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Coupled flexural-torsional vibration of beams in the presence of static axial loads and end moments
Journal of Sound and Vibration (1984) 92(4), 583-589
COUPLED IN THE
FLEXURAL-TORSIONAL PRESENCE
OF
VIBRATION
STATIC
AXIAL
LOADS
OF
BEAMS
AND
END
MOMENTS A. JOSHI AND
S. SURYANARAYAN
Department of Aeronautical Engineering, Indian Znstitute of Technology, Bombay, India (Received
12 January 1983, and in revised form 20 April 1983)
The problem of coupled flexural-torsional vibration of a deep rectangular beam in the presence of a static axial load and an end moment is studied. Closed form analytical solutions are obtained for simply supported boundary conditions. Numerical results are obtained for the coupled frequencies and mode shapes (in terms of the location of axes of rotation of the cross-section) for different values of the load and the geometry parameters. The results show that the predominantly flexural frequencies of coupled flexuraltorsional vibration can be obtained as uncoupled flexural frequencies of an equivalent beam-column by defining an effective axial load, and that by defining an equivalent moment parameter the presentation of the results for the predominantly flexural mode can be made in a form independent of the slenderness of the beam in the depth direction.
1. INTRODUCTION It is well known that static axial loads affect the transverse vibration of beams. Studies of the vibration of beam-columns [l] bring out this effect clearly. It has also been shown that static end moments couple transverse flexure and torsion of beams. The lateral stability of deep beams [2] is an example of this effect. Though the general formulation for the coupled vibration of thin-walled beams subjected to eccentric axial loads has already been given by Vlasov [3], numerical studies bringing out the influence of various parameters do not seem to have been attempted until recently. The effect of eccentricity of the axial load on the coupled vibration of a thin tensioned flat strip has been examined in some detail in reference [4]. The present paper describes a complete study of the coupled flexural-torsional vibration of a narrow rectangular beam subjected to a static axial load which may be either tensile or compressive and equal end moments. The ends of the beam are taken to be simply supported and the loads are assumed to remain constant over its span.
2. FORMULATION
AND SOLUTION
Figure 1 shows a uniform narrow rectangular beam which is made up of homogeneous, isotropic and linearly elastic material. The beam is acted upon by a static axial load P and a static moment M at the two ends. The governing equations for the coupled flexural (.y) and torsional (0) motion of the beam can be written as (a list of nomenclature is given in the Appendix) -Ezxx a4vo/az4+Pa2v,/az2+Ma20/az2-pA GJ a2e/az2+ 0022-460X/84/040583+07
(PzJA)
$03.00/O
a2e/az2+M 583
a2~o/a72=0,
a2vo/a22-pZp
a2e/aT2 = 0.
(1)
@ 1984 Academic Press Inc. (London) Limited
584
Figure
1. Geometry
For simply supported
and co-ordinate
system
of a narrow
rectangular
ends the mode shapes may be assumed vO(2, 7) = 6, sin (7rt/L)
beam.
as
sin WT,
O(z, 7) = #sin (572/L) sin WT, and the governing
equation
(1) rewritten (A -Ay) -MIF;EI, [
where L is the wavelength i, is the radius of gyration
(2)
as -M/ftEA
&,
(A-A,)
(3)
11 e 1 =O,
of the mode, A is the dimensionless frequency in the y direction. A and Y,,are given by
A =
7; = I,,/ rr’AL2.
p2L2/En2i2,,
and A0 correspond to the uncoupled flexural respectively, when M = 0 and are given by
A,
A,=l+P,
and torsional
frequency
A,=s+Ji
for the coupled
frequencies
values
of A,
(5)
with respect to the torsional frequency
S=[24/7r2(1+p)](L/d)‘. equation
and (4)
where P( =P/P,,) is the tensile axial load non-dimensionalized magnitude of the Euler buckling load (PC,), and S is the uncoupled when both P and M are zero, which can be expressed as
The characteristic
parameter
(6) can be obtained
from equations
(3) as A’-(A,+AB)A+AyA,-&f’s=&
(7)
where n;i(=M/M,,) is the static end moment non-dimensionalized with respect to the critical moment for lateral instability (MC,). Solution of the quadratic equation (7) gives the values of A for coupled flexural and torsional frequencies as A\,=A,+c, A, =A,,-c, (8) where
c is a correction
factor given by c=[(S-1)/2][{1+41\?2S/(S-1)2}“2-1].
(9)
COUPLED
FLEXURAL-TORSIONAL
VIBRATION
585
A1 and A2 correspond to the predominantly flexural and predominantly torsional modes, respectively. The corresponding mode shapes can be defined in terms of the position of the axes of the rotation of the cross-section as 7jr = nJ(&)
=M/c,
jj2 = &S/r,
=-c/A%,
(IO)
where rP is the polar radius of gyration and n = -i&l 8.
3. NUMERICAL
RESULTS
FOR
VARIOUS
SLENDERNESSES
Figures 2-5 present numerical results bringing out the influence of load and geometry parameters, namely p, A? and d/L, on the coupled frequencies and the mode shapes of the narrow rectangular beam. Though the analytical model adopted here is the EulerBernoulli beam, to show the limited extent of the effect of the slenderness on the frequency and mode shape parameters for the predominantly flexural mode a value of 0.25 is chosen for the slenderness parameter, d/L. Figure 2(a) shows that hr is a linear function of p and the slope is independent of A?. The line corresponding to A? = 0 represents uncoupled flexural vibration. It intersects the horizontal axis at P = -1, which corresponds to the case of Euler buckling, and the vertical axis at A1= 1, which represents uncoupled flexural vibration when P = M = 0. The line corresponding to k = 1 passes through the origin (AI = P = 0), which corresponds
Figure 2. Predominantly flexural frequency parameter parameter, KY’. -, d/L=O; ---, d/L=0.25.
A1 versus (a) axial load parameter
P and (b) moment
586
A. JOSH1
AND
S. SUKY/ZhAKAYAN
L---Y-F
Figure 3. Predominantly torsional d/L =0.25; load parameter I? -,
frequency parameter - - -, d/L =O.OOl.
A, versus (a) moment
parameter
(b)
-3.
0
’
’ 0.4
’
’ 0.8
’
’ I.2
’
1 I.6
’
1 2.0
’
in?’ and (b) axial
-
1 2.4
i7
Figure d/L=O.l;
4. The location of the axes of rotation, ---, dlL=0.25.
ql and ij2, versus moment
parameter
M. --,
d/L = 0; - - -.
to lateral instability. The A, versus P lines show a parallel horizontal shift towards higher values of Pas A? increases. Therefore the horizontal intercept of these lines corresponding to A, = 0, which represents the case of lateral instability, also increases. This means that lateral instability is also possible for tensile values of p if iI? is high enough.
COUPLED
FLEXURAL-TORSIONAL I
7
VIBRATION
”
II
’
r
587 1
(a)-
o-01 -
_----, -
0.005-
.' ‘\
,' 7 1s
- *' __--Odd
‘\_.
____-----_____ /v-z3
T---l._ '\\
‘1,
0.005-
I.0 -
Is? 0*5-
0
0.5
I.5
B
-,
Figure 5. The equivalent moment parameter h;i, and deviation dJL=Q; ---, d/L=O*l; ---, d/L=O.25.
(Me -k)
versus the moment
parameter
M.
Figure 2(b) shows that the variation of A, with &f” is nearly linear for finite values of d/L and becomes linear for d/L =O. It can also be seen from Figures 2(a) and (b) that d/L has only a marginal effect on the predominantly flexural frequency, Ai. Figures 3(a) and (b) show that the predominantly torsional frequency, AZ,also varies linearly with A?f2and p. It can be seen that A2 increases with increase in the moment and also shows a strong dependence on d/L, unlike the predominantly flexural frequency. Figure 4 shows the dependence of the coupled mode shapes on the end moment A?, in terms of the positions of the axes of rotation of the cross-section in dimensionless form (?ji and jj2), plotted as functions of A? for various values of d/L. It can be seen that when d/L = 0, fjl varies as 1/1\;1and ij2 as --&?I.For fi = 0, iii =co and +j2= 0, representing uncoupled modes. When A? takes a finite value both ii1 and jj2 assume finite values but jj2 is negative, indicating that the centres of rotation for the predominantly flexural and predominantly torsional modes are on opposite sides of the centre of the cross-section. Both +r and jj2 are independent of P (as given by equation (lo)), indicating that the coupling is entirely due to the applied static end moment. It can also be seen that the effect of non-zero d/L on the modes is marginal.
4.
SIMPLIFIED
RESULTS
FOR
PREDOMINANTLY
When d/L tends to zero the coupled frequencies equations (8) as A,=A,-ti2,
FLEXURAL Al
A2=Ae+ti2.
and
A2
FREQUENCY
can be obtained from (11)
588
A
JOSH1
For beams with finite slenderness. indicates that one can write.
AND
S.
SUKY.ANAKAY/\N
a comparison
/4,=/i-M’
c3
of equation
( 1 1) and equations
A,-=A,,+~M;,
R;if =c=[(s-1)/2)][{1+4fi.‘,%!&l)~}’
(8)
(12) ‘-I],
(13)
It is evident that && -+ &!! as d/L + 0. Thus a unique relationship exists between I@, and ti for various values of d/L as shown in Figure 5. It can be seen that ii& does not differ very much from &?, and for values of &f up to 2.0 the deviation is less than 2% for d/L = 0.1. This deviation is also shown on an expanded scale in Figure 5. If one considers the equivalent moment parameter &$ instead of the applied moment ti then A, can be treated as independent of the slenderness ratio d/L. If one replaces &? by I’&, in Figures 2(a) and (b) and 4 the curves for A, and ?j for finite value of d/L, as given by the broken lines, will merge into the corresponding solid lines. Thus the solutions for Ai for all finite values of d/L become the same as the solutions for d/L = 0 if &? is replaced by M,. In fact the A, versus fi curve for d/L = 0 is also the curve A, versus &Z, for all values of d/L. However the predominantly torsional frequency AZ does not collapse in this way because the uncoupled torsional frequency At1is itself a function of d/L (see Figure 3). A look at Figure 2(a) and equation (12) suggests that by defining an effective axial load parameter,
P,,=P-Mf, it is possible
to represent
the predominantly
flexural
A, = l+p,r.
(14) frequency
A, as
(1%
It can be seen that equation (15) corresponds to the equivalent beam column problem. If this transformation is incorporated and the horizontal axis in Figure 2(a) is considered to represent Per, then all the solid lines in Figure 2(a) collapse to a single line corresponding to fie = 0. Therefore it can be observed that the predominantly flexural frequency solution represented by equation (15) is valid for all values of d/L and applied moments.
5. CONCLUSIONS
A study of the problem of coupled vibration of narrow rectangular beams subjected to the combined action of a static axial load and equal end moments has been described. Closed form analytical solutions have been obtained for the case of simply supported ends. Numerical results are presented which bring out the influence of the load and geometry parameters on the coupled frequencies and mode shapes of the narrow rectangular beam. An equivalent moment parameter is defined which enables the results for the predominantly flexural frequency to be represented in a form independent of the slenderness of the narrow rectangular beam in the depth direction. It is shown that by a suitable definition of an effective axial load parameter the problem can be reduced to that of an equivalent beam-column. The results also bring out the fact that lateral instability can occur even when tensile axial loads are present if sufficiently large end moments are applied.
COUPLED
FLEXURAL-TORSIONAL
VIBRATION
589
ACKNOWLEDGMENT
The work presented in this paper forms a part of a Grant-in-Aid project sponsored and funded by the Aeronautics Research and Development Board, Directorate of Aeronautics, Ministry of Defence, Government of India. The financial support received is gratefully acknowledged. A presentation of the contents of this paper was made at the Twenty-sixth Congress of the Indian Society of Theoretical and Applied Mechanics held at Coimbatore, India from 28 to 31 December 1981. REFERENCES 1. S. TIMOSHENKO 1964 Vibration Problems in Engineering. New York: Van Nostrand. See pp. 374-377. 2. W. F. CHEN and T. ATSUTA 1977 Theory of Beam-Columns-Space Behaviour and Design volume 2. New York: McGraw-Hill Book Co. 3. V. Z. VLASOV 1961 Thin-Walled Elastic Beams. Jerusalem: Israel Programme for Scientific Translation. See pp. 393-397. 4. S. SURYANARAYAN and A. JOSHI 1982 Journal of Applied Mechanics 49,669-671. Coupled flexural-torsional vibration of an eccentrically stretched strip.
APPENDIX: A d : G Zxx,Zyy,Z, J Lo L M M-CT M n;r, p” P_cr P ‘Y ‘P
S t v, 00, 00 x>Y> 7. 77 f/l f/2 A 4 &I Al A2 CL
P 7 0 0
NOMENCLATURE
area of the cross-section of the beam depth of the narrow rectangular beam correction factor for frequencies (equation (9)) Young’s modulus of elasticity shear modulus moments of inertia about x and y axis and polar moment of inertia, respectively St Venant torsional constant span of the narrow rectangular beam wave length of the mode shape applied bending moment =(P/L) (Er,,GJ)“*, critical lateral buckling moment = M/ M,,, dimensionless moment parameter dimensionless equivalent moment parameter (equation (13)) number of waves in the mode shape applied axial load = aZEZx,/AL2, Euler critical buckling load =P/P,,, dimensionless load parameter slenderness parameter in weak (y) direction (equation (4)) =(Z,/A)“*, polar radius of gyration slenderness parameter in depth direction (x) (equation (6)) thickness of the narrow rectangular beam displacement in the y-direction Cartesian co-ordinate system =--fro/8, location of the axes of rotation of the cross-section from the centre dimensionless coupled flexural mode (equation (10)) dimensionless coupled torsional mode (equation (10)) dimensionless frequency parameter (equation (4)) uncoupled flexural frequency parameter (equation (5)) uncoupled torsional frequency parameter (equation (5)) coupled flexural frequency parameter (equation (8)) coupled torsional frequency parameter (equation (8)) Poisson’s ratio of the material mass density of the material time variable torsional displacement radian frequency of vibration
COUPLED IN THE
FLEXURAL-TORSIONAL PRESENCE
OF
VIBRATION
STATIC
AXIAL
LOADS
OF
BEAMS
AND
END
MOMENTS A. JOSHI AND
S. SURYANARAYAN
Department of Aeronautical Engineering, Indian Znstitute of Technology, Bombay, India (Received
12 January 1983, and in revised form 20 April 1983)
The problem of coupled flexural-torsional vibration of a deep rectangular beam in the presence of a static axial load and an end moment is studied. Closed form analytical solutions are obtained for simply supported boundary conditions. Numerical results are obtained for the coupled frequencies and mode shapes (in terms of the location of axes of rotation of the cross-section) for different values of the load and the geometry parameters. The results show that the predominantly flexural frequencies of coupled flexuraltorsional vibration can be obtained as uncoupled flexural frequencies of an equivalent beam-column by defining an effective axial load, and that by defining an equivalent moment parameter the presentation of the results for the predominantly flexural mode can be made in a form independent of the slenderness of the beam in the depth direction.
1. INTRODUCTION It is well known that static axial loads affect the transverse vibration of beams. Studies of the vibration of beam-columns [l] bring out this effect clearly. It has also been shown that static end moments couple transverse flexure and torsion of beams. The lateral stability of deep beams [2] is an example of this effect. Though the general formulation for the coupled vibration of thin-walled beams subjected to eccentric axial loads has already been given by Vlasov [3], numerical studies bringing out the influence of various parameters do not seem to have been attempted until recently. The effect of eccentricity of the axial load on the coupled vibration of a thin tensioned flat strip has been examined in some detail in reference [4]. The present paper describes a complete study of the coupled flexural-torsional vibration of a narrow rectangular beam subjected to a static axial load which may be either tensile or compressive and equal end moments. The ends of the beam are taken to be simply supported and the loads are assumed to remain constant over its span.
2. FORMULATION
AND SOLUTION
Figure 1 shows a uniform narrow rectangular beam which is made up of homogeneous, isotropic and linearly elastic material. The beam is acted upon by a static axial load P and a static moment M at the two ends. The governing equations for the coupled flexural (.y) and torsional (0) motion of the beam can be written as (a list of nomenclature is given in the Appendix) -Ezxx a4vo/az4+Pa2v,/az2+Ma20/az2-pA GJ a2e/az2+ 0022-460X/84/040583+07
(PzJA)
$03.00/O
a2e/az2+M 583
a2~o/a72=0,
a2vo/a22-pZp
a2e/aT2 = 0.
(1)
@ 1984 Academic Press Inc. (London) Limited
584
Figure
1. Geometry
For simply supported
and co-ordinate
system
of a narrow
rectangular
ends the mode shapes may be assumed vO(2, 7) = 6, sin (7rt/L)
beam.
as
sin WT,
O(z, 7) = #sin (572/L) sin WT, and the governing
equation
(1) rewritten (A -Ay) -MIF;EI, [
where L is the wavelength i, is the radius of gyration
(2)
as -M/ftEA
&,
(A-A,)
(3)
11 e 1 =O,
of the mode, A is the dimensionless frequency in the y direction. A and Y,,are given by
A =
7; = I,,/ rr’AL2.
p2L2/En2i2,,
and A0 correspond to the uncoupled flexural respectively, when M = 0 and are given by
A,
A,=l+P,
and torsional
frequency
A,=s+Ji
for the coupled
frequencies
values
of A,
(5)
with respect to the torsional frequency
S=[24/7r2(1+p)](L/d)‘. equation
and (4)
where P( =P/P,,) is the tensile axial load non-dimensionalized magnitude of the Euler buckling load (PC,), and S is the uncoupled when both P and M are zero, which can be expressed as
The characteristic
parameter
(6) can be obtained
from equations
(3) as A’-(A,+AB)A+AyA,-&f’s=&
(7)
where n;i(=M/M,,) is the static end moment non-dimensionalized with respect to the critical moment for lateral instability (MC,). Solution of the quadratic equation (7) gives the values of A for coupled flexural and torsional frequencies as A\,=A,+c, A, =A,,-c, (8) where
c is a correction
factor given by c=[(S-1)/2][{1+41\?2S/(S-1)2}“2-1].
(9)
COUPLED
FLEXURAL-TORSIONAL
VIBRATION
585
A1 and A2 correspond to the predominantly flexural and predominantly torsional modes, respectively. The corresponding mode shapes can be defined in terms of the position of the axes of the rotation of the cross-section as 7jr = nJ(&)
=M/c,
jj2 = &S/r,
=-c/A%,
(IO)
where rP is the polar radius of gyration and n = -i&l 8.
3. NUMERICAL
RESULTS
FOR
VARIOUS
SLENDERNESSES
Figures 2-5 present numerical results bringing out the influence of load and geometry parameters, namely p, A? and d/L, on the coupled frequencies and the mode shapes of the narrow rectangular beam. Though the analytical model adopted here is the EulerBernoulli beam, to show the limited extent of the effect of the slenderness on the frequency and mode shape parameters for the predominantly flexural mode a value of 0.25 is chosen for the slenderness parameter, d/L. Figure 2(a) shows that hr is a linear function of p and the slope is independent of A?. The line corresponding to A? = 0 represents uncoupled flexural vibration. It intersects the horizontal axis at P = -1, which corresponds to the case of Euler buckling, and the vertical axis at A1= 1, which represents uncoupled flexural vibration when P = M = 0. The line corresponding to k = 1 passes through the origin (AI = P = 0), which corresponds
Figure 2. Predominantly flexural frequency parameter parameter, KY’. -, d/L=O; ---, d/L=0.25.
A1 versus (a) axial load parameter
P and (b) moment
586
A. JOSH1
AND
S. SUKY/ZhAKAYAN
L---Y-F
Figure 3. Predominantly torsional d/L =0.25; load parameter I? -,
frequency parameter - - -, d/L =O.OOl.
A, versus (a) moment
parameter
(b)
-3.
0
’
’ 0.4
’
’ 0.8
’
’ I.2
’
1 I.6
’
1 2.0
’
in?’ and (b) axial
-
1 2.4
i7
Figure d/L=O.l;
4. The location of the axes of rotation, ---, dlL=0.25.
ql and ij2, versus moment
parameter
M. --,
d/L = 0; - - -.
to lateral instability. The A, versus P lines show a parallel horizontal shift towards higher values of Pas A? increases. Therefore the horizontal intercept of these lines corresponding to A, = 0, which represents the case of lateral instability, also increases. This means that lateral instability is also possible for tensile values of p if iI? is high enough.
COUPLED
FLEXURAL-TORSIONAL I
7
VIBRATION
”
II
’
r
587 1
(a)-
o-01 -
_----, -
0.005-
.' ‘\
,' 7 1s
- *' __--Odd
‘\_.
____-----_____ /v-z3
T---l._ '\\
‘1,
0.005-
I.0 -
Is? 0*5-
0
0.5
I.5
B
-,
Figure 5. The equivalent moment parameter h;i, and deviation dJL=Q; ---, d/L=O*l; ---, d/L=O.25.
(Me -k)
versus the moment
parameter
M.
Figure 2(b) shows that the variation of A, with &f” is nearly linear for finite values of d/L and becomes linear for d/L =O. It can also be seen from Figures 2(a) and (b) that d/L has only a marginal effect on the predominantly flexural frequency, Ai. Figures 3(a) and (b) show that the predominantly torsional frequency, AZ,also varies linearly with A?f2and p. It can be seen that A2 increases with increase in the moment and also shows a strong dependence on d/L, unlike the predominantly flexural frequency. Figure 4 shows the dependence of the coupled mode shapes on the end moment A?, in terms of the positions of the axes of rotation of the cross-section in dimensionless form (?ji and jj2), plotted as functions of A? for various values of d/L. It can be seen that when d/L = 0, fjl varies as 1/1\;1and ij2 as --&?I.For fi = 0, iii =co and +j2= 0, representing uncoupled modes. When A? takes a finite value both ii1 and jj2 assume finite values but jj2 is negative, indicating that the centres of rotation for the predominantly flexural and predominantly torsional modes are on opposite sides of the centre of the cross-section. Both +r and jj2 are independent of P (as given by equation (lo)), indicating that the coupling is entirely due to the applied static end moment. It can also be seen that the effect of non-zero d/L on the modes is marginal.
4.
SIMPLIFIED
RESULTS
FOR
PREDOMINANTLY
When d/L tends to zero the coupled frequencies equations (8) as A,=A,-ti2,
FLEXURAL Al
A2=Ae+ti2.
and
A2
FREQUENCY
can be obtained from (11)
588
A
JOSH1
For beams with finite slenderness. indicates that one can write.
AND
S.
SUKY.ANAKAY/\N
a comparison
/4,=/i-M’
c3
of equation
( 1 1) and equations
A,-=A,,+~M;,
R;if =c=[(s-1)/2)][{1+4fi.‘,%!&l)~}’
(8)
(12) ‘-I],
(13)
It is evident that && -+ &!! as d/L + 0. Thus a unique relationship exists between I@, and ti for various values of d/L as shown in Figure 5. It can be seen that ii& does not differ very much from &?, and for values of &f up to 2.0 the deviation is less than 2% for d/L = 0.1. This deviation is also shown on an expanded scale in Figure 5. If one considers the equivalent moment parameter &$ instead of the applied moment ti then A, can be treated as independent of the slenderness ratio d/L. If one replaces &? by I’&, in Figures 2(a) and (b) and 4 the curves for A, and ?j for finite value of d/L, as given by the broken lines, will merge into the corresponding solid lines. Thus the solutions for Ai for all finite values of d/L become the same as the solutions for d/L = 0 if &? is replaced by M,. In fact the A, versus fi curve for d/L = 0 is also the curve A, versus &Z, for all values of d/L. However the predominantly torsional frequency AZ does not collapse in this way because the uncoupled torsional frequency At1is itself a function of d/L (see Figure 3). A look at Figure 2(a) and equation (12) suggests that by defining an effective axial load parameter,
P,,=P-Mf, it is possible
to represent
the predominantly
flexural
A, = l+p,r.
(14) frequency
A, as
(1%
It can be seen that equation (15) corresponds to the equivalent beam column problem. If this transformation is incorporated and the horizontal axis in Figure 2(a) is considered to represent Per, then all the solid lines in Figure 2(a) collapse to a single line corresponding to fie = 0. Therefore it can be observed that the predominantly flexural frequency solution represented by equation (15) is valid for all values of d/L and applied moments.
5. CONCLUSIONS
A study of the problem of coupled vibration of narrow rectangular beams subjected to the combined action of a static axial load and equal end moments has been described. Closed form analytical solutions have been obtained for the case of simply supported ends. Numerical results are presented which bring out the influence of the load and geometry parameters on the coupled frequencies and mode shapes of the narrow rectangular beam. An equivalent moment parameter is defined which enables the results for the predominantly flexural frequency to be represented in a form independent of the slenderness of the narrow rectangular beam in the depth direction. It is shown that by a suitable definition of an effective axial load parameter the problem can be reduced to that of an equivalent beam-column. The results also bring out the fact that lateral instability can occur even when tensile axial loads are present if sufficiently large end moments are applied.
COUPLED
FLEXURAL-TORSIONAL
VIBRATION
589
ACKNOWLEDGMENT
The work presented in this paper forms a part of a Grant-in-Aid project sponsored and funded by the Aeronautics Research and Development Board, Directorate of Aeronautics, Ministry of Defence, Government of India. The financial support received is gratefully acknowledged. A presentation of the contents of this paper was made at the Twenty-sixth Congress of the Indian Society of Theoretical and Applied Mechanics held at Coimbatore, India from 28 to 31 December 1981. REFERENCES 1. S. TIMOSHENKO 1964 Vibration Problems in Engineering. New York: Van Nostrand. See pp. 374-377. 2. W. F. CHEN and T. ATSUTA 1977 Theory of Beam-Columns-Space Behaviour and Design volume 2. New York: McGraw-Hill Book Co. 3. V. Z. VLASOV 1961 Thin-Walled Elastic Beams. Jerusalem: Israel Programme for Scientific Translation. See pp. 393-397. 4. S. SURYANARAYAN and A. JOSHI 1982 Journal of Applied Mechanics 49,669-671. Coupled flexural-torsional vibration of an eccentrically stretched strip.
APPENDIX: A d : G Zxx,Zyy,Z, J Lo L M M-CT M n;r, p” P_cr P ‘Y ‘P
S t v, 00, 00 x>Y> 7. 77 f/l f/2 A 4 &I Al A2 CL
P 7 0 0
NOMENCLATURE
area of the cross-section of the beam depth of the narrow rectangular beam correction factor for frequencies (equation (9)) Young’s modulus of elasticity shear modulus moments of inertia about x and y axis and polar moment of inertia, respectively St Venant torsional constant span of the narrow rectangular beam wave length of the mode shape applied bending moment =(P/L) (Er,,GJ)“*, critical lateral buckling moment = M/ M,,, dimensionless moment parameter dimensionless equivalent moment parameter (equation (13)) number of waves in the mode shape applied axial load = aZEZx,/AL2, Euler critical buckling load =P/P,,, dimensionless load parameter slenderness parameter in weak (y) direction (equation (4)) =(Z,/A)“*, polar radius of gyration slenderness parameter in depth direction (x) (equation (6)) thickness of the narrow rectangular beam displacement in the y-direction Cartesian co-ordinate system =--fro/8, location of the axes of rotation of the cross-section from the centre dimensionless coupled flexural mode (equation (10)) dimensionless coupled torsional mode (equation (10)) dimensionless frequency parameter (equation (4)) uncoupled flexural frequency parameter (equation (5)) uncoupled torsional frequency parameter (equation (5)) coupled flexural frequency parameter (equation (8)) coupled torsional frequency parameter (equation (8)) Poisson’s ratio of the material mass density of the material time variable torsional displacement radian frequency of vibration