∎

Random vibrations of stress-driven nonlocal beams with external damping

Preprint of the article published in
Meccanica
DOI: 10.1007/s11012-020-01181-7

Francesco P. Pinnola,
Marzia S. Vaccaro Barretta,
Raffaele Barretta,
Francesco Marotti de Sciarra

https://doi.org/10.1007/s11012-020-01181-7

ⓒ 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0
license http://creativecommons.org/licenses/by-nc-nd/4.0/

11institutetext: F.P. Pinnola 22institutetext: M.S. Vaccaro 33institutetext:
R. Barretta 44institutetext: F. Marotti de Sciarra55institutetext: Department
of Structures for Engineering and Architecture,
University of Naples Federico II,
via Claudio 21, Ed. 6, 80125 - Naples, Italy
55email<EMAIL_ADDRESS>
# Random vibrations of stress-driven nonlocal beams with external damping

Francesco P. Pinnola Marzia S. Vaccaro Raffaele Barretta Francesco Marotti de
Sciarra

(Received: date / Accepted: date)

###### Abstract

Stochastic flexural vibrations of small-scale Bernoulli-Euler beams with
external damping are investigated by stress-driven nonlocal mechanics. Damping
effects are simulated considering viscous interactions between beam and
surrounding environment. Loadings are modeled by accounting for their random
nature. Such a dynamic problem is characterized by a stochastic partial
differential equation in space and time governing time-evolution of the
relevant displacement field. Differential eigenanalyses are performed to
evaluate modal time coordinates and mode shapes, providing a complete
stochastic description of response solutions. Closed-form expressions of power
spectral density, correlation function, stationary and non-stationary
variances of displacement fields are analytically detected. Size-dependent
dynamic behaviour is assessed in terms of stiffness, variance and power
spectral density of displacements. The outcomes can be useful for design and
optimization of structural components of modern small-scale devices, such as
Micro- and Nano-Electro-Mechanical-Systems (MEMS and NEMS).

###### Keywords:

Stochastic dynamics small-scale beams size effects viscous damping stress-
driven nonlocal integral elasticity MEMS/NEMS

##  1 Literature survey, motivation and outline

Methodologies to predict random vibrations in structural systems have reached
over the last century a significant importance in design and optimization of
new-generation composites Pourasghar2019 ; Xia2019 and technological devices,
such as: micro-bridges Mojahedi2017 , nano-switches Moradweysi2018 , nano-
generators Hosseini ; debellis , nano-sensors Mohammadian ; natsuki , energy
harvesters Tran2018 ; Basutkar2019 ; GhayeshFarokhi2020 . It is acknowledged
that continuum mechanics is able to model structural components of small-scale
systems FarajpourReview2018 , but some mechanical expedients are needed to
accurately predict nonconventional phenomena, such as size and damping
effects. It is well-established that inter-atomic forces and molecular
interactions cannot be overlooked in small-scale structures which exhibit
technically significant size effects Bauer ; Kiang ; Xiao ; Zienert ;
Chowdhury ; Tang . Such a phenomenon cannot be captured by the classical
theory of local continuum marB due to lack of internal characteristic scales.
Nonlocal continua, driven by suitably chosen scale characteristic parameters,
are instead appropriate to model micro- and nano-structures 3 ; 6 ; alo16 ;
marA , as confirmed by molecular dynamic simulations Ansari2010 ; Murmu ;
Ansari2012 . Nonlocal methodologies allow for modeling complex mechanical
behaviours avoiding computationally expensive procedures 1 ; 2 .

Nonlocal theory, in its earliest formulation Rogula1965 ; Rogula1982 is based
on the idea that the stress at a point of a continuum depends not only on
elastic strain at that point but involves local responses of the whole
structure. Long-range interactions are thus described by a strain-driven
convolution integral supplemented with an averaging kernel characterized by a
nonlocal length-scale parameter. Such an approach was consistently applied by
Eringen Eringen ; Eringen1 to nonlocal problems involving screw dislocation
and wave propagation, that are formulated in unbounded domains. However,
mathematical difficulties are apparent when the strain-driven strategy is
applied to structural problems which are generally defined in bounded domains.

In fact, assuming that the averaging kernel is the Green function of a
differential operator, from the integral convolution it is possible to obtain
a consequent differential formulation Tricomi1985 ; Polyanin2008 . For bounded
structural domains, to the differential formulation above is needed to add a
proper set of Constitutive Boundary Conditions (CBCs) BarMar . Paradoxical
results and unacceptable nonlocal responses are obtained Challamel2008 ;
ReddyParadoxSolved if CBCs are ignored. Several nonlocal theories have been
adopted to overcome the aforementioned difficulties, such as: two-phase models
7 ; KhodabakhshiReddy2015 , strain and stress gradient theories 4 ;
ChallamelReddy2016 ; CivalekIJES2018 ; CornacchiaJCOMB2019 ;
CornacchiaMAMS2019 , nonlocal gradient techniques LimJMPS2015 ;
BarrettaMar2018 ; ApuzzoBarretta2018 , strain-difference approaches 48 ;
FusPisPol ; marC , displacement-based nonlocal models DiPaola ; pir1 ; alo13 ;
alo15 , stress-driven formulation of nonlocality RomBar ; BarVacc .
Advantageously, the stress-driven approach has been shown to be able to
effectively model the nonlocal behaviour of small-scale structures and
provides exact solutions for problems of applicative interest in Nano-
Engineering 41 ; BarMar2 . The stress-driven nonlocal integral model is
therefore adopted in this paper to significantly tackle size-dependent random
vibrations in inflected elastic small-scale beams. A first effort on the
matter, disregarding random phenomena, was performed in ApuBarMar .

Another import effect in dynamics of micro- and nano-systems concerns damping
phenomena which should be properly modeled in applicative problems of nano-
engineering. Instances are listed as follows: external magnetic force
LeeLin2010 , humidity, thermal and paddling effects Chen2011 and internal
viscous force due to material rheological properties DiPPinVal . Modeling of
internal viscous forces is successfully performed by properly selecting
constitutive formulations DiMDiPPin . Other effects, mentioned above, are
related to surrounding environmental interactions Calleja2012 . In the present
paper, particular attention is paid to capture external damping effects,
useful to analyse modern small-scale structures, such as: sensors inside
viscous fluid, devices under magnetic field, nano-systems for biological
detection. Specifically, a bed of independent dashpots will be considered to
simulate external viscous interactions between nonlocal beam and surrounding
environment.

At micro- and nano-scales, structures can be excited by different kind of
force systems. An example is the effect of environmental thermal and/or
mechanical noises in nano-sensors Mems . Stochastic approaches can be
conveniently exploited to model external loadings Verma ; Spanos ; Crandal ,
effectively representable by random time-process Pirr1 ; Pirr2 ; AloDiPPin .

For the aforesaid reasons, the present research provides a novel strategy for
stress-driven nonlocal analysis of damped vibrations of elastic nano-beams due
to stochastic excitation. Steady-state solutions are established, detecting
thus analytical expressions of power spectral density and stationary variance
of displacements. Closed-form solutions are also evaluated for non-stationary
responses of nonlocal damped beams forced by Gaussian white noise.

The manuscript is organized as follows. Strain- and stress-driven models of
pure nonlocal integral elasticity are recalled and specialized to
Bernoulli–Euler beams in Section 2. Dynamic equilibrium equations of damped
beams are established in Section 3 by using the well-posed stress-driven
nonlocal strategy of elasticity. Mode shape functions and natural frequencies
are analytically detected and an effective methodology to perform dynamic
eigenanalysis is also elucidated. A stochastic analysis of nonlocal damped
beams forced by Gaussian white noise is developed in Section 4. Both
stationary and non-stationary examinations are performed, detecting thus
closed form solutions of displacement variance, power spectral density and
correlation function. Numerical simulations are implemented in Section 5 to
test accuracy of obtained solutions. Analytical stationary and non-stationary
variances are compared with numerical outcomes derived by Monte Carlo
simulations. A parametric study of stochastic responses, in terms of
displacement variances and natural frequencies, is given in Section 5 to study
nonlocal effects. Closing remarks are outlined in Section 6.

##  2 Purely nonlocal integral elasticity

Two purely nonlocal models of elasticity are available in literature:

  1. 1.

strain-driven integral law Rogula1965 , applied to problems of screw
dislocation and wave propagation Eringen ; Eringen1 ;

  2. 2.

stress-driven integral law RomBar applied to nonlocal mechanics of structures.

These theories are preliminarily recalled below for $\,3$-D continua and
specialized to Bernoulli-Euler beams. Eringen’s strain-driven law is based on
the idea that the stress $\boldsymbol{\sigma}$ at a point $\boldsymbol{x}$ of
a nonlocal $\,3$-D continuous body $\mathcal{B}$ is the output of a
convolution between the local response to the elastic strain field
$\boldsymbol{\varepsilon}$ and a scalar kernel $\Phi_{\lambda}$ depending on a
non-dimensional positive nonlocal parameter $\lambda$. That is,

$\boldsymbol{\sigma}(\boldsymbol{x})=\int_{\mathcal{B}}\Phi_{\lambda}(\boldsymbol{x},\bar{\boldsymbol{x}})\boldsymbol{E}(\bar{\boldsymbol{x}})\,\boldsymbol{\varepsilon}(\bar{\boldsymbol{x}})d\bar{\boldsymbol{x}}$
(1)

with $\boldsymbol{E}$ fourth-order local elasticity stiffness tensor.

For a Bernoulli-Euler inflected beam of length $\,L\,$, the nonlocal strain-
driven relation Eq. (1) takes the form

$M(z)=EI\int_{0}^{L}\Phi_{\lambda}(z,\bar{z})\chi(\bar{z})d\bar{z}$ (2)

with $\,M\,$ bending moment, $\,E\,$ Euler-Young modulus, $\,I\,$ cross-
sectional moment of inertia along the bending axis $\,y\,$, $\,z\,$ beam axial
abscissa and $\,\chi\,$ elastic curvature. The integral kernel
$\Phi_{\lambda}$ can be selected among exponential, Gaussian or power-law type
functions and must satisfy properties of positivity, symmetry and limit
impulsivity Eringen1 . A convenient choice for the averaging kernel is the
special bi-exponential function

$\Phi_{\lambda}(z,\bar{z})=\frac{1}{2\lambda
L}\exp\left({-\frac{|z-\bar{z}|}{\lambda L}}\right)$ (3)

where $\lambda L$ is the characteristic length $L_{c}$. With the assumption
above, the integral law Eq. (2) is equivalent to the second-order differential
equation BarMar

$M^{(2)}(z)-\frac{1}{(\lambda L)^{2}}M(z)=-\frac{EI}{(\lambda L)^{2}}\chi(z)$
(4)

supplemented with the following constitutive boundary conditions (CBC)

$\left\\{\begin{split}&M^{(1)}(0)=\frac{1}{\lambda L}M(0)\\\
&M^{(1)}(L)=-\frac{1}{\lambda L}M(L)\end{split}\right.$ (5)

It is worth noting that, for structural problems of applicative interest, CBCs
Eq. (5) are in contrast with equilibrium requirements BarMar . Incompatibility
between equilibrium and constitutive conditions reveals that Eringen’s
nonlocal model leads to ill-posed structural problems, generally formulated in
bounded domains. Such an obstruction can be overcome by using the stress-
driven approach RomBar which is well-posed. Exact nonlocal structural
solutions can be found in BarMar2 .

In stress-driven mechanics, the elastic strain $\boldsymbol{\varepsilon}$ at a
point $\boldsymbol{x}$ of $\mathcal{B}$ is obtained by convoluting the local
stress $\boldsymbol{\sigma}$ with an averaging kernel $\,\Phi_{\lambda}\,$

$\boldsymbol{\varepsilon}(\boldsymbol{x})=\int_{\mathcal{B}}\Phi_{\lambda}(\boldsymbol{x},\bar{\boldsymbol{x}})\boldsymbol{C}(\bar{\boldsymbol{x}})\,\boldsymbol{\sigma}(\bar{\boldsymbol{x}})d\bar{\boldsymbol{x}}$
(6)

with $\boldsymbol{C}=\boldsymbol{E}^{-1}$ local elastic compliance. The
stress-driven model for Bernoulli-Euler beams is governed by the following
moment-curvature relation

$\chi(z)=\frac{1}{EI}\int_{0}^{L}\Phi_{\lambda}(z-\bar{z})M(\bar{z})d\bar{z}$
(7)

The integral formulation Eq. (7), with the special kernel Eq. (3), is
equivalent to the second order differential equation RomBar

$\chi^{(2)}(z)-\frac{1}{(\lambda L)^{2}}\chi(z)=-\frac{1}{EI(\lambda
L)^{2}}M({z})$ (8)

equipped with the following constitutive boundary conditions

$\left\\{\begin{split}&\chi^{(1)}(0)=\frac{1}{\lambda L}\chi(0)\\\
&\chi^{(1)}(L)=-\frac{1}{\lambda L}\chi(L)\end{split}\right.$ (9)

The stress-driven approach Eqs. (8), (9) provides exact solutions in both
static and dynamic structural problems and is exploited in the present study
to analytically tackle size-dependent random vibrations of slender elastic
beams.

##  3 Dynamical analysis of nonlocal beams

Let us consider a nonlocal beam of length $L$ and cross-sectionial area
$A_{t}$ subjected to a transverse loading per unit of length $q(z,t)$, see
Figure 1. $(x,y,z)$ is the adopted system of orthogonal coordinates, $v(z,t)$
denotes the transverse displacement and $\rho(z)$ is the mass density.

Figure 1: Bernoulli-Euler beam with external damping Figure 2: Free-body
diagram of a beam differential element

The stress-driven formulation Eqs. (8), (9) is used as nonlocal elasticy law
while the effect of external damping is introduced and modeled as a bed of
dashpots with viscosity $\eta$. This kind of viscous interaction is able to
model a possible external interaction between nonlocal beam and viscous fluid.
Damping can be also simulated as a material effect adik ; adik2 ; Failla ;
alo18 ; PirDim ; AlottaA by a viscoelastic law.

In this paper, following the Newtonian approach, we consider a beam
differential element of length $dz$, whose free-body diagram is represented in
Figure 2. The equilibrium equation along the $y$-direction, involving external
loading, inertial and damping forces, and bending and shearing fields, writes
as

$\frac{\partial T(z,t)}{\partial z}dz+q(z,t)dz=\eta dz\frac{\partial
v(z,t)}{\partial t}+\rho Adz\frac{\partial^{2}v(z,t)}{\partial
t^{2}},\;\;\;0<z<L$ (10)

By ignoring second-order terms in $dz$ and neglecting mass moment of inertia
and angular acceleration, the rotational equilibrium along the $x$-axis gives

$\frac{\partial M(z,t)}{\partial z}-T(z,t)=0,\;\;\;\;\;0<z<L$ (11)

Combining the equilibrium equations along $x$\- and $y$-directions we get the
partial differential equation

$-\frac{\partial^{2}M(z,t)}{\partial z^{2}}+\eta\frac{\partial
v(z,t)}{\partial t}+\rho A\frac{\partial^{2}v(z,t)}{\partial
t^{2}}=q(z,t),\;\;\;0<z<L$ (12)

By introducing the nonlocal stress-driven relation in Eq. (8), from Eq. (12)
we get

$-EI\left[\frac{\partial^{2}\chi(z,t)}{\partial z^{2}}-(\lambda
L)^{2}\frac{\partial^{4}\chi(z,t)}{\partial z^{4}}\right]+\eta\frac{\partial
v(z,t)}{\partial t}+\rho A\frac{\partial^{2}v(z,t)}{\partial t^{2}}=q(z,t)$
(13)

According to Bernoulli-Euler kinematics, curvature $\chi(z,t)$ is related to
the transverse displacement $v(z,t)$ by

$\chi(z,t)=-\frac{\partial^{2}v(z,t)}{\partial z^{2}}$ (14)

and by placing Eq. (14) into Eq. (13) we have that

$\frac{\partial^{4}v(z,t)}{\partial z^{4}}-(\lambda
L)^{2}\frac{\partial^{6}v(z,t)}{\partial z^{6}}+\frac{\eta}{EI}\frac{\partial
v(z,t)}{\partial t}+\frac{\rho A}{EI}\frac{\partial^{2}v(z,t)}{\partial
t^{2}}=\frac{q(z,t)}{EI}$ (15)

which is the partial differential equilibrium equation ruling bending
vibrations of a nonlocal Bernoulli-Euler beam resting on a bed of dashpots.

For vanishing nonlocal parameter $\lambda\to 0^{+}$, Eq. (15) provides the
known differential equation governing forced vibrations of local Bernoulli-
Euler beams resting on a bed of independent dashpots. Moreover, setting
$\eta=0$, the classical formulation of undamped local beams is obtained
Meirovi ; Pirrotta . Solutions of the introduced partial differential equation
may be found by imposing two initial conditions, four standard BCs and two
constitutive BCs of the stress-driven model Eq. (9).

###  3.1 Free vibrations of undamped nonlocal beam

In order to solve Eq. (15), we first consider the undamped nonlocal beam in
free vibration. In other words, we set $q(z,t)=0$ and $\eta=0$. These
assumptions imply that Eq. (15) yields

$\frac{\partial^{4}v(z,t)}{\partial z^{4}}-(\lambda
L)^{2}\frac{\partial^{6}v(z,t)}{\partial z^{6}}+\frac{\rho
A}{EI}\frac{\partial^{2}v(z,t)}{\partial t^{2}}=0,\;\;\;0<z<L$ (16)

which rules free vibrations of undamped nonlocal beam. We suppose that the
displacement function $v(z,t)$ is with separable variables and so it can be
expressed by a product of a space function $\phi(z)$ (mode shape) and a time-
dependent function $y(t)$ that modulates the amplitude of mode shape in time.
That is,

$v(z,t)=\phi(z)y(t)$ (17)

by substituting Eq. (17) into Eq. (16) we get

$y(t)\left[\frac{d^{4}\phi(z)}{dz^{4}}-(\lambda
L)^{2}\frac{d^{6}\phi(z)}{dz^{6}}\right]+\frac{\rho
A}{EI}\phi(z)\frac{d^{2}y(t)}{dt^{2}}=0,\;\;\;0<z<L$ (18)

where partial derivatives have been replaced by total derivatives due to the
assumption in Eq. (17). Moreover, by using Lagrange’s differential notation
for space derivative and Newton’s notation for time derivative we can rewrite
Eq. (18) as

$\frac{EI}{\rho A}\frac{(\lambda
L)^{2}\phi^{(6)}(z)-\phi^{(4)}(z)}{\phi(z)}=\frac{\ddot{y}(t)}{y(t)},\;\;\;\;\;0<z<L$
(19)

where both sides of Eq. (19) must be equal to a constant $\alpha$ that can be
associated with the natural frequency of the oscillation $\omega_{0}$ as

$\alpha=-\omega_{0}^{2}$ (20)

Hence, Eq. (19) can be rewritten as

$\frac{EI}{\rho A}\frac{(\lambda
L)^{2}\phi^{(6)}(z)-\phi^{(4)}(z)}{\phi(z)}=\frac{\ddot{y}(t)}{y(t)}=-\omega_{0}^{2}$
(21)

by considering the left side of Eq. (21), we get the following sixth order
differential equation in the space variable $z$

${(\lambda L)^{2}\phi^{(6)}(z)-\phi^{(4)}(z)}+\omega_{0}^{2}\frac{\rho
A}{EI}{\phi(z)}=0,\;\;\;\;\;0<z<L$ (22)

whose solution must satisfy the four BCs depending on the type of loads and
constraints at the bounds and the two constitutive BCs in Eq. (9). That is,

$\left\\{\begin{split}&\phi^{(3)}(0)=\frac{1}{\lambda L}\phi^{(2)}(0)\\\
&\phi^{(3)}(L)=-\frac{1}{\lambda L}\phi^{(2)}(L)\end{split}\right.$ (23)

The problem to find the constant $\omega_{0}$ and $\phi(z)$ such that Eq. (22)
admits nontrivial solution is known as differential eigenvalue-eigenfunction
problem, where $\omega_{0}$ is the eigenvalue and $\phi(z)$ represents the
corresponding eigenfunction. Eq. (22) admits infinite eigenvalues and then
infinite eigenfunctions. Therefore, the solution in terms of displacement can
be expressed as a sum of infinite products between the modal time coordinates
$y_{j}(t)$ and the mode shapes $\phi_{j}(z)$

$v(z,t)=\sum_{j=1}^{\infty}\phi_{j}(z)y_{j}(t)$ (24)

where the $j$-th eigenfunction $\phi_{j}(z)$ is a solution of the $j$-th
homogeneous sixth-order differential equation in Eq. (22). Specifically,

$\phi_{j}(z)=\sum_{i=1}^{3}\left\\{C_{i}\exp\left[{\sqrt{\gamma_{i}(\omega_{0,j})}z}\right]+C_{i+3}\exp\left[-{\sqrt{\gamma_{i}(\omega_{0,j})}z}\right]\right\\}$
(25)

where $\gamma_{i}(\omega_{0,j})$ with $i=1,2,3$ are the roots of the following
characteristic third degree polynomial

$(\lambda L)^{2}\gamma^{3}-\gamma^{2}+\omega_{0,j}^{2}\frac{\rho A}{EI}=0$
(26)

and the coefficients $C_{i}$ are obtained by imposing that solution in Eq.
(25) satisfies the previous six BCs and the normality condition. In this
manner, each eigenfunction possesses the following orthonormality property

$\int_{0}^{L}\phi_{i}(z)\phi_{j}(z)dz=\delta_{kj}$ (27)

where $\delta_{kj}$ indicates the Kronecker delta defined as

$\delta_{kj}=\left\\{\begin{split}&1\;\;\textrm{if}\;k=j,\\\
&0\;\;\textrm{if}\;k\neq j\end{split}\right.$ (28)

Such solution of differential eigenvalue problem will be used in the next
section to solve the more general case of a forced nonlocal damped beam.

###  3.2 Forced vibrations of damped nonlocal beam

Now, we consider the dynamical problem of a forced beam with external damping.
The solution in terms of displacements expressed in Eq. (24) in function of
mode shapes $\phi_{j}(z)$ is placed into the equilibrium equation (15).
Specifically,

$\begin{split}&\frac{EI}{\rho
A}\left[\sum_{j=1}^{\infty}y_{j}(t)\phi^{(4)}_{j}(z)-\sum_{j=1}^{\infty}y_{j}(t)(\lambda
L)^{2}\phi^{(6)}_{j}(z)\right]+\\\ &+\frac{\eta}{\rho
A}\sum_{j=1}^{\infty}\dot{y}_{j}(t)\phi_{j}(z)+\sum_{j=1}^{\infty}\ddot{y}_{j}(t)\phi_{j}(z)=\frac{q(z,t)}{\rho
A}\end{split}$ (29)

by multiplying both sides of Eq. (29) by the i-th eigenfunction $\phi_{i}(z)$
and integrating on the domain $[0,L]$, Eq. (29), we get

$\begin{split}&\frac{EI}{\rho
A}\left[\sum_{j=1}^{\infty}y_{j}(t)\int_{0}^{L}\phi_{i}(z)\phi^{(4)}_{j}(z)dz\right]+\\\
&-\frac{EI}{\rho A}\left[\sum_{j=1}^{\infty}y_{j}(t)(\lambda
L)^{2}\int_{0}^{L}\phi_{i}(z)\phi^{(6)}_{j}(z)dz\right]+\\\ &+\frac{\eta}{\rho
A}\sum_{j=1}^{\infty}\dot{y}_{j}(t)\int_{0}^{L}\phi_{i}(z)\phi_{j}(z)dz+\sum_{j=1}^{\infty}\ddot{y}_{j}(t)\int_{0}^{L}\phi_{i}(z)\phi_{j}(z)dz=\\\
&\frac{1}{\rho A}\int_{0}^{L}\phi_{i}(z)q(z,t)dz\end{split}$ (30)

Taking into account the eigenfunctions orthonormality property in Eq. (27),
from Eq. (30) we get the second order differential equation in terms of modal
coordinate $y_{i}(t)$ that rules the motion of a forced damped modal
oscillator

$\frac{EI}{\rho A}y_{i}(t)\left[a_{i}-(\lambda
L)^{2}b_{i}\right]+\frac{\eta}{\rho
A}\dot{y}_{i}(t)+\ddot{y}_{i}(t)=\frac{1}{\rho
A}\int_{0}^{L}\phi_{i}(z)q(z,t)dz$ (31)

where the coefficients $a_{i}$ and $b_{i}$ are

$a_{i}=\int_{0}^{L}\phi_{i}(z)\phi^{(4)}_{i}(z)dz,\;\;\;\;\;b_{i}=\int_{0}^{L}\phi_{i}(z)\phi^{(6)}_{i}(z)dz$
(32)

Moreover, we assume that also the load is with separable variables. Therefore,

$q(z,t)=g(z)f(t)$ (33)

this implies that Eq. (31) can be rewritten as

$\frac{k_{\lambda,i}}{\rho A}y_{i}(t)+\frac{\eta}{\rho
A}\dot{y}_{i}(t)+\ddot{y}_{i}(t)=\frac{c_{i}}{\rho A}f(t)$ (34)

where the _nonlocal modal stiffness_ $k_{\lambda,i}$ is defined as

${k}_{\lambda,i}=EI\left[a_{i}-(\lambda L)^{2}b_{i}\right]$ (35)

and the coefficient $c_{i}$ is

$c_{i}=\int_{0}^{L}\phi_{i}(z)g(z)dz$ (36)

Notice that the following term

$\omega_{0,i}=\sqrt{\frac{k_{\lambda,i}}{\rho A}}$ (37)

represents the natural frequency associated to the $i$-th mode $\phi_{j}(z)$.

Solution of Eq. (34) provides the i-th modal time coordinate of the infinite
series in Eq. (24). Truncating the sum to an appropriate number $n$ of
eigenfunctions and modal coordinates leads to the following approximated
solution, that will be used for the numerical applications

$v(z,t)\approx\sum_{j=1}^{n}\phi_{j}(z)y_{j}(t)$ (38)

##  4 Stochastic analysis of nonlocal beams

Now we turn the attention to random vibrations of nonlocal beams, supposing
that the time-dependent part of the transverse load $q(z,t)$ has a stochastic
nature. Specifically, from Eq. (33) we assume that

$q(z,t)=g(z)F(t)$ (39)

where $g(z)$ is a deterministic function, while $F(t)$ is a stochastic
process, the latter is noted by a capital letter to distinguish it from the
deterministic function $f(t)$. Moreover, we assume that $F(t)$ is a stationary
Gaussian process with zero mean $\mu_{F}$ and with an assigned correlation
function (CF) denoted by $R_{F}(\tau)$. Thus, the input process is completely
described by the following time-independent parameters

$\mu_{F}=\mu_{F}(t):=\mathbb{E}\left[F(t)\right]=0$ (40a)
$\begin{split}R_{F}(\tau)=R_{F}(t,t+\tau):&=\mathbb{E}\left[F(t)F(t+\tau)\right]-\mu_{F}^{2}\\\
&=\mathbb{E}\left[F(t)F(t+\tau)\right]\end{split}$ (40b)

where $\mathbb{E}[\cdot]$ is the averaging operator. For $\tau=0$ the CF gives
the value of the variance $\sigma_{F}^{2}$. That is,

$\sigma_{F}^{2}=\mathbb{E}\left[F(t)F(t)\right]=R_{F}(0)$ (41)

Being $F(t)$ a stationary process, by virtue of Wiener-Khinchin theorem, the
power spectral density (PSD), denoted by $S_{F}(\omega)$, is the Fourier
transform of the correlation function. That is,

$\begin{split}S_{F}(\omega):&=\frac{1}{2\pi}\int_{-\infty}^{\infty}R_{f}(\tau)e^{-\mathrm{i}\omega\tau}d\tau\\\
&=\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{F}^{*}(\omega,\mathrm{T})\hat{F}(\omega,\mathrm{T})\right]\end{split}$
(42)

where $\mathrm{i}=\sqrt{-1}$ is the imaginary unit,
$\hat{F}(\omega,\mathrm{T})$ indicates the truncated Fourier transform of the
process $F(t)$ in a finite time interval $[0,\mathrm{T}]$, and
$\hat{F}^{*}(\omega,\mathrm{T})$ denotes its complex conjugate.

By taking into account Eq. (39) and the definition in Eq. (40b), CF of the
loading $q(z,t)$ is given by

$\begin{split}R_{q}(z,\tau)&=\mathbb{E}\left[q(z,t)q(z,t+\tau)\right]\\\
&=g(z)\mathbb{E}\left[F(t)F(t+\tau)\right]g(z)\\\
&=g^{2}(z)R_{F}(\tau)\end{split}$ (43)

Similarly, the PSD of the loading is

$\begin{split}S_{q}(z,\omega)&=\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{q}^{*}(z,\omega,\mathrm{T})\hat{q}(z,\omega,\mathrm{T})\right]\\\
&=g(z)\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{F}^{*}(\omega,\mathrm{T})\hat{F}(\omega,\mathrm{T})\right]g(z)\\\
&=g^{2}(z)S_{F}(\omega)\end{split}$ (44)

and the variance $\sigma_{q}^{2}(z)$ is

$\sigma^{2}_{q}(z)=\mathbb{E}\left[q(z,t)q(z,t)\right]=g(z)\sigma^{2}_{F}g(z)$
(45)

Now we want to characterize the response process in terms of displacement
$v(z,t)$ when the input is a Gaussian stationary process. Being the stochastic
input Gaussian also the stochastic output will be Gaussian, but the response
process will have a stationary part for $t\gg 0$ and a transient non-
stationary one. Therefore, it is needed the evaluation of the evolution in
time of the statistics.

###  4.1 Time-domain response

By observing Eq. (34) we deduce that if the forcing load $f(t)=F(t)$, then
also the time response of the beam in terms of modal coordinate will be a
stochastic process $Y(t)$. Hence, from Eq. (24) we get

$v(z,t)=\sum_{j=1}^{\infty}\phi_{j}(z)Y_{j}(t)$ (46)

where $\phi_{j}(z)$ is a deterministic function that can be analytically
evaluated. While, the stochastic response $Y_{j}(t)$ is solution of the
following stochastic differential equation

$\ddot{Y}_{j}(t)+\frac{\eta}{\rho A}\dot{Y}_{j}(t)+\frac{k_{\lambda,j}}{\rho
A}Y_{j}(t)=\frac{c_{j}}{\rho A}F(t)$ (47)

The forced input $F(t)$ is a Gaussian process and the differential equation in
Eq. (47) is linear. This implies that the output process $Y_{j}(t)$ will be
Gaussian too, and then can be completely characterized by the mean
$\mu_{Y_{j}}(t)$ and the correlation function $R_{Y_{j}}(t,t+\tau)$. Under the
assumption that the beam is quiescent at $t=0$, the two initial conditions
become

$\begin{cases}v(z,0)=0,\,\forall z\in[0,L]\Rightarrow Y_{j}(0)=0,\,\forall
j\in\mathbb{N}^{+}\\\ \dot{v}(z,0)=0,\,\forall
z\in[0,L]\Rightarrow\dot{Y}_{j}(0)=0,\,\forall j\in\mathbb{N}^{+}\\\
\end{cases}$ (48)

The output process is obtained by applying the Duhamel superposition integral

$Y_{j}(t)=\frac{c_{j}}{\rho A}\int_{0}^{t}h_{j}(t-\tau)F(\tau)d\tau$ (49)

being $h_{j}(t)$ a deterministic function which represents the impulse
response of the $j$-th modal oscillator. Such a function is defined as

$h_{j}(t)=\sqrt{\frac{4(\rho A)^{2}}{4k_{\lambda,j\rho
A}-\eta^{2}}}\exp\left(-\frac{\eta}{2\rho
A}t\right)\sin\left(\sqrt{\frac{4k_{\lambda,j}\rho A-\eta^{2}}{4(\rho
A)^{2}}}t\right)$ (50)

We can observe that the term

$\omega_{D,j}=\sqrt{\frac{4k_{\lambda,j}\rho A-\eta^{2}}{4(\rho A)^{2}}}$ (51)

is the damped frequency. Therefore, the impulse response can be rewritten as

$h_{j}(t)=\frac{1}{\omega_{D,j}}\exp\left(-\frac{\eta}{2\rho
A}t\right)\sin\left(\omega_{D,j}t\right)$ (52)

Taking Eq. (40a) into account and by applying the averaging operator to Eq.
(49) we can prove that the mean of the response process is zero. That is,

$\begin{split}\mu_{Y_{j}}(t)&=\mathbb{E}\left[\frac{c_{j}}{\rho
A}\int_{0}^{t}h_{j}(t-\tau)F(\tau)d\tau\right]\\\ &=\frac{c_{j}}{\rho
A}\int_{0}^{t}h_{j}(t-\tau)\mathbb{E}\left[F(\tau)\right]d\tau\\\
&=\frac{c_{j}}{\rho
A}\int_{0}^{t}h_{j}(t-\tau)\mu_{F}(\tau)d\tau=0\end{split}$ (53)

Eq. (53) implies that also the mean of displacements is zero $\mu_{v}(z,t)=0$
for all $t\geqslant 0$, and for $z\in[0,L]$.

CF of the response process $Y_{j}(t)$ can be evaluated taking into account the
assumption in Eq. (40b) and the Duhamel integral in Eq. (49). We apply the
averaging operator to the process $Y_{j}(t)$ considering two different time-
step $t=t_{1}$ and $t+\tau=t_{2}$. That is,

$\begin{split}R_{Y_{j}}(t_{1},t_{2}):=&\mathbb{E}\left[Y_{j}(t_{1})Y_{j}(t_{2})\right]\\\
=&\frac{c_{j}^{2}}{(\rho
A)^{2}}\int_{0}^{t_{1}}\int_{0}^{t_{2}}h_{j}(t_{1}-\tau_{1})h_{j}(t_{2}-\tau_{2})\mathbb{E}\left[F(\tau_{1})F(\tau_{2})\right]d\tau_{1}d\tau_{2}\\\
=&\frac{c_{j}^{2}}{(\rho
A)^{2}}\int_{0}^{t_{1}}\int_{0}^{t_{2}}h_{j}(t_{1}-\tau_{1})h_{j}(t_{2}-\tau_{2})R_{F}(\tau_{2}-\tau_{1})d\tau_{1}d\tau_{2}\end{split}$
(54)

from Eq. (54) the variance $\sigma^{2}_{Y_{j}}(t)$ can be also evaluated by
placing $t_{1}=t_{2}$.

By taking into account the Eq. (46) the CF of the displacement $v(z,t)$ is

$\begin{split}R_{v}(z,t_{1},t_{2}):&=\mathbb{E}\left[v(z,t_{1})v(z,t_{2})\right]\\\
&=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)R_{Y_{j}Y_{i}}(t_{1},t_{2})\phi_{i}(z)\end{split}$
(55)

where $R_{Y_{j}Y_{i}}(t_{1},t_{2})$ is the cross correlation of the modal
response processes $Y_{j}(t$) and $Y_{i}(t)$ defined as

$\begin{split}R_{Y_{j}Y_{i}}(t_{1},t_{2}):=&\mathbb{E}\left[Y_{j}(t_{1})Y_{i}(t_{2})\right]\\\
=&\frac{c_{j}c_{i}}{(\rho
A)^{2}}\int_{0}^{t_{1}}\int_{0}^{t_{2}}h_{j}(t_{1}-\tau_{1})h_{i}(t_{2}-\tau_{2})R_{F}(\tau_{2}-\tau_{1})d\tau_{1}d\tau_{2}\end{split}$
(56)

If $t_{1}=t_{2}=t$ Eq. (55) provides the non-stationary variance of the
displacement $v(z,t)$. That is,

$\begin{split}\sigma^{2}_{v}(z,t):&=\mathbb{E}\left[v(z,t)v(z,t)\right]\\\
&=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\sigma^{2}_{Y_{j}Y_{i}}(t)\phi_{i}(z)\end{split}$
(57)

where $\sigma^{2}_{Y_{j}Y_{i}}(t)$ is

$\begin{split}\sigma^{2}_{Y_{j}Y_{i}}(t):=&\mathbb{E}\left[Y_{j}(t)Y_{i}(t)\right]\\\
=&\frac{c_{j}c_{i}\sigma^{2}_{F}}{(\rho
A)^{2}}\int_{0}^{t}\int_{0}^{t}h_{j}(t-\tau)h_{i}(t-\tau)d\tau d\tau\\\
=&\frac{c_{j}c_{i}\sigma^{2}_{F}}{(\rho
A)^{2}}t\int_{0}^{t}h_{j}(t-\tau)h_{i}(t-\tau)d\tau\end{split}$ (58)

and represents the cross variance of the modal response processes $Y_{j}(t$)
and $Y_{i}(t)$.

####  4.1.1 Monte Carlo simulation

In some cases, Eq. (56) and Eq. (58) cannot be evaluated in closed form and it
is needed a numerical approach to characterized the response process form a
stochastic point of view. In this context, Monte Carlo (MC) method is a
powerful tool that provides a time-domain response with the aid of digital
simulations. Specifically, as a first step, it is needed the generation of a
proper number $N$ of samples (or realizations) of the stochastic input $F(t)$.
The $i$-th realization of the stochastic input process is denoted as
$F^{i}(t)$ and can be generated by harmonic superposition method proposed by
Shinozuka and Deodatis Shino . According to this approach the generic $i$-th
sample of the forced process is given as

$F^{i}(t)=\sqrt{2}\sum_{j=1}^{m}\sqrt{2S_{F}(\omega_{j})\Delta\omega}\cos{\left(\omega_{j}t+\theta_{j}^{i}\right)}$
(59)

where $\omega_{j}=j\Delta\omega$, $\Delta\omega$ is the discretization step in
the frequency domain of the PSD function $S_{F}(\omega)$, $\theta_{j}^{i}$
represents $i$-th realization of the independent random phase with uniform
distributed probability density function between $0$ and $2\pi$.

As second step, it is needed to evaluate the response samples. In particular,
for each input sample $F^{i}(t)$ we need to evaluate the output process
$Y^{i}_{j}(t)$ in Eq. (47) with the aid of the Duhamel superposition integral
in Eq. (49). Thus,

$Y_{j}^{i}(t)=\frac{c_{j}}{\rho A}\int_{0}^{t}h_{j}(t-\tau)F^{i}(\tau)d\tau$
(60)

As last step, after the evaluation of the $n\times N$ response processes
$Y_{j}^{i}(t)$ with $i=1,2\dots,N$, and $j=1,2,\dots,n$, the stochastic
displacement process samples $v^{i}(z,t)$ can be used to evaluate the
statistics numerically.

###  4.2 Frequency domain approach

Steady-state analysis and characterization of stationary responses can be
driven in analytical way. Specifically, we make the truncated Fourier
transform of the stochastic differential equation (47). That is,

$\hat{Y}_{j}(\omega,\mathrm{T})\left[-\omega^{2}+\frac{\eta}{\rho
A}\mathrm{i}\omega+\frac{k_{\lambda,j}}{\rho A}\right]=\frac{c_{j}}{\rho
A}\hat{F}(\omega,\mathrm{T})$ (61)

which transforms the differential equation in time domain to an algebraic
equation in frequency domain. From Eq. (61) we get the solution
$\hat{Y}_{j}(\omega,\mathrm{T})$ as

$\begin{split}\hat{Y}_{j}(\omega,\mathrm{T})&=\frac{1}{-\omega^{2}\hat{+}\frac{\eta}{\rho
A}\mathrm{i}\omega+\frac{k_{\lambda,j}}{\rho A}}\frac{c_{j}}{\rho
A}\hat{F}(\omega,\mathrm{T})\\\ &=H_{j}(\omega)\frac{c_{j}}{\rho
A}\hat{F}(\omega,\mathrm{T})\end{split}$ (62)

where $H_{j}(\omega)$ is the transfer function of the $j$-th modal oscillator.
Once the response $\hat{Y}_{j}(\omega)$ in frequency domain is known, we can
place it in the expression of the cross PSD of the response processes
$Y_{j}(t)$ and $Y_{i}(t)$. That is,

$\begin{split}S_{Y_{j}Y_{i}}(\omega)&=\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{Y}_{j}^{*}(\omega,\mathrm{T})\hat{Y}_{i}(\omega,\mathrm{T})\right]\\\
&=\frac{c_{j}}{\rho
A}H_{j}^{*}(\omega)\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{F}^{*}(\omega,\mathrm{T})\hat{F}(\omega,\mathrm{T})\right]\frac{c_{i}}{\rho
A}H_{i}(\omega)\\\ &=\frac{c_{j}c_{i}}{\left(\rho
A\right)^{2}}H_{j}^{*}(\omega)H_{i}(\omega)S_{F}(\omega)\end{split}$ (63)

Finally, by using Eq. (63), it is possible to evaluate the analytical form of
the PSD of beam displacements as follows

$\begin{split}S_{v}(z,\omega)&=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)\lim_{\mathrm{T}\to\infty}\frac{1}{2\pi\mathrm{T}}\mathbb{E}\left[\hat{Y}_{j}^{*}(\omega,\mathrm{T})\hat{Y}_{i}(\omega,\mathrm{T})\right]\\\
&=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)S_{Y_{j}Y_{i}}(\omega)\\\
&=\frac{S_{F}(\omega)}{(\rho
A)^{2}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z){c_{j}c_{i}}H_{j}^{*}(\omega)H_{i}(\omega)\end{split}$
(64)

The PSD in Eq. (64) allows for evaluating the stationary variance of
transverse displacements $\sigma^{2}_{v}(z)$. That is,

$\sigma^{2}_{v}(z)=\int_{-\infty}^{\infty}S_{v}(z,\omega)d\omega=\frac{1}{(\rho
A)^{2}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)c_{j}c_{i}\int_{-\infty}^{\infty}H_{j}^{*}(\omega)S_{F}(\omega)H_{i}(\omega)d\omega$
(65)

###  4.3 Nonlocal beam under Gaussian white noise

In this section we consider the case in which the input process is a Gaussian
white noise with zero mean denoted by $W(t)$ and characterized by a constant
PSD and a Dirac delta as CF. That is,

$S_{W}(\omega)=S_{0},\;\;\;R_{W}(\tau)=2\pi S_{0}\delta(\tau)$ (66)

With this assumption the present study does not lose generality inasmuch
several real excitation process can be modeled as summation of modulated white
noises.

Moreover, from this considered case analytical solutions in terms of
statistics of the response can be obtained and some useful results about the
time and frequency domain analysis can be drawn. Under the assumptions in Eq.
(66), Eq. (64) and Eq. (65) provide the characterization of the stochastic
output process in terms of PSD and stationary variance. Specifically,

$S_{v}(z,\omega)=\frac{S_{0}}{(\rho
A)^{2}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z){c_{j}c_{i}}H_{j}^{*}(\omega)H_{i}(\omega)$
(67)

and

$\sigma^{2}_{v}(z)=\frac{S_{0}}{(\rho
A)^{2}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)c_{j}c_{i}\int_{-\infty}^{\infty}H_{j}^{*}(\omega)H_{i}(\omega)d\omega$
(68)

However, the quantities in Eqs. (67) and (68) provide a characterization of
the displacement just at steady-state. In order to provide a complete
stochastic characterization of the response it is needed to evaluate the CF in
Eq. (46) and the time-dependent non-stationary variance in Eq. (57). In this
regard, exploiting the properties of the stochastic input, the CF of the
displacement is given by

$R_{v}(z,t_{1},t_{2})=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)R_{Y_{j}Y_{i}}(t_{1},t_{2})\phi_{i}(z)$
(69)

where each term in the summations can be evaluated in closed form. That is,

$\begin{split}R_{Y_{j}Y_{i}}(t_{1},t_{2})&=\frac{c_{j}c_{i}2\pi S_{0}}{(\rho
A)^{2}}\int_{0}^{t_{1}}\int_{0}^{t_{2}}h_{j}(t_{1}-\tau_{1})h_{i}(t_{2}-\tau_{2})\delta(\tau_{2}-\tau_{1})d\tau_{1}d\tau_{2}\\\
&=\frac{c_{j}c_{i}2\pi S_{0}}{(\rho
A)^{2}}\int_{0}^{t_{1}}h_{j}(t_{1}-\tau_{1})h_{i}(t_{2}-\tau_{1})d\tau_{1}\end{split}$
(70)

From Eq. (69) the time-dependent variance of the displacement is

$\begin{split}\sigma_{v}^{2}(z,t)&=R_{v}(z,t,t)=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)R_{Y_{j}Y_{i}}(t,t)\\\
&=\frac{2\pi S_{0}}{(\rho
A)^{2}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}c_{j}c_{i}\phi_{j}(z)\phi_{i}(z)\int_{0}^{t}h_{j}(t-\tau)h_{i}(t-\tau)d\tau\\\
\end{split}$ (71)

Note that both the expressions in Eq. (70) and in Eq. (71) can be evaluated in
analytical form.

##  5 Numerical simulation and parametric study

This section is devoted to the stationary and non-stationary analysis of the
stochastic response of a nonlocal damped Bernoulli-Euler beam forced by a
random load. Such numerical analysis aims to study the influence of the
nonlocal parameter $\lambda$ in the response in terms of CF, PSD and
stationary and non-stationary variance.

We consider a micro-beam of length $L=300$ $\mu$m, rectangular cross section
with width $b=30$ $\mu$m and thickness $h=25$ $\mu$m, made of epoxy
characterized by density $\rho=1.20$ g/cm3 and elastic modulus $E=2.20$ GPa
Mems ; Ashby1 . Damping effects due to surrounding environment are modeled by
the following value of viscosity $\eta=2.00$ cP describing a wide variety of
viscous fluids of technical interest Ashby1 ; Ashby2 . The considered micro-
beam is constrained as a cantilever beam and is forced by a ground motion
acceleration as depicted in Figure 3(a). Without loss of generality, such
imposed acceleration on the basis $z=0$ is a Gaussian white noise

$\ddot{v}(0,t)=a_{g}(t)=W(t)$ (72)

where the white noise $W(t)$ has zero-mean and constant PSD $S_{0}=10^{3}$
N2s, and $v(z,t)$ denotes the relative displacement with respect to the basis.
In this case the partial differential equation (15) becomes

$\frac{\partial^{4}v(z,t)}{\partial z^{4}}-(\lambda
L)^{2}\frac{\partial^{6}v(z,t)}{\partial z^{6}}+\frac{\eta}{EI}\frac{\partial
v(z,t)}{\partial t}+\frac{\rho A}{EI}\left[\frac{\partial^{2}v(z,t)}{\partial
t^{2}}+a_{g}(t)\right]=0,\;\;\;\;\;0<z<L$ (73)

and the BCs $\forall t$ are

$\left\\{\begin{split}&v(0,t)=0,&\;\;\;\;&M(L,t)=0,\\\
&\varphi(0,t)=0,&\;\;\;&T(L,t)=0\end{split}\right.$ (74)

Let us recall that, by virtue Eq.(8) and Eq.(11), bending moment $\,M\,$ and
shear force $\,T\,$ fields can be expressed in terms of elastic curvature and
its derivatives as

$M(z,t)=EI\chi(z,t)-EI(\lambda L)^{2}\frac{\partial^{2}\chi(z,t)}{\partial
z^{2}}$ (75a) $T(z,t)=EI\frac{\partial\chi(z,t)}{\partial z}-EI(\lambda
L)^{2}\frac{\partial^{3}\chi(z,t)}{\partial z^{3}}$ (75b)

To show the effects of the nonlocal parameters in the response, different
cases of Eq. (73) are considered. Specifically, we select three values of the
nonlocal parameter $\lambda$, i.e.,
$\lambda=\left\\{0.1,\,0.2,\,0.3\right\\}$.

(a) Layout of the cantilever micro-beam

(b) First five eigenfunctions for $\lambda=0.2$

Figure 3: Cantilever micro-beam forced by ground motion acceleration

The solution in terms of displacement function is obtained with the aid of Eq.
(38), where each eigenfunction is obtained by solving the differential problem
in Eq. (22) with the two constitutive BCs in Eq. (23) and the four BCs in Eq.
(74). The latter BCs in terms of eigenfunctions are

$\left\\{\begin{split}&\phi(0)=0,&\;&\phi^{(2)}(L)-(\lambda
L)^{2}\phi^{(4)}(L)=0,\\\ &\phi^{(1)}(0)=0,&\;\;\;&\phi^{(3)}(L)-(\lambda
L)^{2}\phi^{(5)}(L)=0\end{split}\right.$ (76)

The first five eigenfunctions for $\lambda=0.2$ are shown in Figure 3(b) and
the first five natural frequencies for different values of $\lambda$ are
reported in Table 1.

$\lambda$ | $\omega_{0,1}$ | $\omega_{0,2}$ | $\omega_{0,3}$ | $\omega_{0,4}$ | $\omega_{0,5}$
---|---|---|---|---|---
0.10 | $4.2323\times 10^{5}$ | $2.8373\times 10^{6}$ | $8.8780\times 10^{6}$ | $1.9924\times 10^{7}$ | $3.7820\times 10^{7}$
0.15 | $4.4551\times 10^{5}$ | $3.1643\times 10^{6}$ | $1.0609\times 10^{6}$ | $2.5278\times 10^{7}$ | $5.0090\times 10^{7}$
0.20 | $4.6795\times 10^{5}$ | $3.5244\times 10^{6}$ | $1.2495\times 10^{7}$ | $3.1001\times 10^{7}$ | $6.2992\times 10^{7}$
0.25 | $4.9002\times 10^{5}$ | $3.9038\times 10^{6}$ | $1.4464\times 10^{6}$ | $3.6905\times 10^{7}$ | $7.6186\times 10^{7}$
0.30 | $5.1192\times 10^{5}$ | $4.2951\times 10^{6}$ | $1.6481\times 10^{7}$ | $4.2909\times 10^{7}$ | $8.9539\times 10^{7}$
Table 1: Natural frequencies in $rad/s$ of cantilever micro-beam for different
values of $\lambda$.

Taking Eq. (73) into account and with the aid of the definition in Eq. (67)
the PSD of the displacement is

$S_{v}(z,\omega)\approx
S_{0}\sum_{j=1}^{n}\sum_{i=1}^{n}\phi_{j}(z)\phi_{i}(z)c_{j}c_{i}H_{j}^{*}(\omega)H_{i}(\omega)$
(77)

where for the present numerical simulations we assume $n=5$.

In Figure 4 the PSD of the displacements are reported for different values of
$\lambda$. Specifically, Figure 4(a) shows the PSDs of the mid-point
displacements, while in Figure 4(b) the PSDs at $z=L$ are shown. We can
observe that the nonlocal parameter influences both the natural frequencies
and the peak amplitudes in the PSD. Specifically, when the nonlocal parameter
increases the amplitude of the peaks decrease and their frequencies increases.

(a) PSD for $z=L/2$

(b) PSD for $z=L$

Figure 4: PSD of a cantilever micro-beam response for different values of
$\lambda$

From the PSD in Eq. (77) and with the aid of Eq. (65) we can also evaluate the
stationary variance of the displacement. Such quantity can be evaluated by the
integration of the PSD in Eq. (77) as follow

$\sigma_{v}^{2}(z)=\int_{-\infty}^{\infty}S_{v}(z,\omega)d\omega\approx
S_{0}\sum_{j=1}^{n}\sum_{i=1}^{n}\phi_{j}(z)\phi_{i}(z)c_{j}c_{i}\int_{-\infty}^{\infty}H_{j}^{*}(\omega)H_{i}(\omega)d\omega$
(78)

In Table 2 stationary variances at $z=L/2$ and $z=L$ for different values of
$\lambda$ are reported.

$\lambda$ |  | $\sigma^{2}_{v}(L/2)$ | $\sigma^{2}_{v}(L)$
---|---|---|---
0.10 |  | $0.0200$ | $0.1975$
0.15 |  | $0.0165$ | $0.1791$
0.20 |  | $0.0144$ | $0.1645$
0.25 |  | $0.0138$ | $0.1548$
0.30 |  | $0.0115$ | $0.1416$
Table 2: Displacement variances in $\mu$m2 at $z=L/2$ and $z=L$, for different
values of $\lambda$.

Taking into account the values in this table we can state that the nonlocal
parameter also influences stationary variances. Specifically, when the
nonlocal parameter $\lambda$ grows up then the displacement stationary
variances decrease.

The PSD in Eq. (77) and the stationary variance in Eq. (78) provide a steady-
state characterization of the displacement process due to a stochastic ground
motion acceleration. However, taking Eq. (73) into account, a characterization
of the non-stationary response can be pursued by using Eq. (69) to evaluate
the CF of the process $v(z,t)$. That is,

$R_{v}(z,t_{1},t_{2})=2\pi
S_{0}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z){c_{j}c_{i}}\int_{0}^{t_{1}}h_{j}(t_{1}-\tau_{1})h_{i}(t_{2}-\tau_{1})d\tau_{1}$
(79)

By virtue of the definition in Eq. (71) the time-dependent variance is

$\begin{split}\sigma_{v}^{2}(z,t)&=R_{v}(z,t,t)=\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\phi_{j}(z)\phi_{i}(z)R_{Y_{j}Y_{i}}(t,t)\\\
&={2\pi
S_{0}}\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}c_{j}c_{i}\phi_{j}(z)\phi_{i}(z)\int_{0}^{t}h_{j}(t-\tau)h_{i}(t-\tau)d\tau\\\
\end{split}$ (80)

From Eq. (80) it is possible to obtain the stationary variance in Eq. (78) by
performing the following limit

$\sigma_{v}^{2}(z)=\lim_{t\to\infty}\sigma_{v}^{2}(z,t)$ (81)

while the stationary CF is given by

$R_{v}(z,\tau)=\lim_{t\to\infty}R_{v}(z,t,t+\tau)$ (82)

the latter equation is related to the PSD in Eq. (77) by Fourier transform
(Wiener-Khinchin theorem).

(a) Displacement variance at $z=L/2$

(b) Displacement variance at $z=L$

Figure 5: Exact stationary (dashed line) and non-stationary (continuous line)
displacement variance in contrast with those obtained by MC simulations

Figure 5 shows the stationary and non-stationary displacement variances for
different values of the nonlocal parameter $\lambda$. Dashed lines represent
the stationary variances, whereas continuous lines are the non-stationary ones
obtained with the aid of Eq. (80). Such exact variances are compared with the
numerical results obtained by means of the MC approach described in Section
4.1.1. Specifically, the numerical variances (dotted lines in Figure 5) are
obtained considering $N=4\times 10^{3}$ samples and by assuming $m=2\times
10^{3}$ and $\Delta\omega=\omega_{0,1}/50$ in Eq. (59). From Figure 5 we can
observe that the steady state is reached in all considered case in a few tens
of microseconds. This is due to the fact that the involved stiffnesses in such
micro-beams are great and the masses are little. Moreover, the nonlocal
parameter influences the duration of the transient state, indeed, if $\lambda$
grows up the variance reaches the stationary value more quickly.

##  6 Concluding remarks

Random vibrations of damped nonlocal Bernoulli-Euler beams due to stochastic
excitation have been investigated in the present research. Two specific
effects have been accurately analyzed: size and damping phenomena,
respectively modeled by stress-driven nonlocal mechanics and external viscous
interactions. A stochastic input for the loading has been assumed to simulate
external actions randomness.

A stochastic differential problem in space and time, governing the motion of
nonlocal beams under stochastic loading, has been formulated.

Exact solutions of power spectral density, correlation function and
displacement variance have been evaluated by differential eigenanalysis.

From the analytical formulation of stationary and non-stationary responses and
with the aid of numerical simulations, it has been highlighted a significant
reduction of stationary variances and duration of the transient state in
responses and an increasing of natural frequencies for increasing nonlocal
scale parameter. The predicted smaller-is-stiffer phenomenon, confirmed
recently in FusPisPol , agrees with most of experimental outcomes associated
with inflected small-scale beams AbazariSensors2015 .

In summary, the nonlocal approach developed to model damped small-scale beams
is able to capture size and damping effects and random excitations. The
methodology provides analytical solutions in terms of statistics of the
response and closed-form natural frequencies. The contributed results can be
exploited for structural design and optimization of smaller and smaller
devices used in modern technological applications, such as: sensors,
actuators, MEMS/NEMS, resonators.

## Acknowledgments

Financial supports from the MIUR in the framework of the Projects PRIN 2015
”COAN 5.50.16.01” (code 2015JW9NJT _Advanced mechanical modeling of new
materials and structures for the solution of 2020 Horizon challenges_) and
PRIN 2017 (code 2017J4EAYB _Multiscale Innovative Materials and Structures
(MIMS)_ ; University of Naples Federico II Research Unit) and from the
research program ReLUIS 2019 are gratefully acknowledged.

## Conflict of interest

The authors declare that they have no conflict of interest.

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