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Identification of non-linear parameter of a cantilever beam model with boundary condition non-linearity in the presence of noise: an NSI- and ANN-based approach

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Abstract

In this study the non-linear system identification (NSI) and parameter estimation of a model of a cantilever beam with non-linear stiffness attached to its free end are investigated. For this purpose, the impulse response of the beam model, in the presence of added measurement noise, is obtained by solving the weak form of the governing equation of motion via Rayleigh–Ritz method. The non-linear interaction model (NIM) including a set of intrinsic modal oscillators (IMOs) is constructed based on intrinsic mode functions (IMFs), which are derived by applying empirical mode decomposition (EMD) on the noise-contaminated response signals. For reducing the effect of noise and increasing the accuracy of extracted IMFs, the EMD-based noise reduction is employed, followed by the modification of the IMF amplitudes by introducing a “beta-factor” criterion. The changes in the amplitudes of the forcing functions associated with IMOs are used to extract features to estimate the non-linear parameter of the system. To this end, an artificial neural network has been trained to establish the non-linear relationship between the non-linearity of the system and the forcing functions of IMOs.

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Abbreviations

A :

Cross-sectional area, \(\hbox {m}^2\)

\(a_k \) :

Acceleration of the \(k\hbox {th}\) coordinate

\(a_k^n \) :

Artificial contaminated acceleration

\(B_m \) :

Amplitude of the masking signal

C:

Constant coefficient

\(c_j \) :

\(j\hbox {th}\) IMF

E :

Young’s modulus, \(\hbox {N}/\hbox {m}^{2}\)

F :

Impulsive force, N

\(\bar{{F}}\) :

Non-dimensional external force

\(\mathbb {F}\) :

Feature

\(f_{11}^q \) :

Elements of the force vector

\(f_{i1}^t \) :

Elements of the force vector

H:

Hilbert transform

I :

Identity matrix

I :

Second moment of cross-sectional area, \(\hbox {m}^4\)

\(k^{qq}_{11} \) :

Stiffness matrix elements

\(k^{tt}_{ij} \) :

Stiffness matrix elements

\(k_{nl} \) :

Non-linear coefficient of the spring, \(\hbox {N}/\hbox {m}^{3}\)

l :

Length, m

\(m^{qq}_{11} \) :

Mass matrix elements

\(m^{tq}_{i1} \) :

Mass matrix elements

N :

Number of modes

\(n_i^j \) :

\(i\hbox {th}\) neuron of the \(j\hbox {th}\) layer

o :

Output

p :

Number of patterns

q :

Time response

R:

Correlation factor

\(R_{m+1} \) :

 Residue

s :

Non-dimensional axial coordinate

s*:

Non-dimensional location of force

sig :

A given signal

\(T_{i}\) :

Time response

\(U_{i}\) :

Mode function of clamped-pinned beam

UII:

Theil parameter

u :

Non-dimensional transverse displacement

\(\mathbf{W}^{k}\) :

weight matrix of \(k\hbox {th}\) layer

w :

Transverse displacement, m

\(w_{i,j}^k \) :

Weight connecting the \(i\hbox {th}\) neuron to the \(j\hbox {th}\) source in the \(k\hbox {th}\) layer

x :

Axial coordinate

x*:

Location of force along the beam, m

\(x_{mask} \) :

 Masking signal

y :

Target

z :

Vertical coordinate

\(\beta \) :

Beta-factor

\(\gamma \) :

Non-dimensional non-linear parameter

\(\Delta \) :

Deviation of the natural logarithms from the calculated line

\(\zeta \) :

Linear modal damping ratios

\(\eta \) :

Generalized coordinate

\(\Lambda _m\) :

Forcing amplitude of the \(m\hbox {th}\) IMO

\(\lambda _m \) :

Damping coefficients of the \(m\hbox {th}\) IMO

\(\mu \) :

Average relative error

\(\rho \) :

Mass density, kg/m3

\({\upsigma }\) :

Standard deviation of error

\(\sigma _a^2 \) :

Variance of the acceleration signal

\(\sigma _n^2 \) :

Variance of the noise signal

\(\tau \) :

Non-dimensional time

\({\varvec{\Phi }} \) :

Modal eigenvector matrix

\(\varphi _{ij} \) :

Elements of matrix \({\varvec{\Phi }}\)

\(\varphi _m \) :

Complex slow-varying amplitude

\(\psi \) :

Static mode deflection

\(\Omega \) :

Eigenvalue of the linear free vibration

\(\omega _j\) :

\(j\hbox {th}\) dominant frequency

\(\varpi _m\) :

Frequency of the masking signal

AEMD:

Advanced empirical mode decomposition

ANN:

Artificial neural network

CPD:

Conjugate-pair decomposition

EMD:

Empirical mode decomposition

FFT:

Fast Fourier transform

IMF:

Intrinsic mode function

IMO:

Intrinsic modal oscillator

LMBPA:

Levenberg–Marquardt backpropagation algorithm

MLP:

Multi-layer perceptron network

MSE:

Mean square error

MSRE:

Mean square relative error

NIM:

Non-linear interaction model

NNM:

Non-linear normal mode

NSI:

Non-linear system identification

POD:

Proper orthogonal decomposition

PSD:

Power spectral density

RMSRE:

Root of mean square relative error

ROM:

Reduced-order model

SNR:

Signal-to-noise ratio

SVD:

Singular value decomposition

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Tabriz, P.O. Box 5166616471, Tabriz, Iran

    Morteza H. Sadeghi & Saeed Lotfan

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  1. Morteza H. Sadeghi
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  2. Saeed Lotfan
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Correspondence to Morteza H. Sadeghi.

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Sadeghi, M.H., Lotfan, S. Identification of non-linear parameter of a cantilever beam model with boundary condition non-linearity in the presence of noise: an NSI- and ANN-based approach. Acta Mech 228, 4451–4469 (2017). https://doi.org/10.1007/s00707-017-1947-8

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  • DOI: https://doi.org/10.1007/s00707-017-1947-8

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