andersen_2002_post-critical behavior of beck's column with a tip mass

7/28/2019 Andersen_2002_Post-Critical Behavior of Beck's Column With a Tip Mass

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*Corresponding author.

International Journal of Non-Linear Mechanics 37 (2002) 135}151

Post-critical behavior of Beck's column with a tip mass

Steen Brahe Andersen, Jon Juel Thomsen*

Department of Solid Mechanics, Building 404, Technical University of Denmark, DK-2800 Lyngby, Denmark

Received 1 May 2000; accepted 2 October 2000

Abstract

This study examines how a tip mass with rotary inertia a!ects the stability of a follower-loaded cantilevered column.

Using nonlinear modeling and perturbation analysis, expressions are set up for determining the stability of the straight

column and the amplitude of post-critical #utter oscillations. Bifurcation diagrams are given, showing how the vibration

amplitude changes with follower load and other parameters. These results agree closely with numerical simulation. It is

found that su$ciently large values of tip mass rotary inertia can change the primary bifurcation from supercritical into

subcritical. This can imply very large motions for follower loads just beyond critical, contrasting the "nite amplitude

motions accompanying supercritical bifurcations. Also, the straight column may be destabilized by a su$ciently strong

disturbance at loads far below the value of critical load predicted by linear theory. A similar change in bifurcation is

found to occur with increased external (as compared to internal) damping, and with a shortening in column length. These

e!ects are not revealed by linear modeling and analysis, which may consequently fail to predict even qualitatively the real

critical load for a column with tip mass. 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

We examine how a tip mass with rotary inertia

a!ects the stability of a follower-loaded canti-

levered column. If there is no tip mass this system

corresponds to the well-known Beck's column [1],

which is known to exhibit soft #utter as the magni-

tude of the follower load exceeds a critical value[2]. We show how this behavior is qualitatively

changed by the rotary inertia of a tip mass, which

can imply that the straight equilibrium of the col-

umn can be destabilized at a forcing lower than

critical by a "nite-sized disturbance.

This work was motivated by theoretical and ex-

perimental studies by Sugiyama et al. [3] on a col-

umn subjected to follower forcing produced by

a solid rocket motor.

Beck's column has a load at the tip that remains

tangent to the deformed column axis. The signi"-

cance of this system is its capability to model and

illustrate, in a simple manner, the basic properties

of systems that can display #utter vibrations due to

follower loading. Such loads are inherentlynon-conservative, acting along lines that move

with the deformation or displacement of a struc-

ture. They occur with many real structures,

e.g., rocket driven aerospace structures, aircraft

wings, wind turbine blades, and pipes carrying

#uid #ow. The present study sheds further light

on possible behavior of follower loaded structures

for cases where the follower load is delivered by

a device whose mass and rotary inertia cannot be

ignored.

0020-7462/01/$- see front matter 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 1 0 2 - 5


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Beck's initial work [1] on determining the criti-

cal force of the follower-loaded column has been

followed by numerous studies of di!erent aspects

and variants of the system. For example, Herrmannand Jong [4] and Bolotin [5] described the seem-

ingly paradoxical destabilizing e!ect of internal

damping (the #utter-critical force in the presence of

vanishing internal damping being about half of the

value for zero damping), as contrasted by the

stabilizing e!ect of external damping [6]. P#uKger

[7] seems to be the "rst to examine the stability of

a Beck's column with a point mass, whereas more

recently Sugiyama et al. [3] paid consideration to

the e!ects of damping and of a tip mass with rotary

inertia in their combined theoretical and experi-mental study of a clamped column subjected to

follower forcing from a rocket motor; see also Lan-

gthjem and Sugiyama [8].

Most studies of Beck's column concerns the de-

termination of critical loads. This can be accomp-

lished using only linear modeling and analysis.

However, for studying post-critical behavior one

needs to incorporate in the model those nonlinear

e!ects that are thought to be responsible for limit-

ing the initially exponentially growing motions at

post-critical loads. To the authors knowledge only

Kolkka [2] and Chen [9] have employed nonlinearmodeling for Beck's column, showing that the bi-

furcation describing stationary oscillation ampli-

tudes at near-critical loads is supercritical, that is:

for post-critical loads the column performs stable

oscillations at a constant "nite amplitude (so-called

`softa #utter).

The primary motivation for the present work

was recent laboratory experiments conducted by

Sugiyama and Langthjem [10], indicating a critical

load much lower than the one predicted by linear

modeling. One possible explanation for this couldbe a change of the type of bifurcation at the (lin-

early predicted) critical force from supercritical into

subcritical, perhaps due to the inertial properties of

the rocket motor. As will be explained further be-

low the existence of such a subcritical bifurcation

implies that the straight con"guration of the col-

umn can lose stability at loads lower than (linearly)

critical, if the disturbance is su$ciently large. We

therefore investigate how the rigid body properties

of a tip mass (mass, rotary inertia, center of gravity)

a!ects the stability properties and types of bifurca-

tions for Beck's column.

The paper is organized as follows: Section 2 de-

scribes the physical model, and the setup of thedi!erential equation governing transverse motions

with consideration to geometrical nonlinearities.

Using modal expansion, this equation is discretised

into a set of nonlinear ordinary di!erential equa-

tions for use in subsequent analysis. Section 3 de-

scribes how these equations are solved numerically

and also shows numerical solutions illustrating

typical system behavior. Section 4 presents a per-

turbation analysis of the nonlinear equations of

motions, leading to analytical expressions for the

stationary solutions and their stability. In this anal-ysis, nonlinear terms up to "fth order are retained,

in order to be able to predict possible secondary

bifurcations associated with a primary subcritical

bifurcation. If a secondary bifurcation exists, it

means that stable oscillations could exist at loads

greater than critical. In this case, the third-order

solution would be insu$cient to describe the dy-

namic of the system. Section 5 then employs these

expressions to present bifurcation diagrams, show-

ing how the solutions and their stability change

with the magnitude of the follower load and para-

meters describing damping and inertial propertiesof the tip mass. The perturbation results presented

here are supported by two other types or levels of

analysis: Numerical time integration of a "nite ele-

ment discretisation of the underlying physical

model, and numerical time integration of the mode

shape discretisation. Section 6 "nally sums up the

"ndings and the main results: the rotary inertia of

a tip mass on Beck's column can change the type

of bifurcation at the critical load from supercritical

to subcritical, and increased external damping as

compared to internal damping can have the samee!ect.

2. Model system and equation of motion

In this section a geometric nonlinear model is set

up, which describes motion of the column for loads

larger than the critical force. The model can be used

to examine the nonlinear e!ects from the mass, its

rotary inertia and the location of the center of

136 S.B. Andersen, J.J. Thomsen /International Journal of Non-Linear Mechanics 37 (2002) 135}151


3/17

Fig. 1. System model.

gravity on the oscillations of the column. If the

magnitude of the follower load exceeds the critical

force at which #utter is initiated, linear stability

theory predicts that the oscillation amplitude willgrow unbounded. In reality, nonlinear e!ects will

either stabilize the #utter motion into a periodic

oscillation with constant amplitude (a limit cycle)

or it will result in amplitudes growing even more

violently than predicted by the linear model (ex-

plosive #utter).

Fig. 1 shows the model. A rocket motor is moun-

ted at the tip of the cantilevered column. The rocket

motor produces a thrust P, which is the follower

load applied to the column. The motor is assumed

a rigid body, with mass m and rotary inertia J. The

center of gravity is located at the distance a fromthe tip of the column. The column is assumed to be

viscoelastic and uniform, and has mass per unit

length A, length , moment of inertia I, andYoung's modulus E.

A curvilinear coordinate system is used, where

y"y(s, t) denotes the transversal deformation of

the column, t denotes time, and s denotes the cur-

vilinear position on the column axis. The inextensi-

bility condition for an in"nitesimal column element

gives the relation

(x)#(y)"1, (1)

where primes denote di!erentiation with respect to

position s.

Hamilton's extended principle is employed for

setting up the equation of motion. Hamilton's prin-

ciple (e.g. [11]), which holds only for conservative

systems, states that

H"0 where H,R

R

Ldt, L,T!V, (2)

where H de"nes the Hamiltonian of the system,

L is the Lagrangian,T andV are the kinetic and

potential energy, t

and t

are arbitrary instants of

time, and H is the variation ofH.Hamilton's extended principle states that the vir-

tual work W done by non-conservative forcesadded to the variation of the Hamiltonian should

vanish (e.g. [12,13])

H#R

R

Wdt"0. (3)

The kinetic and potential energy, T

and V

, for

a column element is

T"

A(y #x ), (4)

V"

1

2EI"

1

2EI

y

1!y, (5)

where dots denote di!erentiation with respect to

time t, and where the non-linear curvature measure

for the curvilinear coordinate s has been used. (This

di!ers from the curvature in rectangular coordi-

nates, see e.g. [14, p. 46]). The Lagrangian L and

the Lagrangian density E are introduced as

L"

*

(T!V

) ds"

*

E ds. (6)

The Hamiltonian can be divided in two:

H"H#H

, whereH

is the Hamiltonian for

the column and H

is the Hamiltonian for the

mass. The variation of the HamiltonianH

for the

column is

H"

R

R

*

*E

*yy#

*E

*yy

#*E

*yy#

*E

*xx dsdt. (7)

By using the inextensibility condition (1) and integ-

ration by parts, (7) can be written as

H"!

R

R

*

*

*t

E

*y#

*

*s

E

*y!

*

*s

E

*y

!y1#1

2y

*

*t

*E

*x

#y1#3

2y

*

Q

*

*t

*E

*xdsyds dt. (8)

S.B. Andersen, J.J. Thomsen /International Journal of Non-Linear Mechanics 37 (2002) 135}151 137


4/17

The kinetic energy T

of the mass has contribu-

tions from plane rotation and translations, i.e.

T"T

G#T

V#T

W. By using Dirac's delta-

function K, the energies per unit length can bewritten as

TG"

JK(s!s

)

Q"JK(s!s

)(1#y)y , (9)

TV"

mK(s!s

)

d

dt(x#a cos(

))

"mK(s!s

)(x#ax ), (10)

TW"

mK(s!s

)

d

dt(y#a sin(

))

"mK(s!s

)(y#ay ), (11)

where s

is the location of the mass on the column-

axis and

denotes the rotation of the column. The

center of gravity of the mass is located at the dis-

tance a from s

in tangential direction. Later s

is

set equal to . The variation of the Hamiltonian

H

for the mass can now be written as

H"

R

R

*

T

ds dt

"R

R

*

TG

yy#

TG

y y #

TV

x x

#T

Vx

x#T

Wy

y #T

Wy

y dsdt.(12)

By using the inextensibility condition (1) and integ-

ration by parts this can be transformed to

H"!

R

R

*

*

*t

TW

*y#

*

*s

TG

*y

!*

*s *tT

W#T

G*y !y1#

1

2y

;!*

*t *s

*TV

*x#

*

*t

*TV

*x #y1#3

2y

;*

Q!*

*t*s

*TV

*x#

*

*t

*TV

*x dsy dsdt.(13)

The follower force P has x- and y-components

PV

, PW

:

P"(PV , PW )"(!P cos( ()),!P sin( ())), (14)

where

() is given by

()"arcsin(y()) whichgives that

PV"!P(1!y()

+!P(1!y()!

y()), (15)

PW"!P sin(arcsin(y()))"!Py(). (16)

The virtual work of the non-conservative follower

force can be divided in two: W"WV#W

W,

where WV

and WW

are the virtual work in the x-and y-directions, respectively. By using (15) and (16)

these can be written as

WV"P

Vu"P(1!

y()!

y())u, (17)

WW"P

Wy()"!Py()y(). (18)

Here u denotes the tip displacement in the negative

x-direction. By considering an in"nitesimal column

element and integrating over the column length

, u can be found as

du"1!dx

ds ds"(1!(1!y) ds, (19)

u"*

(1!(1!y) ds"*

f(y) ds. (20)

The variation u become

u"*

f(y) ds"*

*f(y)

*yy ds

K*

y#12y#3

8yy ds. (21)

Inserting (21) into (17), the virtual work in the

x-direction becomes

WV"

*

P1!1

2y()!

1

8y()

;y#1

2y#

3

8yy ds. (22)



5/17

Integrating (22) by parts and applying Dirac's

delta-function to (18), the total virtual work from

the force P can be written as

W"*

!PyK(s! )

!Py#y3

2y!

1

2y()

#y15

8y!

3

4y()y

!1

8y()yds. (23)

By inserting (8), (13) and (23) in (3) and requiring

that (3) is ful"lled for arbitrary admissible variation

in y, one obtains in dimensionless form

K###K(!1)

#(!

)(K#K )!(K(!

)K )

!(K(!

)(K#K ))

#R#'#R

#'#R

#R

#R

.

#R.#RK#RK"0, (24)

where the nondimensional quantities are given by

"y

, "

x

, "

EI

At, "

P

EI,

"J

A, "

m

A, "

a

(25)

and the dots and primes denote di!erentiation with

respect to dimensionless time and position . This

equation describes the transverse displacements(, ) of the column. The "rst and second term in(24) represent linear inertia and sti!ness from the

column, the third and fourth term represent a linear

sti!ening e!ect from the follower force and the "fth

to the seventh term represent linear inertia from the

tip mass, its rotary inertia and the location of the

center of gravity. The functions R#'

, R#'

, R

,

R

, R.

, R.

, RK

and RK

represent the nonlinear

e!ects and are given in the appendix. They repres-

ent terms of third and "fth order and describe,

respectively, sti!ness of the columnR#'

, R#'

, axial

inertia of the column R

, R

, e!ects from the

follower force R.

, R.

, and e!ects from the mass

RK , RK .A mode shape expansion is now applied to trans-

form the partial di!erential equation (24) into an

approximating set of ordinary di!erential equa-

tions. The expansion

(, )",H

H()

H() (26)

is used, where H

are the mode shapes of a canti-

levered column with a tip mass, and H

are the

unknown time varying modal coe$cients. By use of

Galerkin's method (e.g. [15]) one obtains

M$#CQ#S#g(,Q ,$)#g(,Q ,$)"0

(27)

where M is the mass matrix, C is the damping

matrix, and S is the sti!ness matrix. The vectors

g and g contain nonlinear terms of third and "fthorder, respectively:

g",

HIJ

((HIJ#

HIJ)

H

I

J)

#,

HIJ

((HIJ#

HIJ)

H(

I$

J#Q

IQ

J)), (28)

g",

HIJKL

((IHIJKL#I

HIJKL)

H

I

J

K

L)

#,

HIJKL

((I?AAHIJKL#I?AA

HIJKL)

H

I

J

K$

L)

#,

HIJKL

((ITCJHIJKL#ITCJ

HIJKL)

H

I

JQ

KQ

L).

(29)

The constants HIJ

, HIJ

, HIJ

, HIJ

, IHIJKL

, IHIJKL

,

I?AAHIJKL

, I?AAHIJKL

, ITCJHIJKL

and ITCJHIJKL

are computed from

the expansion functions and can be found in the

appendix.

The constants HIJ

and HIJ

, which correspond to

the nonlinear e!ects of third order from the sti!ness

and the axial inertia of the column, respectively,

have also been found and used by e.g. Semler et al.

[12], who set up the equation of motion for a pipe



6/17

Fig. 2. Tip displacement and frequency spectrum for a column with no tip mass for a force "13.53 which is greater than the critical

force "12.93. Parameters: "0, "0, "0, "0.1, "0.01. (a): Tip displacement as function of time for four di!erent initial

conditions, where the tip amplitude is given and the velocity is zero; (b): Frequency spectrum. &*' nonlinear analytic model, &- - -' Finite

element model.

conveying #uid. To the author's knowledge, the set

up of an equation of motion including "fth-order

terms by use of Galerkins method for a column

with a tip mass subjected to a follower force is notreported in the literature. The remaining constants

other than HIJ

and HIJ

are therefore new results.

Damping has been added to (27) in the form of

internal and external viscous damping, where the

elements in C are given by

cGH"

GH

d#

GH

d. (30)

Here is the external damping factor and is theinternal damping factor. The elements in M and

S are given by

mGH"

GH

d, (31)

sGH"

(GH!

GH) d#

G(1)

H(1). (32)

Eq. (27) along with (28)}(32), describes a set of

coupled nonlinear ordinary di!erential equations

where the time-dependent modal coe$cients H

are

the unknowns. The equation is now ready to be

solved, and this is done numerically and by a per-

turbation method in the following two sections.When the modal coe$cients are found the behavior

of the column in time can be described by using (26)

to calculate the transverse de#ection of the col-umn.

3. Numerical simulation

Numerical simulations of (27) are performed

with the purpose to verify pertubation solutions,

and to verify the model itself by comparing the

numerical solutions with solutions obtained by

a "nite element model. Furthermore, the transient

behavior and a frequency spectrum can be ob-

tained.

Eq. (27) is solved numerically using the trapezmethod (e.g. [16]), which is a "nite-di!erence

method. The solution gives the displacement of the

column during time. Fig. 2(a) shows the tip dis-

placement as function of time for four di!erent

initial tip displacements, for a situation where the

load is greater than critical for #utter. As can be

seen the post-transient vibration amplitude is the

same for the four simulations. A frequency spec-

trum for the simulations in Fig. 2(a) is shown in Fig.

2(b), along with a frequency spectrum from a "nite

element simulation. It appears that the column

vibrates at its "rst eigenfrequency and a higherharmonic. The "nite element simulation gives both



7/17

Fig. 3. Finite tip vibration amplitude

as function of number

of shape functions. Critical force"12.93, "0, "0, "0,

"0.1, "0.01.

qualitative and quantitative the same results as the

model presented here.

Fig. 3 shows the dependency of the number N of

expansion functions H

on the post transient vibra-

tion amplitude

. It is necessary to use from four

to nine expansion functions to ensure that the solu-

tion is independent of the discretisation. It is neces-

sary to use even more expansion functions near

critical loads.The geometric nonlinear dynamic "nite element

model, used above for comparison was set up using

64 2D beam elements with two nodes and six de-

grees of freedoms each. The nonlinearities have

been applied by using a method of updating the

coordinates [17]. Lin and Tsai [18] have used this

method in a nonlinear dynamic analysis of a canti-

levered column conveying #uid. The method here is

used similarly, to simulate the dynamics of a canti-

levered column with a tip mass and a follower force

at the tip. The follower force and the e!ects fromthe tip mass have been applied as external forces at

the "nite element node corresponding to the col-

umn tip.

Numerical di$culties (lack of convergence) have

been encountered with the "nite element model in

cases where the tip mass is large or rotary inertia is

present. Implementation of another numerical

method, which takes into account the external for-

ces in the tangential sti!ness matrix might solve

this problem.

The numerical solutions of (27) and the "nite

element solutions described above are used for

verifying the pertubation solutions to be described

next.

4. Perturbation solutions

At loads greater than the critical, #utter occurs,

which can evolve into periodic oscillations about

the previously stable equilibrium. This change can

be described as a supercritical Hopf-bifurcation. In

this case, the nonlinear e!ects are stabilizing. An-

other possibility is that the oscillation amplitude

grow unbounded, which can be described as a sub-critical Hopf-bifurcation. In this case, the non-lin-

ear e!ects are destabilizing on the oscillations.

To obtain insight into the system behavior at

near-critical loads, and to analyze the e!ect of

changing system variables, a perturbation analysis

is performed. A multiple scales method is employed

for this (e.g. [19,20]). The procedure presented here

is equivalent to the one used by Thomsen [21] and

Jensen [22].

Eq. (27) can be written as a system of"rst-order

equations

x"A()x#f(, x, x )#fI(, x, x ), (33)

where x"+ Q ,2 and the parameter is introduc-ed as a bookkeeping parameter to indicate the

smallness of non-linearities. The matrix A and the

vectors f and fI are given by

A"0 I

!M\S !M\C, f"0

!M\g,

fI"

0

!M\g

, (34)

where I is the identity matrix. Two time scales are

introduced: a fast scale " for describing

motions at frequencies comparable to the natural

frequencies, and a slow scale " describing

slow modulation of oscillation amplitudes.

Solutions x to (33) are approximated through

uniformly valid expansions of the form

x"x

(

,

)#x

(

,

)#O(), (35)



8/17


9/17

;

(d

HIJKL!

fHIJKL

)

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL

!

eHIJKL

uHuIuJuKuL#

uHuIuJuKuL!

uHuIuJuKuL!uHuIuJuKuL#

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL#

uHuIuJuKuL

, (51)

q"aL

,

HIJ

(bHIJuHuIuJ!2

cHIJuHuIuJ)

#aaL,

HIJKL

(52)

;

(d

HIJKL!

fHIJKL

)

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL

!

eHIJKL

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL

q"aL

,

HIJKL

(dHIJKL

uHuIuJ!

(e

HIJKL

#fHIJKL

)uHuIuJuKuL

), (53)

where uH

is the jth element of u. Since i

is an

eigenvalue of A

, the solution of (50) will contain

secular terms proportional to

exp(i

) unless

q

is orthogonal to the left eigenvector v of A

(e.g.

[23]), which means that v2q"0 where v is given by

(A2!i

)v"0. (54)

Substituting q

of (51) into v2q"0 yields the

solvability condition

da

d

!a!aa!aa"0, (55)

where , , and are given by

"v2A

u

v2u,

0#i

'(56)

"v2L,

HIJ(b

HIJ(u

HuIuJ#u

HuIuJ#u

HuIuJ)

v2u

!2

cHIJuHuIuJ)

v2u,

0#i

'(57)

"v2L

v2u

,

HIJKL

;(dHIJKL!fHIJKL)uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL#

uHuIuJuKuL



10/17

!

eHIJKL

uHuIuJuKuL#

uHuIuJuKuL!

uHuIuJuKuL!uHuIuJuKuL#

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL!

uHuIuJuKuL#

uHuIuJuKuL ,0#i' . (58)

Here (0

, '), (

0, ') and (

0, ') denotes the real and

imaginary parts of , and . The function a()can be expressed in terms of two real-valued func-

tions A(

) and (

):

a(

)"A(

) exp(i(

)), A, 3R. (59)

Inserting (59) in the solvability condition (55), this

can be written as two modulation equations, de-

scribing evolutions of the slowly changing oscilla-

tion amplitude A and frequency :

dA

d"

0A#0A

#0A

, (60)

d

d

A"'A#

'A#

'A. (61)

Solutions corresponding to oscillations with con-

stant amplitude can be found by letting dA/d"

0. Below, one solution is found for the third- and

"fth-order expansion, respectively. It is possible

that the solution for the "fth-order expansion gives

a secondary bifurcation that does not show up

using only the third-order expansion [22].

4.1. Third-order solution

For this approximation terms of"fth order in the

modulation equations are ignored. Eq. (60) then

has two singular points: A"0 which corresponds

to the zero-solution, and A"(!0

/0

which

corresponds to a stationary periodic solution for

!0

/0'0 (a limit cycle). Combining Eqs. (35),

(48), (59) and (61) and setting "1, the limit cycle

approximation becomes

x"x#x

#O()

"2Re[A expiu exp(i)]#O()

"2!0

0

"u"cos((#

'!(

'/0

)0

))#O().

(62)

The post-transient tip amplitude

for the column

can now be found by inserting the amplitudes

max(G)"max(x

G), i)N of the modal coordi-

nates into the modal expansion (26):

" ,

G

G(1) max(x

G)"2!

00

,G

G(1)"u

G",

(63)

where G

is the ith eigenmode for the cantilevered

column with a tip mass and uG

is the ith component

of the eigenvector u from (49).

4.2. Fifth-order solution

Retaining the "fth-order terms in (60), the non-

zero singular points become

A"!0$(0!40020

.(64)

The tip amplitude

can now be found as above by

inserting (62) into (26)

"2!0$(0!4002

0

,G

G(1)"u

G".

(65)

4.3. Stability of stationary solutions

The stability of the singular point A is evaluated

by inserting it into the gradient of (60)

(A)"0#3

0A#5

0A. (66)

If this value is negative, then A is stable, otherwise

A is unstable.

Now, the stationary solutions and their stability

are calculated for particular parameters of the

column. These solutions are valid for loads in a

vicinity of the critical force. This means that



11/17

Fig. 4. Bifurcation diagrams. (a) No tip mass "0.0; (b) With tip mass "0.2. Damping: "0.1, "0.01. * Finite element; numerical solution to (27); &*' stable, and &- - -' unstable 3rd and 5th order pertubation solutions.

a bifurcation diagram can be made from the solu-

tions and their stabilities, and hereby the e!ects

from the nonlinearities can be evaluated.

5. Bifurcations

The nonlinear oscillations of the column at loads

near the critical force will now be examined. The

nonlinear e!ects from the sti!ness of the column,

the tip mass, its rotary inertia, and the location of

the center of gravity will be examined separately.

For loads greater than critical the nonlinear ef-

fects can either stabilize the #utter into oscillations

with constant amplitude, or they can destabilizeeven further so the #utter becomes explosive. This

change in response as the load exceeds the critical

value corresponds to a supercritical or a subcritical

bifurcation, respectively.

If the bifurcation is supercritical only the stable

zero-solution exists at loads smaller than the criti-

cal force. The stability limit of the zero-solution can

therefore be found by linear theory. If the bifurca-

tion is subcritical, however, the nonlinear e!ects

can destabilize the zero-solution if the column ex-

periences a disturbance of su$cient strength. In

this case, the behavior of the system is thereforeradically changed and linear stability theory is in-

su$cient to determine the real critical force for

disturbances of "nite magnitude. It is therefore of

great importance to know which kind of bifurca-

tion there will exist at particular values of para-

meters.

5.1. Column with no tip mass

Fig. 4(a) shows the bifurcation diagram for a col-

umn with no tip mass. The bifurcation is supercriti-

cal, which means that the nonlinear e!ects stabilize

the #utter into a limit cycle. This is in agreement

with Kolkka [2], who shows mathematically that

the bifurcation is supercritical for Beck's column.

As appears, the numerical simulations of Eq. (27)

match very well with the "nite element simulations

and the approximative perturbation solution. Theperturbation solutions of third and "fth order show

no signi"cant di!erences.

5.2. Column with a tip mass

A tip mass with a mass of 20% of the column

mass has been applied at the tip-end of the column.

The bifurcation diagram in Fig. 4(b) shows that the

bifurcation is still supercritical. The critical force is

smaller than for the column with no tip mass. The

shape of the bifurcation is comparable with theone for the column with no tip mass, indicating that

the mass does not qualitatively change the dynam-

ics of the column.



12/17

Fig. 5. Bifurcation diagrams for a column having rotary inertia at the tip. (a) "0.002; (b) "0.2. Damping: "0.1, "0.01. Numerical solution to (27); &*' stable, and &- - -' unstable 3rd and 5th order pertubation solutions.

5.3. Column with rotary inertia at the tip

To shed light on the isolated e!ect of rotary

inertia, we consider here the (unrealizable) caseof a zero tip mass having nonzero rotary inertia.

Fig. 5(a) and (b) shows the bifurcations for a col-

umn with two di!erent values of inertia at the

tip-end. The bifurcation is still supercritical as long

as the rotary inertia is very small. The shape of the

branching bifurcation becomes steeper, indicating

that the nonlinear e!ects from the rotary inertia are

destabilizing. This indication is supported by the

bifurcation in Fig. 5(b), where the rotary inertia is

even larger. First, it should be noted that the larger

rotary inertia decreases the critical load signi"-cantly. Secondly the bifurcation has changed and

has become subcritical, meaning that the system

behaves in a completely di!erent way. At loads

smaller than critical, it is possible to destabilize the

zero solution by disturbing the column su$ciently

so the amplitude becomes greater than the ampli-

tude of the unstable limit circle. The "fth-order

perturbation solution shows the same as the solu-

tion of third order: there are no stable solutions at

loads greater than the critical. At loads greater than

the critical, motions are therefore not limited by the

nonlinearities included in the present model; fora real model other nonlinearities such as plastic

deformation or fracture will limit the response. In

numerical simulations explosive growths in ampli-

tudes were observed until numerical instability oc-

curred at very large deformations.

5.4. Column with a tip mass having rotary inertia

Fig. 6(a) shows the bifurcation for a column with

a tip mass having rotary inertia and Fig. 6(b) shows

the bifurcation for a case where the center of gravity

of the mass is located at a distance from the tip. The

"gures show that the shape of the bifurcations are

steeper compared to the bifurcation for the column

with no tip mass. This is due to the rotary inertia

and the location of the center of gravity, which

gives a resulting moment of inertia at the tip. The

numerical simulations match very well the solu-tions found by the third- and "fth-order perturba-

tion solutions. If the rotary inertia is su$ciently

large, the bifurcation will become subcritical, and

the #utter will be explosive. This was con"rmed by

numerical simulations.

5.5. Change of bifurcation

For a given tip mass having rotary inertia, Fig. 7

shows if the bifurcation at the critical load is sub- or

supercritical. As appears, when the rotary inertia is

su$ciently large the type of bifurcation changesfrom super- to subcritical, we call this value the

critical rotary inertia. When the tip mass is about

the same as the mass of the column or larger ('1)



13/17

Fig. 6. Bifurcation diagrams for a column with a tip mass having rotary inertia , the center of gravity beeing located a distance fromthe tip. (a) "2.0, "0.2, "0.0; (b) "2.0, "0.2, "0.1. Damping: "0.1, "0.01. Numerical solution to (27); &*' stable,and &- - -' unstable 3rd and 5th order pertubation solutions.

Fig. 7. Type of bifurcation as function of tip mass , tip massrotary inertia , and position of center of gravity . Damping:"0.1, "0.01. Solid lines mark the boundaries between

supercritical and subcritical bifurcations.

there is an approximative linear relationship be-

tween the mass and the critical rotary inertia

(/"m/J"constant). As also appears a sub-critical bifurcation can occur only when the

rotary inertia is non-zero. Finally, it should

be observed, that for a given tip mass having

rotary inertia, a shortening of column length can

cause a change in bifurcation from super- to sub-

critical.

5.6. Dependency of damping

Fig. 8 shows the signi"cants of the ratio of ex-

ternal to internal damping / on the type of bifur-cation. As appears the bifurcation type is highly

dependent on this ratio. The damping coe$cients

and can be very di$cult to estimate correctlyfor real systems, and thus it may be di$cult toestimate which kind of bifurcation will actually

occur. It should also be noted that if the energy

dissipation is dominated by external damping

(/


14/17

Fig. 8. Type of bifurcation as function of the ratio of tip mass to

tip mass rotary inertia, /, and the ratio of external to internaldamping /.

for near-critical loads. This lead to analytical ex-

pressions for the post-critical solutions and their

stability. The results were subsequently employed

to establish bifurcation diagrams, showing how the

stationary vibration amplitude depends on the

magnitude of the follower load and other para-

meters. Again, results were veri"ed by numerical

simulation.The main result of this work is the observation

that the rotary inertia of a tip mass on a Beck's

column can change the type of bifurcation at the

critical load. At su$ciently large values of rotary

inertia, this bifurcation will change from supercriti-

cal to subcritical. This implies two phenomena that

are not revealed by linear analysis: First, at su$-

ciently large values of rotary inertia there will be no

stable small-amplitude motions for follower loads

just beyond the critical; only motions with very

large amplitude are then possible. Secondly, thestraight con"guration of the column may be de-

stabilized by a su$ciently strong disturbance at

loads below the critical load predicted by linear

theory. Thus, the presence of large rotary inertia

may make stability analysis based only on linear

modeling insu$cient. The same observation holds

true even for a tip mass with negligible rotary

inertia, if its center of gravity is displaced rigidly

beyond the column tip, or if a column with a given

tip mass and rotary inertia is shortened.

Increasing the external damping of the system (as

compared to internal damping) also has this e!ect

of changing the primary bifurcation from super-

critical to subcritical. Since with real systems thecharacter and magnitude of damping e!ects are

rarely known to any great precision, this implies

that predictions based only on linear theory might

fail simply due to a wrong estimate of damping

coe$cients.

The above theoretical results will need experi-

mental veri"cation. It would be relevant to system-

atically examine critical loads and post-critical

behavior for a follower-loaded clamped column as

a function of changing rotary inertia of a tip mass.

For example, tests similar to those described inSugiyama et al. [3] and Langthjem and Sugiyama

[8] could be performed with two additional masses

at the rocket motor, mounted on a rigid beam

perpendicular to the column and with variable sep-

aration.

Acknowledgements

The authors wish to thank Mikael Langthjem for

suggesting this work, and for many useful com-

ments and suggestions.

Appendix A

A.1. Dimensionless functionals

R#'"#4# (A.1)

R#'"6#8#, (A.2)

R"

K

(

#K )d!

K K

(

#K ) d d, (A.3)

R"

K

3

2 #

1

2K d

#1

2

K

( #K ) d

!

K K

3

2 #

1

2K d d



15/17

!3

2

K K

( #K ) d, (A.4)

R."

3

2!

1

2(1), (A.5)

R."

15

8!

3

4(1)!

1

8(1), (A.6)

RK"!(K(!

)(K # ))

!K(!)K

( #K ) d

#( #K )!K(!

)

K

(

#K ) d#( #K )!

K

( #K ) d#( #K ),(A.7)

RK"!(K(!

)(K #2 ))

!K(!)K

3

2 #

1

2K d

#3

2 #

1

2K

!K(!

)

K

3

2

#1

2

K

d

#3

2 #

1

2K

!

2K(!)

K

(

#K ) d#( #K )

!K(!

)K

( #K ) d

#( #K )

!

K

3

2 #

1

2K d

#3

2(

) (

)

#1

2(

)K (

) (A.8)

!

3

2

K

(

#K ) d

#( #K ). (A.9)A.2. Non-linear coezcients

GHIJ"

3

2

GHIJ

d

!

1

2

GHdI (1)J(1)

, (A.10)

GHIJ"

GHIJ

d#4

GHIJ

d

#

GHIJ

d, (A.11)

GHIJ"

GHK

IJ

d d

!

GH

K

K

IJ

d d d, (A.12)

BGHIJ"

G(1)

H(1)

I(1)

J(1)

#(G(1)

H(1)#(

G(1)

H(1)#

G(1)

H(1)))

;

IJ

d#I

(1)J(1)

!

GH

d

IJ

d#I

(1)J(1),

(A.13)



16/17

IGHIJKL"

15

8

GHIJKL

d

!34H(1)

I(1)

GJKL

d (A.14)

!1

8H(1)

I(1)

J(1)

K(1)

GL

d,

(A.15)

IGHIJKL"6

GHIJKL

d

#8

GHIJKL

d, (A.16)

#

GHIJKL

d, (A.17)

ITCJGHIJKL"

3

2

GHK

IJKL

d d

#1

2

GHIJK

KL

d d

!3

2

GH

K

K

IJKL

d d d

!3

2

GHIJ

K K

KL

d d d,

(A.18)

BITCJGHIJKL"2

G(1)

H(1)

I(1)

J(1)

K(1)

L(1)

#3

2G(1)H(1)#(G(1)H (1)

#G(1)H (1))!

GH d

;

IJKL

d

#I

(1)J(1)

K(1)

L(1)) (A.19)

#1

2(

G(1)

H(1)

I(1)

J(1)

#(G(1)

H(1)

I(1)

J(1))

!3

GHIJ

d;

KL

d#K

(1)L

(1), (A.20)I?AA

GHIJKL"

1

2

GHK

IJKL

d d

#1

2

GHIJK

KL

d d

!1

2

GH

K K

IJKL

d d d

!32

GHIJ

K K

KL

d d d

(A.21)

BI?AAGHIJKL"

G(1)

H(1)

I(1)

J(1)

K(1)

L(1)

#1

2G(1)H(1)#(G(1)H (1)

#G(1)

H(1))!

GH

d;

IJKL d

#I

(1)J(1)

K(1)

L(1)) (A.22)

#1

2G(1)H(1)I (1)J (1)

#(G(1)

H(1)

I(1)

J(1))

!3

GHIJ

d;

KL d#K (1)L(1). (A.23)

References

[1] M. Beck, Die Knicklast des einseitig eingespannten tan-

gential gedruKckten Stabes, Zeitscrift fuKr Angewandte

Mathematik und Physik 3 (1952) 225}228.

[2] R.W. Kolkka, On the non-linear Beck's problem with

external damping, Int. J. Non-Linear Mech. 14 (6) (1984)

497}505.



17/17

[3] Y. Sugiyama, K. Katayama, S. Kinoi, Flutter of canti-

levered column under rocket thrust, J. Aerospace Engng.

8 (1) (1995) 9}15.

[4] G. Herrmann, I.C. Jong, On the destabilizing e!ect of

damping in nonconservative elastic systems, J. Appl.

Mech. 32 (1965) 592}597.

[5] V.V. Bolotin, Nonconservative Problems of the Theory of

Elastic Stability, Pergamon Press Ltd., Oxford, 1963.

(Translated by T.K. Lusher).

[6] R.H. Plaut, E.F. Infante, The e!ect of external damping on

the stability of Beck's column, Int. J. Solids Struct. 6 (1970)

491}496.

[7] A.P#uKger, Zur StabilitaK t des tangential gedruKckten Stabes,

Zeitscrift fuKr Angewandte Mathematik und Mechanik 35

(5) (1955) 191.

[8] M.A. Langthjem, Y. Sugiyama, Optimum shape design

against #utter of a cantilevered column with an end-mass

of"nite size subjected to a non-conservative load, J. SoundVib. 226 (1) (1999) 1}23.

[9] M. Chen, Hopf bifurcation in Beck's problem, Nonlinear

Anal. Theory, Methods Appl. 11 (9) (1987) 1061}1073.

[10] M.A. Langthjem, Personal communication, November

(1998).

[11] C. Lanczos, The Variational Principles of Mechanics, 2nd

Edition, University of Toronto Press, Toronto, 1962.

[12] C. Semler, G. Li, M.P. PamKdoussis, The non-linear equa-

tions of motion of pipes conveying #uid, J. Sound Vib. 169

(1994) 577}599.

[13] J.S. Jensen, Fluid transport due to nonlinear #uid-

structure interaction, J. Fluids Struct. 11 (1997)

327}344.

[14] V.V. Bolotin, The Dynamic Stability of Elastic Systems,

Holden-Day, San Francisco, CA, 1964.

[15] L. Meirovitch, Analytical Methods in Vibrations, Macmil-

lan, New York, 1967.

[16] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Ap-

plications of Finite Element Analysis, Wiley, New York,

1989.

[17] P.C. Kohnke, Large de#ection analysis of frame structures

by "ctitious forces, Int. J. Numer. Methods Engng. 12

(1978) 1279}1294.

[18] Y.H. Lin, Y.K. Tsai, Nonlinear vibrations of Timoshenko

pipes conveying #uid, Int. J. Solids Struct. 34 (23) (1996)

2945}2956.

[19] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley-

Interscience, New York, 1979.[20] J.J. Thomsen, Vibration and Stability, Order and Chaos,

McGraw-Hill, London, 1997.

[21] J.J. Thomsen, Chaotic dynamics of the partially follower-

loaded elastic double pendulum, J. Sound Vib. 188 (3)

(1994) 385}405.

[22] J.S. Jensen, Articulated pipes conveying #uid pulsating

with high frequency, Nonlinear Dyn. 19 (1999) 171}191.

[23] A.H. Nayfeh, B. Balachandran, Applied nonlinear dynam-

ics: Analytical, Computational, and Experimental Meth-

ods, Wiley, New York, 1995.


andersen_2002_post-critical behavior of beck's column with a tip mass

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