Pd-1-Bernoulli's Transformation of the Response of an Elastic Body and Damping

8/6/2019 Pd-1-Bernoulli's Transformation of the Response of an Elastic Body and Damping

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BERNOULLIS TRANSFORMATION OF THE

RESPONSE OF AN ELASTIC BODY AND DAMPING

Vasilios Nikitas1, Alexandros Nikitas2 and Nikolaos Nikitas3*

1Head of the Department of Building Structures

Prefecture of Drama

Drama 66100, Greece

E-mail: [email protected]

2 Faculty of the Built Environment

University of the West of England

Frenchay Campus, Bristol BS16 1QY, UK


3Department of Civil Engineering

University of Bristol

Queens Building, University Walk, Bristol BS8 1TR, UK


*Corresponding author


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Abstract

Bernoulli's transformation and his method of separation of variables for solving a partial dif-

ferential equation as classically applied to the dynamics of an elastic continuous system will

always conclude an undamped response. However, this conclusion lacks reliability, since the

underlying analysis assumes either integrandwise differentiability (i.e. differentiating an inte-

gral equals differentiating its integrad) or termwise differentiability (i.e. differentiating an in-

finite series equals differentiating its terms) for Bernoulli's transformation, which not only is

responsible for the undamped response but also is arbitrary.

This paper, using Bernoulli's transformation and either an integration or a differentiation ap-

proach to separating variables, examines a uniform elastic column subjected to an axial ex-

ternal force at its free end or to a free axial vibration, and shows that actually the two under-

lying differentiability assumptions not only are equivalent but also constitute a limitation to

the class of the response functions. Only on this limitation, damping appears to be inconsis-

tent with elasticicity. Removing the limitation results in a damped response of the elastic

column, which indicates that damping actually complies with elasticity.

Keywords: Bernoullis transformation, differentiability assumptions about Bernoullis

transformation, elastic response and damping.


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Contents

Abstract

Nomenclature

1. Introduction

2. Partial differential equation of motion of an elastic column

3. Transformation of an inhomogeneous boundary value into a homogeneous one

4. Uncoupling the partial differential equation of motion into undamped vibrations

5. The classical undamped solution of the partial differential equation of motion

6. A critique of the classical undamped solution

7. Equivalence of the assumptions of integrandwise and termwise differentiability

8. Uncoupling the partial differential equation of motion by termwise differentiability

9. Integrandwise andtermwise differentiability as a limitation to the response

10. The damping effect of integrandwise and nontermwise differentiability

11. The really general solution of the partial differential equation of motion

12. The case of free axial vibration

13. Conclusions

References


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Nomenclature

A cross-sectional area of the column

c velocity of wave propagation, constant

E Youngs modulus of elasticity

ng t generalized acceleration

L length of a column

N z ,t internal axial force of the column at its level z and at time t

n , natural numbers

n t scalar magnitude representing damping ratio varying with time

n constant damping ratio

P t external axial force at the free end of a column

mass density

nq t , ny t generalized displacements

t time

X integrand variable of time

tR z, t remainder term of a nontermwise differentiation with respect to t

zR z, t remainder term of a nontermwise differentiation with respect to z

u z, t displacement at the level z of an elastic column

u z,t transformed displacement at the level z of an elastic column

z level of cross-section in an elastic column

n z mode

n t generalized acceleration

n natural angular frequency


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1. Introduction

According to Daniel Bernoulli, the displacement u z ,t of the cross-section level z of a line

continuous system at time t may be transformed into an infinite series of products each one

of which consists of two single-variable terms as below [1 p.502-522], [2 p.488-509]

1

n n

n

u z, t q t zNg

!

! , (1.1a)

2 0

0

1L

n nL

n

q t u z, t z dz

z dz

N

N

!

, (1.1b)

where L denotes the length of the line continuous system,

nq t denotes ageneralized displacement,

n zN represent a family oforthogonal functions for 0,1,2,3,n ! K , i.e. a family of

functions possessing the property [2 p.488-489]

0

0 for

0 for

L

n

n z z dx

cons tan t n

{

! { !

(1.2)

Bernoullis transformation (1.1) is nowadays referred to merely as a Fourier series of u z, t

with respect to the system of orthogonal functions n zN [2 p.489], [3 p.74-75]. The conver-

gence of the infinite series (1.1a) seems to obey criteria analogous to those for the common

Fourier series [2 p.489].

For homogeneous (zero) boundary values of the response function u z,t or its derivative


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u z,t zx x , classical analysis assumes that the corresponding boundary values of the or-

thogonal functions n zN are zero, and the functions n zN may be derived from an ordinary

differential equation such as [4 p.418-419, p.425]

2 42 2

2 2 4 20 or 0 with

n nn n nn n

d z d zz z constant

cdz c dz c

N N[ [ [N N ! ! ! . (1.3)

Such orthogonal functions n zN are classically known as natural modes [5 p.236].

By means of Bernoullis transformation (1.1) with appropriate natural modes n zN , classical

analysis can uncouple the partial differential equation of motion of an elastic continuous sys-

tem into an infinite number of independent ordinary differential equations, each one of which

has only one generalized displacement nq t as unknown. This is the methodofseparation

ofvariables for solving a partial differential equation of motion [4 p.418]. As we shall see

later, the ordinary differential equations can represent undamped vibrations only on the arbi-

trary assumption oftermwise differentiability of Bernoullis transform (1.1a), i.e.

2 2

2 20

nn

n

u z, t d q t z

t dt

g

!

x!

x , (1.4a)

2 22 2

0

nn

n

u z, t d zq t

z dz

g

!

x!

x , (1.4b)

or equally on the arbitrary assumption of integrandwise differentiability of Bernoullis trans-

form (1.1b), which means thatdifferentiating the integral in Bernoullis transform (1.1b) to

the second order with respect to time t is carried out by differentiating its integrand, viz.


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2 2

2 22 0

0

1L

nnL

n

d q t u z, tz dz

dt tz dz

N

N

x!

x

, (1.5)

for which a sufficient condition is the continuity of the derivative 2 2u z,t tx x for all values

of z and t[2 p.286, Leibnitzs rule], [3 p.348]. As a consequence of these equivalent as-

sumptions, the classical solution of the partial differential equation of motion of an elastic

continuous system seems to be undamped. An arbitrary generalization of this specific conse-

quence, which, however, can only be valid within the frame of the above equivalent assump-

tions, is what leads to the classical view that damping is inconsistent with the elastic continu-

ous systems. Exactly this arbitrary view has been the focal point of my critique originated in

a peripheral way by a congress paper of 1996 examining the response of an initially at rest

elastic uniform column subjected to an axial external force at its free end [6]. As a matter of

fact, the equivalent assumptions (1.4) and (1.5) impose a limitation to the class of the re-

sponse functions u z,t , which deprives the classical undamped solution for the partial dif-

ferential equation of motion of an elastic continuous system of any reliability. If we remove

this limitation by replacing the termwise differentiation rules of Bernoullis transform (1.1a)

with nontermwise differentiation rules, we arrive at a damped solution, which verifies that ac-

tually, damping complies with elasticity.

In a lot of instances, the failed attempts to convey new ideas across academics point out that

even the reasoning of a simple analysis may not be clear enough and additional details, in-

itially considered as self-evident, must be subsequently discussed and elucidated. From this

viewpoint, the present paper may be seen as a further elaboration on the main idea of the

mentioned published paper of 1996 [6] that aims at a real insight into the topic.


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2. Partial differential equation of motion of an elastic column

The continuum model of a uniform elastic column used in my published paper of 1996 [6]

has been sketched in the following Fig. 1

Fig. 1: Uniform continuum model of an elastic column

The symbols used in Fig. 1 and also in the ensuing analysis are specified as below

z stands for the level variable that refers to the initial state of static equilibrium of the

elastic column. By definition, 0z ! corresponds to the fixed end and z L! corresponds to

the free end of the column, with L standing for the initial (at-rest) length of the column.

t stands for the time variable.

P t stands for the axial external time-varying force imposed on the elastic column,

at its free end.

N z ,t stands for the internal elastic axial force developed at the level z of the elastic

column, at the time moment t.

A stands for the uniform cross-sectional area of the elastic column.

0

z

zdz

P t N z ,t

N z ,t dzz

x

x

N z, t

z=0

z=L


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The partial differential equation of motion of the uniform elastic column subjected to the ex-

ternal force P t at its free end z L! has the form

2 0c u z ,t u z,tdd !&& , (2.1)

where u z,t and c stand for the displacement response of the cross-section level z of the

elastic column at time t, and a constant, respectively. The constant c represents the velocity

of propagation of longitudinal waves along the column and is equal to [4 p.408]

c

V! , (2.2)

with E and V standing for the uniform modulus of elasticity (Youngs modulus) and the uni-

form mass density of the column, respectively.

Dots and primes over functions [e.g. u z ,t&& and u z ,tdd ] stand for differentiation with

reference to the time t and to the level z , respectively.

The partial differential equation (2.1) has been derived from Newton's second axiom as ap-

plied to the dynamics of an infinitesimal length of the elastic column at level z ,

N z ,tA u z,t

z

Vx

!

x

&& , (2.3)

and the generalized Hooke's law as a linear relation between the stresses and the strains at the

same point [7 p.8], [8 p.58] defining the internal elastic axial force N z ,t as follows


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N z ,t EA u z,td! . (2.4)

It is assumed that the elastic column continuum was at rest before subjected to the external

force P t , and hence, its initial conditions 0u z, and 0u z,& must be zero

0 0 0u z, u z,! !& . (2.5)

The influence of the external force tP on the dynamics of the elastic column is accounted

for through the inhomogeneous boundary stress condition

P t N L,t EA u L,td! ! , (2.6a)

while the fixed end at 0z ! is represented by the homogeneous boundary condition

0 0u ,t ! . (2.6b)

3. Transformation of an inhomogeneous boundary value into a homogeneous one

The classical approach to the partial differential equation of motion of the elastic column of

Fig. 1 requires homogeneous (i.e. zero) boundary conditions. Thus, a transformation of the

response function in the partial differential equation (2.1) is sought so that the inhomogene-

ous boundary condition (2.6a) will be transformed into a homogeneous one [9 p.435-436].

Such a transformation of the response function u z,t into a new one, say u z,t , is given as

below


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P t

u z, t u z,t zEA

! . (3.1)

On account of the above transformation, the partial differential equation (2.1), the initial con-

ditions (2.5) and the boundary conditions (2.6), we conclude that the following partial differ-

ential equation in u z,t must hold

2 222 2

u z,t u z,t P tc z

E Az t

x x !

x x

&&

, (3.2)

with the homogeneous boundary conditions

0 0 u ,t u L,t z! x x ! , (3.3)

and the initial conditions

0 0 0

0 andP u z, P

u z, z zE A t E A

x! !

x

&

. (3.4)

The transformed partial differential equation of motion (3.2) actually represents an equivalent

continuum dynamics problem of the elastic column, which is characterized by the transfor-

mation of the concentrated external force tP of the original problem into the external body

force P t E A zV && distributed all over the column.


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4. Uncoupling the partial differential equation of motion into undamped vibrations

The response u z ,t of the elastic column can be expressed as a Fourier series with respect to

the system of orthogonal functions of modes

1 2

nz sin n z LN T!

as follows [2 p.489

eq. (7-40), eq.(7-41)]

0

1 2nn

u z, t q t sin n z LTg

!

! , (4.1a)

0

21 2

L

n q t u z,t sin n z L dzL

T! , (4.1b)

with nq t standing for a Fourier coefficient. The modes 1 2n z sin n z LN T! are de-

rived from the first of the eigenvalue equations (1.3) and the homogeneous boundary condi-

tions 0 0n n LN Nd! ! corresponding to the homogeneous boundary conditions (3.3). The

Fourier series (4.1) is but Bernoullis transformation of the response u z,t [1 p.502-522].

By means of Bernoullis transformation (4.1), classical analysis uncouples the partial diffe-

rential equation of motion (3.2) into ordinary differential equations each one of which repre-

sents an undamped vibration in terms of a generalized displacement nq t .

Indeed, multiplying the partial differential equation (3.2) by 1 2sin n z LT and integrat-

ing from 0z ! to z L! gives

2 22

2 2

0 0

1 2 1 2

L L u z, t u z,tc sin n z L dz sin n z L dz

z t

x x !

x x

0

1 2

LP t

z sin n z L dzEA

T! &&

. (4.2)


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The first integral of the left-hand member after performing an integration by parts becomes

2

2

0

1 2

L u z, tsin n z L dz

z

x !

x

0

1 2 1 2 1 2

L u z, t u z, tsin n z L n L cos n z L dz

z z z

x xx ! ! x x x

0

1 2 1 2 1 2

Lu z,t

sin n z L u z,t n L cos n z Lz

x ! x

2 2

2

0

1 21 2

Ln

u z,t sin n z L dz

L

. (4.3)

Owing to the homogeneous boundary conditions (3.3), the inside brackets quantity in the last

right-hand member of equation (4.3) proves to be zero, and thus, by inserting Bernoullis

transform (4.1b), equation (4.3) may be rewritten as

2 2

20

1 2 1 22

L

n

u z, t Lsin n z L dz n L q t

z

x ! x

. (4.4)

The integral in the right-hand member of equation (4.2), by virtue of the equality [10 p. 164

eq. 216]

20

11 2

1 2

nL

z sin n z L dz

n L

TT

!

, (4.5)

becomes equal to


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20

11 2

1 2

nLP t P t

z sin n z L dzEA EA n L

TT

!

&& &&

. (4.6)

Now, it is classically assumed that differentiating the integral in Bernoullis transform (4.1b)

to the second order with respect to time t equals differentiating its integrand with respect to

time t (namely, assumption ofintegrandwise differentiability), viz.

2

2

0

21 2

L

n

u z,tq t sin n z L dx

L tT

x!

x&& , (4.7)

which, however, is only ensured by the continuity of the derivative 2 2u z ,t tx x for all val-

ues of z and t [2 p.286, Leibnitzs rule], [3 p.348]. And then, combining with equations

(4.2), (4.4) and (4.6), the classical analysis gets the ordinary differential equations [5 p.212]

2

2 2

2 1

or 0,1, 2,3,1 2

n

n n n

P t L

q t q t n=EA n[ T

!

&&

&& K , (4.8)

where n[ represents a natural angular frequency of the elastic column equal to

1 2n n c L[ T! . (4.9)

Operating similarly, but without using equation (4.7), on the initial conditions (3.4) yields

2 22 2

0 2 1 0 2 10 and 0

1 2 1 2

n n

n n

P L P Lq q

EA EAn nT T

! !

&

& . (4.10)


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It is emphasized that each one of the ordinary differential equations (4.8) describes a forced

vibration of a single-degree-of-freedom system in a generalized displacement nq t , and in

view of the initial conditions (4.10), is easy to be solved for a known external force tP .

Moreover, the lack of a damping term in the left-hand member of each ordinary differential

equation (4.8) suffices for theundamped nature of the forced vibration in nq t .

Since each ordinary differential equation (4.8) is equivalent to the partial differential equation

of motion (3.2) on the assumption of integrandwise differentiability (4.7), its undamped char-

acter indicates the undamped character of the partial differential equation of motion (3.2) on

the assumption of integrandwise differentiability (4.7). Only on this assumption, the classical

view that the response of an elastic continuous system is undamped seems to be reasonable.

5. The classical undamped solution of the partial differential equation of motion

As exposed above, on the assumption of integrandwise differentiability (4.7), the classical so-

lution of the partial differential equation (3.2) must be undamped. Let us evaluate this solu-

tion.

Making use of Duhamels integral (convolution integral), the general solution of the ordinary

differential equation (4.8) may be written as [4 p.234 eq. (5.22)]

0

0n

n n n n

n

qq t sin t q cos t[ [

[

! &

2 2

0

2 11 1

1 2

n t

nn

LP sin t d

EAnX [ X X

[ T

&& . (5.1)

Inserting the initial conditions (4.10) and performing the integration by parts, equation (5.1)


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becomes

2 20

2 1 12 0

1 2

n t

n n n n

Lq t P t P cos t P sin t d

E An[ [ X [ X X

T

!

, (5.2)

which combined with Bernoullis transform (4.1a) and equation (4.9) leads to

2

20

122 2 0

n

n

n n

cu z, t P t P cos t

L E A[

[

g

!

!

0

t

nn nP sin t d sin z

c

[[ X [ X X

. (5.3)

On account of Bernoullis transformation (4.1) and equations (4.5) and (4.9) it follows

0 0

21 2 1 2

L

n

z z sin n z L dz sin n z LL

T Tg

!

! !

2 2

0

121 2

1 2

n

n

Lsin n z L

nT

T

g

!

! !

22

0

12n

n

n n

csin z

L c

[

[

g

!

! , (5.4)

and combining with equations (3.1) and (5.3), we conclude the undamped classical solution

of the partial differential equation of motion (2.1)

2

20

122 2 0

n

n

n n

cu z, t P t P cos t

L EA[

[

g

!

!


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0

t

nn nP sin t d sin z

c

[[ X [ X X

. (5.5)

The classical solution (5.5) in u z,t and the ordinary differential equation (4.8) in the gener-

alized displacement nq t , which all represent undamped vibrations for the reasons exposed

in par. 4, have been considered for a long time as the general ones for the vibration of an ini-

tially at rest elastic uniform column subjected to an axial external force at its free end.

6. A critique of the classical undamped solution

There are three major points of dispute over the reliability of the classical undamped solu-

tions (5.2) and (5.5) as the general solutions of the problem:

1. The undamped vibrations (4.8), and hence, the undamped character of the responses

u z,t and u z ,t , have been derived from the partial differential equation of motion (3.2)

exclusively on the assumption of integrandwise differentiability (4.7).

2. The integrandwise differentiability (4.7) can only be assured if the second-order derivative

2 2u z ,t tx x is continuous for all values ofz and t, which, however, constitutes an arbi-

trary limitation to the response functions u z,t and u z,t , beyond the need for existence

of the second-order derivative 2 2u z ,t tx x dictated by the partial differential equation

(3.2). Thus, the assumption of integrandwise differentiability (4.7) is but an arbitrary limi-

tation to the elastic column continuum dynamics, which deprives the undamped vibrations

(4.8) of any reliability as equivalent to the partial differential equation (3.2), and hence,

the related classical general solutions (5.3) and (5.5) cannot be reliable.

3. The assumption of integrandwise differentiability (4.7) proves to be equivalent to the as-


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sumption of termwise differentiability of the infinite series in Bernoullis transform (4.1a).

And hence, the really general solution to the elastic column continuum dynamics requires

that the termwise differentiation rules be replaced by nontermwise differentiation rules.

The ensuing analysis elucidates these points and finally concludes that the really general so-

lution to the elastic column continuum dynamics must represent a damped vibration.

7. Equivalence of the assumptions about integrandwise and termwise differentiability

As shown in par. 4, each one of the ordinary differential equations (4.8) is equivalent to the

partial differential equation (3.2) on the assumption of integrandwise differentiability (4.7).

With the aim to use this postulate, we can multiply equation (4.8) by 1 2sin n z LT and

get the equivalent form

21 2 1 2n n nq t sin n z L q t sin n z LT [ T ! &&

2 2

2 11 2

1 2

nP t L

sin n z LEA n

TT

!

&&

, (7.1)

which, taking into account that the second-order derivative of 1 2sin n z LT and equa-

tion (4.9) allow the substitution

2

2

21 2 1 2n

dsin n z L c sin n z L

dzT [ T ! , (7.2)

becomes


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2

2

21 2 1 2n n

dc q t sin n z L q t sin n z L

dzT T ! &&

2 2

2 11 2

1 2

nP t L

sin n z LEA n

TT

!

&&

. (7.3)

Summing up all equations (7.3) for 0 1 2 3n , , , ,! K entails

2

2

20 0

1 2 1 2n nn n


dzT T

g g

! !

! &&

2 2012 1 21 2

n

n

Pt L sin n z L

E A nT

T

g

!

!

&&

, (7.4)

which in view of equation (5.4) results in

22

20 0

1 2 1 2n nn n

P tdc q t sin n z L q t sin n z L z

EAdzT T

g g

! !

! &&

&& . (7.5)

Equation (7.5) as a superposition of the ordinary differential equations (4.8) must be equiva-

lent to the partial differential equation of motion (3.2), on the assumption of integrandwise

differentiability (4.7). This equivalence entails the termwise differentiability of Bernoullis

transform (4.1a), viz.

2

20

1 2nn

u z, tq t sin n z L

t

g

!

x!

x && , (7.6a)

2 2

2 20

1 2nn

u z,t dq t sin n z L

z dz

g

!

x!

x . (7.6b)


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Indeed, applying Bernoullis transformation (4.1) to the function 2 2u z,t tx x , with this

function replacing the function u z, t in the transformation, gives

2

20

1 2nn

u z,tg t sin n z L

tT

g

!

x!

x , (7.7a)

2

20

21 2

L

n

u z, tg t sin n z L dz

L tT

x!

x , (7.7b)

with ng t standing for a new Bernoullis coefficient replacing nq t . Further, on account of

the assumption of integrandwise differentiability (4.7), Bernoullis transform (7.7b) yields

n ng t q t ! && , (7.8)

and substituting in Bernoullis transform (7.7a) gives the termwise differentiation rule (7.6a),

which combined with equation (7.5) and the partial differential equation of motion (3.2) re-

sults in the termwise differentiation rule (7.6b). And inversely, the assumption of termwise

differentiability (7.6a) via Bernoullis transformation (7.7) assures equality (7.8), and proves

to be equal to the assumption of integrandwise differentiability (4.7).

On this base therefore, Bernoullis transformation (4.1) and the assumption of integrandwise

differentiability (4.7) entail the termwise differentiability (7.6). And inversely, Bernoullis

transformation (4.1) and the termwise differentiability (7.6) entail the assumption of inte-

grandwise differentiability (4.7), which implies that the assumptions ofintegrandwise differ-

entiability (4.7) and termwise differentiability (7.6) are equivalent. Thus, recalling that the

assumption of integrandwise differentiability (4.7) is the prerequisite for the undamped char-

acter of the partial differential equation (3.2), it follows that the undamped character of the


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partial differential equation (3.2) requires the termwise differentiability (7.6). That is, with-

out the assumption of termwise differentiability (7.6), the partial differential equation (3.2)

cannot describe any undamped motion, notwithstanding conventional wisdom.

8. Uncoupling the partial differential equation of motion by termwise differentiability

The equivalence of the assumptions of integrandwise differentiability (4.7) and termwise dif-

ferentiability (7.6) can be verified by showing that the latter assumption in combination with

the partial differential equation of motion (3.2) and Bernoullis transformation (4.1) can result

in the classical ordinary differential equations (4.8) like the former assumption.

Indeed, inserting the termwise differentiation rules (7.6) in the partial differential equation of

motion (3.2), multiplying the resulting equation by 1 2sin z LT for 0,1,2,3, ! K , in-

tegrating all over the column length L , and after having made use of the differentiation prop-

erty (7.2) of the trigonometric function 1 2sin n z LT and its orthogonality property

0

0 for

1 2 1 2

for2

Ln

sin n z L sin z L dzL

n

{

! !

(8.1)

we can derive the classical ordinary differential equations (4.8) directly from the partial dif-

ferential equation of motion (3.2) on the assumption of termwise differentiability (7.6).


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9. Integrandwise and termwise differentiability as a limitation to the response

The analysis in par. 4 made quite clear that the undamped ordinary differential equations

(4.8) and their equivalence with the partial differential equation of motion (3.2), which suf-

fices for the undamped character of the motion, is due to the assumption of integrandwise dif-

ferentiability (4.7), viz.

2

20

21 2

L

n

u z,tq t sin n z L dz

L tT

x!

x&& . (4.7)

Actually, Bernoullis transform (4.1b) ensures the above assumption only on the condition of

a continuous derivative 2 2u z ,t tx x for all values of z and t[2 p.286, Leibnitzs rule], [3

p.348], which, however, is beyond the requirement of the existence of the derivative

2 2u z,t tx x for the formation of the partial differential equation of motion (3.2). Moreover,

by transformation (3.1), the derivative 2 2u z,t tx x must be discontinuous at any time t

where the surface traction p t or its rate p t& is discontinuous. Besides, the distribution of

the acceleration 2 2u z,t tx x along z cannot be a continuous function of time at all instants

t, due to the theoretical possibility of abruptly applying an infinitesimal additional velocity

distribution at a finite number of instants t without affecting the forced vibration of the col-

umn as a linear stable system. To explain, this possibility is realized by abruptly applying a

finite additional acceleration distribution of infinitesimal duration at a finite number of in-

stants t, which indicates that the acceleration distribution must vary discontinuously with

time t. That is, a discontinuous acceleration distribution cannot be ignored and must be

taken into account as a legitimate condition for the dynamic problem of the elastic column.

In short, the continuity of the acceleration 2 2u z,t tx x sufficing for the integrandwise diffe-


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rentiability (4.7), and hence, for the validity of the classical ordinary differential equations

(4.8), cannot be taken for consistent. Thus, the assumption ofintegrandwise differentiability

(4.7) is arbitrary and in fact constitutes a limitation to the class ofthe response functions

u z, t . This limitation deprives the classical ordinary differential equations (4.8) of any reli-

ability as the real ordinary differential equations in the generalized displacements nq t that

can be derived from the partial differential equation of motion (3.2). For this reason, the

classical view that damping is inconsistent with an elastic continuum, which is based on the

assumption of integrandwise differentiability (4.7), is deprived of any reliability.

As its equivalence with the assumption of integrandwise differentiability (4.7) indicates, the

assumption oftermwise differentiability (7.6) also is arbitrary[3 p.24-29]and in factconsti-

tutes a limitation to the class ofthe response functions u z,t . Actually, only the uniform

convergence of the infinite series in equations (7.6) can suffice for their derivation from Ber-

noullis transform (4.1a), that is, for the validity of the termwise differentiability of Ber-

noullis transform (4.1a) [2 p.407-417], [3 p.24-30]. And of course, assuming this uniform

convergence is but an arbitrary limitation to the class of the response functions

u z,t sought.

10. The damping effect of nonintegrandwise and nontermwise differentiability

Bernoullis transformation (4.1) itself does not require the assumption of integrandwise diffe-

rentiability (4.7) or the equivalent assumption of termwise differentiability (7.6). Either of

these equal assumptions actually constitutes a limitation to the class of the response functions

u z, t . So, in seeking the really general solution of the partial differential equation of mo-

tion (3.2), we have to remove this limitation and accordingly uncouple the partial differential

equation (3.2) into ordinary differential equations. This task can be done in two ways:


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24

1) By replacing the integrandwise differentiability assumption (4.7) with the nonintegrand-

wise general differentiation rule

2

2

2

0

21 2

L

n n n

u z, tq t sin n z L dz y t

L tT [x!

x&& , (10.1)

where 2n ny t[ stands for a generalized acceleration representing the difference between

the actual magnitude nq t&& and the integral in the above equation. Since 2n[ denotes a natu-

ral angular frequency, it follows that ny t must denote a generalized displacement.

Further, combining the nonintegrandwise general differentiation rule (10.1) with equations

(4.2), (4.4) and (4.6) yields the ordinary differential equations

2

2 2

2 1for 0,1, 2, 3,

1 2

n

n n n n

P t Lq t q t y t n =

E A n[

T

!

&&

&& K (10.2)

In view of equations (4.8) and the analysis of par. 4 and 8, the generalized displacement

ny t can be zero only on the integrandwise differentiability assumption (4.7), that is,

0ny t int egrandwise differentiability assumption (4.7)! . (10.3a)

Equations (10.2) are the actual ordinary differential equations in the generalized displace-

ments nq t that can in general be derived from the partial differential equation of motion

(3.2). It is quite clear that the ordinary differential equations (10.2) are equivalent to the par-


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tial differential equation of motion (3.2). And hence, the undamped or damped character of

the ordinary differential equations (10.2) indicates the undamped or damped, respectively,

character of the partial differential equation of motion (3.2). Let us examine whether the ac-

tual ordinary differential equations (10.2) represent undamped or damped vibrations.

Evidently, each ordinary differential equation (10.2) describes a forced vibration of a single-

degree-of-freedom system in a generalized displacement nq t , with the system internal force

2n n nq t y t [ having magnitude dependent not only on the displacement nq t of its

application point, which implies that its work does not depend only on the displacement

nq t . This latter assures that the internal force 2n n nq t y t [

must be nonconservative

[11 p.90], [12 p.3-4], which verifies that the actual ordinary differential equation (10.2) must

represent a damped vibration of the single-degree-of-freedom system.

On this base therefore, the partial differential equation ofmotion (3.2) must in general repre-

sent a damped motion. Only assuming the integrandwise differentiability (4.7), which im-

plies 0ny t ! , the internal force becomes equal to 2n nq t[ , and hence, conservative. Ac-

cordingly, the actual ordinary differential equations (10.8) are reduced to the classical ordi-

nary differential equations (4.8), thereby representing undamped vibrations. However, this

outcome is deprived of any reliability because of the arbitrariness of the integrandwise and

termwise differentiability assumptions (4.7) and (7.6) shown in par. 9.

2) By replacing the termwise differentiability assumption (7.6) with the nontermwise general

differentiation rules

2

20

1 2n tn

u z, tq t sin n z L R z, t

tT

g

!

x!

x && , (10.4a)


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2 2

2 20

1 2n zn

u z, t dq t sin n z L R z, t

z dzT

g

!

x!

x , (10.4b)

where t

z, t and z R z,t stand for remainder terms that cannot be described by only

nq t and n z sin n 1 2 z LN T! and their derivatives.

Inserting equations (10.5) in the partial differential equation of motion (3.2) will give

2

2

20 0

1 2 1 2n nn n


dzT T

g g

! !

&&

2

2 20

121 2

1 2

n

z t

n

P t Lc R z,t R z,t sin n z L

E A nT

T

g

!

!

&&

, (10.5)

which after using equation (7.2) and rearranging terms becomes

_ a 2 20

1 2n n n t zn

q t q t sin n z L R z, t c R z, t[ Tg

!

! &&

2 2

0

121 2

1 2

n

n

P t Lsin n z L

EA nT

T

g

!

!

&&

. (10.6)

Assuming that the series in equations (10.4) merely converge instead of classically assuming

that they uniformly converge, and on account of equation (10.6), which assures that the sum

2t z R z ,t c R z,t takes on the same homogeneous boundary values at the boundaries 0z !

and z L! as the system of orthogonal functions 1 2sin n z LT , we can express the sum

2t z R z,t c R z,t by means of Bernoullis transformation (4.1) as follows


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2 20

1 2t z n nn

R z, t c R z, t y t sin n z L[ Tg

!

! , (10.7a)

_ a 2 20

21 2

L

n n t zy t R z, t c R z, t sin n z L dzL

[ T ! , (10.7b)

where 2n ny t[ stands for a generalized acceleration, and hence, ny t denotes a new gen-

eralized displacement in addition to the generalized displacement nq t . Notwithstanding

that the generalized acceleration 2n ny t[ appears as a product, it actually represents a sin-

gle Bernoullis coefficient, say, 2n n ny t t [ ] ! .

Inserting Bernoullis transform (10.7a) in equation (10.6) yields

_ a 20

1 2n n n nn

q t q t y t sin n z L[ Tg

!

! &&

2 20

12 1 2

1 2

n

n

P tL sin n z L

EA nTT

g

!

!

&&

. (10.8)

Multiplying equation (10.8) by 1 2sin z LT with 0,1,2,3, ! K , integrating the result-

ing equation all over the length L of the column, interchanging the order of integration and

summation, and taking into account the orthogonality property (8.1), we can finally conclude

that equation (10.8) is equivalent to the ordinary differential equations

2

2 2

2 1or 0,1, 2, 3,

1 2

n

n n n n

P t Lq t q t y t n=

EA n[

T

!

&&

&& K (10.2)


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In view of equations (4.8) and the analysis of par. 4 and 8, the generalized displacement

ny t can be zero only on the termwise differentiability assumption (7.6), that is,

0ny t termwise differentiability assumption (7.6)! . (10.3b)

Thus, all conclusions derived from using the nonintegrandwise general differentiation rule are

also derived from using the nontermwise general differentiation rules.

11. The really general solution of the partial differential equation of motion

The velocity magnitude n ny t[ can be related to the generalized velocity nq t& as below

2n n n ny t t q t [ \ ! & , (11.1)

where n t\ denotes a scalar coefficient varying with time so that it ensures equation (11.1).

Actually, n t\ is a function of both the generalized magnitudes ny t and nq t& .

Inserting substitution (11.1) in the actual ordinary differential equation (10.2) yields

22 2

2 12 for 0,1, 2, 3,1 2

n

n n n n n nP t Lq t t q t q t n=EA n

\ [ [T

!

&&

&& & K , (11.2)

which discloses that the scalar coefficient n t\ is but the damping ratio of the thn mode.

Putting the damped general solutions of the ordinary differential equations (11.2) in Ber-


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noullis transformation (4.1) defines the really general solution of the partial differential

equation (3.2) as a damped response. And finally, using transformation (3.1) defines the

really general solution of the partial differential equation (2.1) as a damped response too. In

fact, the damped character of the general solutions to the partial differential equations (3.2)

and (2.1) reflects nothing but the equivalence of the differential equations (11.2) and (3.2).

In the case of an almost constant damping ratio n\ or a mean damping ratio n\ , the actual

differential equation (11.2) may be simplified to [13 p.12-15, p.48]

2

2 2

2 12 for 0,1, 2,3,

1 2

n

n n n n n n

P t Lq t q t q t n=

EA n\ [ [

T

!

&&

&& & K , (11.3)

whose general solution for the underdampedcase, i.e. 0 1n\ , equals [4 p.234 eq. (5.22)]

2 2

2

0 0

0 1 11

n n n n n nt

n n n n n n

n n

q q

q t e q cos t sin t \ [ \ [

\ [ \ [\ [

!

&

22 22

0

2 11 11

1 21

n n

n tt

n n

n n

LP e sin t d

E An

\ [ XX \ [ X XT\ [

&& . (11.4)

On account of the classical initial conditions (4.10), which are independent of the assumption

of integrandwise or termwise differentiability, the general solution (11.4) becomes

2

2 2

2 1 10 1

1 2

n n

n

tn n n

Lq t e P cos t

E An

\ [ \ [T

!


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30

22

0 01

1

n nn n

n n

P Psin t

\ [\ [

\ [

&

2

20

1

11

n n

tt

n nn n

P e sin t d

\ [ X

X \ [ X X\ [

&&

. (11.5)

Substituting in Bernoullis transform (4.1a) and using equation (4.9) gives

2

2

20

120 1n n

n

tn n

n n

cu z, t e P cos t

L EA

\ [ \ [[

g

!

!

22

0 01

1

n nn n

n n

P Psin t

\ [\ [

\ [

&

22

0

11

1

n n

tt n

n n

n n

P e sin t d sin zc

\ [ X [X \ [ X X\ [

&& , (11.6)

which is the really general solution of the partial differential equation (3.2).

Combining equations (11.6), (3.1) and (5.4), we conclude a really general solution of the par-

tial differential equation of motion (2.1) as below

2

2

20

120 1n n

n

tn n

n n

cu z, t P t e P cos t

L E A

\ [ \ [[

g

!

!

22

0 01

1

n nn n

n n

P Psin t

\ [\ [

\ [

&


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31

22

0

11

1

n n

tt n

n n

n n

P e sin t d sin zc

\ [ X [X \ [ X X\ [

&& , (11.7)

which represents a superposition of damped vibrations of the elastic column, thereby repre-

senting a damped response of the elastic column. Only on the arbitrary assumption of inte-

grandwise differentiability (4.7) or its equivalent assumption of termwise differentiability

(7.6), equations (10.3) and (11.1) make the damping ratio n\ be zero, thereby reducing the

really general solution (11.7) to the undamped solution

2

20

1 020 0

n

n nnn n

Pcu z, t; P t P cos t sin t

L E A\ [ [

[[

g

!

! !

&

0

1t

nn

n

P sin t d sin zc

[X [ X X

[

&& , (11.8)

which after performing the integration by parts becomes equal to the classical solution

2

20

120 2 2 0

n

n

n n

cu z, t; P t P cos t

L E A\ [

[

g

!

! !

0

tn

n nP sin t d sin zc

[[ X [ X X

. (5.5)

It is worth emphasizing that the really general solution (11.7) has been derived on the as-

sumption of constant damping ratios n\ , which limits the actual ordinary differential equa-

tions (11.2) to linear equations only. For the general case of varying damping ratios n t\ ,

equations (11.2) become nonlinear, thereby representing nonlinear damped vibrations in the


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generalized displacements nq t , and hence, the really general solution of the partial differ-

ential equation (2.1) becomes a superposition of these nonlinear damped vibrations.

12. The case of free axial vibration

A free axial vibration of the column corresponds with zero external force, 0P t ! , and

hence, with zero external force derivatives 0P t P t! !& && , which implies that u z,t u z ,t! ,

and is excited by the general initial conditions 0u z, and 0u z,& . In this case, the partial

differential equation of motion (2.1), due to its homogeneous boundary values, does not re-

quire any transformation, and Bernoullis transformation (4.1) can equally be rewritten as

0

1 2nn

u z, t q t sin n z LTg

!

! , (12.1a)

0

21 2

L

nq t u z,t sin n z L dz

L

T! . (12.1b)

Owing to the existence of the second-order derivative 2 2u z ,t tx x , Leibnitzs rule [2 p.286],

[3 p.348] can apply to Bernoullis transform (12.1b) and conclude the initial conditions

0

2

0 0 1 2

L

nq u z, sin n z L dzL T!

, (12.2a)

0

20 0 1 2

L

nq u x, sin n z L dzL

T! & & . (12.2b)


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All preceding equations also hold true for a free axial vibration of the elastic column by in-

serting the above general conditions. For example, the actual ordinary differential equations

in the generalized displacements nq t for the free axial vibration of the elastic column, after

putting 0P t !&& in equations (10.3), prove to be

2 0 for 0,1, 2,3,n n n nq t q t y t n =[ !&& K (12.3)

which, for the reasons explained in par. 10, represent damped vibrations of single-degree-of-

freedom systems. By means of substitution (11.1) the above equations become equal to

22 0 for 0,1, 2, 3,n n n n n nq t t q t q t n =\ [ [ !&& & K (12.4)

whose general solution for n nt constant\ \! ! equals

2 2

2

0 00 1 1

1

n nn n n nt

n n n n n n

n n

q qq t e q cos t sin t

\ [ \ [\ [ \ [\ [

!

&, (12.5)

with the initial values 0nq and 0nq& given by equations (12.2).

Now, inserting the general solution (12.5) in Bernoullis transform (12.1a) and using equation

(4.9) gives the following really general solution of the free axial vibration in u z,t

20 0

20 1n n

Lt n n

n n

n

u z, t e sin z u z, sin z dz cos tL c c

\ [ [ [ \ [g

!

!


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34

22

0

10 0 1

1

L

nn n n n

n n

u z, u z, sin z dz sin t c

[\ [ \ [

\ [

& , (12.6)

which represents a superposition of damped free axial vibrations of the elastic column,

thereby representing a damped response of the elastic column. Only on the arbitrary assump-

tion of integrandwise differentiability (4.7) or its equivalent assumption of termwise differen-

tiability (7.6), equations (10.3) and (11.1) make the damping ratio n\ be zero, thereby reduc-

ing the really general solution (12.6) to an undamped solution.

It is worth emphasizing that the above really general solution has been derived on the as-

sumption of constant damping ratios n\ , which limit the actual ordinary differential equations

(12.4) to linear equations only, thereby depriving them of their general validity as equations

representing nonlinear damped vibrations in the generalized displacements nq t .

13. Conclusions

The above analysis has justified V. Nikitas early criticism of 1996 that the classical applica-

tion of Bernoullis transformation to the dynamics of an elastic continuous system obeying

the generalized Hookes law is insufficient. To specify, by means of Bernoullis transforma-

tion, classical analysis can uncouple the partial differential equation of motion of an elastic

continuous system into ordinary differential equations representing undamped vibrations of

independent single-degree-of-freedom systems. So, the response of the elastic continuous

system appears to be undamped as a superposition of undamped vibrations, thereby appearing

to suggest the inconsistency of elasticity with damping.

However, classical uncoupling and conclusions are founded on two arbitrary equivalent as-


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sumptions about integrandwise and termwise differentiability of Bernoullis transformation.

Besides, both of these arbitrary assumptions prove not only to be responsible for the un-

damped character of the elastic response deduced from the partial differential equation of mo-

tion but also to form an arbitrary limitation to the class of the response functions.

To remove this limitation and obtain the general solution of the partial differential equation of

motion of an elastic continuous system obeying the generalized Hookes law, the integrand-

wise and termwise differentiability of Bernoullis transformation are replaced by noninte-

grandwise and nontermwise differentiability. As a result, the partial differential equation of

motion of the elastic continuous system is uncoupled into ordinary differential equations rep-

resenting damped vibrations of independent single-degree-of-freedom systems. And hence,

the general solution of the former is a superposition of the general solutions of the latter,

thereby being representative of a damped response. This general solution is reduced to the

classical undamped solution only on the arbitrary limitation of the classical assumption of in-

tegrandwise and termwise differentiability of Bernoullis transformation. This outcome indi-

cates that the elastic stresses ruled by the generalized Hookes law are responsible for damp-

ing, thereby indicating the consistency of elasticity with damping, notwithstanding conven-

tional wisdom.

References

[1] M. Kline,M

athematical Thoughtfrom

An

cient to

Modern Times, Oxford University

Press, 1972.

[2] W. Kaplan, Advanced Calculus, 2nd edition, Addison-Wesley, 1973.

[3] I. S. Sokolnikoff& R. M. Redheffer, Mathematics ofPhysics andModern Engineering,

2nd edition, McGraw-Hill, 1966.


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[4] A. Dimarogonas, Vibration forEngineers, 2nd edition, Prentice Hall, 1996.

[5] L. Meirovitch, Elements ofVibration Analysis, 2nd edition, McGraw-Hill, 1986.

[6] V. Nikitas, The Dynamics ofContinuous Systems Criticized, 2nd National Congress on

Computational Mechanics, Crete, Greece, p. 653-658, 1996.

[7] S. Timoshenko, Theory ofElasticity, 3rd edition, McGraw-Hill, 1970.

[8] I. S. Sokolnikoff, Mathematical Theory ofElasticity, 2nd edition, McGraw-Hill, 1956.

[9] C. Lanczos,Linear Differential Operators, Van Nostrand, 1961.

[10] L. Rde and B. Westergren, Mathematics Handbook, 4th edition, Springer, 1999.

[11] T. von Karman & M. A. Biot, MathematicalMethods in Engineering, McGraw-Hill,

1940.

[12] H. Goldstein, ClassicalMechanics, Addison-Wesley, 1980.

[13] A. Chopra,Dynamics ofStructures: Theory andApplications toEarthquake Engineer-

ing, Prentice-all, 2001.

Pd-1-Bernoulli's Transformation of the Response of an Elastic Body and Damping

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Transcript of Pd-1-Bernoulli's Transformation of the Response of an Elastic Body and Damping