natural frequencies of beams under tensile axial loads

Journal of Sound and Vibration (1990) 142(3), 481-498

NATURAL FREQUENCIES OF BEAMS UNDER TENSILE AXIAL LOADS

A. BOKAIANt

Nuclear Design Associates, Canute Court, Toft Road, Knutsford WA16 ONL, England

(Received 18 September 1989, and in revised form 4 January 1990)

The effect of a constant axial tensile force on natural frequencies and mode shapes of a uniform single-span beam, with different combinations of end conditions, is presented. Numerical observations indicate that the variati_on of normalized natural frequency parameter, 0, with normalized tension parameter, U, is almost the same for clamped-pinned and pinned-free, and similarly for clamped-clamped and clamped-sliding beams; the variation of the sliding-free beam is only slightly different from that of the latter pair and the free-free beam. For pinned-pinned, pinn -

,‘;h” Tiding and sliding-sliding beams, this

variation may exactly be expressed as 0 = 1 + i?. This formula may be used for beams with other types of end constraints when the beam vibrates in a third mode or higher. It also gives the upper bound approximation to the fundamental natural frequency of a pinned-free beam. For beam wi h other types of boundary conditions, this approximation may be expressed as 0 = F 1 + yU (y < l), where the coefficient y depends only on the type of the end constraints.

It is found that when the dimensionless tension parameter U is greater than about 12, then U can be expressed as an analytical function of 0/ CJ, where R is the dimensionless natural frequency parameter. For such a beam in the first few modes, the natural frequency is independent of the flexural rigidity and the beam behaves like a string. The string solution gives a lower bound approximation to the natural frequency.

1, INTRODUCTION

Lateral vibrations of beams under tensile axial loading is of practical interest, and has wide application in civil, mechanical and aerospace engineering. In the deign of certain spacecraft structural components, for example, it sometimes becomes necessary to deter- mine the natural frequencies and mode shapes of beam-type components which are in a state of preload or prestress. Current designs for large flexible solar arrays [l] are such that the boom which supports the array is in a state of prestress due to the tension that must be maintained in the solar cell substrate.

Another example is a marine riser or a tether (tendon) of a tension leg platform. Both structures have to be maintained in tension to prevent buckling due to self-weight. They derive their lateral stiffness primarily from their axial tensile force. Large vibrational stresses are normally associated with a resonance that exists when the frequency of the imposed force, from whatever source, is tuned to one of the natural frequencies of these structures.

A marine riser or tether can be modelled as a long and extremely slender vertical prismatic tube. It may have a wide variety of end constraints [2-41. The case when one end of the tubular member is free, for example, corresponds to a hanging riser during installation or removal phase, or to a standing riser during emergency conditions [5]. This case may also be found in OTEC cold water pipes [6] and in mining engineering [7]. At present these structures are most commonly analyzed by large complex finite

t Present address: Brown & Root Vickers Ltd., 150 The Broadway, Wimbledon, London SW19 1RX.

481 0022-460X/90/210481 + 18 %03.00/O @ 1990 Academic Ress Limited

482 A. BOKAIAN

element programmes [8], which are cumbersome and expensive to use. A few simpler linear techniques are also available [9, lo]. One technique ignores the pipe bending stiffness. The other takes the stiffness into account but simplifies the calculation by assuming a uniform tension, which is equal to the tension at the bottom ball joint plus one-half of the pipe weight [ll], along the pipe length. These techniques need to be further clarified, both as an aid to the better understanding of marine riser structural behaviour, and to provide a tool for performing simple initial design calculations without resorting to large computer codes.

This paper is concerned with natural frequencies and mode shapes of a uniform beam under a constant tension with various end constraints. It is an extension of the author’s earlier work [ 121. The numerical results and the novel closed form solution presented in this manuscript provide designers and analysts with a quick simple estimate of natural frequencies of beams under tensile loads.

2. ANALYTICAL CONSIDERATIONS

Column 1 of Table 1 depicts uniform beams of length I with different combinations of end conditions, subjected to a constant tensile force T. On the assumption that the beam material is linearly elastic, and the shear deformation and rotary inertia are negligible, the differential equation for small deflection is

EZ d4Y(x)/dx4- T d*Y(x)/dx*-pAo*Y(x) =O, (1)

where Y(x) is the transverse displacement of a typical segment of the beam located at the distance x from the left-hand end, E the Young’s modulus, Z the second moment of are,a, A the cross-sectional area, p the mass density of the beam material and w the circular natural frequency. By introducing the dimensionless beam co-ordinate 5 = x/ 1, where OS 56 1, the solution of equation (1) may be written as

Y(x)= c, sinh Ml+c,cosh Ml+c, sin Nl+c,cos N& (2)

in which c, , c2, c3 and c, indicate constant coefficients, and M and N are defined as

M=I{(T/2EZ)+[(T/2EZ)*+(pA/EZ)w2]”*}”’=(U+~)”*,

N=I{-(T/2EZ)+[(T/2EZ)Z+(pA/EZ)w2]”2}”’=(-U+~)”2, (3)

where U = TZ*/2 EZ is the dimensionless tension parameter, f2 = wl*/ CY is the dimensionless natural frequency parameter and (Y = m (a list of symbols is given in the Appendix).

Consider the clamped-pinned beam for which the boundary conditions are Y(0) = 0, d Y(O)/dx = 0, Y(Z) = 0 and d* Y(Z)/dx’ = 0. Implementing these conditions on equation (2) will result in the following relationships:

c* + cq = 0, MC, + Nc3 = 0,

(sinh M)c,+(cosh M)c,+(sin N)c,+(cos N)c4=0,

(M’sinh M)c,+(M*cosh M)c,--(N’sin N)c,-(N’cos N)c,=O. (4)

The characteristic equation associated with these relationships is

(U+m)“‘cosh(U+m)“*sin(-U+m)“‘-

(-U+~)““sinh(U+~)“‘cos(-U+~)“r=O. (5)

TAB

LE

1

Cha

ract

eris

tics

of b

eam

s un

der

tens

ile

axia

l lo

ads

Slid

ing-

free

Cla

mpe

d-fr

ee

T-‘-g

_T

Pinn

ed-f

ree

Pinn

ed-p

inne

d

Cla

mpe

d-pi

nned

Cla

mpe

d-cl

ampe

d

Cla

mpe

d-sl

idin

g r-

@-g

-T

Slid

ing-

pinn

ed

Slid

ing-

slid

ing

Free

-fre

e J-

-T

Bou

ndar

y co

nditi

ons

Bea

m

desc

riptio

n

(1)

Left-

hand

en

d

(2)

dY(O

)=o

dx

’

Y(0

) =

0,

Y(0

) =

0,

Y(0

) =

0,

Y(0

) =

0,

Y(0)

= 0,

Y(0

) =

0,

dYoz

O

dx

’

dYoE

O

dx

’

d2W

)_0

dx’

’

dZ Y(

O) _

o

dx’

dYo=

O

dx

d2 Y(

O) _

o

dx2

d* Y

(0)

_ o

dx’

dYoz

O

dx

!$?E

!LO

dx

%!X

O

dx

d3 Y(

O) _

o

dx3

d7 Y

(0)

-=O

dx’

d3Y

(0)

7 dY

(0)

dx’+

+-=O

E

l dx

d2W

)_0

dx*

’

dZW

_O

dx=

’

dZW

)_O

dx

= ’

Y(I

) =

0,

Y(l)

=O,

Y(I

)-0,

CL!

!=0

dx

’

Y(/

)=O

,

!!wO

dx

’

d2W

_0

dx=

’

Rig

ht-h

and

end

(3)

d3 Y

(I)

T

dY(I

) ---

----_

O

dx3

El

dx

d’Y

(U

T dY

(O_O

~-

- dx3

El

dx

d3Y

(U

T dY

(UEO

dx’

El

dx

d2Y

(0_0

dx’

d2Y

(0_0

dx

’

d3 Y

(l)

_ o

dx’

dZY

(O_O

dx

2

dZY

(/)

dx’

~+J~

dYo=

O

dx’

El

dx

_

TAB

LE

l-con

tinue

d

Cha

ract

eris

tic

equa

tion

u ,,I

, (4

) (5

) PC

, (6

) 0;

(7

)

Var

iatio

n of

n

with

iJ

(8)

(‘d)

Mod

e sh

ape

coef

ficie

nts

cf;,

(2)

(U+~

)3’2

cosh

(U+~

)1/2

xs

in(-

lJ+J

UL+

RZ)

1’2

+(-c

J+Ji

Xzy

xc

os(-l

J+m

y

xsin

h(U

+m”‘

=O

R2+

RU

si

nh

(U +

m)“

’ xs

in(-

U+J

FGF)

‘/2

+(2U

2+f1

2)co

sh(U

+~)“

z

(-U

+~)3

’2co

sh(U

+~)1

/2

sin

(-U

+JiF

ZF)“

* -(

u+vv

Tir

y xs

inh

( U+J

U~+

~~~)

‘/~

(2;

- 1)

2mZ

T2E

I

8 41

2

(2i

- 1)

*7r*

?r

’EI

8 41

2

[ 1 2

(4i-1

): -

[ 1

(Zi-1

); 2

-

i27r

2 7T

2EI

-[

1 2 2

I2

(4i+

l);

i2?r

2 T

T2E

I

2 I’

(i7

r)’

fi=vG

E 0

M2

sinh

M

+ M

N

sin

N

1 -M

2cos

hM+N

Zco

sN

0

0 N

si

nh

M

M

sin

N

M

M2

sinh

M

+ M

N

sin

N

-- N

M

2 ca

sh

M +

N2

cos

N

M2

sinh

M

-- N

’ si

n N

1

0 0

2 9

zi2

N

‘A + ._ z

00

N ,. : “c: + ”

? c +

3

TAB

LE

l-con

tinue

d

Cha

ract

eris

tics

equa

tion

for

a la

rge

U

(13)

$ ta

n[U

(-l+

JZJ]

“*

Mod

e sh

ape

Mod

e sh

ape

coef

ficie

nts

coef

ficie

nts

for

a la

rge

U i

n fir

st

for

a la

rge

Cl

few

mod

es

I .

Val

ue o

f R

\

Upp

er

boun

d Lo

wer

bou

nd

elm

=

,, c,

, fo

r fir

st f

ew m

odes

c;

m

c;,

c;,

valu

e fo

r d

valu

e fo

r R

(1

4)

(15)

(1

6)

;y)

(18)

(1

9)

(20)

(2

1)

(24)

(2

5)

tan[

U(-

l+J$

)]“*

n’j

u3

=(-

l+Jl

+R

Z/U

y

1 -1

-x

1

2(2i

-1)J

D

I 2f

iJ-D

-1

_~

7r(2

i - 1

)

---

- $(

Zi+

l)JB

-

- -

0 01

0

0 1

0.92

5 Jl

+

0.92

5 ij

1 _

0.92

6 Jl

+O

.926

ii

- I.

13

0

I I

I I

I I

0 0

0

0

0

0

0 0

487

488 A. BOKAIAN

It is assumed that c, = 1. From equations (4) it is deduced that cZ = -tanh M, c3 = -M/N and cq = (Ml N) tan N. These coefficients vary with CJ in a complicated fashion.

Column 6 indicates the exact critical buckling load in the first mode, while column 5 shows the critical buckling load Umi corresponding to mode i [12]. Column 7 indicates Ri = oil*/a, where wi is the circular natural frequency of beam under no axial force in mode i and ai is its corresponding dimensionless parameter. The variation of 0 with U was obtained by solving equation (5) numerically.

The characteristic equation may be rewritten in the following form:

tanh (U+m)“’

(

u+JiKZ ‘I2

tan (-U+vZGZ)‘i2= -U+JiZZ 1 ’ (6)

When the value of U is large (U b 12) the numerator of the term on the left side of equation (6) approaches unity, and the above expression is simplified to

(7)

The solution of equation (7) is

U = tan-’ I (

-1+m

fllU 1 1 +i7r *I[--1 +m]. (8)

The variation of 0 with U was readily obtained by giving a fixed value to 01 U and solving equation (8) for U. The constant coefficients of mode shape in equation (2) are simplified as clm=l, czm=-l, Cam= -M/N and c,,=(M/N)tan N. Note that for a very large value of U, the computer was unable to calculate the hyperbolic functions of equation (5) as they became exceedingly large. Instead, equation (8) was employed. The observations for ten vibration modes are plotted in Figure 1, with the normalized natural frequency parameter b = n/n; versus the normalized tension parameter U = U/ U,,,i.

The results indicate that when the value of U is large, the value of 0 in the first few modes is considerably smaller than U. In this case the characteristic equation is further reduced to tan (n/m)=@/U-tan [(0/2U)+irr] or 0-27riU/(m-11)~ &Tin, which is equivalent to w = (h/l)-. The mode shape coefficients are further simplified as c;, = -2 U/0 = -m/(ni) and c;, = (2 U/n) tan (a/m) = (2U/R) tan k-O(c{,= ci,= 1). The above observations are conveniently tabulated in line 5 of Table 1.

Shown in Figure 2 is the effect of axial loading on the mode shapes of the beam. This figure depicts the normalized amplitude K defined as the vibration amplitude divided by the maximum amplitude along the beam length, against the beam co-ordinate ratio 5 for the first three modes, when the tension T = I’,.,. The corresponding results for beams with other types of boundary conditions are similarly presented in Table 1 and drawn in Figures 3- 11.

3. DISCUSSION OF THE RESULTS

For the pinned-pinned, sliding-pinned and sliding-sliding beams, the equation fi = m defines the variation of the normalized natural frequency parameter with the normalized tension parameter for all modes. The mode shape coefficients of these beams are constant values, those of the last two beams being identical. These are also the case when the beams are under a compressive load but u in the above formula should be replaced by - I? [ 121. As expected, a tensile force has the effect of increasing the motion frequency. Figures 2, 10 and 11 indicate that despite the heavily applied tension (equal to the critical buckling load P,,) the tension effect on the mode shapes is small.

FREQUENCIES OF AXIALLY TENSIONED BEAMS 489

0

0

omc~ononru~o . . . . . . . -000600000

4

490 A. BOKAIAN


Figure 6. Variation of ii with ti for a pinned-pinned, a pinned-sliding and a sliding-sliding beam.

9-

7-

'0 II,,,,,,,, IO 20 30 40 50 60 70 60 90 100

Figure 5. Variation of ii with u for a pinned-free beam.

IC

7-

'0 IO 20 30 40 50 60 70 80 90 100

492 A. BOKAIAN

FREQUENCIES OF AXIALLY TENSIONED BEAMS

0

493

E m _o”

m+ln*- - - a - - - - -

494 A. BOKAIAN

-0.2 -

-0.4-

-O-6-

-0.61 ’ ’ ’ ’ ’ ’ ’ ’ ’ ll 0 0.2 0.4 0.6 0.6 I.0 0 0.2 0.4 0.6 0.6 I.0

Figure 11. Variation of P with t for a sliding-free beam (I, T = 0; II, T = PC,). (a) i = 1; (b) i = 2; (c) i = 3.

An attempt was made to compare the variation of fi with i? for beams with different end constraints. This was done by replotting the data associated with the first, second, third and the tenth modes of Figures 1 and 3-9 in Figure 12. It is seen that in the fundamental mode, the variation of fi versus u is almost the same for clamped-pinned and pinned-free and similarly for clamped-clamped and clamped-sliding beams; this is also the case when these beams are under a compressive load [ 121. The variation of the sliding-free beam is only slightly different from the latter air and from the free-free beam. The data for the beams fall below the curve fi = se 1 + 0. However, a large scatter

Figure 12. Variation of ji with Q V, Sliding-free; X, pinned-free; 0, clamped-pinned; 0, clamped-clamped; 0, clamped-free; A, clamped-sliding; +, free-free; -, pinned-pinned, or sliding-pinned, or sliding-sliding.


of data is seen in the first mode. In particular, the clamped-free beam data exhibit a considerable deviation from the other data sets. The reason for this can be found in Figure 4, which shows that for this beam, at ii = 0 the value of fi is slightly different from 1. This is because the root of the associated characteristic equation (under no axial load) is somewhat different from the approximate value of [(2i - 1)(7r/2)12. The scatter becomes smaller in the second mode. In higher modes all data fall onto one another and the relationship between d and 0 becomes fi = m (Timoshenko et al. [ 13 ] presented the formula o, = ( i2r2a/ Z')Jl + 77’/ i2EZr’ for a pinned-pinned beam, Shaker [I] numerically calculated the variation of 0 against T/P,, for the first three modes in the range T < PC,, and Gorman [ 141 similarly presented the variation of R versus 77’/( T’EI) for the six modes of clamped-pinned and clamped-clamped beams).

The Rayleigh quotient for the fundamental natural frequency of a beam may be written as

jbEZ(d2Y/dx2)2+Tj;(dY/dx)2dx I:, pA Y2 dx

1+ Tj:,(dY/dx)Zdx 112

I:, EZ(d2 Y/dx2)’ dx 1 = w,Jl+ yT/Pcr, (9)

where PC, = & EZ(d’Y/dx’) dx/lL (d Y/dx)’ d x is the exact critical buckling load of a beam with no vibration, w, = [IA EZ(d2 Y/dx2) dx/]A pA Y2 dx]“’ is the fundamental natural frequency of the beam with no axial force, and y is a coefficient which de ends only on the type of end conditions. Expression (9) can be rewritten in the form d = J”-- 1 + yU and can be evaluated by using the well-known eigenfunctions for a beam with no axial load [15, 161. The value of y is shown in Table 1, and is identical to that in the case when the beam is under a compressive force [17,18].

Suppose Y(x) is the actual (unknown) mode shape for transverse vibration in the presence of the axial force; the Rayleigh quotient then may be written as

w2 = I:, EI(d2 Y/dx2)2 dx T I:, (d Y/dx)’ dx

I’ opAY’dx ’ + j; EZ(d’ Y/dx2)2 dx 1 ’ (10)

This relationship (10) is exact if the exact Y(x) is employed, and is only true if the oscillating mode shape is identical with the buckling mode shape. This is the case for pinned-pinned, sliding-sliding and pinned-sliding beams. Equation (10) may instead be evaluated with vibration and buckling treated separately a different approximation to evaluate the integrals): that is, w2 = w: (1 + T/P,,) or d = J(- I+ u. The static and dynamic mode shapes of most other beams are sufficiently similar for this relationship to be applicable.

To check the accuracy of this prediction, the first mode data of Figures 1, 3-9 are replotted in Figure 13 as dotted lines. In replotting the data of these figures, exact numerical values were used in the calculation of fl and 0 (which affect only the clamped-pinned and clamped-free data). The solid lines above the dotted lines are the upper bound solution to the natural frequent With the exception of the pinned-free beam, this solution is represented by fi = s”- 1-C yU, the value of which is smaller than 8 =x&?? (for most compressed beams, when u is replaced by -u, the former expression also giving the upper bound value while the latter gives the lower bound [ 121). For a pinned-free beam, the latter gives the upper bound solution (see column 24 of Table 1). This is almost certainly due to the dissimilarity of mode shapes in buckling and vibration. It is interesting to note that the clamped-clamped and clamped-sliding beams, which have the same numerical values of d versus 0, have also the same coefficients of y.

496 A. BOKAIAN

Figure 13. Variation of d with 0 in the first mode and its lower and upper bound approximations.

Column 18 of Table 1 shows that for the pinned-pinned, sliding-pinned, sliding-sliding sliding-free, clamped-pinned and clamped-clamped beams, the expression R = fi&Jii’ which is equivalent to wi = (jr/I)-, gives the beam natural frequency provided tha; the value of U is large but the mode number i is not. Furthermore, it indicates that similar relationships hold for beams with other types of end constraints, but with different proportionality constants. The fundamental mode prediction represented by this column is plotted as a solid line in Figure 13, and falls below the dotted line in all cases. A tether is usually long [19]. The above observations clearly indicate that for this structure or a deep-water riser, irrespective of the type of the end constraints, the natural frequency in the first few modes is independent of the flexural rigidity EZ and the structure behaves like a string. It is no surprise that the above prediction gives a lower bound approximation to the natural frequency (see column 25). It is interesting to mention that in his study of the influence of rigidity on static curvature of risers, Sparks [ 191 similarly found that the pipe bending stiffness plays a relatively insignificant role in their behaviour. Note that for a large U the clamped-free, clamped-clamped and clamped-sliding beams have the same shape coeficients. Furthermore, with the exception of a free-free beam, for a large U in the first few modes, the mode shape coefficients are directly proportional to only m.


4. CONCLUDING REMARKS

The variation of the normalized natural frequency parameter fi with the normalized tension parameter 0 for pinned-pinned, pinned-sliding and sliding-sliding beams is d = &%. This expression is found to give the upper bound approximation to the natural frequency of a pinned-free beam. For other beams this approximation is in the form d = m (y < l), where the coefficient y depends only on the type of the end constraints.

The numerical variation of fi with 0 is almost the same for clamped-pinned and pinned-free and similarly for clamped-clamped and clamped-sliding beams; the variation of a sliding-free beam is only slightly different from the latter pair and from that of the free-free beam. The effect of end constraints on natural fre uency is significant only in the first two modes. For higher modes the equation fi = ? 1 + 0 may be used for all beams.

It is found that when a beam is long or heavily tensioned, so that I/ > 12, then U can be expressed as an analytical function of L!/ U. For such a beam, in the first few modes, the parameter 0 is considerably smaller than U. Furthermore, the natural frequency is roughly independent of the flexural rigidity and the beam behaves like a string. For a large U, the cable solution gives a lower bound approximation to the natural frequency.

REFERENCES

1. F. J. SHAKER 1975 NASA Lewis Research Centre Report NASA-TN-8109. Effects of axial load on mode shapes and frequencies of beams. See pp. l-25.

2. W. FISHER and M. LUDWIG 1966 Journal of Petroleum Technology, 272-280. Design of floating vessel drilling riser.

3. M. M. BERNITSAS and T. KOKKINIS 1983 American Society ofMechanical Engineers, Journal of Energy Resources Technology 105, 277-281. Buckling of risers in tension due to internal pressure: nonmovable boundaries.

4. R. D. YOUNG, J. R. FOWLER, E. A. FISHER and R. R. LUKE 1978 American Society of Mechanical Engineers, Journal of Pressure Vessel Technology 100, 200-205. Dynamic analysis as an aid to the design of marine risers.

5. G. W. MORGAN 1980-1985 Petroleum Engineer-International (a series of articles published). Modem production risers.

6. J. R. PAULING 1979 O$shore Technology Conference, Houston, Paper No. OTC 3543. Frequency domain analysis of OTEC CW pipe and platform dynamics.

7. B. E. BENNETT and M. F. METCALF 1977 O@hore Technology Conference, Paper No. OTC 2776. Nonlinear dynamic analysis of coupled axial and lateral motions of marine risers.

8. M. H. PATEL, S. SAROHIA and K. F. NG 1984 Engineering Structures 6(3), 175-184. Finite- element analysis of the marine riser.

9. Y. C. KIM 1986 American Society of Mechanical Engineers, Ftfth Symposium on Offshore Mechanics and Arctic Engineering, 442-449. Natural frequencies and critical buckling loads of marine risers.

10. M. H. PATEL and S. SAROHIA 1982 American Society of Mechanical Engineers, Offshore Deep Seas Symposium, New Orleans, 139-148. The influence of non-linear marine riser behaviour on methods of analysis and design.

11. D. W. DAREING and T. HUANG 1976 Journal of Petroleum Technology, 813-818. Natural frequencies of marine drilling risers.

12. A. BOKAIAN 1988 Journal of Sound and Vibration 126, 49-65. Natural frequencies of beams under compressive axial loads.

13. S. TIMOSHENKO, D. H. YOUNG and W. WEAVER, JR. 1974 Vibration Problems in Engineering. New York: John Wiley. 4th edition; see pp. 453-455.

14. D. J. GORMAN 1975 Free Vibration Analysis of Beams and Shafts. New York: John Wiley. See pp. 359-381.

15. D. YOUNG and R. P. FELGAR 1949 The University of Texas, Publication No. 4913. Tables of characteristic functions representing normal modes of vibration of a beam.

498 A. BOKAIAN

16.

17.

18. 19.

20.

A

R. P. FELGAR 1950 The University of Texas, Circular No. 14. Formulas for integrals containing characteristic functions of vibrating beams. N. G. STEPHEN 1989 Journal of Sound and Vibration 131, 345-350. Beam vibration under compressive axial load-upper and lower bound approximation. A. BOKAIAN 1989 Journal of Sound and Vibration 131, 351. Author’s reply. A. BOKAIAN 1988 American Society of Mechanical Engineers, Seventh Symposium on O$shore Mechanics and Arctic Engineering. Estimation of natural frequencies of marine risers and TLP tendons. C. P. SPARKS 1980 American Society of Mechanical Engineers, Journal of Energy Resources Technology 102,214-222. Mechanical behaviour of marine risers: mode of influence of principal parameters.

APPENDIX: LIST OF SYMBOLS

cross-sectional area Cl 9 c2 9 c3 9 c4 mode shape coefficients elm, cZm, c,,, c,, mode shape coefficients for a large U clla?, c&J, c;,, c:cc mode shape coefficients for a large U in the first few modes E Young’s modulus I second moment of area

vibration mode number beam length parameters as defined in text axial tensile force

M N T PW lJ U Uln, X

Y(X)

F

critical buckling load in the first mode dimensionless tension parameter, 772/2EI normalized tension parameter, U/ U,,, dimensionless critical buckling load for vibration mode i distance from the left-hand end of the beam beam deflection normalized vibration amplitude dimensionless parameter, m dimensionless natural frequency -parameter, w12/ (Y dimensionless natural frequency parameter of beam under no axial force in vibration mode i normalized natural frequency parameter, R/R, circular natural frequency of beam under axial load circular natural frequency of beam under no axial force in vibration mode i mass density of beam material 3.1415927 dimensionless beam co-ordinate, x/ 1 a coefficient

natural frequencies of beams under tensile axial loads

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