Closed-form response of a linear fractional visco-elastic oscillator under arbitrary
stationary input
V. Denoëla,
aStructural Engineering Division, Faculty of Applied Sciences, University of Liège, Liège, Belgium
Abstract
This paper studies the structural response of a single degree-of- freedom structure including a fractional deriva-tive constituderiva-tive term. Unlike usual existing models for this kind of structure, the excitation is not necessarily a Markovian process but it is slowly varying in time, so that a timescale separation is used. Following the general formulation of the Multiple Timescale Spectral Analysis [1], the solution is developed as a sum of background and resonant components. Because of the specific shape of the frequency response function of a system equipped with a fractional viscoelastic device, the background component is not simply obtained as the variance of the loading divided by the stiffness of the system. On the contrary the resonant component is expressed as a sim-ple extension of the existing formulation for a viscous system, at least at leading order. As a validation case, the proposed solution is shown to recover similar results (in the white noise excitation case) as former studies based on a stochastic averaging approach [2, 3, 4]. A better accuracy is however obtained in case of very small fractional exponent. Another example related to the buffeting analysis of a linear fractional viscoelastic system demonstrates the accuracy of the proposed formulation for colored excitation.
Keywords: Caputo fractional derivative, Riemann-Liouville fractional derivative, perturbation analysis, stochastic averaging, background component, resonant component
1. Fractional Calculus in Mechanics
1
Fractional calculus has attracted considerable at-tention over the last decades in the mechanical and structural engineering community. In some early works [5] dealing with the modeling of visco-elastic damping, a fractional derivative Maxwell model was used. An experimental demonstration [6] has con-firmed the appropriateness of this model and fur-ther trigger scientific curiosity about fractional cal-culus in mechanics. This model is based on the Rie-mann—Liouville definition of the fractional deriva-tive operator Dαy(t) = 1 Γ(1 − α) t ˆ 0 ˙ y(˜t) (t − ˜t)αd˜t (1)
where α ∈ [0, 1] is the fractional exponent. Some
2
other experimental investigations and identification
3
techniques have shown that this model is able to
accu-4
rately capture the dynamics of visco-elastic systems
5
[7, 8, 9]. Also the linearity of this operator (the
frac-6
tional derivative of a linear combination is the linear
7
combination of the fractional derivatives) makes it
8
rather attractive to combine viscoelastic devices with
9
others existing features of linear stochastic dynamics
10
Email address: v.denoel@ulg.ac.be (V. Denoël)
[10, 11, 2, 12, 13]. The nonstationary solution of
lin-11
ear systems might be expressed in closed form [14],
12
which constitutes another reason to use this type of
13
model. As soon as nonlinear stochastic dynamical
14
systems are considered, the exact solution is usually
15
not obtained in closed form, even for wide-band
(usu-16
ally white noise) excitation. Approximations similar
17
to or derived from the stochastic linearization and
18
stochastic averaging methods [15, 16, 17, 18, 19], or
19
those based on a Fokker-Planck equation of the
pro-20
cess envelope [20] appear to be the most classical ways
21
to deal with such problems. Narrow band
excita-22
tions and complex dynamical interactions can be
sim-23
plified with similar multiple scales approaches [21].
24
Other more realistic types of loadings [22] or even
25
earthquake loadings of linear and nonlinear systems
26
equipped with viscoelastic devices have also been
con-27
sidered in [23, 24].
28
Beside these approximations of the exact solution
29
of the problem, other numerical techniques have been
30
proposed to deal with the structural analysis of
sys-31
tems with fractional derivatives. In particular, Monte
32
Carlo simulation methods are consistently used to
33
validate approximations. Also there exist ad hoc
sim-34
ulation methods [14] which are computationally
effi-35
cient, and methods based on Wiener path integral
36
approaches [25, 26, 27, 28]. These techniques,
to-37
gether with exact assembling procedures of structural
38
analysis [29], or finite element approaches [30, 4, 12],
make it possible to study more realistic structures
40
composed of several beams and columns, and several
41
viscoelastic devices.
42
This review of the literature reveals two trends. On one side, there are simple dynamical systems with linear (or linearized) behaviour and white noise ex-citation, which possess closed form solutions. On the other side, there are numerical techniques to deal with more realistic loadings (sometimes nonstation-ary), more realistic structures and/or slight nonlin-earities. The missing gap in-between is related to the understanding (via simple analytical solutions) of the behavior of complex structures subjected to more re-alistic loadings. As a first step towards this goal, we consider the stochastic analysis of a linear system subjected to a low frequency loading specified by its arbitrary power spectral density. More precisely, we consider the oscillator governed by
x00(τ ) + 2ξDαx(τ ) + x(τ ) = u(τ ) (2) where ξ is the dimensionless fractional coefficient and
43
u(τ ) is a stationary random process with an arbitrary
44
power spectral density Su(Ω) with the restriction
45
that its characteristic timescale is slow compared to
46
the dimensionless natural frequency of the oscillator
47
(which is equal to unity). This problem is a
dimen-48
sionless version of a classical singe degree-of-freedom
49
system with given mass, stiffness and damping, see
50
[31] for the details of the scaling. This equation is for
51
instance encountered in the analysis of wind-excited
52
structures with visco-elastic devices, where the
ob-53
jective is to determine the statistics (the variance,
54
mainly) of the structural response. The fact that the
55
exogenous loading u(τ ) could in principle be a non
56
Markovian process is a specific contribution of this
57
study.
58
This problem could be studied by means of the
59
usual time domain multiple scales approach, or the
60
stochastic averaging approach, but would require the
61
consideration of three interacting timescales as well as
62
fractional models —similar to those used to simulate
63
realizations of wind fields [32],— for the augmented
64
state coloring the excitation. This seems possible to
65
solve the problem in this way, although no track
ev-66
idences of this type of problem has been found in
67
the literature. Instead, we take advantage of the
lin-68
earity of the problem to derive a simplified solution
69
in a frequency domain. It is based on the Multiple
70
Timescale Spectral Analysis [1] which seeks the same
71
objectives as the stochastic averaging, with slightly
72
more versatility as shown next.
73
2. The Multiple Timescale Spectral Analysis
74
In 1961, Davenport proposed a method to
com-75
pute the response of structures to the buffeting
ac-76
tion of wind [33], which is known today as the
77
Background/Resonant decomposition [34]. In this
78
method, the total variance of the response, which
cor-79
responds to the area under the power spectral density
80
of the response a sketched in Figure 3, is approached
81
by the sum of two terms. The first one, the
back-82
ground component, corresponds to the quasi-static
83
response of the structure and therefore depends on
84
the stiffness of the system. The second one, the
reso-85
nant component, corresponds to the resonance of the
86
structure and is also governed by the damping in the
87
system. It is possible to prove that this
decomposi-88
tion is valid in the necessary and sufficient condition
89
that the two timescales associated with the problem
90
of buffeting, i.e. the slow action of wind and the fast
91
dynamics of the structure, are well separated. With
92
the usual rules of perturbation theories, this
trans-93
lates into one order of magnitude at least [35].
94
In fact, it turns out that the decomposition
pro-95
posed by Davenport in 1961 is just a particular case
96
of the Multiple Timescales Spectral Analysis, which
97
has been recently formalized by the author [1].
Un-98
der this general terminology, one can find an ad hoc
99
version of the computation of integrals with small
pa-100
rameters, as typically depicted in Pertubation
Meth-101
ods textbooks, e.g. [35]. Indeed, in many applications
102
of stochastic dynamics, the integrals to be computed
103
(in the frequency domain) feature several small
pa-104
rameters. In particular, as soon as the ratio of two
105
of the timescales of a problem is a small number, the
106
problem is said to feature well separated timescales.
107
Although this ratio is the most important small
num-108
ber, there are several additional small numbers in
en-109
gineering applications: small damping, small
nonlin-110
earity, small stochasticity (see for instance [36] where
111
four small numbers are identified and their smallness
112
is exploited to construct very accurate approximate
113
solutions of the problem).
114
The Multiple Timescale Spectral Analysis (MTSA)
115
is a method to develop simple analytical solutions, or
116
at least to reveal the influence of the different
prob-117
lem parameters to the various contributions to the
118
stochastic response of a system. As stated by its
119
name, the MTSA requires the existence of various
120
well separated timescales in the problem. We notice
121
that this does not really limit the possible field of
ap-122
plication and corresponds anyways to usual
assump-123
tions formulated in deterministic and stochastic
av-124
eraging [37]. The MTSA consists in recognizing the
125
existence of these different timescales and consider
126
the problem with all these timescales, successively,
127
and provide local approximations in the frequency
128
domain for the statistical moments associated with
129
all of them. The method is rather general and is not
130
limited to two timescales, nor to the estimation of the
131
variance of the response. It has already been applied
132
to the estimation of covariances of modal responses
133
in case of multi-degree of freedom structures [38]
(al-134
though this was before the general method has been
135
fully formalized). In this context, the MTSA provides
Aerodynamic Force
Spectral density
Mechanical
Admittance
Response
Spectral density
Resonant (fast timescale) Background (slow timescale)Figure 1: Sketch illustrating the decomposition of the response to buffeting into a slow background component and a fast resonant component [33].
a much simpler but very accurate solution for the
es-137
timation of the cross-correlation terms in a Complete
138
Quadratic Combination, much simpler than the
ex-139
act correlation coefficients [39], which are still widely
140
used today despite their complexity. The MTSA also
141
provides a very simple way to manage non
propor-142
tional damping in a deterministic or stochastic
con-143
text [31]. It has also given very concise and accurate
144
results in the context of non Gaussian loading [40],
145
nonstationary loading [41] slightly nonlinear systems
146
[36], and oscillators featuring some visco-elastic
com-147
ponents [31]. It is not limited to the variance of the
148
response. Extensions to the estimation of the
skew-149
ness [40] and excess [42] of the response have been
150
developed. It is sometimes believed that averaging
151
in the frequency domain simply consists in replacing
152
the power spectral density of the load by a constant
153
value, taken as the ordinate of the power spectral
den-154
sity of the load at natural frequency. This is fine for
155
most of the problems dealing with the second-order
156
moments, as confirmed again in this paper. However,
157
this is not true for higher order statistics [40], and this
158
is naturally and deductively shown with the Multiple
159
Timescale Spectral Analysis. It therefore unifies
un-160
der the same format, the necessary steps to deal with
161
efficient and accurate solution of stochastic dynamics
162
problems in the frequency domain.
163
3. Solution of The Problem
164
In this paper, the Multiple Timescales Spectral Analysis is applied to the considered problem, which demonstrates the ability of the method to deal with visco-elastic dampers, but also to provide an engi-neering insight into the different components of the structural response of a stochastically excited oscil-lator in the presence of dissipative devices modeled with fractional derivatives. To do so, let us first con-sider the Fourier transform of (2)
1 + 2ξCΩ|α− Ω2− 2ξiSΩ|α X(Ω) = U (Ω) (3)
where C = cosαπ2 and S = sinαπ2 , so that the power spectral density of the response is given by
Sx(Ω) =
Su(Ω)
(1 + 2ξC |Ω|α− Ω2)2+ (2ξS |Ω|α)2
:= K(Ω)Su(Ω). (4)
This equation defines the kernel K(Ω). The only
as-165
sumption required for the method to be applicable
166
is that the central frequency β in the loading Su(Ω)
167
be much smaller that unity, β 1 i.e. that the
168
stochastic loading is slower that the dynamics of the
169
system. This kernel is illustrated in Figure 2 for
sev-170
eral values of α. It has some peculiarities: (i) the
171
resonance peak located near Ω = ±1 in the viscous
172
case (α = 1) regularly moves to higher frequencies as
173
α → 0, i.e. as the fractional derivative term tends to
174
correspond to a stiffness term. In the limiting case
175
α = 0, the fractional derivative corresponds to a usual
176
stiffness term and the peak is located at abscissa
177
Ωp =
√
1 + 2ξ ' 1 + ξ; (ii) the frequency response
178
function passes trough a common crossing point, at
179
abscissa Ω = 1, no matter the fractional exponent α;
180
(iii) the intercept is K(0) = 1 provided α 6= 0. As
181
α → 0, a short boundary layer, whose extent is of
or-182
der α, develops in the neighborhood of the origin and
183
creates the transition from the upper bound K(0) = 1
184
to the lower bound K(Ω) ' (1+2ξ)1 2. For α → 0, the 185
size of this transition zone tends to zero; for α = 0,
186
there is no transition anymore and K(0) = (1+2ξ)1 2. 187
As a result of the fractional powers of Ω appearing in K(Ω), the response of the system at second order, its variance, defined as
σ2x=
+∞
ˆ
−∞
Sx(Ω)dΩ, (5)
is unfortunately not available in a simple closed form,
188
even for simple forms of Su(Ω). This is all the more
189
valid for complex expressions of Su(Ω) corresponding
190
non Markovian processes.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10-1 100 101 102 103 singularity
Figure 2: Representation of the kernel K(Ω) for various values of the fractional exponent α. Other parameters: ξ = 0.2.
Figure 3 shows some examples of the power
spec-192
tral density of the structural response given by (4).
193
This function features two distinct types of peaks:
194
one in the low-frequency range around Ω ' 0 and
195
over a domain whose extent is of order β (the
back-196
ground component) and the other in the order-one
197
frequency range (the resonant component). They are
198
well distinct because β 1.
199
Application of the Multiple Timescale Spectral
200
Analysis consists in successively focusing on the
dif-201
ferent contributions to the response. In this
applica-202
tion, there are only 2 components.
203
First, the background component is evaluated by rescaling the frequency axis Ω with the stretched co-ordinate ζ defined as Ω = βζ. With this scaling, the background component develops over a domain of or-der 1 and, consior-dering the separation of timescales β 1, the kernel K [Ω (ζ)]can be approximated by
ˆ
K [Ω (ζ)] = 1
1 + 4ξCβα|ζ|α+ 4ξ2β2α|ζ|2α. (6)
This is the frequency response function of a lowpass
204
fractional filter [43]. The expression Su(Ω) ˆK (Ω) is
205
therefore a local approximation of Sx(Ω) in the
neigh-206
borhood of the origin, for Ω ∼ β 1. This
approxi-207
mation is represented by dotted lines in Figure 3, for
208
α = 0.25 and ξ = 0.2. Using this approximation, the
209
background component of the response is expressed
210 as 211 σx,b2 = +∞ ˆ −∞ Su(Ω) ˆK (Ω) dΩ = +∞ ˆ −∞ Su(Ω) 1 + 4ξC |Ω|α+ 4ξ2|Ω|2αdΩ (7)
which is the lowpass fractional filtered energy in the
212
loading. For α ' 1 and ξ 1, the frequency response
213
function of this filter tends to unity and
214 Background -2 -1 0 1 2 10-3 10-2 10-1 100 101 -2 -1 0 1 2 10-4 10-2 100 102 Resonant
Figure 3: Examples of the power spectral density of the struc-tural response for various values of the fractional exponent α. Other parameters: ξ = 0.2, β = 0.05.
lim
α→1,ξ→0σ 2
x,b= 1 (8)
which is the well-known result from linear stochastic
215
dynamics.
216
Second, the resonant component needs to be devel-oped. To do so a remainder is constructed by sub-tracting this first approximation ˆK (Ω) from the orig-inal function to integrate, that is
r1= +∞ ˆ −∞ Sx(Ω) − Su(Ω) ˆK (Ω) dΩ = +∞ ˆ −∞ Su(Ω) K (Ω) − ˆK (Ω)dΩ. (9)
The function to be integrated features two
symmetri-217
cal peaks which will equally contribute the resonant
218
part of the response. So we only only focus on the
219
positive peak, then multiply by two. It is possible
220
to prove [31] that the peaks (in absolute value) are
221
located close to abscissa
222 Ωp= 1 + Cξ − α + 1 2− α C2 ξ2+ O(ξ2). (10)
The position of the peak is a perturbation of 1 (the
223
dimensionless natural frequency) and
• for α ' 1, the fractional derivative resembles a
225
viscous effect, C = cosαπ2 1 and the position
226
of the peak is very close to 1 − ξ2, the peak
po-227
sition of the viscously damped system.
228
• for α ' 0, C ' 1 and the position of the peak is
229
located close to abscissa 1 + ξ; this is consistent
230
with existing results obtained with a stochastic
231
averaging approach, at least at leading order in
232
ξ [18].
233
The natural stretched coordinate to focus on the peak
234
in the positive region is therefore η defined as
235
Ω = 1 + Cξ + ξη2 ⇐⇒ η =
Ω − 1 − Cξ
ξ . (11)
Using this stretching, the integrand in (9) becomes, after some simplifications [31]
Su(Ω) K (Ω) − ˆK (Ω)' ˆK(η) := Su(1 + Cξ) 4ξ2 1 c2η2+ c1η + c0 (12) where the coefficients c0 = S2(1 + 2ξαC), c1 =
236 1 2ξ(1 + (1 − 4α) (1 − 2S 2)) and c 2= 1 + 2ξ(1 − α)C. 237
This approximation is also represented by dotted
238
lines in Figure 3, for α = 0.25 and ξ = 0.2. The
ap-239
proximation of the remainder r1, multiplied by two is
240
the resonant contribution to the response. It reads
241 σx,r2 = 2 +∞ ˆ −∞ ˆ K(η)ξdη = πSu(1 + Cξ) 2ρξ (13) where ρ = p4c0c2− c21 = [4 (1 + 2ξC) S 2 + 242
ord ξ2]1/2. Truncating ρ to its leading order terms
243
for consistency with the previous orders of
approx-244
imations, the resonant contribution to the response
245
finally finally simplifies into
246
σ2x,r= πSu(1 + Cξ)
2Sξ√1 + 2ξC. (14)
247
To summarize, the background/resonant decompo-sition of the variance of a linear oscillator with frac-tional derivatives is given by
σx2= σx,b2 + σx,r2 = (15) +∞ ˆ −∞ Su(Ω) 1 + 4ξC |Ω|α+ 4ξ2|Ω|2αdΩ + πSu(1 + Cξ) 2Sξ√1 + 2ξC. (16) This expression regularly extends the well-known background resonant decomposition in case of inte-ger derivative (α = 1, C = 0, S = 1), which is
lim α→1σ 2 x= 1 + πSu(1) 2ξ .
It also shows that, at leading order, the response
248
of a fractionally damped oscillator depends on the
249
power spectral density of the loading, computed for
250
a unique value of the frequency: Ω = 1 + Cξ. This is
251
the only way the response depends on the power
spec-252
tral density of the loading. With this approximation,
253
we show that the Markovianity of the input of this
254
system is secondary; in other words, the proposed
255
solution is valid no matter the shape of the power
256
spectral density (rational fractions of Ω or not).
257
4. Validation, illustrations and discussion
258
In order to validate the proposed solution, we
de-259
termine the response of the oscillator subjected to a
260 loading specified by 261 Su(Ω) = 0.546 β1 + 1.64|Ω|β 5/3 (17)
with β 1 is a small dimensionless
characteris-262
tic frequency. In wind engineering applications, it
263
is related to the slow turbulence, compared to the
264
natural frequency of the structure. The accuracy of
265
the proposed formulation is assessed by comparison
266
with a reference result obtained by accurate
numer-267
ical integration of the exact power spectral density
268
of the response. Integration is performed with the
269
adaptive algorithm proposed in Wolfram
Mathemat-270
ica [44], with default integration parameters of
Ver-271
sion 11.0.1.0.
272
Figures 4 and 5 show the variance of the response
273
obtained with the proposed formulation (MTSA) and
274
by numerical integration of the exact analytical
for-275
mulation (Exact). The variance is represented as a
276
function of α for given values of ξ, and as a function
277
of ξ for given values of α. It is also given for two
278
values of β, namely β = 0.01 and β = 0.1. These two
279
small numbers correspond to typical values of this
280
parameter in buffeting applications.
281
In all cases, the proposed formulation (MTSA)
pro-282
vides very accurate results, when compared to the
283
reference solution (Exact). The smaller β, the more
284
accurate. This is consistent with the methodology
285
to develop the approximate solution and with the
in-286
herent assumptions in the Multiple Timescale
Spec-287
tral Analysis. The same observation also holds for ξ
288
which also needs to be a relatively small number. In
289
fact, this relative smallness can be discussed with this
290
example. Indeed, the comparison shows that the
pro-291
posed method is very accurate for values of ξ which
292
are as large as 1.
293
In both figures, the background component σ2 x,b is
294
shown with dashed lines. Where the total variance is
295
similar to background component σ2
x,b, the resonant
296
counterpart to the response is negligible and the
re-297
sponse is actually quasi-static. This happens for large
298
values of the damping (see Figure 4) or, for given ξ,
10-3 10-2 10-1 100 10-1 100 101 10-3 10-2 10-1 100 10-1 100 101 MTSA Background Exact quasi-static response dynamic (resonant) response
Figure 4: Variances of the response of the system subjected to the buffeting type excitation, for β = 0.01 (top) and β = 0.1 (bottom). Represented as a function of the fractional coeffi-cient ξ. and for various values of the fractional exponent α (Please see online version for colors)
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 MTSA Background Exact quasi-static response dynamic response
Figure 5: Variances of the response of the system subjected to the buffeting type excitation, for β = 0.01 (top) and β = 0.1 (bottom). Represented as a function of the fractional exponent α and for various values of the fractional coefficient ξ. (Please see online version for colors)
for large values of α, i.e. as the behaviour of the
300
fractional damper tends to viscosity (see Figure 5).
301
5. Conclusions
302
In this paper, we have applied the Multiple
303
Timescale Spectral Analysis to the structural
anal-304
ysis of a linear system equipped with a viscoelastic
305
device. The proposed formulation extends the well
306
known background/resonant decomposition which is
307
usually applied in the field of wind engineering. It
308
shows that
309
• the background component can be determined
310
without any prior knowledge of the dynamics of
311
the system; it consists in a fractional filtered
ver-312
sion of the input;
313
• the power spectral density of the loading just
314
enters in the solution of this problem through the
315
value of the power spectral density at a frequency
316
equal to Ω = 1 + cosαπ2 ξ;
317
• the resonant component of the response σ2 x,r is
318
proportional to the power spectral density;
319
• the resonant component of the response σ2 x,rdoes
320
not scale with the inverse of the damping ratio
321
anymore (as what would be obtained in the
in-322
teger derivative case). It rather scales with the
323
inverse of ξ√1 + 2ξC. In the limit case α → 1
324
(viscous damping), C → 0, the usual scaling is
325
recovered
326
• the resonant component scales with S−1 =
327
cscαπ2 , which tends to infinity as α → 0. This
328
results from the fact that there is no damping
329
anymore in this limit case, and the dynamic
re-330
sponse is unbounded.
331
These preliminary results are very promising. Future
332
works should combine the developments summarized
333
in this paper with other contexts of application of the
334
Multiple Timescale Spectral Analysis and consider in
335
this way slightly nonlinear systems (with the help of
336
a Volterra model) or to multiple degree-of-freedom
337
structures equipped with fractional derivative
dissi-338
pative devices.
339
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340
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![Figure 1: Sketch illustrating the decomposition of the response to buffeting into a slow background component and a fast resonant component [33].](https://thumb-eu.123doks.com/thumbv2/123doknet/6205605.160275/3.892.145.767.84.306/illustrating-decomposition-response-buffeting-background-component-resonant-component.webp)
