to 5. Note that (a!) is maximum at the central frequency ! = ! 0 =a and the Morlet wavelet can be viewed as a linear bandpass lter whose bandwidth is proportional to 1=a or to the central frequency. Thus, the value of the dilatation parameter a at which the wavelet lter is focused on the wavelet frequency ! can be determined from

a = ! 0 =!: (35)
For a given value of the dilatation parameter a, the spectrum of the Morlet wavelet has a xed bandwidth. If the analysed frequency is important, the dilatation parameter becomes small and the spectrum of the Morlet wavelet function is wide. There is then a bad spectral resolution. An alternative is to modify the Morlet wavelet function introducing a parameter N which controls the shape of the basic wavelet: this parameter balances the time resolution and the frequency resolution of the Morlet wavelet. The modied Morlet wavelet function used in this paper is (t)=e j!0t e -t 2 =N (36)

with N¿0 and whose dilated version of its Fourier transform is

(a!)== √ N e -N=4(a!-!0) 2 : (37)
An important value of N gives a narrower spectrum allowing a better resolution of closely spaced modes, but at the expense of time resolution. Indeed, increasing N will increase the frequency resolution but it decreases the time resolution. This modied Morlet wavelet function oers a better compromise in terms of localisation, in both time and frequency for a signal, than the traditionally Morlet wavelet function.

6. Applications

Simulated results

In order to show the usefulness of the wavelet transform method in modal analysis, a simulation is performed with a 2 degrees of freedom system viscously damped. The parameters of the system are chosen as follows: f 1 =25Hz;f 2 =30Hz; 1 =0:02; 2 =0:06. The free time response and the frequency response functions of this system are given in Fig. 1. The natural frequencies can be obtained directly by inspection of peaks of resonance. To estimate the damping coecients we use two dierent methods: the autocorrelation method [START_REF] Kullaa | System identication of the tower of a wind turbine[END_REF] and the wavelet transform method presented in this paper. In the autocorrelation method, each mode is isolated and ltered around its natural frequency. The autocorrelation function is then computed for each isolated mode and the damping ratio is estimated by

= 1 2k ln R x () R x ( + k) ; (38)
where R x () is the value of the autocorrelation function at time and k is the number of cycles. Autocorrelation functions for the two dierent modes are given in Fig. 2.

In order to estimate damping ratios i (i =1; 2) from the wavelet transform, we have to determine the dilatation parameter a i for each eigen-mode. This can be done by plotting the variations of the scale factor in time. If we use the traditionally Morlet wavelet function where N = 2, like Fasana et al. [START_REF] Fasana | A road bridge dynamic response analysis by wavelet and other estimation techniques[END_REF] and Staszewski [START_REF] Staszewski | Identication of damping in MDOF systems using time-scale decomposition[END_REF], the results presented in Fig. 3 (upper) are not exploitable: we cannot extract rigorously the coecients a i and present modes are very dicult to observe. The resolution of the wavelet transform is bad to separate the two dierent modes. Although, increasing N in Eqs. (36)o r( 37), we obtain for N = 17 horizontal bands (Fig. 3 below) representing appropriate dilatation parameters and the two present modes can be easily observed. The resolution of the wavelet transform is now accurate enough. Once the coecients a 1 and a 2 have been obtained, we plot for each a i the logarithm of the wavelet modulus as a function of time, and use Eq. (31). Thus the damping ratio 1 of the rst mode can be estimated from the slope of the straight line of the wavelet modulus, as shown in Fig. 4. The damped eigenfrequency in Hz (f d1 =! d1 =2) is obtained from Eq. ( 32) and plotted: we obtain practically a horizontal line corresponding to f d1 . The damping ratio 2 of the second mode can be obtained similarly from Fig. 5. Natural frequencies and damping ratios using the autocorrelation method and the wavelet transform method are given in Table 1.

Application to real data

Fig. 6 shows the main structure of the TV tower in the city of Nanjing in China. This tower is 310 m high and the acceleration response of the structural system measured under ambient conditions is used to identify its dynamic characteristics. System identication using ambient vibration measurements is a challenge requiring the use of special identication techniques, which can deal with very small magnitudes of ambient vibration contaminated by noise in the situation where input forces cannot be measured. The accelerometers are installed on the tower at two sets of dierent locations, as shown in Fig. 6, to measure the ambient vibrations of the system. The sensors at the rst set of locations are concentrated on the upper part of the structure since this part is more exible resulting in more vibration than the lower part, while those at the second set of locations are distributed along the height of the whole structure. The accelerometers are placed as close as possible to the centre of the cross section of the tower in order to minimise the response component due to torsional vibration. Acceleration records are obtained simultaneously in one direction each time, with a sampling time interval of 0:03125 s and a total recording time of approximately 600 s. Fig. 7 shows some time histories (400 s) recorded.

The wavelet estimation technique operates on the free response of the analysed system. A wellestablished method to convert random responses to free decay responses is the random decrement technique [START_REF] Feng | Identication of a dynamic system using ambient vibration measurements[END_REF][START_REF] Ibrahim | Random decrement technique for modal identication structures[END_REF]. Its basic concept is that the acceleration response y(t) measured on the structure can be decomposed into free vibration component and forced vibration component. The free vibration component contains an impulse or step response, while the forced vibration component represents response to the random wind load. The free vibration component can be obtained by a special averaging procedure of measurements which remove the random part, leaving its deterministic part. The randomdec signal (or free response) x() of the measured signal is dened by

x()= 1 N N m=1 y(t m + ); ( 39
)
where N is the number of time samples averaged, is the free response time length and t m is determined from original data as all the time instants satisfying initial conditions; t m 's are taken as those of zero crossing time with positive slope on the acceleration time history record, we have a The total length of the randomdec signal is 4096 points. In Fig. 8, the time histories of four randomdec signals are plotted and are similar to free vibration responses. Their spectra are plotted in Fig. 9 and natural frequencies can be estimated directly from these plots. Autocorrelation functions for the two rst modes are plotted in Fig. 10 and used to determine their damping ratios from Eq. (38). Similar results are obtained if we consider other modes and are not reported in the paper. Using the wavelet transform to estimate i from Eq. (31), we have to determine the dilatation parameter a i for each eigen-mode. This can be obtained by plotting the variations of the scale factor in time. We use the traditionally Morlet wavelet function with N = 2. The results presented in Figs. 11 (upper) and 12 (upper) are not exploitable: we cannot extract rigorously the coecients a i and present modes are impossible to observe. The resolution of the wavelet transform is not sharp enough to separate dierent modes of the system. Although, by increasing N in the modied Morlet wavelet function, we obtain for N = 15 horizontal bands (Fig. 11) representing appropriate dilatation parameters and present modes can be easily observed. The resolution of the wavelet transform is now accurate enough. This can be conrmed if we plot the amplitude of the wavelet transform for dierent values of the dilatation parameter as shown in Fig. 12. Six values of the dilatation parameter are predominant in this plot and they correspond to the number of modes present in the frequency band. Once the coecients a i (i =1;:::;6) have been obtained, we plot for each a i the logarithm of the wavelet modulus as a function of time, and use Eq. (31). Thus the damping ratio i can be estimated from the slope of the straight line of the wavelet modulus, as shown in Figs. 131415, where only the three rst modes have been considered (similar results are obtained if we use other modes). The damped angular frequency ! di (or damped frequency in Hz) is given by Eq. (32) and plotted in these gures: we obtain practically horizontal lines, since the ! di 's should be constant in time. Natural frequencies and damping ratios using an average over accelerometers are given in Table 2.

Results using the autocorrelation function are conditioned to the choice of a bandpass lter to isolate the mode under consideration. Here, we have taken a 6th-order Butterworth bandpass lter and results obtained are not optimal. These results vary if we use other bandpass lters. The wavelet transform technique does not use any external bandpass lter and the results obtained can be considered as optimal for the modied Morlet wavelet function, once the value of N has been properly chosen. When the free time responses recorded from several points of the structure are available, phase and amplitude relationships between the dierent degrees of freedom of the system can be obtained through the wavelet transform analysis. The ith mode shape of the structure can be estimated by evaluating the wavelet transform of the time signals from all measured points, at the corresponding ith frequency, that is for a = a i . Let W j (a i ;b) be the wavelet transform of the signal obtained from the accelerometer positioned at point j and let W r (a i ;b) be the wavelet transform of the signal obtained from the accelerometer of reference, positioned at point r. The quantity ij = W j (a i ;b)=W r (a i ;b) (41)

represents the jth component of the ith complex mode shape of the structure, referred to point r.

The two rst experimental mode shapes of the tower are shown in Fig. 16 and are in concordance with the mode shapes of a traditionally clamped beam. Errors are due to the exact positioning of accelerometers, to measurement noise, to the estimate of eigenfrequencies and damping ratios of dierent modes in the real structure, to the values of Young's module, density and moment of inertia in the theoretical clamped beam.

Conclusion

An approach to estimate modal parameters in time domain, from output data only, using the wavelet transform has been presented. The eciency of the method has been demonstrated using a numerical example and experimental results. The wavelet transform method is very suitable for the analysis of mechanical systems excited by random forces and has been applied to real data. Further work is continuing to determine the optimal value of N , used in the modied Morlel wavelet function. Note that other methods have been used to obtain modal parameters from output only measurements based on the spectral decomposition of a transition matrix [START_REF] Lardies | Modal analysis of random vibrating systems from multi-output data[END_REF][START_REF] Peeters | Reference based stochastic subspace identication for output-only modal analysis[END_REF][START_REF] Kullaa | System identication of the tower of a wind turbine[END_REF]. A comparison of these methods with the wavelet transform method presented here is under investigation.

Fig. 1 .

 1 Fig. 1. Time and frequency response of a two degrees of freedom system.

Fig. 2 .

 2 Fig. 2. Autocorrelation functions ltered around the 1st and 2nd mode.

Fig. 3 .

 3 Fig. 3. Time scale for N = 2 and for N = 17.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Wavelet transform envelope and damped frequency for the rst mode.

Table 1 Fig. 6 .Fig. 7 .Fig. 8 .

 1678 Fig. 6. Main structure of the tower and sensor locations.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Frequency response of randomdec acceleration signals.

Fig. 11 .

 11 Fig. 11. Time-scale for N = 2 (upper) and for N = 15.

Fig. 12 .

 12 Fig. 12. Amplitude of the wavelet transform for N = 2 (upper) and for N = 15.

Fig. 13 .

 13 Fig. 13. Wavelet transform envelope and damped frequency for the rst mode with N = 15, using channel 1.

Fig. 14 .

 14 Fig. 14. Wavelet transform envelope and damped frequency for the second mode with N = 15, using channel 1.

Fig. 15 .

 15 Fig. 15. Wavelet transform envelope and damped frequency for the third mode with N = 15, using channel 1.

Fig. 16 .

 16 Fig. 16. Mode shapes of the tower: (•) Set 1 location of accelerometers, ( * ) Set 2 location of accelerometers, continue: theoretical mode shapes of a clamped beam.

Table 2

 2 Estimation of frequencies and damping ratios of the tower

	Mode Frequency (Hz)	Damping ratio by the	Damping ratio by the
			autocorrelation method	wavelet transform
	1	0.234	0.0141	0.0137
	2	0.734	0.0062	0.0028
	3	1.273	0.0014	0.0064
	4	1.594	0.0036	0.0030
	5	2.726	0.0030	0.0044
	6	3.468	0.0012	0.0020

Acknowledgements

The authors wish to thank Prof. Maria Q. Feng for some helpful contributions to the applications to real data.