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Stability boundaries of a rotating cantilever beam with end mass under a transverse follower excitation
Journal ofSound
and Vibration (1992)
STABILITY
154(l), 81-93
BOUNDARIES
OF A ROTATING
CANTILEVER BEAM WITH END MASS UNDER TRANSVERSE FOLLOWER EXCITATION R. C. KAR
AND
A
T. SUJATA
Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India (Received 4 April 1990, and infinalfirm
18 February 1991)
The dynamic stability of a rotating cantilever beam with end mass under a transverse follower parametric excitation has been studied. The regions of parametric resonance have been obtained by using the method of multiple scales. The effects of end mass, rotationalspeed, hub radius and warping rigidity on these zones have been investigated.
1. INTRODUCTION
The problem of a rotating, cantilever beam with an end mass under a transverse follower force arises in the design of certain helicopter rotor systems in which a time independent transverse force and tip mass represent the impulse and inertia of a constant thrust producing device at the free end of the rotor blade. Como [I] investigated the lateral buckling of a cantilever beam under a transverse follower force. Wohlhart [2] dealt with the lateral instability of an elastic beam of narrow rectangular cross-section with an intermediate concentrated mass subjected to a direction-controlled transverse follower force. Anderson [3] studied the stability of a rotating cantilever subjected to aerodynamic, dissipative and transverse follower forces. In a later work [4], he obtained the influence of tip mass, hub radius, rotational speed, warping rigidity and internal damping on the critical flutter load of a viscoelastic cantilever beam under the action of a transverse follower load. While all the works reported above are concerned with transverse loads of constant magnitude, Dugundji and Mukhopadhyay [5], and Dokumaci [6,7] dealt with the dynamic stability of beams under lateral pulsating (time-varying) forces. The present work deals with the study of simple and combination resonances of a rotating cantilever beam with an end mass subjected to a transverse follower parametric excitation. The method of multiple scales is used to-obtain the resonance zones of first and second order for various values of the system parameters.
2. FORMULATION
OF THE PROBLEM
In Figure 1 is shown a cantilever beam of length L, set off a distance COfrom the axis of rotation and rotating at a uniform angular velocity J2 about a vertical +-axis. A pulsating transverse follower force P(t) = I’,,+ PI cos wt is applied at the tip mass MO capping the free end of the beam at xl = L. The equations of motion for the coupled 81 0022-460X/92/070081 + 13 %03.00/O
0 1992 Academic Press Limited
82
R. C. KAR AND T. SUJATA
Figure 1. Configuration of the system.
flexural and torsional deformations
described by W(x, , t) and q5(x, , t) are
P~W,,-pG3W,,,,+~~33W,,,,,+(~33/~)[~(X,)~,,,l.,,+P~33.n2W.,, - [W,)
W,l,, - fTO[(L
~~23~,,,-~~~,,,,,+~~~,,,,, - (~23/~)[~(~I)~,Il.,
- x,M,,l.,
+ no4,,
+(~/4w(X,)~,,,l,,,
= 0,
(1)
+w~*-Pw,,,
+ p(122 - 133).n2$ - P(t)[@‘-
x,) w,,],, - P(t) w.1 = 0,
(2)
subject to the boundary conditions WO, O=
Wl(O,
0=6(0,
{~,W,,+p~33W,,,-~I,,W,,,
0
=
4.l@,
0
(3)
=o,
-(~33l~mww,,l.,
+~~~~*~-P~33~*lw,,}lx,=L=~,
(4)
{(Mo~ol~o)W,,,+~33[~+~(x,)IAlW,,)l,,=L=0, {wo~ol~oM,tt+
N-4.1rt-
Er#,lll-
(5)
(~I4w(~lM,lll,,
+[~~+(~23/~)~(~l)-~~*~l~,,}~x,=r.=~,
(6)
{~~o~l~o~~,,rr+~t~+~~~,~I~l~,,,}l~,=~=~~
(7)
where the longitudinal force iV(x,) is given by N(x,)=M~s22(C,+L)+pA~2[Co(L-x,)+~(L2-X:)],
(8)
with the notation W,, = 8 W/ax, , W., = 8 W/at, etc., p being the density of the material of the beam, A is its cross-sectional area, Z2*and G3 respectively the moments of inertia about xi- and x3-axes, I23 = Z2* + Z3,, E is the Young’s modulus, ET and ~1C are respectively the warping and torsional rigidity of the beam, A0 is the cross-sectional area of the tip mass and lo is its moment of inertia. Upon introducing the dimensionless variables 4 =x,/L, q= W/L, 5 =ct (where c2= EZ,3/pAL4) and 0 = CO/C,and defining the parameters co= Co/L, pA02L4/EZ33, r, = Z33/AL2, r,=Z23/AL*, Y=ET//.~CL*, Q=pC/EZ33, m=Mo/pAL, J=Zo/AoL*, G= r/AoL4,
wt=
83
PARAMETRICINSTABILITYOF ROTATING BEAM po=Pd2/EZn,p~ as
= PIL~/EZU andp(r)=po+pI
cos Or, equations (l)-(7)
can be rewritten
~-r~~“+f7”“+r,O~[No(~)ft”]“+r,w~~”-o~[No(~)~’l’ (9)
-p(7)W --5M’l’+P(~M’=o9 r,t$- rSQycj”+ Qyd”” + r,Qyo&Vo( 5 )#J“I”+ (r,Qy& +(~g-2~s)&$-P(wl
- QM ”- rg@@o(
6 )# ‘I’
-5h’l’-P(~bl’=0,
(10)
q(O, 7) = ry(0, 7) = f#l(O,7) = 4 ‘(0, 7) = 0,
(11)
{m~+r,~‘-~“‘-r,w~[No(~)~“]‘+w~[No(~)-~,l17’}l~=,=o,
(12)
whereNo(~)=m(l+~o)+~o(l-~)+~(l-~2), ( )‘=a( )/a{ and (‘)=a( Equations (9)-( 15) admit non-trivial solutions of the form
)/ar, etc.
and
(16)
where J(r), g,(r) are unknown functions of time and nl( {), &( 5) are co-ordinate functions chosen to satisfy as many of the boundary conditions in equations (1 1)-( 15) as possible. It is further assumed that n,( 0, +,( 5) can be approximated by
17~(5)=5’+‘(a,,-2Bl,5+y,,52)and where alI, /3,,, ylr, a2,, /L and ~2~(r= !,2,. at,=(r+2)(r+3){(r+ PII=(r+3)(r+ y,,=(r+
l)(r+2){r(r+
l)+w$n(l
Pzl=(r+3)(r+l)(yr(r+2)+g0},
4r(t)=Sr+‘(a2,-
+
Y2r52), (17)
. . , N) are given bY
l)(r+2)+w&l l){r(r+2)+oh(l
+cg)},
v2r5
+c0)}. +c0)),
a2,=(r+2)(r+3){y(r+ y2r=(r+
l)(r+2){yr(r+
l)(r+2)+g0}, l)+goj,
with go = I+ r,&n( I+ co)/Q, which satisfy the boundary conditions obtained by deleting the terms containing p( 2) and time derivatives from equations (1 1)-( 15). Substitution of equations (16) and (17) in equations (9)-( 15) and application of the extended Galerkin method yields the matrix equations (18)
(19)
84
R. C. KAR
AND
T. SUJATA
where
H4k.j) = ’[~14j-(l-t)~i$jl s0 &(i,j)=
s0
G
’ [di$--(1-5MIrlrl
G- Vi(l)#j(l), i,j=1,2
,...,
N.
(20)
Equations (18) and (19) can be rewritten as a single matrix equation of the form
where14 = Wk>
I’-
3. DETERMINATION OF THE INSTABILITY REGIONS Equation (21) can be further written as {~}+(A)(X)+2
&cos Br[B]{X} = {O},
(22)
with
The linear transformation {X} = [Z]{ u } is introduced in equation (22), where [Z] is a non-singular matrix such that [Z]-‘[A][Z] is a Jordan canonical form. Multiplying the result by [Z]-’ yields ~~,+o;u.+~Ecos
8r ; b,u,=O, III=1
n=1,2 ,...,
N,
(23)
in which b,,, is a real constant, w. is the nth natural frequency of the system having distinct and positive eigenvalues, and E is assumed to be less than unity. The transition curves bounding the regions of instability for various resonant combinations of frequencies obtained using the following conditions [8] can be derived by the method of multiple scales.
PARAMETRIC
INSTABILITY
OF ROTATING
BEAM
85
Case(i): 8ZWi+Wj The boundaries of the regions of instability in the vicinity of (wi+ Oj) exist if Ati>0 and are given by
where 2W$j= i bi~ri[(wi+80)2-oP]-‘+ r=l A,= [bi,b,i/tiimj] Case (ii): i>j The transition curves in the neighbourhood the sign of Wj in equations (24) and (25).
C bi~li[(wi-8,)2-W~]-‘, r+j
and
O,=(Wi+Wj).
(25)
ONOli-Wj,
of (oi - Oj) are obtained simply by changing
Case (iii) : 28 N Oj+ The boundaries between the stable and unstable zones near (Wi+ Wj)/2 exist if pqpji>O and are described by Wj
8=~(Wj+Wj)+&2{~(Xi+Xj)f(~~~ji)"2},
(26)
where 2WiXi= i bi,b,i[{(wi+80)2-w~}-1 1=l 2~#,=
i bi,b,[(~j-Bo)‘-w~J-‘,
r=l
Case (iv): 28ZWi_Oj, i>j The transition curves in the vicinity of (Oi_Oj)/2 Oj in equations (26) and (27).
4.
+ {(oi-&)‘-w3}-‘], &=(Wi+Oj)/2.
(27)
are obtained by changing the sign of
RESULTS AND DISCUSSION
Numerical calculations have been carried out on a Cyber 180/840A Computer System for a beam with p0=0.5, G=O and Q= 1 to obtain the boundaries of the regions of instability of simple and combination resonances of kth order in the neighbourhoods of 2oi/k and (w(* Wj)/k (i>j, i= 1,2,3,4, j= 1,2,3, k= 1,2). In view of the fact that E= pl/2 and is less than unity, the maximum value Ofpj is restricted to 1.5 as a representative measure. The critical transverse follower loads obtained for relevant values of the system parameters have been compared, and found to be in good agreement, with those of Anderson [4]. The effects of rotational speed parameters wo, hub radius co, end mass parameter m, inertia parameter J of the end mass and the warping rigidity parameter y on the regions of dynamic instability of the beam are shown in Figures 2-13. Second order resonance zones of negligible widths have been omitted from the figures for the sake of greater clarity.
86
R. C.
KAR
AND
T. SUJATA
The instability regions with t-,=3.33 x 10-4, r,=O-33 x 10V4,y=O, m=0*2, J=O, co=0 and w. = 0, 5 and 15 are plotted in Figures 2-4 respectively. Both sum- and differencetype combination resonances are found to occur in all the three cases, and those associated with u94 are predominant. The first order region of sum-type combination resonance near l-5
‘G
VC
^,
‘: 3 I
3 +
s
;
4’
0.5
OC 16
4
36
53
Figure 2. Instability regions for r,=3.33~ Unstable.
56
72
81
86
10M4,r,=O.33~ 10m4, y=O, m=0.2,
J-O,
cO=O, oO=O.
?? .
1.5-
l,ON 2 a’
5 I .i
0 5-
o.o_
7
Figure 3. Instability regions for r,=3.33x Figure 2.
IO-‘, r,=O.33x
lo-‘, r=O, m=0.2, J=O, q=O, m0=5. Key as
PARAMETRIC
INSTABILITY
OF ROTATING
87
BEAM
1.:
1.c ,-
?
-2 d
d
I
0.5
0.c
I_
13
-
Figure 4. Instability Figure 2.
-i regions
for r,=3.33
x 10m4, r,=0.33
x 10-4, y=O, m=0.2,
J=O,
co=O, wg= 15. Key as
0 = (03 + w,) appearing in Figures 2 and 3 has been replaced by a difference-type one at 8 N (ox - ol) in Figure 4. While the increase in the value of w. from 0 to 5 has widened the zone near 8 = 2w2, the change from 5 to 15 has narrowed it. In both cases, most of the other regions have reduced in width as a result of an increase in the value of wo. Moreover, the rise in the rotational speed parameter has caused shifting of the region in the vicinity of (w4- wi) to a lower excitation frequency, and repositioned the others at higher frequencies. In Figures 3, 5 and 6 is depicted the effect of variation of m on the stability diagrams t-,=0.33x 10m4,y=O, J=O, co=0 and wo=5. The instability region with r,=3*33x 10w4, in the neighbourhood of (w4- w3), which is relatively significant with m=O, does not appear for non-zero values of m. The rise in the value of the end mass parameter has not only shifted the region near 8 = 20~ to a lower forcing frequency and those at 8 N (w4 - w,) and w4 to higher frequencies, but also reduced the spans of most of the resonance zones. On the other hand, while the increase in the value of m from 0 to 0.2 has shifted the other instability regions to lower frequencies, the change from 0.2 to 0.9 has repositioned most of the zones at higher frequencies and widened the one at 9 N w4. The boundaries of the instability regions for J=O, 0.001 and O-05 with r,=3.33 x 10P4, r,=0*33x 10M4,y= 0, co=O, wo= 5 and m =0*2 have been plotted in Figures 3, 7 and 8 respectively. For J= 0X)01, the combination resonance regions at 0 N (03 - w ,), (04 + o ,) and (w4- w2) have replaced those near 6 = (Ok + w,), (w4 - w,) and (w4 + 02) of Figure 3, and a new zone has appeared at the low forcing frequency in the neighbourhood of (w4- w3). The change of the inertia parameter from 0 to 0.001 has widened the regions at 8 NW~, and (w2+ Ok) and reduced the widths of others. A further rise in the value of J to 0.05 has caused disappearance of the regions near 8 = w4, (a4 - w2) and (~0~- wj) and reappearance of the one at 8 N (04 - oi). Moreover, there is a widening of the regions in the vicinities of (WZ- wi) and 2w1, and a narrowing of the others. A rise in J over the
88
R. C. KAR
AND
T. SUJATA
15-
1 .a N \ 0’
3’ I
7 N
3 +
i
I
.s
5 0 5-
o-o__ 8
s A./ 13
19
Figure 5. Instabili Figuie 2.
tY r egions
for i-,=3,33x
10m4
m=O,
J=O, q,=O, coo=5
Key as
j-
1.0
,-
5 3
s’
;
r
3‘ +
s
0.5
0.0
I
I
“87
&X0
*
I 112
v
lJG--I 117 li
e
Figure 6. Instability regions for Figure 2.
r,=3.33
x
10m4,r,=O.33 x 10-4, y=O, m=O.9, J=O, c,,=O, w,,=S. Key as
specified values has made the beam more sensitive to* periodic forces by shifting all the instability zones to lower excitation frequencies. In Figures 3, 9 and 10 are shown the regions of parametric instability for co=O.O, 0.2 and 0.9 with rK=3.33 x 10e4, t-,=0-33 X 10P4, y=O, oo=5, m=0.2 and J=O. The increase
PARAMETRIC
OF ROTATING
89
BEAM
r
A--
1.5-
INSTABILITY
1 o-
a’
s I 3”
5
l-4
0 5-
0.0 L 0
8 G 12
5 I
s
16
3”
3”
22
N
34
45
50
82
113
8
Figure 7. Instability regions for r,=3.33 x 1Om4,r,=O.33 x 10-4, ~‘0, m=O.2, ~=0.001, c,,=O, 00=5. beg as Figure 2.
Figure 8. Instability regions for r,=3.33 x 10m4,r,=O,33 x lO-4, y=O, m=0.2, Figure 2.
J=O.OS, co=O, 00=5. Key as
in the value of co from 0.2 to 0.9 has widened the resonance zone at f3=m4. Moreover, the changes in co from 0 to 0.2 and from 0.2 to O-9 have shifted the region near t?= (04 - w,) to lower frequencies. On the other hand, both cases of increase in hub radius
90
R. C. KAR
AND
T. SUJATA
1
1,
cu \ ^,
p'
:
3 0.
-
0,
s\ 5
Figure 9. Instability regions for r,= 3.33 x 10. “, r,=O,33x Figure 2.
10-.4, y=O,
m=0.2,J=O,
co=0.2, 0,,=5. Key as
1.5-
vo-
9'7
c
5
:
I
3
<
R.
;
s
53
64
.?
3 +
6
0.5-
o,o6
-
16
IS+-27
72
J-4 76
112
117
127
130
B
Figure IO. Instability regions for r,=3.33 Figure 2.
x 10e4, r,=O.33 x 10-4, y=O, m=0.2, J=O,
~0~0.9,
00~5.
Key as
have reduced the widths of most of the other zones and repositioned them at higher forcing frequencies. The effects of variation of the warping rigidity parameter y on the instability regions with r,=2*67 x 10e4, r,=O*39 x 10e4, CO= 0, wo= 5, m =0*2 and J=O are displayed in
PARAMETRIC 1,:
INSTABILITY
OF ROTATING
BEAM
r
1 .c
p’
5
5 +
I
,
.?
s
-2 3 +
3”
3”
N
0.5
0.0
+k
+i
Figure 11. Instability regions for r,=2.67~ Figure 2.
P-5 10e4, r,=O.39x
lo-‘, 7=0, m=O.2, J=O, ~~0,
~~5.
Key as
1.:
1.c
c 4’
3’ I
3
-
4
. :
3 +
2
‘;
0.5
0.0
Figure 12. Instability regions for r,=2.67 x lo-‘, r,=O.39 x 10m4,7=O.O2, m=0.2, J=O, c,,=O, cu,,=S. Key as Figure 2.
Figures 1l-l 3. The increase in the value of y from 0 to 0.02 has caused the appearance of a new resonance zone in the neighbourhood of (w, - a@. A further rise in the value of y to 0.1 has resulted in the disappearance of the regions near 8 = (a)4 - ~3) and (ci2,- o ,). While the increments in y have reduced the spans of most of the resonance zones, they
92
R. C. KAR AND T. SUJATA
-
1,:
1.c
3
d
3”
3 +
N
.?
0.5
0.0
;“”
+ik-
1
0
0
Figure 13. Instabilityregionsforr,=2.67~ Figure 2.
10-4, r,=O.39~ 10m4,y=0~1,m=0~2,J=0,c0=0,
00=5. Keyas
have relocated only the regions associated with w4 at higher excitation frequencies, thus showing negligible effects on the positions of the other regions along the frequency axis.
5. CONCLUSIONS
The parametric instability of a rotating beam with an end mass under a transverse follower excitation has been studied. The results reveal that first order combination resonances of sum- and difference-type are predominant in almost all cases. In addition, higher tip mass and inertia parameters may either stabilize or destabilize the system. On the other hand, increase in rotational speed, hub radius and warping rigidity make the beam less sensitive to periodic forces.
REFERENCES 1. M. COMO 1966 Internafionaf Journal of Solids and Structures 2, 515-523. Lateral buckling of a cantilever subjected to a transverse follower force. 2. K. WOHLHART 1971 Zeitschiift fir Hugwissenschuften 19, 291-298. Dynamische kippstabilitPt eines plattenstreifens unter folgelast. 3. G. L. ANDERSON 1975 Journal of Sound and Vibration 39, 55-76. Stability of a rotating cantilever subjected to dissipative, aerodynamic and transverse follower forces. 4. G. L. ANDERSON 1976 Meccnnica 11, 89-97. The influence of tip mass on the stability of a rotating cantilever subjected to dissipative and transirerse follbwer forces. 5. J. DUGUNDJI and V. MUKHOPADHYAY 1973 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 40, 693-698. Lateral bending-torsion vibrations of a thin beam under parametric excitation.
PARAMETRIC INSTABILITY OF ROTATING BEAM 6. E. DOKUMACI 1978 Journalof Sound and Vibration !%, 233-238. Pseudo-coupled
93
bending-torsion vibrations of beams under lateral parametric excitation. 7. E. DOKUMACI 1979 ASME Paper No. 79-DET-91. Dynamic stability of pre-twisted blades under lateral parametric excitation. 8. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: John Wiley.
and Vibration (1992)
STABILITY
154(l), 81-93
BOUNDARIES
OF A ROTATING
CANTILEVER BEAM WITH END MASS UNDER TRANSVERSE FOLLOWER EXCITATION R. C. KAR
AND
A
T. SUJATA
Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India (Received 4 April 1990, and infinalfirm
18 February 1991)
The dynamic stability of a rotating cantilever beam with end mass under a transverse follower parametric excitation has been studied. The regions of parametric resonance have been obtained by using the method of multiple scales. The effects of end mass, rotationalspeed, hub radius and warping rigidity on these zones have been investigated.
1. INTRODUCTION
The problem of a rotating, cantilever beam with an end mass under a transverse follower force arises in the design of certain helicopter rotor systems in which a time independent transverse force and tip mass represent the impulse and inertia of a constant thrust producing device at the free end of the rotor blade. Como [I] investigated the lateral buckling of a cantilever beam under a transverse follower force. Wohlhart [2] dealt with the lateral instability of an elastic beam of narrow rectangular cross-section with an intermediate concentrated mass subjected to a direction-controlled transverse follower force. Anderson [3] studied the stability of a rotating cantilever subjected to aerodynamic, dissipative and transverse follower forces. In a later work [4], he obtained the influence of tip mass, hub radius, rotational speed, warping rigidity and internal damping on the critical flutter load of a viscoelastic cantilever beam under the action of a transverse follower load. While all the works reported above are concerned with transverse loads of constant magnitude, Dugundji and Mukhopadhyay [5], and Dokumaci [6,7] dealt with the dynamic stability of beams under lateral pulsating (time-varying) forces. The present work deals with the study of simple and combination resonances of a rotating cantilever beam with an end mass subjected to a transverse follower parametric excitation. The method of multiple scales is used to-obtain the resonance zones of first and second order for various values of the system parameters.
2. FORMULATION
OF THE PROBLEM
In Figure 1 is shown a cantilever beam of length L, set off a distance COfrom the axis of rotation and rotating at a uniform angular velocity J2 about a vertical +-axis. A pulsating transverse follower force P(t) = I’,,+ PI cos wt is applied at the tip mass MO capping the free end of the beam at xl = L. The equations of motion for the coupled 81 0022-460X/92/070081 + 13 %03.00/O
0 1992 Academic Press Limited
82
R. C. KAR AND T. SUJATA
Figure 1. Configuration of the system.
flexural and torsional deformations
described by W(x, , t) and q5(x, , t) are
P~W,,-pG3W,,,,+~~33W,,,,,+(~33/~)[~(X,)~,,,l.,,+P~33.n2W.,, - [W,)
W,l,, - fTO[(L
~~23~,,,-~~~,,,,,+~~~,,,,, - (~23/~)[~(~I)~,Il.,
- x,M,,l.,
+ no4,,
+(~/4w(X,)~,,,l,,,
= 0,
(1)
+w~*-Pw,,,
+ p(122 - 133).n2$ - P(t)[@‘-
x,) w,,],, - P(t) w.1 = 0,
(2)
subject to the boundary conditions WO, O=
Wl(O,
0=6(0,
{~,W,,+p~33W,,,-~I,,W,,,
0
=
4.l@,
0
(3)
=o,
-(~33l~mww,,l.,
+~~~~*~-P~33~*lw,,}lx,=L=~,
(4)
{(Mo~ol~o)W,,,+~33[~+~(x,)IAlW,,)l,,=L=0, {wo~ol~oM,tt+
N-4.1rt-
Er#,lll-
(5)
(~I4w(~lM,lll,,
+[~~+(~23/~)~(~l)-~~*~l~,,}~x,=r.=~,
(6)
{~~o~l~o~~,,rr+~t~+~~~,~I~l~,,,}l~,=~=~~
(7)
where the longitudinal force iV(x,) is given by N(x,)=M~s22(C,+L)+pA~2[Co(L-x,)+~(L2-X:)],
(8)
with the notation W,, = 8 W/ax, , W., = 8 W/at, etc., p being the density of the material of the beam, A is its cross-sectional area, Z2*and G3 respectively the moments of inertia about xi- and x3-axes, I23 = Z2* + Z3,, E is the Young’s modulus, ET and ~1C are respectively the warping and torsional rigidity of the beam, A0 is the cross-sectional area of the tip mass and lo is its moment of inertia. Upon introducing the dimensionless variables 4 =x,/L, q= W/L, 5 =ct (where c2= EZ,3/pAL4) and 0 = CO/C,and defining the parameters co= Co/L, pA02L4/EZ33, r, = Z33/AL2, r,=Z23/AL*, Y=ET//.~CL*, Q=pC/EZ33, m=Mo/pAL, J=Zo/AoL*, G= r/AoL4,
wt=
83
PARAMETRICINSTABILITYOF ROTATING BEAM po=Pd2/EZn,p~ as
= PIL~/EZU andp(r)=po+pI
cos Or, equations (l)-(7)
can be rewritten
~-r~~“+f7”“+r,O~[No(~)ft”]“+r,w~~”-o~[No(~)~’l’ (9)
-p(7)W --5M’l’+P(~M’=o9 r,t$- rSQycj”+ Qyd”” + r,Qyo&Vo( 5 )#J“I”+ (r,Qy& +(~g-2~s)&$-P(wl
- QM ”- rg@@o(
6 )# ‘I’
-5h’l’-P(~bl’=0,
(10)
q(O, 7) = ry(0, 7) = f#l(O,7) = 4 ‘(0, 7) = 0,
(11)
{m~+r,~‘-~“‘-r,w~[No(~)~“]‘+w~[No(~)-~,l17’}l~=,=o,
(12)
whereNo(~)=m(l+~o)+~o(l-~)+~(l-~2), ( )‘=a( )/a{ and (‘)=a( Equations (9)-( 15) admit non-trivial solutions of the form
)/ar, etc.
and
(16)
where J(r), g,(r) are unknown functions of time and nl( {), &( 5) are co-ordinate functions chosen to satisfy as many of the boundary conditions in equations (1 1)-( 15) as possible. It is further assumed that n,( 0, +,( 5) can be approximated by
17~(5)=5’+‘(a,,-2Bl,5+y,,52)and where alI, /3,,, ylr, a2,, /L and ~2~(r= !,2,. at,=(r+2)(r+3){(r+ PII=(r+3)(r+ y,,=(r+
l)(r+2){r(r+
l)+w$n(l
Pzl=(r+3)(r+l)(yr(r+2)+g0},
4r(t)=Sr+‘(a2,-
+
Y2r52), (17)
. . , N) are given bY
l)(r+2)+w&l l){r(r+2)+oh(l
+cg)},
v2r5
+c0)}. +c0)),
a2,=(r+2)(r+3){y(r+ y2r=(r+
l)(r+2){yr(r+
l)(r+2)+g0}, l)+goj,
with go = I+ r,&n( I+ co)/Q, which satisfy the boundary conditions obtained by deleting the terms containing p( 2) and time derivatives from equations (1 1)-( 15). Substitution of equations (16) and (17) in equations (9)-( 15) and application of the extended Galerkin method yields the matrix equations (18)
(19)
84
R. C. KAR
AND
T. SUJATA
where
H4k.j) = ’[~14j-(l-t)~i$jl s0 &(i,j)=
s0
G
’ [di$--(1-5MIrlrl
G- Vi(l)#j(l), i,j=1,2
,...,
N.
(20)
Equations (18) and (19) can be rewritten as a single matrix equation of the form
where14 = Wk>
I’-
3. DETERMINATION OF THE INSTABILITY REGIONS Equation (21) can be further written as {~}+(A)(X)+2
&cos Br[B]{X} = {O},
(22)
with
The linear transformation {X} = [Z]{ u } is introduced in equation (22), where [Z] is a non-singular matrix such that [Z]-‘[A][Z] is a Jordan canonical form. Multiplying the result by [Z]-’ yields ~~,+o;u.+~Ecos
8r ; b,u,=O, III=1
n=1,2 ,...,
N,
(23)
in which b,,, is a real constant, w. is the nth natural frequency of the system having distinct and positive eigenvalues, and E is assumed to be less than unity. The transition curves bounding the regions of instability for various resonant combinations of frequencies obtained using the following conditions [8] can be derived by the method of multiple scales.
PARAMETRIC
INSTABILITY
OF ROTATING
BEAM
85
Case(i): 8ZWi+Wj The boundaries of the regions of instability in the vicinity of (wi+ Oj) exist if Ati>0 and are given by
where 2W$j= i bi~ri[(wi+80)2-oP]-‘+ r=l A,= [bi,b,i/tiimj] Case (ii): i>j The transition curves in the neighbourhood the sign of Wj in equations (24) and (25).
C bi~li[(wi-8,)2-W~]-‘, r+j
and
O,=(Wi+Wj).
(25)
ONOli-Wj,
of (oi - Oj) are obtained simply by changing
Case (iii) : 28 N Oj+ The boundaries between the stable and unstable zones near (Wi+ Wj)/2 exist if pqpji>O and are described by Wj
8=~(Wj+Wj)+&2{~(Xi+Xj)f(~~~ji)"2},
(26)
where 2WiXi= i bi,b,i[{(wi+80)2-w~}-1 1=l 2~#,=
i bi,b,[(~j-Bo)‘-w~J-‘,
r=l
Case (iv): 28ZWi_Oj, i>j The transition curves in the vicinity of (Oi_Oj)/2 Oj in equations (26) and (27).
4.
+ {(oi-&)‘-w3}-‘], &=(Wi+Oj)/2.
(27)
are obtained by changing the sign of
RESULTS AND DISCUSSION
Numerical calculations have been carried out on a Cyber 180/840A Computer System for a beam with p0=0.5, G=O and Q= 1 to obtain the boundaries of the regions of instability of simple and combination resonances of kth order in the neighbourhoods of 2oi/k and (w(* Wj)/k (i>j, i= 1,2,3,4, j= 1,2,3, k= 1,2). In view of the fact that E= pl/2 and is less than unity, the maximum value Ofpj is restricted to 1.5 as a representative measure. The critical transverse follower loads obtained for relevant values of the system parameters have been compared, and found to be in good agreement, with those of Anderson [4]. The effects of rotational speed parameters wo, hub radius co, end mass parameter m, inertia parameter J of the end mass and the warping rigidity parameter y on the regions of dynamic instability of the beam are shown in Figures 2-13. Second order resonance zones of negligible widths have been omitted from the figures for the sake of greater clarity.
86
R. C.
KAR
AND
T. SUJATA
The instability regions with t-,=3.33 x 10-4, r,=O-33 x 10V4,y=O, m=0*2, J=O, co=0 and w. = 0, 5 and 15 are plotted in Figures 2-4 respectively. Both sum- and differencetype combination resonances are found to occur in all the three cases, and those associated with u94 are predominant. The first order region of sum-type combination resonance near l-5
‘G
VC
^,
‘: 3 I
3 +
s
;
4’
0.5
OC 16
4
36
53
Figure 2. Instability regions for r,=3.33~ Unstable.
56
72
81
86
10M4,r,=O.33~ 10m4, y=O, m=0.2,
J-O,
cO=O, oO=O.
?? .
1.5-
l,ON 2 a’
5 I .i
0 5-
o.o_
7
Figure 3. Instability regions for r,=3.33x Figure 2.
IO-‘, r,=O.33x
lo-‘, r=O, m=0.2, J=O, q=O, m0=5. Key as
PARAMETRIC
INSTABILITY
OF ROTATING
87
BEAM
1.:
1.c ,-
?
-2 d
d
I
0.5
0.c
I_
13
-
Figure 4. Instability Figure 2.
-i regions
for r,=3.33
x 10m4, r,=0.33
x 10-4, y=O, m=0.2,
J=O,
co=O, wg= 15. Key as
0 = (03 + w,) appearing in Figures 2 and 3 has been replaced by a difference-type one at 8 N (ox - ol) in Figure 4. While the increase in the value of w. from 0 to 5 has widened the zone near 8 = 2w2, the change from 5 to 15 has narrowed it. In both cases, most of the other regions have reduced in width as a result of an increase in the value of wo. Moreover, the rise in the rotational speed parameter has caused shifting of the region in the vicinity of (w4- wi) to a lower excitation frequency, and repositioned the others at higher frequencies. In Figures 3, 5 and 6 is depicted the effect of variation of m on the stability diagrams t-,=0.33x 10m4,y=O, J=O, co=0 and wo=5. The instability region with r,=3*33x 10w4, in the neighbourhood of (w4- w3), which is relatively significant with m=O, does not appear for non-zero values of m. The rise in the value of the end mass parameter has not only shifted the region near 8 = 20~ to a lower forcing frequency and those at 8 N (w4 - w,) and w4 to higher frequencies, but also reduced the spans of most of the resonance zones. On the other hand, while the increase in the value of m from 0 to 0.2 has shifted the other instability regions to lower frequencies, the change from 0.2 to 0.9 has repositioned most of the zones at higher frequencies and widened the one at 9 N w4. The boundaries of the instability regions for J=O, 0.001 and O-05 with r,=3.33 x 10P4, r,=0*33x 10M4,y= 0, co=O, wo= 5 and m =0*2 have been plotted in Figures 3, 7 and 8 respectively. For J= 0X)01, the combination resonance regions at 0 N (03 - w ,), (04 + o ,) and (w4- w2) have replaced those near 6 = (Ok + w,), (w4 - w,) and (w4 + 02) of Figure 3, and a new zone has appeared at the low forcing frequency in the neighbourhood of (w4- w3). The change of the inertia parameter from 0 to 0.001 has widened the regions at 8 NW~, and (w2+ Ok) and reduced the widths of others. A further rise in the value of J to 0.05 has caused disappearance of the regions near 8 = w4, (a4 - w2) and (~0~- wj) and reappearance of the one at 8 N (04 - oi). Moreover, there is a widening of the regions in the vicinities of (WZ- wi) and 2w1, and a narrowing of the others. A rise in J over the
88
R. C. KAR
AND
T. SUJATA
15-
1 .a N \ 0’
3’ I
7 N
3 +
i
I
.s
5 0 5-
o-o__ 8
s A./ 13
19
Figure 5. Instabili Figuie 2.
tY r egions
for i-,=3,33x
10m4
m=O,
J=O, q,=O, coo=5
Key as
j-
1.0
,-
5 3
s’
;
r
3‘ +
s
0.5
0.0
I
I
“87
&X0
*
I 112
v
lJG--I 117 li
e
Figure 6. Instability regions for Figure 2.
r,=3.33
x
10m4,r,=O.33 x 10-4, y=O, m=O.9, J=O, c,,=O, w,,=S. Key as
specified values has made the beam more sensitive to* periodic forces by shifting all the instability zones to lower excitation frequencies. In Figures 3, 9 and 10 are shown the regions of parametric instability for co=O.O, 0.2 and 0.9 with rK=3.33 x 10e4, t-,=0-33 X 10P4, y=O, oo=5, m=0.2 and J=O. The increase
PARAMETRIC
OF ROTATING
89
BEAM
r
A--
1.5-
INSTABILITY
1 o-
a’
s I 3”
5
l-4
0 5-
0.0 L 0
8 G 12
5 I
s
16
3”
3”
22
N
34
45
50
82
113
8
Figure 7. Instability regions for r,=3.33 x 1Om4,r,=O.33 x 10-4, ~‘0, m=O.2, ~=0.001, c,,=O, 00=5. beg as Figure 2.
Figure 8. Instability regions for r,=3.33 x 10m4,r,=O,33 x lO-4, y=O, m=0.2, Figure 2.
J=O.OS, co=O, 00=5. Key as
in the value of co from 0.2 to 0.9 has widened the resonance zone at f3=m4. Moreover, the changes in co from 0 to 0.2 and from 0.2 to O-9 have shifted the region near t?= (04 - w,) to lower frequencies. On the other hand, both cases of increase in hub radius
90
R. C. KAR
AND
T. SUJATA
1
1,
cu \ ^,
p'
:
3 0.
-
0,
s\ 5
Figure 9. Instability regions for r,= 3.33 x 10. “, r,=O,33x Figure 2.
10-.4, y=O,
m=0.2,J=O,
co=0.2, 0,,=5. Key as
1.5-
vo-
9'7
c
5
:
I
3
<
R.
;
s
53
64
.?
3 +
6
0.5-
o,o6
-
16
IS+-27
72
J-4 76
112
117
127
130
B
Figure IO. Instability regions for r,=3.33 Figure 2.
x 10e4, r,=O.33 x 10-4, y=O, m=0.2, J=O,
~0~0.9,
00~5.
Key as
have reduced the widths of most of the other zones and repositioned them at higher forcing frequencies. The effects of variation of the warping rigidity parameter y on the instability regions with r,=2*67 x 10e4, r,=O*39 x 10e4, CO= 0, wo= 5, m =0*2 and J=O are displayed in
PARAMETRIC 1,:
INSTABILITY
OF ROTATING
BEAM
r
1 .c
p’
5
5 +
I
,
.?
s
-2 3 +
3”
3”
N
0.5
0.0
+k
+i
Figure 11. Instability regions for r,=2.67~ Figure 2.
P-5 10e4, r,=O.39x
lo-‘, 7=0, m=O.2, J=O, ~~0,
~~5.
Key as
1.:
1.c
c 4’
3’ I
3
-
4
. :
3 +
2
‘;
0.5
0.0
Figure 12. Instability regions for r,=2.67 x lo-‘, r,=O.39 x 10m4,7=O.O2, m=0.2, J=O, c,,=O, cu,,=S. Key as Figure 2.
Figures 1l-l 3. The increase in the value of y from 0 to 0.02 has caused the appearance of a new resonance zone in the neighbourhood of (w, - a@. A further rise in the value of y to 0.1 has resulted in the disappearance of the regions near 8 = (a)4 - ~3) and (ci2,- o ,). While the increments in y have reduced the spans of most of the resonance zones, they
92
R. C. KAR AND T. SUJATA
-
1,:
1.c
3
d
3”
3 +
N
.?
0.5
0.0
;“”
+ik-
1
0
0
Figure 13. Instabilityregionsforr,=2.67~ Figure 2.
10-4, r,=O.39~ 10m4,y=0~1,m=0~2,J=0,c0=0,
00=5. Keyas
have relocated only the regions associated with w4 at higher excitation frequencies, thus showing negligible effects on the positions of the other regions along the frequency axis.
5. CONCLUSIONS
The parametric instability of a rotating beam with an end mass under a transverse follower excitation has been studied. The results reveal that first order combination resonances of sum- and difference-type are predominant in almost all cases. In addition, higher tip mass and inertia parameters may either stabilize or destabilize the system. On the other hand, increase in rotational speed, hub radius and warping rigidity make the beam less sensitive to periodic forces.
REFERENCES 1. M. COMO 1966 Internafionaf Journal of Solids and Structures 2, 515-523. Lateral buckling of a cantilever subjected to a transverse follower force. 2. K. WOHLHART 1971 Zeitschiift fir Hugwissenschuften 19, 291-298. Dynamische kippstabilitPt eines plattenstreifens unter folgelast. 3. G. L. ANDERSON 1975 Journal of Sound and Vibration 39, 55-76. Stability of a rotating cantilever subjected to dissipative, aerodynamic and transverse follower forces. 4. G. L. ANDERSON 1976 Meccnnica 11, 89-97. The influence of tip mass on the stability of a rotating cantilever subjected to dissipative and transirerse follbwer forces. 5. J. DUGUNDJI and V. MUKHOPADHYAY 1973 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 40, 693-698. Lateral bending-torsion vibrations of a thin beam under parametric excitation.
PARAMETRIC INSTABILITY OF ROTATING BEAM 6. E. DOKUMACI 1978 Journalof Sound and Vibration !%, 233-238. Pseudo-coupled
93
bending-torsion vibrations of beams under lateral parametric excitation. 7. E. DOKUMACI 1979 ASME Paper No. 79-DET-91. Dynamic stability of pre-twisted blades under lateral parametric excitation. 8. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: John Wiley.