Sabatier Agrawal Machado Advances in Fractional Calculus Episode 11 doc

Damping behavior of materials, if eled using linear, constant coefficient differential equations, cannot include the longmod-imated by fractional order derivatives.. However, sufficiently gre

Size of the cluster number of particles involved) DLA-

procedure

0.1743

( 0.0044)

0.9996 ( 0.0045)

2.0235 ( 0.0702)

0.5427 ( 0.0034)

1.8419 ( 0.0059)

1,9807 ( 0.0651)

0,5399 ( 0.0034)

1,858 ( 0.0087)

2.1802 ( 0.1379)

0.6046 ( 0.0046)

1.6396 ( 0.0103)

2.0153 ( 0.0841)

0.6087 ( 0.0059)

1.6458 ( 0.0109)

1.8826 ( 0.0922)

0.6172 ( 0.0047)

1.6312 ( 0.0105)

2.1936 ( 0.1043)

0.612 ( 0.0047)

1.6183 ( 0.0097)

2.0733 ( 0.105)

0.6096 ( 0.0059)

1.6335 ( 0.0109)

2.2237 ( 0.1929)

0.609 ( 0.006)

1.6266 ( 0.008)

2.2119 ( 0.094)

0.6096 ( 0.0055)

1.6229 ( 0.0111)

2.2872 ( 0.216)

0.6054 ( 0.0101)

1.631 ( 0.0173)

2.1654 ( 0.1483)

0.6081 ( 0.0084)

1.6331 ( 0.0188)

of the stdev) (diameter,

of the stdev)

(with values

of the stdev)

D (with values

of the stdev)

Size of the cluster number of particles involved) Random

1.8358 ( 0.0035) Random

1.9497 ( 0.0004) Random

lattice

0.7053

( 0.0069)

0.9867 ( 0.002)

4.5213 ( 0.0909)

0.5075 ( 0.001)

1.9442 ( 0.0003)

2 Nigmatullin RR, Alekhin AP (2005) Realization of the Riemann-Liouville Integral

on New Self-Similar Objects In: Books of abstracts, Fifth EUROMECH Nonlinear

Dynamics Conference August 7–12, pp 175–176 Prof Dick H van Campen (ed.), Eindhoven University of Technology, The Netherlands

4 Nigmatullin RR, Le Mehaute A (2005) J Non-Cryst Solids, 351:2888

5 Nigmatullin RR (2005) Fractional kinetic equations and universal decoupling

of a memory function in mesoscale region, Physica A (has been accepted for publication)

6 Fractals in Physics (1985) The Proceedings of the 6th International

Sym-posium, Triest, Italy, 9–12 July; Pietronero L, Tozatti E (eds.), Elsevier Science, Amsterdam, The Netherlands

Geometrie Fractale, Hermez, Paris (in French)

including models for viscoelastic damping Damping behavior of materials, if eled using linear, constant coefficient differential equations, cannot include the long

mod-imated by fractional order derivatives The idea has appeared in the physics erature, but may interest an engineering audience This idea in turn leads to an infinite-dimensional system without memory; a routine Galerkin projection on that

lit-material may have little engineering impact.

Fractional-order derivatives appear in various engineering applications

microstructural disorder can lead, statistically, to macroscopic behavior well memory that fractional -order derivatives require However, sufficiently great

approx-infinite-dimensional system leads to a finite dimensional system of ordinary tial equations (ODEs) (integer order) that matches the fractional-order behavior over user-specifiable, but finite, frequency ranges For extreme frequencies (small

differen-or large), the approximation is podifferen-or This is unavoidable, and users interested in such extremes or in the fundamental aspects of true fractional derivatives must take note

of it However, mismatch in extreme frequencies outside the range of interest for a particular model of a real

Keywords

Fractional-order derivatives have proved useful in the modeling of viscoelastic

will show that sufficiently disordered (random) and high-dimensional nal integer -order damping processes can lead to macroscopically observablefractional-order damping This suggests that such damping may be theoret-

inter-© 2007 Springer

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

in Physics and Engineering, 389–402

Abstract

Damping, fractional derivative, disorder, Galerkin, finite element

unknown to engineering audiences (this discussion may be found in [2]) Wemay be found in the physics literature (e.g [1]) but which seems largely

389

Satwinder Jit Singh and Anindya Chatterjee

approximations can be developed for the fractional derivative term, so that

be accurately approximated by finite dimensional systems without memory.Otherwise-motivated finite dimensional approximations have been obtainedaccessible to some audiences Results of finite element formulations based onthis Galerkin projection will also be presented The approximations developedhave approximately uniform and small error over a broad and user-specifiedfrequency range Our basic approach, though differently motivated, has strongsimilarities with an approximation scheme developed in [5] That scheme hasrecently been critiqued [6], and some of that criticism (concerned with some

0

x(τ )(t − τ )αdτ ,where 0 < α < 1, and Γ represents the gamma function Observe that

1

Γ (1 − α)

ddt

and has initial conditions x(t) ≡ 0 for t ≤ 0, and if h(t) is an impulse at zero,

For simplicity, we consider an equation relevant to a “springpot”:

decay in time

Rubber molecules presumably cannot remember the past Linear modelsfor rubber should therefore involve linear differential equations with constantcoefficients Such systems have exponential decay in time Why the power law?

part to develop a Galerkin procedure Using this, accurate finite-dimensionalinfinite dimensional and memory-dependent fractionally damped systems can

before (e.g [3] and [4]), but we think our approach is new, direct, and more

By Eq (1), the strain in a sample obeying Eq (2) can have power law

short-time and high-frequency asymptotics) applies to our work as well We willdiscuss those asymptotic issues and their engineering relevance at the end

of this paper The latter part of this paper has material that may also be

Fig 1 One dimensional viscoelastic model.

Consider the model sketched in Fig 1 An elastic rod of length L has adistributed stiffness b(x) > 0 Its axial displacement is u(x, t) The internal

derivatives The free end of the rod is displaced, held for some time, andreleased Subsequent motion obeys

We will now discuss how sufficient complexity (randomness) in b and c canlead to power law decay

A solution for the above is sought in the form

residuals [8] Defining symmetric positive definite matrices B and C by

Let us study a random B Begin with A, an n×n matrix, with n large Let

is symmetric positive definite with probability one We will solve

Solution is done numerically using, for initial conditions, a random n × 1

are shown in Fig 2

k/n

λ k

n=250 n=400

Fig 3 Eigenvalues of B for n = 250 and 400.

The solutions, though they are sums of exponentials, decay on average like

eca

a straight line on a log-log scale A fitted line has slope −0.24 ≈ −1/4.

x against time 30 individual solutions (thin lines) as well as their RMS values (thic gray) Right: RMS value of norm(x) against time is

Left: norm(x) = x

k

The answer lies in the eigenvalues of B The spectra of random matricescomprise a subject in their own right Here, we use numerics to directly obtain

k = 1, 2, · · · , n, be its eigenvalues in increasing order Figure 3 shows

=

λk

nplotted against k/n

Superimposed are the same quantities for n = 400 The coincidence tween plots indicates a single underlying curve as n → ∞ That curve passesthrough the origin, and can be taken as linear if we restrict time to values

be-t ≫ O (1/n), by when solube-tion componenbe-ts from be-the large eigenvalues havedecayed to negligible values Then

eigenvectors of the latter, are taken as random, i.i.d., and with zero expectedvalue The variance is then (upon scaling the initial condition suitably)

2

k2

t/n Define ξ =

microstruc-k = 1

th element of x is of the form

coefficient system starting from rest Let us denote that linear system by theNow if we replace the forcing δ(t) in Eq (8) with some sufficientlywell-behaved function x˙(t), then the corresponding response r(t) of the same

have replaced an α -order derivative by the following operations:

Prompted by the above, consider the PDE (or ODE in t with a free meter

para-There is no approximation so far We have replaced one infinite sional system (fractional derivative) with another The advantage gained isthat we can now use a Galerkin projection to obtain a finite system of ODEs.

where the T superscript denotes matrix transpose

For the Galerkin projection, we assume that Eq (10) is satisfied by

the Galerkin procedure for Eq (10)

Substituting the approximation for u(ξ, t) in Eq (10), we define

Equation (12) constitute n ODEs, which can be written in the form

During numerical solution of (say) a second-order system including both

we can solve Eq (13) numerically to obtain the a Note that

is in fact c , the ith element of c in Eq (13) above It follows that

4 Finite Element Approximation

discretization of the interval [0, 1] into a given finite element mesh, the sponding points on the frequency axis are independent of α

ξ1/α.This affects the choice of our last element’s shape function Suppose we

β >α

1

2.The above is always satisfied if we take β = 1 (because 0 < α < 1), and

we take β = 1 (independent of α) in this paper

To perform the Galerkin projection, we use the “hat” functions defined asfollows (see Fig 4):

Notice that, for large values of ξ, Eq (14) becomes

all integrals involved in Eq (12) (i.e., in the Galerkin approximation cedure) are bounded if

where “logspace” is shorthand for n−1 points that are logarithmically equally

1 /α i

to get an (n − 1) × 1 array of nonuniformly spaced points in the interval (0,1);add two more nodes at 0 and 1; and get an (n+1)×1 array of nodal locations

We now come to an interesting point regarding the choice of mesh points

in the nonuniform finite element discretization While the map from ξ to η isα-dependent, the choice of mesh points can be made using

2 i

i

Fig 4 Hat-shape functions.

with no negative consequences (see Eq (15) with α = 1/2) The advantage

is that the frequency range of interest can be specified easily in this way

Now a Galerkin projection is performed after changing the integrationvariable to η, giving

In Fig 5, we present the comparisons in FRFs for α = 1/3, α = 1/2 and

α = 2/3 15 nonuniform finite elements were used The performance is verygood for all cases over a significant frequency range The percentage error inmagnitude and phase angle for α = 1/3, α = 1/2 and α = 2/3 are shown

in Fig 6 The errors are below 1% for more than seven orders of magnitude

of frequency Calculations for other values of α were also done, and similarresults were obtained (not presented here) Similarly, we have also verifiedthat taking more elements gives smaller errors over the same frequency range

5 Modeling Issues and Asymptotics

the discussion of [5] in [6].This unavoidable feature may, however, have low implications for engi-neering practice

Consider some real material whose experimentally observed damping

be-of course, also describe this behavior using a large number be-of (integer order)dashpot combinations may be difficult to estimate robustly in experiments,however, as explained below

It is observed in [7] that the Galerkin procedure gives very good imations to fractional order derivatives for many different choices of meshthe real material can be described by many different combinations of integer-order or classical spring-dashpot combinations; these combinations will do

approx-an experimentally indistinguishable job of capturing the experimental data,sical integer-order approach requires identification of many parameters that

which will always span only a finite-frequency range In this way, the

clas-for m = 1, 2, · · · , n Equation (17) constitute n ODEs, which can be written inthe form of Eq (13) On combining them with the ODE at hand, we get

an we get an initial value problem which can be solved numerically in O(t)

No matter how many elements we take in the finite element (FE) mesh, thematch in the frequency response function (FRF) will be good only over some

points In other words, the same approximately fractional-order behavior of

A Maple-8 worksheet to compute the matrices A , B, and c is available on [9].

10ư4 10ư2 100 102 10428

29 30 31 32

Frequency

(f)

15 Nonưuniform size elements

Fig 5 Magnitude and phase angle comparison in FRFs Plots (a) and (b): 15 nonuniform hat elements and α = 1/3 Plots (c) and (d): 15 nonuniform hat elements and α = 1/2 Plots (e) and (f): 15 nonuniform hat elements and α = 2/3.

cannot really be uniquely determined The parameter estimation problem istherefore not only bigger, but more ill-posed In contrast, a model involv-where data exists; and will also involve identification of fewer parameters in

a better-posed problem For this reason, description of damping should beparameter identification easier for any individual experimenter; but, more im-portantly, it allows different experimenters in different laboratories to obtainthe same parameter estimates, without which material behavior cannot bestandardized for widespread engineering use

ing fractional-order derivatives may match the data over the frequency range

done, wherever indicated, using such fractional-order derivatives This makes

Fig 6 Percentage errors in the magnitude and the phase angle for α = 1/3, α = 1/2 and α = 2/3.

identified and standardized, simulations using that model can use differentapproximation techniques; it matters little what the approximation scheme is,provided it is good enough The only issue for a given calculation is whetherthe final computed results are accurate enough

But what is accuracy?

For the numerical analyst, accuracy means correspondence with the

origi-be good over all frequencies and time scales that are important in the lation If the results are not reliable for some very high frequency, the analystnotes it, but uses the reliable part of the results anyway This is the same spirit

calcu-in which reentrant corners and cracks calcu-in elastic bodies are often modeled ing finite element codes: the technique is not invalidated simply because evenvery small finite elements cannot exactly capture the singularities Rather, acareful analyst keeps a watch on how far from the singularity one must gobefore the numerical results are reliable

us-For the engineer, in addition to the numerical issue, accuracy also meanscorrespondence with the behavior of the original real material we started with.Any difference between exact and approximate mathematical solutions, inbehavior regimes where there is no experimental data, are academic curiositieswithout practical implication in many cases

However, once a suitable model with fractional-order derivatives has been

nal and exact fractional-order derivative behavior The approximation should

Finally, if the engineer believes (as we propose early in the paper) that

an artifact of many complex internal dissipation mechanisms, each withoutmemory, then the very-low and very-high (outside the fitting range) frequencybehavior In other words, the asymptotic regime where the Galerkin approxi-mation fails to match the exact fractional derivative may also be the regimewhere the fractional order derivative fails to match the material behavior

6 Discussion

Many materials with complex microscopic dissipation mechanisms may

macro-any such material Numerical solution of differential equations that involvesuch terms by direct methods requires evaluation of an integral for every time

This is prohibitively large for large n With the Galerkin projection presentedhere (as also the similar method of [5]), the approximated numerical solu-tion can be computed in O(n) time, which is a big improvement The readermay also be interested in the approach of [10], which has O(n ln n) complex-ity, i.e., is almost as good as O(n); however, that approach is algorithmicallymore complicated, because it involves evaluating the integral (required for thefractional derivative) after breaking the interval (0, t) into a large number ofcontiguous intervals of exponentially varying size In contrast, the approachexcellent accuracy over user-specifiable frequency ranges, O(n) complexity,and a system of ODEs that can be tackled using routine methods and readilyavailable commercial software

Some final words of warning The present Galerkin-based approximationscheme, in addition to the asymptotic mismatches referred to by [6], is not fullyunderstood at this time What we have presented so far amount to numericalobservations, and formal studies of convergence may provide useful insights

in the future Moreover, there is as yet no consensus on which of the severalapproximation schemes for fractional derivatives (e.g., the present work aswell as [3] and [4]) work best, and by which criterion; or even what a goodcriterion for evaluating a discretization/approximation scheme should be

the fractional-order derivative behavior observed in experiments is actually

behavior of the material may actually not match the fitted fractional-order

scopically show fractional-order damping behavior Damping models that usesuch fractional-order terms may involve relatively fewer fitted parameters for

presented here, especially if extended to higher-order finite elements, can give

3 Oustaloup A, Levron F, Mathieu B, Nanot F (2000) IEEE Trans Circ Syst I:

Fundamental Theory and Applications 47(1):25–39

4 Chen Y, Vinagre BM, Podlubny I (2004) Nonlinear Dynamics 38:155–170

5 Yuan L, Agrawal OP (2002) J Vib Acoust 124:321–324

6 Schmidt A, Gaul L (2006) Mech Res Commun 33(1):99–107

7 Singh SJ, Chatterjee A (2005) Nonlinear Dynamics (in press)

8 Finlayson BA (1972) The Method of Weighted Residuals and Variational Principles

Academic Press, New York

9 http://www.geocities.com/dynamics_iisc/SystemMatrices.zip

10 Ford NJ, Simpson AC (2001) Numer Algorithms 26:333–346

References

1 Vlad MO, Schönfisch B, Mackey MC (1996) Phys Rev E 53(5):4703–4710

2 Chatterjee A (2005) J Sound Vib 284:1239–1245

AND EXPERIMENTAL IDENTIFICATION

OF VISCOELASTIC MECHANICAL SYSTEMS

equivalent damping ratio valid for fractional derivative models is introduced, making it possible to test their ability in reproducing experimentally obtained damping estimates A numerical procedure for the experimental identification of the parameters of the Fractional Zener rheological model is then presented and vibrations

DIEM, Department of Mechanics, University of Bologna, V iale del Risorgimento 2, 40136 mail.ing.unibo.it

DIEM, Department of Mechanics, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy;

mail.ing.unibo.it

applied to a high-density polyethylene (HDPE) beam in axial and flexural

relaxation behaviour to high-frequency vibrations, by means of a minimum

© 2007 Springer

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

in Physics and Engineering, 403–416

Giuseppe Catania and Silvio Sorrentino

Bologna, Italy; Tel: +39 051 2093447, Fax: +39 051 2093446, E-mail: giuseppe.catania@

Tel: +39 051 2093451, Fax: +39 051 2093446, E-mail: silvio.sorrentino@

2

means of a minimum number of parameters is first discussed in comparisonwith classical integer order derivative models A technique for evaluating an

403

In the present study some differential linear rheological models are

flexural vibrations Structural and hysteretic damping laws are not included in the analysis, since they lead to non-causal behaviour [2]

Classical integer order differential models are compared to fractional differential ones, which are considered to be very effective in describing the

fractional calculus to viscoelasticity yielding physically consistent stress-strain constitutive relations with a few parameters, good curve fitting properties and causal behaviour [7]

Since with fractional derivative models the evaluation of closed form expressions of an equivalent damping ratio n does not seem an easy task, a

the evaluation of time or frequency response from a known excitation can still be obtained from the equations of motion using standard tools such as modal

established, since the current methods do not seem to easily work with

complex stress-strain relationship parameters related to the material The for testing its accuracy, and then to experimental inertance data

2

In the present study the uniform, rectangular cross-section, straight axis HDPE

Average density 954 Kg×m-3

Young’s modulus 0.2 to 1.6 GPa

considered, discussing their effectiveness in solving the above-mentioned problem, in relation to a high-density polyethylene (HDPE) beam in axial and

different approach is proposed [8], based on the standard circle-fit technique [9] When using fractional derivative models the solution of direct problems, i.e.,

analysis [10, 11, 12], but regarding the inverse problem, i.e., the identification from measured input–output vibrations, no general technique has so far been

In the present study a frequency-domain method is thus proposed for the experimental identification of the fractional Zener model, also known as fractional standard linear solid [5], to compute the frequency-dependent procedure is first applied to numerically generated frequency-response functions

Selection of a Rheological Model

linear viscoelastic dynamic behaviour of mechanical structures made of poly- mers [3] Extensive literature exists on this topic [4, 5, 6], the application of

differential operators of non-integer order [1]

and Table 1 some HDPE material typical values [13]

beam shown in Fig 1 is considered, Table 2 showing its geometrical parameters

Section moment of inertia I zz = 1.1328×10-7 m4

Section moment of inertia I yy = 1.8125×10 -6 m 4

Total mass M = 2.346 Kg

According to data available in the literature, an appropriate model for the

HDPE beam should yield a creep compliance J(t) (response to the unit stress

G(t) (response to the unit strain step) reaching 5% of its initial value after

response functions), thus reproducing the experimentally found behaviour of the damping ratio n as a function of the natural angular frequencies n, as shown for example in Fig 2

step) reaching 95% of its final value after 100 ÷ 500s and a relaxation modulus

10 ÷ 50s [13] On the other hand, the same model should accurately fit the responses of the system under analysis (in the case considered herein, frequency-

Subsequently, several different integer order and non-integer order derivative

rheological models, depicted in Fig 3, are considered and compared, discussing

their ability to satisfy the above mentioned requirements

2.1 Integer order derivative models

(Fig 3a), whose constitutive equation is:

c Series of 2 Kelvin-Voigt

d Fractional Kelvin-Voigt

e Fractional Zener

E1

E2C

b Zener

Eq (3) is incompatible with experimental results like those shown in Fig 2,

9for the HDPE static Young’s modulus and n = 0.05 at a frequency of 200 Hz,

The simplest real, causal, and linear viscoelastic model is the Kelvin–Voigt

symbols.)

due to flexural vibrations of free-free HDPE beams Assuming E = 1.5 10 N/m