Analytical modeling of flexible structures for health monitoring under environmentally induced loads

Abstract

Wireless structural health monitoring (SHM) represents an emerging paradigm in non-destructive testing evaluation, which is regularly performed within the framework of structural maintenance of critical infrastructure. While traditional cable-based SHM strategies have largely relied on centralized structural response data collection and processing, in wireless SHM the sensor nodes essentially operate as stand-alone processing units. Furthermore, the on-board processing capabilities of wireless sensor nodes have been widely exploited for decentralizing SHM tasks, thus avoiding power-consuming wireless transmission of entire structural response data sets. Evidently, on-board processing requires approaches tailored to the limited computational resources of wireless sensor nodes, particularly for computationally intensive, state-of-the-art strategies for SHM that rely on artificial intelligence (AI). Specifically, in model-based SHM/AI strategies, AI algorithms are trained by running the so-called what-if scenarios using numerical models of monitored structures. However, the use of numerical models in a decentralized wireless SHM scheme might be prohibitive from a computational resources point of view. To overcome this limitation, analytical modeling techniques using elastic waveguides are developed here that carry a low computational burden. These are specifically tailored for flexible structures of variable cross section, which comprise typical components of critical infrastructure such as masts, antennas and pylons, and can be used for simulating axial, torsional and flexural vibrations. The accuracy and efficiency of the proposed analytical modeling approach are then demonstrated through comparisons with conventional numerical models based on the finite element method.

Appendix

In this Appendix, we list a typical software package developed in the Python programming environment [26] for axial vibrations. It computes the eigenvalue problem and the transient response based on modal analysis for the tapered pylon.

# Axial Vibrations for Tapered Pylons

# Python External Libraries

import numpy as np

# Building of Local Libraries

from Boundaries import *

from NR_solver import *

from Normalization import *

from Plot_Eigenfunction import *

from Gen_Values import *

from Plot_u_xt import *

from Plot_u_xf import *

# Compute cross-sectional area

def A(x):

$\hbox {Aa} = 2$*np.pi*Ra*t

return Aa*(x/a)**2

# Compute mass and stiffness $\rho \mathrm{A}$, $\mathrm{E}\mathrm{A}$

def wg(x):

return d*A(x), E*A(x)

# Compute constants appearing in each eigenfunction as C2/C1

$\hbox {C}=[]$

for k in k:

C.append(-np.tan(k*b))

return C

# Computation of eigenfunctions

def f(c,C,k,x):

$\hbox {return c/x *(np.sin(k*x)}+\hbox {C*np.cos(k*x))}$

# Computation of first spatial derivative of eigenfunctions

def df(c,C,k,x):

return (

$(-\hbox {c}/\hbox {x**2})\hbox {*}(\hbox {np.sin(k*x)}+\hbox {C}*\hbox {np.cos(k*x))}$

$+(\hbox {c}*\hbox {k/x}*(\hbox {np.cos(k*x)}-\hbox {C*np.sin(k*x)))\quad )}$

# Computation of the characteristic equation

def Eq(x):

$\hbox {return 1/a*np.sin(x*L)}+\hbox {x*np.cos(x*L)}$

# Computation of the first derivative of the characteristic equation

def dEq(x):

$\hbox {return 1/a*L*np.cos(x*L)}+\hbox {np.cos(x*L)-x*L*np.sin(x*L)}$

if __name__$==$”__main__”:

# Input Values

$\hbox {E} = 44.4$*10**6

$\hbox {d} = 2.55$

$\hbox {Ra} = 0.2150$

$\hbox {Rb} = 0.3375$

$\hbox {t}= 0.0875$

$\hbox {L}= 10.0$

# The slope at the boundaries requires the value of the radius at the top Ra

# at the bottom Rb, and the length computed from $\hbox {a}=39.54$ and $\hbox {b}=49.54$

# For constant cross section then $\hbox {a}=0 \kappa \alpha \iota \hbox {b}=\hbox {L}$

a, $\hbox {b}=\hbox { boundaries(Ra, Rb, L)}$

# Routine roots() uses the Newton–Raphson method to solve the characteristic equation

# Requires the characteristic eqn and its first derivative plus 4 values close to

# the roots. Solves for the first 4 wave numbers

$\hbox {k} =\hbox { roots(Eq, dEq, [0.20, 0.50, 0.80, 1.10])}$

# Routine constant()requires the wave number and returns constant

# C equal to C2/C1

$\hbox {C} =\hbox { constant(k)}$

# Routine norm() requires eigenfunction f, constants C2/C1,

# the length as either b-a or L, and the wave numbers k

# It returns constants c for normalizing the eigenfunction to unity

$\hbox {c} =\hbox { norm(f, a, C, k)}$

# Routine plot_3d_Eig() requires the eigenfunction f, its first derivative,

# the waveguide ends (a,b), normalization constants c, ratios C2/C1, C3/C1, etc.,

# wavenumbers k, the input loading function and the radii at top and bottom as

# Ra, Rb. It plots the first 4 eigenfunctions as x-y plots and as 3D plots

plot_3d_Eig(f,df, (a,b), c, C, k, ’axial’, Ra, Rb)

# Routine gen_Values() requires the eigenfunction f, its first derivative, the

# weight functions wg, the end coordinates (a,b), the normalization constants c,

# constants C from the list C, and the wavenumbers k

# It returns values for the generalized mass M, generalized stiffness K,

# the generalized external force P, and the eigenvalues W

${\hbox {W,M,K,P}} ={\text {gen\_Values(f, df, wg, (a,b), c, C, k)}}$

# Routine plot_u_xt_, requires the generalized mass $\mathrm{M}$, the generalized force P,

# the eigenfrequencies W, and the external load frequencies. Here we use values of

# [100 Hz, 300 Hz, 500 Hz, 700 Hz]. Output are Displacement versus Time curves

plot_u_xt_(M, P, W, [100, 300, 500, 700])

# Routine plot u_xf is similar and plots Displacement versus Frequency [0–800 Hz]

# There is a 0.20 mm cut-off point for the displacement u, to avoid division

# by zero when we have tuning

plot_u_xf_(K, P, W, [800, .20])