Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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98 Y. Huang and X.W. Tangpong

Fig. 9.2 Force-deﬂection

relation of a slip element

k/N

A−A

Fig. 9.3 Distribution

function '

••• •••

CNT

Matrix

Fig. 9.4 Frictional contact model of a CNT and polymer matrix

where '.f



/df



is the fraction of elements satisfying f



 f



 f



C df



.The

force developed on other parts of the hysteresis loop is derived in a similar manner.

The second term in (9.2) vanishes for large x, and the total slip force becomes



'.f



/df



: (9.3)

The distribution function can be prescribed in analytical form [25, 28, 29], or it can

be determined from experimental data. The distribution function is closely related to

the topography of the contact interface, the contact compliance, and the coefﬁcient

of friction. In this chapter, the distribution function ' is expressed in an analytical

form as illustrated in Fig. 9.3, similar to the one considered in [25].

The frictional interaction between a single CNT and the polymer matrix is

described in Fig. 9.4. The CNT, matrix, and the friction interface between the two

9 A New Model for Evaluating Energy Dissipation 99

are each discretized into a large number of elements in the longitudinal direction.

The frictional interface between each pair of CNT/matrix elements is modeled by

Iwan’s distributed friction concept (depicted in Fig. 9.1). The equation of motion of

the ith CNT element is

D K

x

iC1

 K

x

C F

; (9.4)

where K

is the axial stiffness of one CNT element, x

is the length of the ith

element, and the friction force F

is determined by (9.2). The total energy dissipation

in one cycle of vibration is calculated by

E D

iD1

. Py

Px

/dt; (9.5)

and the total number of elements .N / is determined by convergence study of the

energy dissipation. The matrix’s motion is speciﬁed in a sinusoidal form as

y.x; t/ D y

sin



2n



cos.!t/; (9.6)

where y

is the amplitude, ! is the excitation frequency, n is the spatial wave

number and L is the CNT’s length. The friction force is, therefore, driving the CNT’s

motion, but not the matrix’s. In this study, the matrix’s motion is speciﬁed, and there-

fore, the stiffness between matrix elements (parameter K

) does not come into the

equations of motion. The matrix’s response to dynamic loads will be included in a

future work by the authors, and in that case, the friction force will also inﬂuence

the matrix’s motion. It should be noted that the stiffness between the CNT’s two

ends and the matrix is represented by a separate parameter K

.Thisstiffnesscomes

from the binding strength between the CNT’s ends and the matrix, and the binding

mechanism is yet to be discovered [30,31].

9.3 Results and Discussion

The following nondimensionalized parameters are considered in this chapter.



;D

;tD

;eD



;

ˇ D

f

;MD

; (9.7)

where m is the elemental mass of the CNT and K

is the tangential stiffness of

the interface. Parameter e characterizes the ratio of the CNT’s end binding stiffness

to its axial stiffness. Parameter ˇ is considered in the distribution function ' (see

Fig. 9.3). Parameter M is introduced here to describe the relative stiffness of the

CNT and the contact interface, and it is discussed in detail in Sect. 9.3.3.

100 Y. Huang and X.W. Tangpong

9.3.1 Excitation Frequency and Binding Strength

The energy dissipation per cycle is calculated for a range of excitation frequency for

different values of stiffness ratio e (Fig. 9.5). By varying the value of e,theaxial

stiffness of the CNT is ﬁxed; therefore, larger values of e correspond to stronger

interaction or stronger binding of the CNT’s ends with the matrix. As shown in

Fig. 9.5, higher energy dissipation takes place around the natural frequencies of

the CNT. Due to the symmetry of the model considered in this study, only the odd

modes appear. It can also be observed that as the value of e increases, the CNT’s

natural frequencies shift higher due to larger binding stiffness of the ends.

At the same excitation frequency, the energy dissipation decreases as the CNT

binds with the matrix stronger, or as the value of e increases (Fig. 9.6). As the inter-

action between the CNT’s ends and the matrix becomes stronger, the relative motion

at the interface has smaller amplitude under the same excitation, and therefore, less

energy dissipation takes place.

Fig. 9.5 Frequency response

of energy dissipation per

cycle; ˇ D 0:8; t D 0:2,

M D 200; y



D 0:05

0 1 2 3

Frequency ratio,

Energy dissipation per cycle, E*

e = 0.1

e = 0.5

0.5

Fig. 9.6 Energy dissipation

as a function of CNT’s

binding stiffness with the

matrix; ˇ D 0:8; t D 0:2,

M D 200; y



D 0:05

0 0.1 0.2 0.3 0.4 0.5

0.03

0.032

0.034

0.036

0.038

Stiffness ratio, e

Energy dissipation per cycle, E*

=1.2

=0.3

=0.7

9 A New Model for Evaluating Energy Dissipation 101

9.3.2 Excitation’s Amplitude

Energy dissipation at the CNT–matrix interface is investigated as a function of

excitation’s amplitude (Fig. 9.7). When the matrix’s displacement has small ampli-

tude, the CNT–matrix interface experiences a combination of stick and slip motions

since the matrix’s displacement is not large enough to activate full slip at the in-

terface. The energy dissipation in the low excitation amplitude range, therefore,

exhibits a nonlinear relationship as indicated in Fig. 9.7. As the matrix’s ampli-

tude increases, the relative motion at the interface gradually progresses to full slip

and energy dissipation changes linearly with respect to the excitation’s amplitude.

Similar observation was made experimentally in [32], where energy dissipation in

the CNT-based composite material was found to change with the strain amplitude

of the composite nonlinearly in the low amplitude range, and then linearly as the

strain amplitude increases (data with circle markers). The theoretical predictions

presented in Fig. 9.8 [32], however, did not capture the nonlinear relationship in low

strain amplitude range.

9.3.3 Compliance Number M

In deﬁning the parameter M in (9.7), F

is the total slip force of the CNT–matrix

contact interface, and F

=L can be considered as the linear density of slip force

along the interface. The slip force F

is closely related to the surface topography

of the contact interface, the contact pressure, and the coefﬁcient of friction. For

one CNT, its stiffness K

is ﬁxed, and the parameter M can be viewed as a com-

pliance number that quantiﬁes how easily full slip motion can be activated at the

interface. The larger the compliance number is, the easier it is for full slip motion

to take place at the interface. Figure 9.9 depicts how energy dissipation changes

Fig. 9.7 Energy dissipation

as a function of excitation’s

amplitude; t D 0:1;  D 0:3,

e D 0:2; ˇ D 0:8; M D 200

0 0.05 0.1 0.15 0.2 0.25

0.1

0.2

0.3

0.4

0.5

Excitation's amplitude, y*

Energy dissipation per cycle, E*

102 Y. Huang and X.W. Tangpong

Fig. 9.8 Effect of shear strain amplitude on damping. Original ﬁgure used with permission of

Nature Publishing Group [32]

Fig. 9.9 The effect

of compliance number

to energy dissipation;

ˇ D 0:8; t D 0:2;  D 0:5,

e D 0:2; y



D 0:05

0 500 1000 1500 2000 2500

0.2

0.4

0.6

0.8

x 10

−4

Compliance number, M

Energy dissipation per cycle, E*

with the compliance number. An optimal value of M exists for maximum energy

dissipation. In the study shown in Fig. 9.9, the CNT’s stiffness K

is ﬁxed while

the slip force F

is varied. When the value of M is small, F

is large, and it is not

easy to activate slip at the interface; therefore, the energy dissipation is small. As

the value of M increases, a combination of stick and slip motions begin to emerge,

and the dissipation increases. At large values of M ,orwhenF

is small, the inter-

face experiences full slip motion; however, since the slip force has small magnitude,

the total dissipation is again not optimal. Similar study has been done to keep F

ﬁxed while changing the value of K

, and the resulting energy dissipation shows

9 A New Model for Evaluating Energy Dissipation 103

the same trend as depicted in Fig. 9.9 and an optimal value of M exists. In the light

of these studies, damping of the nanocomposite can be varied by choosing different

CNT ﬁllers (single walled, multiwalled, different aspect ratios, etc.) for the polymer

matrix. The value of F

for various nanoﬁllers will be investigated experimentally

in a later study.

9.4 Summary

A new vibration model has been developed to evaluate the energy dissipation at the

CNT–polymer matrix interface. This model takes into account spatially distributed

friction at the tube–matrix interface in a statistical sense, and it is capable of cap-

turing the nonlinear stick/slip phenomenon. Energy dissipation has been found to

have larger values around the CNT’s natural frequencies. At the same excitation fre-

quency, energy dissipation decreases as the binding of CNT’s ends with the matrix

becomes stronger. Strong end interaction of the CNT with the matrix constraints the

motion of the CNT, limits the relative motion, and therefore lowers the energy dis-

sipation. The model also predicts that the energy dissipation grows nonlinearly with

the increase of excitation’s amplitude in small amplitude range, and the relationship

becomes linear when the amplitude is large. Under low magnitude of excitation, the

motion at the CNT–matrix interface is a combination of stick and slip motions, and

under large excitation, full slip at the interface is activated. The model developed in

this work is capable of capturing such transition of nonlinear to linear relationship

of energy dissipation with excitation’s amplitude. Such results are consistent with

some experimental observations in the literature. A compliance number (M )isalso

introduced to quantify how easily full slip motion can be activated at the interface,

and an optimal value of M exists for maximum energy dissipation.

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Chapter 10

Nonlinear Response in a Rotor System

With a Coulomb Spline

C. Nataraj and Karthik Kappaganthu

Abstract This chapter deals with the nonlinear analysis of a system with two rigid

rotors connected by a spline coupling and mounted on isotropic bearings. The

coupling has Coulomb friction. The system is found to have three unstable ﬁxed

points and a limit cycle above a certain critical speed. The response of the system

at the limit cycle indicates transient chaos, which is usually an artifact of the crisis.

The crisis route to chaos has been analyzed and a strange attractor is found.

10.1 Introduction

Figure 10.1 illustrates a typical spline coupling used to drive a rotating shaft from a

motor or other prime mover. Under insufﬁcient lubrication conditions, such a cou-

pling has been found to exhibit Coulomb friction. We will hence call this spline a

“Coulomb spline.” Past analyses of such systems are very limited. Nataraj et al. [1]

discovered that such a system can exhibit limit cycles under certain conditions. In

fact, this was veriﬁed with a practical installation as well.

In this chapter, the nonlinear dynamic behavior of the system is studied further.

We will assume a rigid rotor model that simpliﬁes some of the dynamics; such a

model is valid for rotors operating below the ﬁrst critical speed.

First, the ﬁxed points of the system and their stability are analyzed, then the

possibility of creation or destruction of ﬁxed points with changes in the spin speed

is studied. A limit cycle is identiﬁed using analytical methods. This limit cycle exists

only above a certain critical spin speed; this is the bifurcation point in the system.

The numerical simulation of this limit cycle suggests the presence of transient

chaos. Transient chaos and its cause is explained in [2]. In a paper published later

by the same authors [3], they discuss the dependence of average lifetime of the

system on the parameter. They suggest that transient chaos is one of the effects of

crisis. They analyzed crisis and transient chaos in three different continuous systems

C. Nataraj (



)

Department of Mechanical Engineering, Villanova University, Villanova, PA, USA

e-mail: nataraj@villanova.edu

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

10,

 Springer Science+Business Media, LLC 2010

105

106 C. Nataraj and K. Kappaganthu

Fig. 10.1 Spline coupling

including the Lorenz system. Transient chaos has also been observed in many other

systems like Sine-Gordon system [4], Gear-Rattling model [5], Plasmas [6], and

experimentally in [7].

In the next section, the mathematical model of the system is described, and the

following section discusses the nonlinear phenomenon observed.

10.2 Mathematical Modeling

The system being considered consists of two rigid rotors coupled by a Coulomb

spline. A schematic of the system is shown in Fig. 10.2. The equations of motion of

the system as derived in [8]are

M Rq C .D  ˝G/Pq C Kq D F; (10.1)

where M , D, G,andK are the mass, damping, gyroscopic, and stiffness matrices

respectively, given by:

M D





D D





G D



0 I



10 Nonlinear Response in a Rotor System With a Coulomb Spline 107

l1 l2

Fig. 10.2 Schematic diagram of the system

K D





q D ŒV W 

is the vector of displacements of the point of interconnection along

y and z axes respectively.

The parameters of the system are:

 m: Equivalent mass

 I

: Equivalent polar moment of inertia

 c: External viscous damping coefﬁcient

 k

and k

: Equivalent stiffness in y and z directions respectively

 ˝: Spin speed of the rotor

The force acting due to the spline coupling is

F D

.1 C l



W  ˝V /

C .

V  ˝W /



1=2



V C ˝W

W  ˝V



(10.2)

T

sin 

(10.3)

where  is the coefﬁcient of friction l

are the lengths of the shaft, and 

is the

angular position of kth spline.

The equations should be nondimensionalized to identify key parameters.

The nondimensional form of the equation is