International Journal of Non-Linear Mechanics 46 (2011) 114–124

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International Journal of Non-Linear Mechanics

0020-74

doi:10.1

n Corr

E-m

tamavray De1 ht

journal homepage: www.elsevier.com/locate/nlm

Dynamic motion of a conical frustum over a rough horizontal plane

Ioannis Stefanou n,1, Ioannis Vardoulakis y, Anastasios Mavraganis

Department of Applied Mathematics and Physics, National Technical University of Athens, P.O. Box 15780, Athens, Greece

a r t i c l e i n f o

Article history:

Received 24 June 2009

Received in revised form

4 March 2010

Accepted 28 July 2010

Keywords:

Conical frustum

Dynamic motion

Wobbling

Rocking

Ancient classical columns

62/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijnonlinmec.2010.07.008

esponding author. Tel.: +30 697 7 212 890; f

ail addresses: istefanou@mechan.ntua.gr (I. St

@central.ntua.gr (A. Mavraganis).

ceased.

tp://geolab.mechan.ntua.gr.

a b s t r a c t

An analytical and numerical study of the dynamic motion of a conical frustum over a planar surface is

presented resulting to a non-linear system of ordinary differential equations. Wobbling and rocking

components of motion are discussed in detail concluding that, in general, the former component

dominates the latter. For small inclination angles an asymptotic approximation of the angular velocities

is possible, revealing the main characteristics of wobbling motion and its differences from rocking.

Connection is made of the analysis with the behavior of the ancient classical columns, whose three

dimensional dynamic response challenges the accuracy of the two dimensional models, usually applied

in practice. The consideration of such discrete-blocky systems can benefit from the present study,

through qualitative results and benchmarks for more complicated numerical methods, like the Distinct

Element Method.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The non-holonomic problem of a symmetric body by revolu-tion, rolling on a planar surface, was first formulated by Routh in1868 [1]. Since then, a significant number of papers appeared onthis subject, focusing mostly on the study of the motion of a thindisk on a horizontal plane. The elaboration of the problem of thethin disk is presented in most classical textbooks of Dynamics[2–5] providing to the readers a typical example of non-holonomic motion. Noticing the early works of Appell [6] in1900 (cf. also Korteweg [7]) and Gallop [8] in 1904, where analyticsolutions are given in terms of Gauss hypergeometric andLegendre functions, we pass to the corps of papers of the currentdecade. The papers of O’Reilly [9], Kuleshov [10], Paris and Zhang[11], Kessler and O’Reilly [12], Borisov et al. [13], Le Saux et al.[14] provide a deep insight to the dynamic behavior of the thindisk. Equally important for the present study are also the papersof Koh and Mustafa [15] and Batista [16], which discuss themotion of a disk of finite thickness on a planar surface. In thelatter papers the equations of motion of a cylindrical drum arederived and numerical simulations are performed.

In the present paper we deal with the case of a conical frustum,rolling on a rough horizontal surface. Using for the description ofmotion the Lagrange formulation, we distinguish between thewobbling and the rocking of the frustum and comment exten-sively on these components of motion. Stability analysis reveals

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efanou),

the pure three dimensional character of the motion, while furtherapproximations of the angular velocities under small inclinationangles are elaborated to examine the main characteristics of themotion of the frustum. Finally, an attempt is made to interpret thedynamical behavior of ancient classical columns considering themas conical frustums with slightly different radii.

2. Equations of motion of a conical frustum on a roughhorizontal plane

The formulation of the problem is based on the followingassumptions:

a.

The body is a homogeneous, rigid conical frustum. b. The contact with the horizontal plane is assumed punctual.

Notice that Kessler and O’Reilly [12] introduced a contactmoment for simulating a ‘flat’ contact. This additional complica-tion is not considered here, because rolling friction is disregarded.

c.

At any given time the body is in contact with its horizontal planarbase and only smooth transitions in time are considered.

2.1. Formulation of the system

The position of the body in the inertial frame O(XYZ) isdetermined by the coordinates of the contact point P(XP, YP) andby the Euler angles (j, y, c), where j is the precession angle, ythe inclination (nutation) angle and c the rotation about z-axis(Fig. 2). For y¼0 the frustum comes into contact with thehorizontal plane by whole base. Hence, the motion is restricted inthe interval yA 0,p=2

� �.

C

R

r

a

h

r

R

C

l

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124 115

If the frustum rolls without sliding then the velocity of thecontact point P(XP, YP) is:

VP ¼�R _c ð1Þ

where R is the radius of the base of the drum. Applying theFrobenius criterion, it may be easily proven that constraint (1) isnon-holonomic.

Ground accelerations can also be considered by introducingthe additional inertia terms

_X PþR _c cosj¼ agrX_ugrðtÞ, _Y PþR _c sinj¼ agr

Y_ugrðtÞ ð2Þ

where agrX , agr

Y are two scalar quantities, constant in time, thatexpress the direction of the ground acceleration €ugrðtÞ.

Given the frictional law of the materials in contact (eg. Coulombfriction), the estimation of the sliding velocity is feasible bycombining the velocity of the point P, regarded as a point of thefrustum, with the frictional forces developed at the contact.However, this formulation extends the limits and the scope of thepresent paper and it will not be pursued further hereafter.Numerical and parametric studies that include sliding are, of course,important for practical applications, as they supply quantitativeinformation to be used for design purposes, but add little to thequalitative understanding of the basic dynamics of the system.

The angular velocity components of the body relative to C(x Zz) are

ox ¼_y

oZ ¼ _j siny

oz ¼ _j cosyþ _c ð3Þ

whilst the components relative to the central principal axessystem Cðx y zÞ are

ox ¼_y coscþ _j sinysinc

oy ¼�_y sincþ _j sinycosc

oz ¼ _j cosyþ _c ð4Þ

Notice the coincidence of oz and oz, because z¼ z.The coordinates of the center mass of the conical frustum in

O(X Y Z) are given in terms of the contact point coordinatesP(XP,YP), by the following relations:

Xc ¼ XP�‘cosðaþyÞsinjYc ¼ YPþ‘cosðaþyÞcosjZc ¼ ‘sinðaþyÞ ð5Þ

where ‘¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2þð1=4Þk2

1h2q

, tan a¼(k1h/2R), k1 ¼ 2ðzcm=hÞ,

zcm ¼ ðð3b2þ2bþ1Þ=ðb2

þbþ1ÞÞðh=4Þ, b¼r/R,r and R are, respec-tively, the radii of the upper and lower rim of the conical frustumand h is its height (Fig. 1).

The velocity VC of the center of the mass of the frustum yields

_X C ¼_X Pþ‘ _y sinðaþyÞsinj�‘ _j cosðaþyÞcosj

_Y C ¼_Y P�‘ _y sinðaþyÞcosj�‘ _j cosðaþyÞsinj

_ZC ¼ ‘ _y cosðaþyÞ ð6Þ

2.2. Dynamic equations of motion

The kinetic and the potential energy of the drum are

T ¼1

2mV2

Cþ1

2xTICx

V ¼mgzC ð7Þ

where IC is the inertia tensor relative to Cðx y zÞ

IC ¼ IP�

1

4k1mh2þmR2 0 0

01

4k1mh2þmR2 0

0 0 mR2

0BBBB@

1CCCCA,

IP ¼

IPx

0 0

0 IPy

0

0 0 IPz

0BB@

1CCA

IPz ¼

1

2k3þ2ð ÞmR2, IP

x ¼ IPy ¼

1

4k3þ4ð ÞmR2þ

1

12k2þ3k2

1

� �mh2

k2 ¼9

20

b4þ4b3

þ10b2þ4bþ1

b2þbþ1

� �2and k3 ¼

3

5

b4þb3þb2þbþ1

b2þbþ1

The inertia tensor IP expresses the inertia moments of the bodyat the contact point P. For cylindrical drums it holdsk1¼k2¼k3¼b¼1.

Introducing the generalized coordinates q1¼j, q2¼y, q3¼c,q4¼XP and q5¼YP the general form of the Lagrange equations fornon-holonomic systems are

d

dt

@ðT�VÞ

@ _qi

�@ðT�VÞ

@qi�X2

j ¼ 1

ljBji ¼ 0 ð8Þ

With li we denote the Lagrange multipliers, while

Bji ¼@ non-holonomic constraint equation 0j0� �

@ _qi

, resulting to :

fBjig ¼1 0 0 0 Rcosf0 1 0 0 Rsinf

!:

For convenience we introduce the following dimensionlessquantities:

h¼h

R, t¼ t

ffiffiffig

R

r, X ¼

XP

R, Y ¼

YP

R, ugr ¼

ugr

R, ð:Þu�

dð:Þ

dt,

Ik ¼Ik

mR2, ok ¼ok

ffiffiffiR

g

s, T ¼

T

mgR, V ¼

V

mgR, E¼

E

mgR

ð9Þ

where g is the acceleration of gravity and E the total energy of thesystem.

According to Eq. (8), the equations of motion are written inmatrix notation

AUU¼ B ð10Þ

Fig. 1. The conical frustum: 3D and 2D view.

Fig. 3. Mobilized friction coefficient and normalized inclination angle of the

column presented in Fig. 4. The energy dissipated during the short time intervals

of sliding (peaks of the mobilized friction coefficient) generally results to a

practically smooth collisionless contact.

C

(� – �)

O

XY

Z

x

y�

�

�

�

P(XP, YP)

�, z

Fig. 2. Coordinate systems and Euler angles. O(X, Y, Z) is the inertial frame,

P(XP, YP) the contact point with the horizontal plane, y the inclination angle, j the

precession angle and c the rotation about z-axis.

2 In the experiment of a ‘‘wobbling’’ coin on a table this is sensed by a high

frequency noise that is produced towards the last phase of the motion.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124116

whereU¼ j00,y00,c00,l1,l2,x00,y00

� �and A, B are matrices given in

Appendix A.The determinant of matrix A is

detðAÞ ¼1

4IP

y 4 IP

y�1� �

IP

z�h2k2

1

sin2y

¼1

2883k2

1þk2

� �h

2þ3 k3þ4ð Þ

3k3k21þk2 k3þ2ð Þ

� �h

2þ3 k3þ2ð Þ

sin2y40

ð11Þ

for y40.For y¼0, det(A)¼0 and the system of Eq. (10) is singular. In

the limit of y-0+, a smooth full contact of the drum withthe horizontal plane is reached. This collision is not trivial in thesense that the impact is taking place between surfaces in thethree dimensional space. Consequently, the impact involvesimpulse reaction forces and torques, including torsion. Thehypotheses usually made for the dynamics and the contact pointjust after the impact [9,17,18] permit the application of theangular momentum principle and the calculation of a uniquerestitution coefficient. However, these hypotheses are notstraightforward, because of the spin of the body at the instantof the impact, which may influence the dynamics of the collisioneven in the frictionless limit. Besides, as it is will be shown in thenext sections, impact happens only when a special initialcondition is satisfied.

Eq. (10) describe the three dimensional motion of a frustumover a shaking rough horizontal plane. These equations are non-linear and for ya0 they become

j00 ¼ g11ðyÞjuþg13ðyÞcu� �

yuþg14ðyÞu00 gr

y00 ¼ g21ðyÞjuþg23ðyÞcu� �

juþg22ðyÞþg24ðyÞu00 gr

c00 ¼ g31ðyÞjuþg33ðyÞcu� �

yuþg34ðyÞu00 gr

X00 ¼jucusinj�c00 cosjþagr1 u00 gr

Y 00 ¼ �jucucosj�c00 sinjþagr2 u00 gr

ð12Þ

where

g11 ¼2 I

P

z�1� �

2coty IP

z�hk1

� �4 I

P

y�1� �

IP

z�h2k2

1

�2coty; g13 ¼4cscy I

P

z�1� �

IP

z

4 IP

y�1� �

IP

z�h2k2

1

g21 ¼IP

y�IP

z�1� �

sin2y�hk1 cos2y

2IP

y

; g22 ¼hk1 siny�2cosy

2IP

y

;

g23 ¼�2 I

P

z þ1� �

sinyþ hk1 cosy

2IP

y

g31 ¼

4hk1 cosy�4 IP

y�IP

z�1� �

sinyh i

IP

z�1� �

þ2 4IP

y IP

z�2IP2

z �2IP

z�h2k2

1

�cscy

4 IP

y�1� �

IP

z�h2k2

1

;

g33 ¼�2 I

P

z�1� �

2IP

z coty�hk1

� �4 I

P

y�1� �

IP

z�h2k2

1

;

g14 ¼2hk1 I

P

z�1� �

cscy

h2k2

1�4 IP

y�1� �

IP

z

agr1 cosjþagr

2 sinj� �

g24 ¼2sinyþ hk1 cosy

2IP

y

�agr1 sinjþagr

2 cosj� �

g34 ¼4 I

P

y�1� �

þ2 IP

z�1� �

coty�h2k2

1

4 IP

y�1� �

IP

z�h2k2

1

agr1 cosjþagr

2 sinj� �

We observe that the cot y and csc y terms of Eq. (12) result intoa singularity for j00 and c00 that is carried over y00. The unphysicalunlimited angular accelerations predicted by this model for y-0+

imply that the assumption we made here of frictionless contact istoo strong for this limit, since ‘‘large’’ angular accelerations wouldlead to strong tangential reaction forces. These forces would inturn violate the imposed non-sliding constraint. In Fig. 3 we plot:(a) the normalized inclination angle and (b) the mobilized frictioncoefficient as functions of time. The mobilized friction coefficientis defined here as the ratio of the magnitude of the total tangentialforce over the magnitude of the normal force developed at thecontact point. From this figure follows that in certain occasions ashort duration slip would occur leading to energy losses, becauseof the increased mobilized friction coefficient. The energydissipated in these short time intervals of sliding should lead toa high-frequency stick and slip mechanism2 depending on the

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124 117

assumed friction law [19], that will result eventually to apractically smooth collisionless contact. This remark enforcesthe conclusion that in the problem at hand, and in general, impactis unattainable.

For zero ground accelerations the system is autonomous andthe Jacobi’s integral exists and reduces to the total energy of thesystem [20]

E¼1

2IP

yyu2þ

1

2IP

z cos2yþsiny IP

y�1� �

siny�hk1 cosyh ih i

ju2þ

1

2IP

zcu2

þ IP

z cosy�1

2hk1 siny

�jucuþ

1

2e

¼1

2IP

yo2xþ

1

2IP

y�1� �

o2Zþ

1

2IP

zo2z�

1

2hk1oZozþ

1

2e ð13Þ

In the general case of non-autonomous systems, as the onedescribed above, the equations of motion can only be numericallyintegrated. The numerical scheme that was applied here wascompared and validated by extending the analytical solutionproposed for cylindrical drums in [17] to conical frustums. Thevalidation is presented in Appendix B.

4m

0.4

0.5

Fig. 4. Geometry of the column considered for the numerical examples.

3. Three dimensional character of the motion

The initial conditions define the trajectory of the frustum onthe horizontal plane. For certain conditions this trajectory may bea circle, a line or even a point. In the particular case, where thetrace of the contact point is stationary and cu0 ¼fu0 ¼ 0, themotion degenerates into a two-dimensional rocking in the verticalplane. In this case the behavior of the system is described by thefollowing equation only:

y00 ¼hk1 siny�2cosy

2IP

y

ð14Þ

Rocking is described by Eq. (14) with the additional assump-tion that when the block turns back to the vertical position (y¼0)it collides with the horizontal plane (i.e. it does not cross-over).Then, the contact point changes abruptly to the other side of thecircular base, the angular momentum is preserved and therotation continues about the new contact point until a maximuminclination angle is reached. Therefore, rocking could be seen as aparticular case of the general three dimensional motion (Eq. (12))between the time interval of two full contacts with the horizontalplane. This consideration makes the stability analysis of rockingmeaningful for our study.

3.1. Linear stability analysis of rocking

Introducing the following small perturbations of the depen-dent variables:

j-j0þ ~jðtÞ, y-yðtÞþ ~yðtÞ, c-c0þ~cðtÞ ð15Þ

and by linearizing Eqs. (12), we obtain

~j00 ¼ g11~juþg13

~cuh i

yu

~y 00 ¼ g22~y

~c 00 ¼ g31~juþg33

~cuh i

yu ð16Þ

Setting f0

¼x1 and c0

¼x3 Eq. (16)a and c become in matrixform

xu1

xu3

!¼ yu

g11 g13

g31 g33

!U

x1

x3

!ð17Þ

Let r1,2 be the eigenvalues of system (17). Then

r1r2 ¼4 I

P

z�1� �

hk1 cotyþ IP

z þ1� �

4 IP

y�1� �

IP

z�h2k2

1

yu2

¼�6k3 2hk1 cotyþk3þ4

� �3k3k2

1þk2 k3þ2ð Þ� �

h2þ3k3 k3þ2ð Þ

yu2o0 ð18Þ

Therefore, the system has two real distinct eigenvalues withone of those positive, which means that in any interval betweentwo collisions of the drum with the horizontal plane, small out-of-plane perturbations of the motion grow exponentially in time(saddle point). Consequently, rocking is an unconditionallyunstable motion, independently of the slenderness, h, and theconicity, b, of the frustum. As a result, the three dimensionalcharacter of the problem prevails. This is not an astonishing resultbecause rocking could be seen as an inverted pendulum motion.Pendulum motions are also extremely sensitive to out-of-planeperturbations as this is well-known. A numerical exampleconfirms the above result from linear stability analysis; considerthe column depicted in Fig. 4. The column is a conical frustumwith h¼4m, r¼0.4m, R¼0.5m, b¼0.8 and h¼ 8. The equations ofmotion are integrated numerically for the following three cases ofinitial conditions:

IC1 : j0 ¼ 0, y0 ¼ 13, c0 ¼ 0, ju0 ¼ 0, yu0 ¼ 0, cu0 ¼ 0 ð19Þ

IC2 : j0 ¼ 0, y0 ¼ 13, c0 ¼ 0, ju0 ¼ 0:01, yu0 ¼ 0, cu0 ¼ 0 ð20Þ

IC3 : j0 ¼ 0, y0 ¼ 13, c0 ¼ 0, ju0 ¼ 0:1, yu0 ¼ 0, cu0 ¼ 0 ð21Þ

Initial condition IC1 refers to rocking. Notice that even forsmall values of j00 (IC2 and IC3) the contact point changesposition (Fig. 5) and it is not limited in the plane of the twodimensional rocking motion, where XðtÞ ¼ 0 (IC1).

Fig. 5. Coordinate, X, of the contact point for initial conditions IC1, IC2 and IC3.

Notice that for rocking (IC1) the XP coordinate of the drum remains constant

(zero), while for IC2 and IC3 is not. This means that the contact point is not limited

in the plane of the two dimensional rocking motion.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124118

3.2. Wobbling motion

An animation of the above three dimensional motion revealsan interesting kind of motion that it is evolving for smallinclination angles of the drum. In this paragraph, we distinguishthis special kind of motion, which is not limited in the verticalplane of rocking, but it is rather evolving out-of-plane, in the threedimensional space, and is observed when the drum rolls on itsedge under small inclination angles (0oy51).We call this threedimensional motion for small inclination angles wobbling.Another example of this particular motion could also be a coinwobbling on a desk. The main difference of rocking from wobblingis that the pole of rotation of the body is not kept constant in timeand that the motion of the frustum needs three angles to bedescribed, i.e. the Euler angles, instead of one. Assuming smallinclination angles, y51, Eq. (12) are linearized as follows:

j00 ¼ f11ðyÞjuþ f13ðyÞcu� �

yuþ f14ðyÞu00 grþOðyÞ

y00 ¼ f21ðyÞjuþ f23ðyÞcu� �

juþ f22ðyÞþ f24ðyÞu00 grþOðy2Þ

c00 ¼ f31ðyÞjuþ f33ðyÞcu� �

yuþ f34ðyÞu00 grþOðyÞ

X00 ¼jucusinj�c00 cosjþagr1 u00 gr

Y 00 ¼ �jucucosj�c00 sinjþagr2 u00 gr

ð22Þ

where

f11 ¼�1

y

4 IP

z�1� �

IP

z

h2k2

1�4 IP

y�1� �

IP

z

þ2

264

375þ 2hk1 I

P

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

;

f13 ¼�1

y

4IP

z IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

f21 ¼2y I

P

y�IP

z�1� �

�hk1

2IP

y

; f22 ¼ yhk1

2IP

y

�1

IP

y

; f23 ¼�2yI

P

z þ hk1

2IP

y

f31 ¼ 21

y

2 �2IP

yþ IP

z þ1� �

IP

z þ h2k2

1

h2k2

1�4 IP

y�1� �

IP

z

�4hk1 I

P

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

;

f33 ¼1

y

4IP

z IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

�2hk1 I

P

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

f14 ¼�1

y

2hk1 IP

z�1� �

4 IP

y�1� �

IP

z�h2k2

1

agr1 cosjþagr

2 sinj� �

f24 ¼ y1

IP

y

þhk1

2IP

y

0@

1A �agr

1 sinjþagr2 cosj

� �

f34 ¼4I

P

y�h2k2

1�4

4 IP

y�1� �

IP

z�h2k2

1

þ1

y

2hk1 IP

z�1� �

4 IP

y�1� �

IP

z�h2k2

1

264

375 agr

1 cosjþagr2 sinj

� �

It is worth mentioning that in the above linearization theapproximation is not uniform, because if we truncate the O(y)terms in Eq. (22)b an equilibrium point is lost.

4. Free wobbling

For zero ground accelerations we derive the rotationalvelocities of wobbling as functions of the inclination angle y(see Appendix C). These velocities are expressed as follows:

ju¼�c11

peð1=sÞy 1

y�

s

2

1

y2

�c2eð�1=sÞy 1

y2

cu¼ c11

peð1=sÞy �pþ

1

y�

s

2

1

y2

þc2eð�1=sÞy 1

y2ð23Þ

where p¼ ð2hk1=IP

z Þ, s¼ ðð4ðIP

y�1ÞIP

z�h2k2

1Þ=ð2hk1ðIP

z�1ÞÞÞ and c1

and c2 are constants specified by the initial conditions

c1 ¼�eð�y0=sÞ ju0þcu0� �

c2 ¼ eðy0=sÞ 2y0�s

pju0þcu0� �

�ju0y20

ð24Þ

Adding Eq. (23)a and (23)b yields to

juþcu¼ ju0þcu0� �

eðy�y0Þ=sÞ ð25Þ

4.1. Approximations of rotational velocities

The behavior of the drum for yoy051 is approximated byexpanding Eq. (23) in power series. Neglecting terms of O(y/y0)¼O(e1/2) and Oðy3

0=y2Þ ¼Oðe1=2Þ that is for y0¼O(e3/2), y¼O(e2)

and 0oe51, we obtain

ju��2

psju0þcu0� �

þðpsþ2Þju0þ2cu0

ps

y0

y

�2

¼Oðe�1Þ

cu�psþ2

psju0þcu0� �

�ðpsþ2Þju0þ2cu0

ps

y0

y

�2

¼Oðe�1Þ ð26Þ

In Eq. (26) the second term on the right hand side is dominantand of O(e�1) whereas the first term is a constant independent ofe. Eq. (26) can be reduced to the approximation suggested bySrinivasan and Ruina [21] if one neglects the constant term andassumes that f00 and c00 are quite small. This approximation isindeed satisfactory in the near collision state, but fails toreproduce the global response of the wobbling motion. Thenumerical integration of the equations of motion corroboratesthe current asymptotic approximations for ju0,cu0a0 and thecomparison of the asymptotic with the fully numerical solution is

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124 119

presented in Figs. 6–8. For the numerical comparison, the samegeometrical parameters were used with Section 2. The initialvalues were: f0¼0, y0¼(p/18)E0.17, c0¼0, fu0 ¼ 0:1, y00¼0,c00¼�0.2; notice that the range of y is within the limits of theabove approximation.

Fig. 7. On the left j0 versus y is plotted for (a) the numerical solution (solid line) and (b

approximation is presented, which is less than 6%.

Fig. 8. On the left c0 versus y is plotted for (a) the numerical solution (solid line) and (b

approximation is presented, which is less than 6%.

Fig. 6. Precession velocity, j0 , and inclination y versus t. The precession velocity

obtains large values for small inclination angles.

Both angular velocitiesj0 and c0 depend on the slenderness h, andthe conicity b of the frustum because of the term ps�1 in Eq. (26). InFig. 9 we present the contour plot of psðh,bÞ. From the contours weobserve that ps�1, and consequently f0 and c0, increase fordecreasing h and increasing b with bo0.5. For b40.5, the termps�1 is practically not affected by b. This observation would supportthe approximation of typical classical column drums (bE0.8) bycylinders. However, this statement must be checked with numericalanalysis of the multi-drum system response, which is out of the scopeof the present paper.

Eq. (26), show that j0 and c0, are quadratically dependenton the ratio y0/y. Yet, their sum is practically constant (seeEq. (25)). Combining Eqs. (3) and (25), the spin of the drum isapproximated by

oz �juþcu�ju0þcu0 ð27Þ

From the above approximation we infer that forsmall inclination angles the spin of the frustum, oz, isdefined by the initial conditions, it is independent of thegeometrical parameters and it is practically constant in time. InFig. 10 we plot oz(t) for: f0¼0, y0¼p/18, c0¼0,f0 ¼ 0:1, yu0 ¼ 0, cu0 ¼ 0:5.

) the approximation of Eq. (26) (dashed line). On the right the relative error of the

) the approximation of Eq. (26) (dashed line). On the right the relative error of the

Fig. 10. Spin, oz, of the body (solid line). It holds oz �ju0þcu0.

Fig. 11. Oscillation of the conical frustum between ymax¼y0¼p/18 and ymin

(y00¼0).Fig. 9. Contour plot ofpsðh,bÞ as function of the slenderness h and the conicity b of

the conical frustum.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124120

4.2. Impact with the horizontal plane

In the following we derive the condition for impact to takeplace. As mentioned above, at impact the drum is spinning, whichshows that the dynamics of the collision are complex since theyinvolve impulse reaction forces and torques, including torsion.This complication is pursued further here. Eq. (23) shows that theprecession velocity and the spin increase and become infinite fory-0+. Introducing Eq. (23) into the total energy equation of thesystem, Eq. (13), we obtain

E¼IP

y�1� �

2p2

sc1�pc2

y

� �2

þOsc1�pc2

y

� �ð28Þ

However, the total energy must be finite in the limit y-0+.This implies that impact occurs only when the following conditionis met:

sc1 ¼ pc2 ð29Þ

Using Eqs. (24) the impact condition becomes

cu0ju0

�imp

¼py2

0

eð2y0=sÞ�1� �

sþ2y0�1¼

ps

2�1þ

p

3y0þ

p

18sy2

0þO y30

� �ð30Þ

First we note that impact is not affected by the angularvelocity y00. By the same token, we note that the choice of theinitial inclination angle, y0, plays a subordinate role in thecriterion for impact and in view of Eq. (27), we get fromEq. (30) that impact takes place if the (practically constant) spinof the frustum is

oz �ps

2ju0 ð31Þ

Eq. (31) shows that the impact condition is not affected by theconicity of the frustum, for b40.5 (Fig. 9). If the initial conditionssatisfy exactly the impact condition, Eq. (29), the energy equation(13) can be solved for the inclination velocity at the instant of theimpact, yielding to a non-zero value that is compatible withimpact

yuimp ¼�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E0�I

P

z c21þ hk1

IP

y

vuuut a0 ð32Þ

On the contrary, if for given initial conditions the body doesnot overturn and the aforementioned impact condition is not met,the body oscillates between a maximum ymax and a minimuminclination angle ymin (Fig. 11). The values of ymax and ymin can bespecified by introducing Eq. (23) into the total energy of thesystem, Eq. (13), leading to rather complicated algebraicexpressions for ymax,min.

The fact that in the general case of initial conditions the conicaldrum does not collide with the horizontal plane is one morefundamental difference of the dynamics of rocking (2D) fromwobbling (3D).

Fig. 12. Mechanical damage of a column of Parthenon (see also Bouras et al. [24]).

The relative displacements and rotations of the column drums corroborate the

wobbling motion and the three dimensional dynamic response of these articulated

systems.

3 All the analytical calculations in the present paper have been performed

with the symbolic language mathematical package Mathematica. The Mathema-

tica files are available to the reader upon request to the corresponding author.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124 121

5. Application to ancient classical columns

The last decades an increased interest arouse in the dynamicresponse of ancient classic and Hellenistic temples. In certaincases this kind of structures may undergo intense earthquakeactions without collapsing. A particular element of these monu-ments is the multi-drum column. Each column is made bysometimes astoundingly fitted stone drums, which are placedwithout mortar on top of each other on a perfect fit [22]. From themechanics point of view, the dynamic response of the classicalcolumns, seen as rigid-body assemblies, is definitely non-linearand it involves rocking, sliding and wobbling of the drums(Fig. 12). This response has also little in common with thedynamic response of modern structures, which exhibit ‘tensegrity’(tension+integrity) in the sense that they can bear tensilestresses. The stability and resistance of modern columns to axialand lateral loads and moments is owed to the development ofinternal tensile forces, while the stability and resistance ofclassical columns is owed only to their self-weight [23] andgeometric characteristics. This fundamental difference makesinapplicable most of the available structural theories andclassical computational tools.

In the frame of a simplified mechanical model, bothmonolithic classical columns and single column drums can beseen as rigid conical frustums with slightly different radii. Thevertical flutes that are often sculptured on the faces of thecolumns are of small extent and can be ignored in a firstapproach. Under these assumptions, the results that wereexposed in the previous paragraphs can be used for thequalitative understanding of the dynamic behavior of a classicalcolumn.

According to Section 3, wobbling dominates rocking andtherefore the drums of classical columns wobble under dynamicexcitations. Consequently, the two dimensional analyses oftenperformed [25–28] should fail to capture the response of thesearticulated systems, as the out-of-plane motion cannot beignored. In parallel, the mechanism of energy dissipation atwobbling is different from rocking. In particular, the dissipation ofenergy at wobbling is attributed to the frictional forces that arebeing developed at the joints during the stick and slip motion ofthe drums, while at rocking the dissipation is attributed to the

impact with the horizontal plane. But even if impact is realized atthe last moments of the wobbling motion, the spin of the drumsinvolves frictional torsion at the collision, leading to an additionalfactor of energy dissipation. Fig. 12, corroborates the abovearguments, showing the relative rotations of the drums of acolumn of Parthenon.

The presented dynamic model can be used directly tostudy the dynamic behavior of classical monolithic columns3.However, for the study of multi-drum columns and colonnades,more sophisticated numerical tools have to be applied. TheDistinct Element Method (DEM) seems to be a promising choicefor the study of such systems and it has already been used for themodeling of multi-drum columns. The results were quitesatisfactory [29]. However, the inherent theoretical assumptionsof the above three dimensional numerical method that concernthe contact laws, the contact detection and the integration of theequations of motion, question its reliability and accuracy for themodeling of such systems. The authors are not aware of anycomparison of the above mentioned numerical method with thephysical model proposed here (Eqs. (12), (19)). However, thiscomparison exceeds the scope of the current paper and it shouldbe followed in a different work.

6. Conclusions

The objective of the present paper was the study of thewobbling motion of a conical frustum on a rough horizontal plane.For the analysis, the equations of motion were derived usingLagrangian formulation. The system is finally described by a set ofnon-linear equations that cover both the in-plane (rocking) andout-of-plane (wobbling) motion of the conical frustum. Linearstability analysis shows that rocking is unconditionally unstableand that wobbling is the dominant motion for frustums. In thegeneral case of initial conditions, the frustum oscillates between amaximum and a minimum inclination angle. Impact takes placeunder certain initial conditions, while at the instant of the impactthe spin of the frustum is not zero resulting to impact reactionforces and torques, including torsion. Practically, for smallinclination angles, the spin of the body remains constant duringthe motion. On the contrary, large angular velocities appear as theinclination angle takes small values. The energy dissipation of thesystem is attributed to this instantaneous increase of the angularvelocities, leading to a stick-slip motion of the drum, dependingon the friction law. The aforementioned fundamental character-istics of the dynamic response of frustums enable us to get aninsight to the dynamic behavior of classical multi-drum columns.In the particular case of single monolithic columns the proposedapproach is directly applicable and may be treated as benchmarkfor other general purpose numerical tools, often used in the studyof blocky systems.

Acknowledgements

The authors Ioannis Stefanou and Professor Anastasios Mavra-ganis would like to acknowledge the partial financial support ofthe present research by the Basic Research Program of NationalTechnical University of Athens (NTUA), PEVE 2007.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124122

Appendix A

A.1. Matrices of the equations of motion A and B

A¼

IP

z cos2yþsiny IP

y siny�siny�cosyhk1

� �0 cosy I

P

z�1� �

0 01

2f cosj 1

2f sinj

0 IP

y 0 0 01

2esinj �

1

2ecosj

cosy IP

z�1� �

0 IP

z�1 �cosj �sinj 0 0

1

2f cosj 1

2esinj 0 �1 0 1 0

1

2f sinj �

1

2ecosj 0 0 �1 0 1

0 0 cosj 0 0 1 0

0 0 sinj 0 0 0 1

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

ðA:1Þ

B¼

� IP

y�IP

z�1� �

sin2yyujuþ hk1 cos2yyujuþsiny IzI�1

� �yucu

1

2sin2y I

P

y�IP

z�1� �

ju2�hk1 cos2yju

2�2siny I

P

z�1� �

cujuþ f� �

siny IP

z�1� �

yuju

1

2fyu2 sinj�2ejuyucosjþ fju

2 sinj� ��

1

2fyu2 cosj�ejuyusinj�1

2fju

2 cosj

agr1 u00 grþjucusinj

agr2 u00 Ygr�jucucosj

0BBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCA

ðA:2Þ

where e¼ 2sinyþ hk1 cosy and f ¼ de=dy¼ hk1 siny�2cosy.

Appendix B

B.1. Validation of the numerical integration scheme

As was first proposed by Appel [6] and Korteweg [7], Eqs. (12) can be transformed as follows:

d2oz

dy2þcoty

doz

dy�

e

siny

2 IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

oz ¼ 0 ðB:1Þ

oZ ¼h

2k2

1�4 IP

y�1� �

IP

z

h2k2

1�4IP

yþ4

doz

dyþ

2hk1 IP

z�1� �

h2k2

1�4IP

yþ4oz ðB:2Þ

In the case of an infinite thickness disk (h¼ 0 and k1¼k2¼k3¼b¼1) Eq. (B.1) reduces to the one derived by Appell. Setting q¼cot y,Eq. (B.1) yields

q2þ1� �2 d2oz

ds2þq 1þq2� � doz

dsþ4ðbþcqÞoz ¼ 0 ðB:3Þ

where b¼ ððIP

z�1Þ=ðh2k2

1�4ðIP

y�1ÞIP

z ÞÞ and c¼ 12 hk1ððI

P

z�1Þ=ðh2k2

1�4ðIP

y�1ÞIP

z ÞÞ.Eq. (B.3) is a special case of the Riemann–Papperitz equation with two singular points. Its solution is given in terms of hypergeometric

functions below (Batista, [17])

ozðyÞ ¼ A1TDðyÞþA2T�DðyÞ ðB:4Þ

where

A1, A2 are complex constants that depend on the initial conditions of oz and its derivative at y¼y0, TDðyÞ ¼q�iqþ i

� �ð1=4Þþ ðD=2ÞF 1þDþD�

2 , 1þD�D�2 ;1þD; q�i

qþ i

� �, D¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14 þ4ðbþ icÞ

q, D� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14 þ4ðb�icÞ

qand F(a, b; c; z) is the Gauss hypergeometric

function [30].Combining Eqs. (13), (B.2) and (B.4) the equations of motion are integrated for given initial conditions. The integration involves the

calculation of the Gauss hypergeometric function, which finally is performed numerically. For that reason, the result is obtained througha semi-analytical approach.

The validity of the numerical integration scheme that was currently applied was juxtaposed with the abovementioned semi-analyticalsolution. The solutions obtained were identical, with an average relative error, ðoan

Z �onumZ Þ=oan

Z , of order of magnitude 10�8.

I. Stefanou et al. / International Journal of Non-Linear Mechanics 46 (2011) 114–124 123

Appendix C

C.1. Analytical solution of the wobbling ODEs

Eq. (22)a and c are linear in respect of f0 and c0and are written in matrix form

Xu¼ LX ðC:1Þ

where

X¼ju

cu

!

and

L¼1

3

yuy

�26I

P2

z � 12IP

yþ3hk1y�6� �

IP

z þ hk1 3hk1þ3y� �

4 IP

y�1� �

IP

z�h2k2

1

12 IP

z�1� �

IP

z

4 IP

y�1� �

IP

z�h2k2

1

�12I

P2

z þ2 �12IP

y�6hk1yþ6� �

IP

z þ6h2k2

1þ12hk1y

4 IP

y�1� �

IP

z�h2k2

1

2 IP

z�1� �

3hk1y�6IP

z

� �4 I

P

y�1� �

IP

z�h2k2

1

0BBBBBBBBB@

1CCCCCCCCCA

The eigenvalues and eigenvectors of L are

l1 ¼2hk1 I

P

z�1� �

yu

4 IP

y�1� �

IP

z�h2k2

1

, l2 ¼2hk1 I

P

z�1� �

h2k2

1�4 IP

y�1� �

IP

z

�2

y

0B@

1CAyu

S¼�

2IP

z IP

z�1� �

2 �2IP

yþ IP

z�yhk1þ1� �

IP

z þ hk1 2yþ hk1

� � �1

1 1

0BBB@

1CCCA ðC:2Þ

with

X¼ SY ðC:3Þ

Replacing (C.3) into (C.1) and after some algebra we obtain

Yu¼ ðD-S�1SuÞY ðC:4Þ

where D¼l1 0

0 l2

!.

Eq. (C.4) can be solved for Yand the solution of (C.1) is obtained using Eq. (C.3)

X¼

e

�

2yhk1 IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z c1 IP

z 4 IP

y�yhk1�1� �

IP

z þ hk1 4y�hk1

� �h i4y2h

2k2

1 IP

z�1� � �

e

2yhk1 IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z c2

y2

e

2yhk1 IP

z�1� �

h2k2

1�4 IP

y�1� �

iP

z c2

y2þ

e

�

2yhk1 IP

z�1� �

h2k2

1�4 IP

y�1� �

IP

z c1 hk1 1�4y2� �

hk1�4yh i

IP

z þ �4IP

yþ4yhk1þ4� �

IP2

z þ4y2h2k2

1

4y2h2k2

1 IP

z�1� �

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

ðC:5Þ

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