A Time History Method for Analysing Operational Piping Vibrations

doi:10.4236/wjm.2012.26038

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World Journal of Mechanics, 2012, 2, 325-333

doi:10.4236/wjm.2012.26038 Published Online December 2012 (http://www.SciRP.org/journal/wjm)

A Time History Method for Analysing Operational Piping

Vibrations

Subrata Saha

Department of Piping, Reliance Ports & Terminals Ltd.—Engineering Division, Reliance Refinery, Jamnagar, India

Email: [email protected]

Received October 2, 2012; revised November 4, 2012; accepted November 16, 2012

ABSTRACT

Vibration failure of piping is a serious problem and a matter of concern for safety and reliability of plant operations.

Fatigue is the main cause of such failures. Due to the complexity of the phenomenon no closed form design solutions

are available. In our study an analytical technique based on the theory of vibrations in the time domain has been pre-

sented. Using the inverse theory, the problem has been reduced to a system of Volterra Integral equations to be solved

simultaneously at every time step. The solution of the inverse problem may be used in the conventional method to cal-

culate stresses and end reactions which are important from the perspective of engineering design and condition moni-

toring. The method is robust, simple and can b e easily adopted by practicing engineers.

Keywords: Vibration; Inverse Problem; Direct Problem; Time History; Fatigue Failure; Vibration Screening Criterion;

Integral Equation; Frequency; Damping

1. Introduction

Piping witnesses various vibratory loads throughout its

life cycle. These vibrations if not controlled will lead to

fatigue failures at points of high stress intensity or could

even damage the supports. All these could lead to plant

outage or even have more severe consequences like fire

and loss of human lives [1]. Thus it is imperativ e that the

piping system is to be safeguarded against s uc h h aza rd s.

For the standpoint of engineering design adequacy

check, dynamic analysis has to be carried out for the

piping system for which the forcing function has to be

known. This is the conventional method of analysis,

which is also termed mathematically as the direct prob-

lem [2-4]. But the major difficulty in the dealing with the

vibration problems lies in the estimation of the forcing

function. If the exciting forces can be quantified pre-

cisely, the system response can be determined with great

accuracy by the existing analytical methods. Thus the

estimation of the forcing function is essential for carrying

out the dynamic analysis and subsequent engineering

design check.

Unfortunately this is not readily possible in most cases

since the vibrations in an operating pipeline are flow in-

duced. The complexity of flow patterns and the mecha-

nism of force coupling render the determination of the

forcing function extremely difficult. In such a scenario

data in the form of field vibration measurements in con-

junction with some analytical methods can provide a

basis for estimating the dynamic force and stress [5-9].

The theory of Inverse Problems [2-4] invariably forms

the basic theoretical framework for such studies. Inverse

Theory has found wide applications in the fields of engi-

neering and mathematics. It has become indispensible

where the problems are ill-posed in absence of data. In

this sense Inverse Theory has got tremendous practical

value.

The vibration problems can be studied in both the fre-

quency and the time domains. In the frequency domain

the frequency response of the system is studied for the

determination of various parameters of interest [7-10].

For a recent work in the frequency domain for a closely

related application, one may refer to [8]. In the time do-

main the response time history of different points in the

form of observations are taken as the input for the study.

Some good amount of work has been done in the ap-

plication of inverse problems in the field of hyperbolic

partial differential equations or wave equations [5,11-

14]. The determination of point sources from observa-

tions has been the main theme of their studies. The pre-

sent study may be considered as a special case of such

applications. However there are some major differences.

Concentrated forces in the form of point loads at interior

points in the domain have been considered in the previ-

ous works, whereas in our case the point sources are the

end moments at the boundaries. Also we have considered

damping in the system which simulates the real case and

thus the treatment is to a great extent different from the

S. SAHA

326

previous ones. According to us such study in the time

domain has not been done. A few references [7] can be

found wherein dynamic stresses have been computed

from displacement measurements in the time domain.

The method for computing stresses from the displace-

ment measurements has been shown. However the me t hod

of force estimation is not provided. Also the topics on

existence and uniqueness of the solution which are the

key issues for inverse problems have not been addressed.

In our paper we shall present the theory in the time

domain and also the numerical scheme for the problem.

The numerical algorithm is simple and it can be easily

built into any of the common spreadsheet programs with

the help of macros. This we be lieve will find wide appli-

cation amongst the practicing engineers.

2. Current Practice—Vibration Screening

Criteria

The current practice is the vibration screening criteria

method. In this method the vibration response parameters

like velocity or displacements are measured in situ and

compared against some acceptance criteria. These are in

the form of graphs known vibration severity charts [15].

For refinery and petrochemical industries, these charts

are being extensively used. They are normally found to

be conservative.

Another widely used criterion is of ASME OM Code

[16] a standard followed for nuclear piping. Here the

vibration velocity for a piping span between nodes is the

criterion. The limiting valu e of the ve locity is d etermined

by the empirical relationship involving coefficients which

depend on several parameters like weld arrangements,

mass lumping etc. When the peak value of velocity is

less that 12.7 mm/sec, it may be assumed that the piping

has sufficient dynamic capacity. If the vibration exceeds

this level, the guide recommends reviewing the same

with more information on the potential reasons of vibra-

tions and improving the vibration levels.

It is seen that all the above methods are conservative

and provide a cook book or a go/no-go approach. They

only tell us whether the vibrations are within the accept-

able levels or not. It is not possible to have a quantitative

estimation of the forcing function and the actual stress

levels which are essential fo r a design check. In our work

this problem has been studied on the framework of In-

verse Theory as mentioned earlier.

3. Mathematical Background

It has been shown that for a simply supported pipe [vide

Figure 1] the response at any location in the span may be

determined by the vibration measurements at two distinct

points in the span. For the straight span, the excitation

source is through moments from the adjoining segments

as there are no points of excitation by forces in the span.

A distinguishing feature of this method is that no infor-

mation is required on the natural B.C’s. This is remark-

able since in the direct formulation, the B.C’s govern the

solution, whereas in th is case they are not playing a role.

This is also significant for the fact that practically it is

impossible to measure the B.C’s.

3.1. Notations

In this section we will describe the notations used in the

sequel. Please refer to Table 1.

3.2. Problem Formulation

The basic configuration is shown in Figure 1 in which

the simply supported pipe is excited by moments

()

and

()

t at the ends. The length of the span is L.

Considering Bernoulli-Euler formulation and viscous

damping, the dynamic equation of motion in the time

domain [17,18] is as follows:

() ()()

,, ,0

cEI

u xtu xtDu xt

 

 =



 

  (1)

Table 1. Nomenclature.

Symbol Description

x Space variable.

T Time variable.

T Total time.

L Length of the pipe span.

 Spatial derivative operator.

 Time derivative of f (t).

Shape functions.

m Mass per un it leng th .

c Viscous d ampi ng co effi cient .

E Modulus of elasticity.

I Moment of inertia.

()

,uxtTotal displacement variable.

()

,vxt Dynamic displacement variable.

θθ

 Rotational accelerations.

MEn d mo me n t s .

Undamped natural frequency (nth mode) .

Damped frequency (nth mode).

Mode shape (nth mode).

Modal damping.

()

Generalized modal displacement.

nnL

ΓΓ Modal participation factors.

[

]

0,T Closed interval between 0 to T.

[

]

0,CT Space of continuous functions in [0, T].

 Acceleration measurement time history.

KKernel of the integral equation.

TH. Time History.

S. SAHA

327

Figure 1. Piping configuration.

Boundary Conditions (B.C’s):

()( )

;0, 0, 0ut uLt==

(2)

()()( )()

0;0,, L

EI utMtEI uLtMt==DD (3)

Equation (1) pertains to vibrations without any exter-

nal loading in the span. It is similar to the free vibration

equation. However for our case the excitations are thro ugh

end moments. This is shown in B.C’s (3). In absence of

any forcing function in the span the sources of vibrations

are through the ends. As mentioned earlier, this is a sig-

nificant development, since in the earlier studies the

point sources of excitation forces have been dealt with.

Our study is aimed at the determination of the end mo-

ments by observation of the response of some internal

points. Then the response at any point in the span can be

determined. It also assumed that the system starts from

rest (i.e. it has zero initial conditions).

We now express the total displacement function in

terms of dynamic and quasi-static components as below.

() ()()

,,,uxtvxt gxt=+ (4)

The quasi-static part can be written in terms of the

shape or pa rticipation f un ct i ons

()

and

()

()() ()()()

,LL

xtx txt

θθ

(5)

The function

()

(respectively)

()

is de-

fined as the displacement of the points in the span with a

unit positive rotation at end at 0x= and

L= re-

spectively. Since the system starts from rest we haves

() ()

000 00

θθ

(6)

() ()

000 00

θθ



(7)

The following can be easily verified.

()( )

0, 0, 0vt vLt==

(8)

()( )

0, 0, 0vt vLt==DD

(9)

() ()

001 01

(10)

() ()

000

xx==

DD (11)

() ()

,0 0,0 0vx vx==

 (12)

Using (7) to (10) we can recast (1) as follows:

() ()()

() ()

()

,, ,

cEI

vxtvxtDvxt

ttgxt

θθ

 

 



 



+−



−



 

 

(13)

It is customary to consider damping in terms of dy-

namic displacements only and hence the last term in (13)

may be dropped . Equation (13) represen ts a forced vi bra-

tion problem with a distributed loading for a pipe with

clamped ends. Equation (12) represents the initial condi-

tions. However this being an inverse problem, the forcing

function 0

 and

 (the rotational accelerations) be-

come the unknown quantities which are to be determined.

Once they are found out, the problem is transformed into

a direct problem and is solvable using commonly used

numerical methods. The modal superposition method

will be the basis of our study in the sequel.

In line with the modal superposition theory the dy-

namic displacement may be expressed as the sum of

modal components as below:

() ()()

,nn

vxtxq t

∞

=; (14)

Here

()

is the Eigen-function for the th

n mode

for the clamped pipe.

()

satisfies the B.C’s (7) and

(8). In addition we have the orthonormal properties:

ord;f0

φφ

=≠

 (15)

nmx

φφ

 (16)

Further we define :

() ()

=Γ 

(17)

() ()

Ln Ln

=Γ 

(18)

With the above properties we get modal equation from

(12) as below:

()() ()

() ()

nnnnnn

nnLL

qtqt qt

ωξ ω

θθ

=Γ +Γ

 

  (19)

Equation (19) is the differential equation for the gen-

eralized modal displacement

()

. This is a second

order differential in time variable. Two initial conditions

are required for its solution. In our case we have zero

displacement and zero velocity at time 0t=. The solu-

tion for

()

is:

S. SAHA

328

()

() ()

()

expsin d

ndnnnLL

nn dn

ωθτθτ

ωτω ττ

=Γ+Γ

⋅− −−

 

(20)

Here 2

dn nn

ωω

=−

. It is also known as the

damped natural frequency.

It is clearly seen that we need to get estimates of the

rotational accelerations to obtain

()

. The modal ac-

celeration is obtained differentiating (20) twice and using

the below identity:

() ()()

,d, t

ut utut

ττ τ



 (21)

We now de fin e the following te rms:

()() ()

00nnLL

τθτθτ

=Γ +Γ

 

(22)

()

22 2

1,sin

nnndndn

ft t

ωωω τ

=− −

(23)

()( )

()

2,2 cos

nnndn dn

ft t

ωωω τ

=−− (24)

() ()

()

() ()

()

,e ,,

nnn

th ftft

ξω τ

ττττ

−−

(25)

ndn

αω

= (26)

The expression for generalized modal acceleration is

()( )()

nnn

qttht

αψττ



 (27)

()()()()

,,d

nnn

vxtxt ht

φαψττ









 (28)

The total acceleration which is a sum of dynamic and

quasi-static components can be written as

() ()()()()()

,, LL

uxtvxtxtxt

θθ

=+ +

 

(29)

Substituting (19) in (20) for

()

,vxt we have

()( )()

() ()()()()

nnn

xtht

txtuxt

φαψττ

θθ







++ =



 

(30)

Equation (30) is the fundamental equation for our

study. It is an integral equation of the second kind [11,

19]. The right hand side (RHS) quantity represents the

acceleration which is the observation. The left hand side

(LHS) contains the unknown forcing functions in form of

rotational accelerations. Our study will focus on the

method of solution for the unknown rotational accelera-

tions.

4. Solution Method

We will now address the aspects of exis tence and un ique -

ness by means of the following propositions.

4.1. Proposition 1

For a system as defined by the governing differential

Equations (13) with B.C’s (8), (9) and initial conditions

(12), the respon se (i.e. displacement, velocity etc.) at any

location x can be obtained from the measurement of ac-

celeration time history at any two interior points.

Proof: We begin with the assumption that

()

 and

()

 belong to the function space

()

0,CT

(i.e. the

space of continuous functions).

It is shown in Appendix A1 that (29) may be reduced

()()

()()()()

,d ,d

Kt Kt

ϕϕ

ττ τ

ττ



 

 (31)

It is seen that (31) represents a pair of Volterra Integral

equations [11,19,20] (one for

()

 and the other for

()

 ).

For the first part of the assertion we need to show that

the trivial solution is the only solution. It has been proved

in Appendix A2 that both the integral equations have

unique fixed points. It is also seen that the trivial solution

exists. Hence it is also the unique solution.

As the existence and uniqueness of (31) has been es-

tablished, we can solve for the individual forcing func-

tions at any time t from the below system:

()()()()()()

11 0101

,, LL

uxt vxtxtxt

θθ

=+ +

 

(32)

()() ()()()()

22020 2

,, LL

uxtvxtxtxt

θθ

=+ +

 

(33)

Equations 32 and 33 represent a system of integral

equations. The numerical solution method for a single

equation may be found in standard texts on numerical

methods [20]. Thus Equations (32) and (33) may be ex-

pressed in the matrix form as

() ()()

tt t

AX U= (34)

In an expanded form (34) can be written as

()

() ()

11 1211

21 2222

AA uh

 

−



=



 

 −



 



 

  (35)

The elements of the matrix are as below

() ()

ijnj niji

=Γ +



(36)

()

() ()

()

,,d

nn nn

Nttt

ininjnj

nnn

ft ft

ωτ

φαθτ

τξτ τ

−−−−

=Γ



(37)

iii

Uuh=−

 (38)

where 1, 2i= and 0,1j= and N is the number of

S. SAHA

329

modes. The solution t

is obtained from (30) as be-

low.

() ()

()

ttt

−

= (39)

(

()

U is the RHS vector of the measurements).

From the existence of the solution we know that op-

erator

()

is invertible and hence

()

is the unique

solution.

4.2. Proposition 2

The response obtained at any point is unique and inde-

pendent of the observation points. This means that if

()

1,vxt

is the response calculated on the basis of obser-

vations for the set of points

()

x and

()

2,vxt

for the set

()

x we have 12

vv=.

Proof: From Proposition 1 we know that the rotational

accelerations are determined uniquely. The response is

calculated on the basis of the solution of the direct prob-

lem (12) which is also unique. The forcing function is the

same for all cases. Hence we have 12

vv=.

4.3. Numerical Method

In this section we shall describe the numerical scheme

for the calculation of acceleration forcing function. Let T

be the total time interval for our study and T

N the num-

ber of time steps and N the number of modes. The objec-

tive of the scheme is to obtain t

for all the time in-

stants t1, t2,···etc. up to T. For convenien ce

()

will be

denoted as i

. The steps are described below:

1) Start with

() ()

XU= (40)

2) For any r

t we define the following quantities

()

{

() ()

()

}

,, e

nn rk

ninjn jk

nrknn rk

Hnij

tftt

ξω τ

φαθτ

τξ τ

−−

=Γ

+Δ



(41)

()

injni

Cij x

=+Γ



(42)

The components of the matrix A in (35) is constructed

()

ij ij

Ai jC==

(43)

The RHS vector U is

()

Uu Hnij

=−



 (44)

(Here i = 1 and 2)

3) Solve for r

from (39).

4) Repeat Steps 2 to 3 till T

rN=

4.4. Determination of Response Variables

We obtain the rotational accelerations as a solution of the

inverse problem. Now we can determine the forcing

function completely. Thus the problem is transformed

into a direct one, which may be solved using existing

methods for determining various response quantities like

displacement, velocity and stress time histories.

For example, Bending Moment, Shear Force and the

Bending Stress are calculated as below.

() ()

xtEIu xt= (45)

() ()

xtEIuxt= (46)

Bendin g Stress

Z= (47)

As a measure of structural integrity a mechanical de-

sign check against fatigue is required to be carried out

using the stress distribution. In the time domain it is cus-

tomary to apply Rain-Flow Counting Method [21] to

determine cumulative usage factor. The value of the us-

age factor should be less than unity which indicates that

the system is safe and no failures from fatigue are ex-

pected to occur in its design life. A value of the factor

greater than one is an indication of a possibility of failure

due to fatigue. However as a crude estimate we may con-

sider maximum zero to peak value of the stress and

compare it with the endurance limit. It should be less

than the endurance limit to designate a system as safe.

The time histories of velocities and the end reactions

can be computed through the direct problem. The end

reaction forces should be used for checking mechanical

design of the support structure. This will ensure integrity

of the pipe supports thereby accounting for an important

hazard of a vibrating piping system.

The velocity at a point may be compared against the

maximum permissible velocity as per common practices

as mentioned earlier. However in view of our detailed

analytical method they are not the essential parameters

and may be taken as an additional piece of inform ation.

4.5. Numerical Simulation and Validation

In order to validate the theory some numerical experi-

ments have been carried out. The problem considered is

as follows.

A simply supported pipe is excited through end mo-

ments. Two cases have been considered. In case 1 the

excitation moment is applied at only one end. In case 2

excitation moments are applied at both ends. For sim-

plicity the harmonic excitation comprising of sine and

cosine terms for a few frequencies have been considered

for the forcing functions. However any continuous time

varying function is permissible. The total time T consid-

ered is 200 seconds. The pipe material is steel, size 219

S. SAHA

330

mm outer diameter (O.D), thickness 8.18 mm and the

span is 8 m. A fixed damping ratio of 1% has been as-

sumed. Five points numbered 1 to 5 have been defined in

the span. Points 1 at x = 0 and 5 at x = L are the boundary

points. Points 2, 3 and 4 are interior points at locations

0.25 L, 0.5 L and 0.75 L respectively. These points have

been defined for the purpose of specifying the input and

output locations.

The direct problem is first solved using the forcing

function as the moments using standard software. The

dynamic analysis time history module of general purpose

Finite Element Analysis (FEA) software has been used

for the direct problem. This analysis model will be

termed as model D in the sequel. The results of the

analysis have been treated as the benchmark. The accel-

erations from model D have been considered as meas-

urements which are the inputs for our proposed method

which is based on Inverse Theory and denoted as model I

for reference. Displacements, stresses and end reactions

have been considered as the response parameters for

comparison with the benchmark.

5. Results and Discussions

The time step interval has been fixed based on the high-

est natural frequency. This is done for the purpose of

minimizing errors due to integration. For the details on

the theory one may refer to standard texts [17,18]. Five

modes have been considered for the problem.

Figures 2 and 8 show the moment time history for

Case 1 and Case 2 respectively. Graph D denotes the

input for direct problem model whereas graph I denotes

the calculated response for the Inverse Problem. It is seen

that the two graphs coincide implying unique correspon-

dence between the Inverse and Direct Problem for our

case.

The observation points are 2 and 4 where the accelera-

tion time histories are measured [see Figures 3 and 9].

The rotational accelerations are calculated from Equation

37 as per inverse theory. It is seen from Figure 4 that the

rotational accelerations are shown at point 1 only. This is

due to the fact that in Case 1 the excitations are applied

at one end. The other response quantities like end reac-

tions, displacements and stresses are shown in Figures 5-

7. In all cases there is no difference between the results

of the two models. In the sequel we shall use the abbre-

viations TH for time history, ATH for acceleration time

history and RTH for rotational time history.

The results for Case 2 are given in Figures 9-13. In

this case we have rotational accelerations for both the

ends unlike Case 1. Also a very close match between the

results of direct and inverse problem is observed similar

to Case 1. This is expected since the theoretical solution

for the two methods is essentially the same. The differ-

ence is basically due to the round off errors.

Plot of End Moments

Figure 2. TH. of end moment excitations (Case 1).

ccln. Measurement

Figure 3. ATH. at measurement points (Case 1).

Plot of Rotational Accln.

Figure 4. RTH. at end points (Case 1).

End Reaction Plot

Figure 5. TH. plot of end reactions (Case 1).

As mentioned earlier, the distinct advantage of the

method over the current ones is that quantitative estimate

of the stresses and the end reactions are obtained in this

method. This is significant from the aspect of condition

monitoring and engineering design. The reaction force

S. SAHA

331

Stress Plot

Figure 6. Stress TH. at interior points(Case 1).

Displacement Plot

Figure 7. Displ. TH. at interior points (Case 1).

Plot of End Moments

Figure 8. TH. of end moment excitation (Case 2).

Accln. Measurement

Figure 9. ATH. at measurement points (Case 2).

estimates will enable us to design the pipe supports,

whereas the stresses and displacements will be useful for

condition monitoring of the system.

Plot of Rotational Accln.

Figure 10. RTH. at end points (Case 2).

Plot of End Reactions

Figure 11. TH. plot of end reactions (Case 2).

Displacement Plot

Figure 12. Displ. TH. at interior points (Case 2).

Stress Plot

Figure 13. Stress TH. at interior points (Case 2).

6. Conclusion

Vibration failure in operational piping is a serious prob-

lem and there is a need for a comprehensive study and

analysis for its remedial measures. In this sense the pro-

posed study has got a tremendous practical value. A

S. SAHA

332

quantitative method with proper mathematical basis has

been provided in con trast to the cook book approach. By

this method it is possible to quantify stresses, velocities

and reaction forces. This gives us a basis for a proper

engineering design. The method being simple can be

easily adopted by engineers involved in trouble-shooting.

Several improvements in the model are in line and

planned for future work. These are like inclusion of

lumped mass in the span or pipe bends. These will widen

the range of application of the method and will be of

greater practical use.

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[1] W. G. Garrison, “Major Fires and Explosions Analyzed

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No. 9, 1988, pp. 115-122.

[2] C. W. Groetsch, “Inverse Problems in Mathematical Sci-

ences,” Vieweg Publishing, Wiesbaden, 1993.

[3] V. Komornik and P. Loreti, “Fourier Series in Control

Theory,” Springer Inc., New York, 2005.

[4] D. D. Ang, R. Gorenflo, V. K. Le and D. D. Trong, “Mo-

ment Theory and Some Inverse Problems in Heat Con-

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S. SAHA

333

Annexure 1:

This section deals with some mathematical details re-

quired for Proposition 1. We define the following terms.

() ()

000

nn n

βαφ

=Γ +



(A-1)

()()

LnnLn

βαφ

=Γ +



(A-2)

() ()

000

ϕβθ

=

 (A-3)

() ()

0LL

ϕβθ

=

 (A-4)

()

()() ()

()

01 2

1,,

nnnnn n

xftft

ωατ τ



=−Γ+





 (A-5)

()

()() ()

()

1,,

nnnnLn n

xftft

ωατ τ



=−Γ+





 (A-6)

Substituting the above in (29) we have

() ()

()()()()

,d ,d

Kt Kt

ϕϕ

ττ τ

ττ



 

 (A-7)

Lemma 1:

The integral equation defined below has a unique triv-

ial solution.

()()( )

,d t

Ktfft

τττ

= (A-8)

(Here f(t) is a continuous function belonging to C (0, T)

and the kernel K (t, τ) is also continuous in the domain

()()

0, 0,tX t with K (t, τ) = 0 for t

<.)

Proof: We will provide the sketch of the proof. For

details on may refer to any standard text on functional

analysis (e.g. [19]).

It can be proved that the operator T defined as

()()()

TftK tf

τττ

= (A-9)

is a contraction mapping. Hence it has a unique fixed

point. Thus (A-8) has a unique trivial solution.