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The natural frequencies of a thin rotating cantilever with offset root
Journal
ofSound and
Vibration
(1979) 65(2), 151-l 58
THE NATURAL FREQUENCIES CANTILEVER C.
Department
OF A THIN ROTATING
WITH OFFSET ROOT
H. J. Fox AND
J. S. BURDESS
of Mechanical Engineering, University of Newcastle upon Tyne, Newcastle upon l)ne NE1 IRU. England (Received 11 December 1978)
In this paper previous analyses of rotating cantilevers are extended to allow consideration of the case where the root offset from the spin axis is such that part of the beam is directed radially inwards towards the axis of rotation. In this situation the fundamental natural frequency is a function of the spin frequency. The form of the functional relationship depends on the value of r, the ratio of root offset to cantilever length. It is shown that, for certain values of r, it is possible to select a value of spin such that the fundamental natural frequency and the spin frequency coincide, the beam then being “tuned”. It is also possible to choose values of r such that the fundamental natural frequency reduces to zero as the spin is increased, and buckling occurs. It is further shown that by suitable choice of beam length for a given root offset the tuning and buckling frequencies can be minimized. The behaviour of higher modes of vibration is also discussed.
1. INTRODUCTION
The transverse vibration of rotating cantilevers has been widely studied in the context of aeroplane propellers and helicopter blades, and a summary of much of the previous work is given in reference [l]. In this previous work the root of the cantilever has been assumed either to lie on the spin axis or to be offset as shown in Figure l(a). Due to the effects of centrifugal loading it is known that the natural frequencies of transverse vibration increase with rotation. For this configuration the critical situation where a natural frequency coincides with the spin frequency cannot arise. L
-++-Irl L
R
4
LIT= q
Y
(b)
(a)
Figure
1. Cantilever
(c) configurations.
151 022%460X/79
140151+08
$02.00/O
c
1979 Academic
Press Inc. (London)
Limited
152
C. H. J. FOX AND J. S. BURDESS
In this present paper the above analysis is extended to permit consideration of the case of a rotating cantilever where either the root and tip are on opposite sides of the spin axis. as in Figure l(b), or are on the same side of the spin axis with the cantilever directed radially inwards, as in Figure l(c). The need to study these new configurations arose in connection with the possible use of a rotating cantilever as the basis for a gyroscopic instrument for measuring angular rates of turn [Z]. For this application it is necessary to tune the spin frequency to the first natural frequency of the cantilever. The problem considered may be regarded as a special case of the vibrations of a beam with a distributed axial force, for which a general theory has recently been presented 131, but this very recent theory was not available when the work reported in this paper was carried out.
2. EQUATIONS
OF MOTION
The equations of motion for the beam configurations shown in Figures 1 may be found in the usual manner by considering the equilibrium of a beam element (see Figure 2) situated at a distance .v from the root. It will be assumed that the central axis of the undeflected cantilever intersects the spin axis at right angles and that only linear vibration parallel to the spin axis can occur. The effects of rotatory inertia and shear deformations are neglected and axial extension of the beam due to centrifugal loading is ignored. The equation of motion for the transverse displacement, z, of a uniform horizontal cantilever of length L, flexural rigidity EZ and mass per unit length ,u, spinning at an angular velocity n about a vertical axis z’z’ a distance R from the root may be written as
22zjat2+ (EI/~L~)i4zpy4 -
n2 a [T(Y) az/ayl/ay
= 0,
(1)
where T(Y) = 4[-2r(l-Y)+(l-Y?],
Y = YlL
Figure
z
2. An element of the beam.
=
z/L,
r = R/L.
(2)
VIBRATION
OF A ROTATING
CANTILEVER
IS3
Because the origin of the beam reference axes, 0.u~~;. is located at the root of the cantilever one should note that the root offset parameter r is positive for the configuration of Figures l(b) and l(c) but is negative for the configuration of Figure l(a). The function T(Y) represents the axial load in the cantilever at position Y due to centrifugal loading. The boundary conditions associated with equation (1) are Z=?Z/CY=O
at
c2ZjiY’
Y=O,
3. SOLUTION
OF EQUATION
= i3Z;i
Y3 = 0
at Y = 1,
(3)
OF MOTION
The presence of the axial load term T(Y) in equation (1) precludes an exact analytical solution in terms of known functions. An approximate solution is therefore constructed by using Ritz’s Second Method [4] (which is also often called Galerkin’s method). A series solution of the form
z, = f
II~&~(Y) sinpt
(4)
k=l
is assumed, where the qk form a finite set of generalized and the &( Y) are the normal modes of the corresponding reference [S] as C,&(Y) = cosh(i.,
co-ordinates, non-rotating
Y) - cos(ik Y) - o,[sinh(l,
I’)-sin(i.,
to be determined, cantilever, given in
Y)],
where, for k = 1, A1 = 18751 and (TV= 0.7341, and, for k = 2, jL2 = 4.6941 and o2 = 1.0185. The solution (4) satisfies the boundary conditions (3) by virtue of this choice of the $k(Y), and substitution into the equation of motion (2) yields a residual which is minimized by selecting the co-ordinates qk such that
,‘-++(Y@]&.(Y)dY Equation
(5) thus yields a system of linear equations [/i-p21
+n’A]q
= 0.
(5)
in II,, of the form = 0,
(6)
from which the natural frequencies pi and the mode shapes qi can be found in the usual way. The matrix n is diagonal and contains as elements the natural frequencies of the non-rotating system: i.e., i,, = p& 6,. The matrix A is symmetric, with aij = + ;[(I-Y’)-ZI(l-Y),g+$dY, s and is associated with the change in beam stiffness, due to centrifugal loading. Equation (6) indicates that the natural frequencies depend upon the spin n and the root offset parameter I’. In contrast to the contiguration previously studied, in which r < 0 and pi > II, it is possible to identify alternative patterns of behaviour. These can be listed as follows. (a) For a range of r (>O) values of spin may be selected so as to coincide with a natural frequency. These funing speeds are found by setting p = n in equation (6) and follow from det(/l+ri*(A-Z))
= 0.
(7)
C.
154
H. J. FOX AND J. S. BURDESS
(b) It is possible to determine values of r( >O) and n which reduce a natural frequency to zero. This implies that the free motion is unstable and that the cantilever is on the point of buckling due to the centrifugal loading. Such values of r and spin follow from equation (6) in the general form det(A + rz”A) = 0.
(8)
Conditions (7) and (8) and the general way in which the natural frequencies vary with spin and root offset are discussed in what follows, on the basis of a detailed examination of the first two modes. In the case of a two mode expansion the frequency equation follows directly from equation (6) and may be written non-dimensionally as
3.1.
FIRST MODE OF VIBRATION
For the beam to function as a gyroscopic instrument it is important to understand how the frequency p1 of the first mode varies with n and r. This variation is plotted nondimensionally in Figure 3 as (pl/pol) versus the non-dimensional spin (n/p,,), for increasing values of r. For a given value of n it is shown that the natural frequency is always reduced as r is increased. For small values of r, p1 increases with spin and, provided r is less than a value r1 x 0.067, is always greater than the spin frequency. The value of r z r1 corresponds to the situation where p, asymptotically approaches the line p1 = n. This condition is expressed by equation (9) on setting (p&t) = 1 as n --f c;r and
(n/p,,)
Figure 3. Variation (n/pa,) for varying r.
of non-dimensional
first natural
frequency
(p,/p,,) with non-dimensional
spin frequency
VIBRATION
OF A ROTATING
lS.5
CANTILEVER
1
a
>Q
c
?
i
4
1
C
a
Figure 4. Variation of non-dimensional for first and second modes of vibration.
tuning
frequency
(n,/pO,) and buckling
frequency
(n,/p,,)
for varying
r
gives r = r, as the solution to 1 - ((111fa,,)
+ aria11 - a:, = 0.
(10)
Two values of r1 satisfy equation (10): rl = 0.067, which applies to the first mode, and rl = 0.657, which applies to the second mode (to be discussed later). If r is further increased it can be seen that the natural frequency still increases with spin but now passes through the critical value pi = n = n,(l) . When the situation pi = n occurs the cantilever is said to be tuned to the spin frequency and this represents the optimum running condition for gyroscopic (angular rate sensing) applications [;?I. The tuning speed n)‘) is readily determined from equation (9) on setting p = n and is given by the roots of
C-J [ 1 +(~y[(ii22-1) + 4
n,
1 -
a11
-
$2
+
alla22
-
a:2
PO1
(~)‘(%-I)]
+ (E)’
= 0.
(11)
Figure 4 shows how the tuning frequency for a cantilever of fixed length L varies with root offset R. Initially a sharp reduction in n, occurs as r is increased beyond r-i. Thereafter n, decreases steadily with increasing r. The root offset and the spin cannot however be increased indefinitely. A value of r is reached above which the natural frequency is not an increasing function of n for all n: e.g., the case r z 0.66 (see Figure 3). For these situations pi will begin to decrease as n is further increased. Ultimately pi becomes zero and the cantilever buckles. The spin nb at which buckling occurs may be determined from equation (9) by setting p = 0 and is given by the solution of
(3J[u’“u22-u:2] +(J$[u22 +(~$4 +0’ =0.
(12)
156
C. H. J. FOX AND J. S. BURDESS
The value of r associated with the limiting condition nj,” + x follows directly from equation (12) as a,,~,,-u:, = 0, and for the first mode gives r = 0.642. Thus as I’ is increased buckling will eventually occur as the spin in increased. The buckling frequency for a cantilever of fixed length L is shown plotted as a function of r by the chain-dotted line of Figure 4. The trend is similar to that associated with the tuning frequency and it will be seen that II!” < I$,‘) with ~1” -+ ni” as the offset R is increased. It is also of interest to consider how the spin frequency at which tuning occurs in the first mode varies with the length of the cantilever for a constant value of root offset R. This variation is plotted non-dimensionally in Figure 5 as (n,/p*), where p* is the first natural frequency of the stationary cantilever when L = R. It can be seen that a minimum value of tuning speed exists. It has been shown for small values of I‘, L $ R, that the natural frequency of the rotating cantilever increases with spin as a result of centrifugal loading. Thus when tuning first occurs, i.e., at I’= Y, = 0.067 and II + ‘%I, it is clear that the centrifugal effects predominate in determining the behaviour of the beam and hence the tuning condition. Therefore since a reduction in cantilever length will, for a fixed speed, cause a reduction in the magnitude of the internal loading due to rotation. tuning will occur at a lower value of spin. For large values of I’, R 9 L, one has a short cantilever and although the inertia loading tends to reduce the natural frequency of the rotating beam the behaviour at tuning is essentially that of the stationary cantilever. A decrease in beam length therefore increases the value of the tuning speed. The variation of tuning frequency with beam length is as follows. As the beam length is reduced from that at I’= r, the reduction in tuning frequency as a result of a fall-off in centrifugal load is arrested by the increase in beam static natural frequency pal. This trend continues until a minimum value of tuning speed is reached at r = rm,,, 2 0.09. For values of r > rmln the increase in pO, becomes the dominating effect and the tuning speed increases. The variation of buckling speed rzi” with beam length is similar and shows a minimum value when r z 1 (see Figure 6). For values of r > 1 the length of the cantilever is less than the offset and consequently stiffer than at r = 1. Furthermore as the beam length decreases the inertia load tending to buckle the beam also decreases. These factors combine and buckling occurs at a higher speed. Now, as the length of the beam is increased beyond that at R = L axial tensile stresses are introduced into the end portion of the beam due to the change in direction of
Figure
5. Variation
of non-dimensional
tuning
frequency
(ni’)/p) with r for constant
root offset K
VIBRATION
OF A ROTATING
157
CANTILEVER
El-
7-
6-
r
5
,9 s
4-
3-
2-
Figure 6. Variation of non-dimensional buckling frequency (ni’),‘p) with r for constant the inertia with sharp
3.2.
loading
increasing increase
in the region
of the tip. Although
L the introduction in buckling
speed
of these
the static natural
tensile
stresses
more
root offset R.
frequency
than
decreases
compensate
and a
is observed.
SECOND MODE OF VIBRATION
The shown Also,
manner in Figure the
in which
the second
7 and is similar
variation
in second
0
natural
frequency
varies
to that of the first natural mode
I
2
tuning
3
and
4
with
buckling
5
6
root
offset
and spin is
frequency. frequencies
are
plotted
in
7
b/P,,)
Figure 7. Variation of non-dimensional quency (n,‘pol)for varying r.
second
natural
frequency
(p2/po1)with non-dimensional
spin fre-
158
C. H. J. FOX AND J. S. BURDESS
Figure 4 and compared with those associated with the first mode. It can be seen that as II is increased (for a given value of r) buckling of the first mode will always occur before the tuning and buckling speeds associated with the second mode are reached. Thus tuning of the second mode has no physical significance since the beam is dynamically unstable once first mode buckling has occurred. From this result it may be anticipated that the same would apply to the higher modes of vibration.
4. CONCLUSIONS
For the free motion of a rotating cantilever with an offset root, it has been shown that the fundamental natural frequency is a function of spin frequency. The form of this functional relationship however depends upon the value of the ratio r = R/L of the root offset (R) to the cantilever length (L). Three distinct regions have been identified. (i) For values of r less than approximately 0.067 the first natural frequency p1 is a monotonic increasing function of spin n and is always greater than the spin frequency. (ii) If r is increased so that it lies in the range 0.067 < r < 0.642 p1 still increases with increasing y1but it is possible to select a value of spin such that n = p, : i.e., tuning is possible. (iii) For root offsets r > 0.642, pi decreases with increasing spin. Tuning is still possible but if the spin frequency is further increased buckling will eventually occur. A similar relationship exists for the second mode. However it has been shown that the first mode will always buckle before the speeds corresponding to second mode tuning (n = pz) and buckling are reached. The speed range for stable operation is therefore limited by the behaviour of the first mode. For a fixed value of root offset R it has been shown that minimum values of tuning (ni”) and buckling (nil)) speeds (n, < nb)may be realized by correct choice of cantilever length L.
ACKNOWLEDGMENTS
The authors acknowledge with thanks the help of Professor L. Maunder in reading and commenting on the manuscript. The authors also acknowledge the financial support of the Science Research Council.
REFERENCES 1. T. H. Lru, G. W. HEMPand R. L. SIERAKOWSKI 1972 6th Southeastern Conferenceon Theoreticaland Applied Mechanics Tampa, Florida. Transverse vibrations of a rotating Rayleigh beam. 2. U.K. Patent Application 7702/78 February 1978. 3. R. E. D. BISHOPand W. G. PRICE 1978 Journal of Sound and Vibration 59,237-244. The vibration
characteristics of a beam with an axial force. 4. S. TIMOSHENKO, D. YOUNG and W. WEAVER,JR. 1974 Vibration Problems in Engineering. New York : Wiley. 5. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge University Press.
ofSound and
Vibration
(1979) 65(2), 151-l 58
THE NATURAL FREQUENCIES CANTILEVER C.
Department
OF A THIN ROTATING
WITH OFFSET ROOT
H. J. Fox AND
J. S. BURDESS
of Mechanical Engineering, University of Newcastle upon Tyne, Newcastle upon l)ne NE1 IRU. England (Received 11 December 1978)
In this paper previous analyses of rotating cantilevers are extended to allow consideration of the case where the root offset from the spin axis is such that part of the beam is directed radially inwards towards the axis of rotation. In this situation the fundamental natural frequency is a function of the spin frequency. The form of the functional relationship depends on the value of r, the ratio of root offset to cantilever length. It is shown that, for certain values of r, it is possible to select a value of spin such that the fundamental natural frequency and the spin frequency coincide, the beam then being “tuned”. It is also possible to choose values of r such that the fundamental natural frequency reduces to zero as the spin is increased, and buckling occurs. It is further shown that by suitable choice of beam length for a given root offset the tuning and buckling frequencies can be minimized. The behaviour of higher modes of vibration is also discussed.
1. INTRODUCTION
The transverse vibration of rotating cantilevers has been widely studied in the context of aeroplane propellers and helicopter blades, and a summary of much of the previous work is given in reference [l]. In this previous work the root of the cantilever has been assumed either to lie on the spin axis or to be offset as shown in Figure l(a). Due to the effects of centrifugal loading it is known that the natural frequencies of transverse vibration increase with rotation. For this configuration the critical situation where a natural frequency coincides with the spin frequency cannot arise. L
-++-Irl L
R
4
LIT= q
Y
(b)
(a)
Figure
1. Cantilever
(c) configurations.
151 022%460X/79
140151+08
$02.00/O
c
1979 Academic
Press Inc. (London)
Limited
152
C. H. J. FOX AND J. S. BURDESS
In this present paper the above analysis is extended to permit consideration of the case of a rotating cantilever where either the root and tip are on opposite sides of the spin axis. as in Figure l(b), or are on the same side of the spin axis with the cantilever directed radially inwards, as in Figure l(c). The need to study these new configurations arose in connection with the possible use of a rotating cantilever as the basis for a gyroscopic instrument for measuring angular rates of turn [Z]. For this application it is necessary to tune the spin frequency to the first natural frequency of the cantilever. The problem considered may be regarded as a special case of the vibrations of a beam with a distributed axial force, for which a general theory has recently been presented 131, but this very recent theory was not available when the work reported in this paper was carried out.
2. EQUATIONS
OF MOTION
The equations of motion for the beam configurations shown in Figures 1 may be found in the usual manner by considering the equilibrium of a beam element (see Figure 2) situated at a distance .v from the root. It will be assumed that the central axis of the undeflected cantilever intersects the spin axis at right angles and that only linear vibration parallel to the spin axis can occur. The effects of rotatory inertia and shear deformations are neglected and axial extension of the beam due to centrifugal loading is ignored. The equation of motion for the transverse displacement, z, of a uniform horizontal cantilever of length L, flexural rigidity EZ and mass per unit length ,u, spinning at an angular velocity n about a vertical axis z’z’ a distance R from the root may be written as
22zjat2+ (EI/~L~)i4zpy4 -
n2 a [T(Y) az/ayl/ay
= 0,
(1)
where T(Y) = 4[-2r(l-Y)+(l-Y?],
Y = YlL
Figure
z
2. An element of the beam.
=
z/L,
r = R/L.
(2)
VIBRATION
OF A ROTATING
CANTILEVER
IS3
Because the origin of the beam reference axes, 0.u~~;. is located at the root of the cantilever one should note that the root offset parameter r is positive for the configuration of Figures l(b) and l(c) but is negative for the configuration of Figure l(a). The function T(Y) represents the axial load in the cantilever at position Y due to centrifugal loading. The boundary conditions associated with equation (1) are Z=?Z/CY=O
at
c2ZjiY’
Y=O,
3. SOLUTION
OF EQUATION
= i3Z;i
Y3 = 0
at Y = 1,
(3)
OF MOTION
The presence of the axial load term T(Y) in equation (1) precludes an exact analytical solution in terms of known functions. An approximate solution is therefore constructed by using Ritz’s Second Method [4] (which is also often called Galerkin’s method). A series solution of the form
z, = f
II~&~(Y) sinpt
(4)
k=l
is assumed, where the qk form a finite set of generalized and the &( Y) are the normal modes of the corresponding reference [S] as C,&(Y) = cosh(i.,
co-ordinates, non-rotating
Y) - cos(ik Y) - o,[sinh(l,
I’)-sin(i.,
to be determined, cantilever, given in
Y)],
where, for k = 1, A1 = 18751 and (TV= 0.7341, and, for k = 2, jL2 = 4.6941 and o2 = 1.0185. The solution (4) satisfies the boundary conditions (3) by virtue of this choice of the $k(Y), and substitution into the equation of motion (2) yields a residual which is minimized by selecting the co-ordinates qk such that
,‘-++(Y@]&.(Y)dY Equation
(5) thus yields a system of linear equations [/i-p21
+n’A]q
= 0.
(5)
in II,, of the form = 0,
(6)
from which the natural frequencies pi and the mode shapes qi can be found in the usual way. The matrix n is diagonal and contains as elements the natural frequencies of the non-rotating system: i.e., i,, = p& 6,. The matrix A is symmetric, with aij = + ;[(I-Y’)-ZI(l-Y),g+$dY, s and is associated with the change in beam stiffness, due to centrifugal loading. Equation (6) indicates that the natural frequencies depend upon the spin n and the root offset parameter I’. In contrast to the contiguration previously studied, in which r < 0 and pi > II, it is possible to identify alternative patterns of behaviour. These can be listed as follows. (a) For a range of r (>O) values of spin may be selected so as to coincide with a natural frequency. These funing speeds are found by setting p = n in equation (6) and follow from det(/l+ri*(A-Z))
= 0.
(7)
C.
154
H. J. FOX AND J. S. BURDESS
(b) It is possible to determine values of r( >O) and n which reduce a natural frequency to zero. This implies that the free motion is unstable and that the cantilever is on the point of buckling due to the centrifugal loading. Such values of r and spin follow from equation (6) in the general form det(A + rz”A) = 0.
(8)
Conditions (7) and (8) and the general way in which the natural frequencies vary with spin and root offset are discussed in what follows, on the basis of a detailed examination of the first two modes. In the case of a two mode expansion the frequency equation follows directly from equation (6) and may be written non-dimensionally as
3.1.
FIRST MODE OF VIBRATION
For the beam to function as a gyroscopic instrument it is important to understand how the frequency p1 of the first mode varies with n and r. This variation is plotted nondimensionally in Figure 3 as (pl/pol) versus the non-dimensional spin (n/p,,), for increasing values of r. For a given value of n it is shown that the natural frequency is always reduced as r is increased. For small values of r, p1 increases with spin and, provided r is less than a value r1 x 0.067, is always greater than the spin frequency. The value of r z r1 corresponds to the situation where p, asymptotically approaches the line p1 = n. This condition is expressed by equation (9) on setting (p&t) = 1 as n --f c;r and
(n/p,,)
Figure 3. Variation (n/pa,) for varying r.
of non-dimensional
first natural
frequency
(p,/p,,) with non-dimensional
spin frequency
VIBRATION
OF A ROTATING
lS.5
CANTILEVER
1
a
>Q
c
?
i
4
1
C
a
Figure 4. Variation of non-dimensional for first and second modes of vibration.
tuning
frequency
(n,/pO,) and buckling
frequency
(n,/p,,)
for varying
r
gives r = r, as the solution to 1 - ((111fa,,)
+ aria11 - a:, = 0.
(10)
Two values of r1 satisfy equation (10): rl = 0.067, which applies to the first mode, and rl = 0.657, which applies to the second mode (to be discussed later). If r is further increased it can be seen that the natural frequency still increases with spin but now passes through the critical value pi = n = n,(l) . When the situation pi = n occurs the cantilever is said to be tuned to the spin frequency and this represents the optimum running condition for gyroscopic (angular rate sensing) applications [;?I. The tuning speed n)‘) is readily determined from equation (9) on setting p = n and is given by the roots of
C-J [ 1 +(~y[(ii22-1) + 4
n,
1 -
a11
-
$2
+
alla22
-
a:2
PO1
(~)‘(%-I)]
+ (E)’
= 0.
(11)
Figure 4 shows how the tuning frequency for a cantilever of fixed length L varies with root offset R. Initially a sharp reduction in n, occurs as r is increased beyond r-i. Thereafter n, decreases steadily with increasing r. The root offset and the spin cannot however be increased indefinitely. A value of r is reached above which the natural frequency is not an increasing function of n for all n: e.g., the case r z 0.66 (see Figure 3). For these situations pi will begin to decrease as n is further increased. Ultimately pi becomes zero and the cantilever buckles. The spin nb at which buckling occurs may be determined from equation (9) by setting p = 0 and is given by the solution of
(3J[u’“u22-u:2] +(J$[u22 +(~$4 +0’ =0.
(12)
156
C. H. J. FOX AND J. S. BURDESS
The value of r associated with the limiting condition nj,” + x follows directly from equation (12) as a,,~,,-u:, = 0, and for the first mode gives r = 0.642. Thus as I’ is increased buckling will eventually occur as the spin in increased. The buckling frequency for a cantilever of fixed length L is shown plotted as a function of r by the chain-dotted line of Figure 4. The trend is similar to that associated with the tuning frequency and it will be seen that II!” < I$,‘) with ~1” -+ ni” as the offset R is increased. It is also of interest to consider how the spin frequency at which tuning occurs in the first mode varies with the length of the cantilever for a constant value of root offset R. This variation is plotted non-dimensionally in Figure 5 as (n,/p*), where p* is the first natural frequency of the stationary cantilever when L = R. It can be seen that a minimum value of tuning speed exists. It has been shown for small values of I‘, L $ R, that the natural frequency of the rotating cantilever increases with spin as a result of centrifugal loading. Thus when tuning first occurs, i.e., at I’= Y, = 0.067 and II + ‘%I, it is clear that the centrifugal effects predominate in determining the behaviour of the beam and hence the tuning condition. Therefore since a reduction in cantilever length will, for a fixed speed, cause a reduction in the magnitude of the internal loading due to rotation. tuning will occur at a lower value of spin. For large values of I’, R 9 L, one has a short cantilever and although the inertia loading tends to reduce the natural frequency of the rotating beam the behaviour at tuning is essentially that of the stationary cantilever. A decrease in beam length therefore increases the value of the tuning speed. The variation of tuning frequency with beam length is as follows. As the beam length is reduced from that at I’= r, the reduction in tuning frequency as a result of a fall-off in centrifugal load is arrested by the increase in beam static natural frequency pal. This trend continues until a minimum value of tuning speed is reached at r = rm,,, 2 0.09. For values of r > rmln the increase in pO, becomes the dominating effect and the tuning speed increases. The variation of buckling speed rzi” with beam length is similar and shows a minimum value when r z 1 (see Figure 6). For values of r > 1 the length of the cantilever is less than the offset and consequently stiffer than at r = 1. Furthermore as the beam length decreases the inertia load tending to buckle the beam also decreases. These factors combine and buckling occurs at a higher speed. Now, as the length of the beam is increased beyond that at R = L axial tensile stresses are introduced into the end portion of the beam due to the change in direction of
Figure
5. Variation
of non-dimensional
tuning
frequency
(ni’)/p) with r for constant
root offset K
VIBRATION
OF A ROTATING
157
CANTILEVER
El-
7-
6-
r
5
,9 s
4-
3-
2-
Figure 6. Variation of non-dimensional buckling frequency (ni’),‘p) with r for constant the inertia with sharp
3.2.
loading
increasing increase
in the region
of the tip. Although
L the introduction in buckling
speed
of these
the static natural
tensile
stresses
more
root offset R.
frequency
than
decreases
compensate
and a
is observed.
SECOND MODE OF VIBRATION
The shown Also,
manner in Figure the
in which
the second
7 and is similar
variation
in second
0
natural
frequency
varies
to that of the first natural mode
I
2
tuning
3
and
4
with
buckling
5
6
root
offset
and spin is
frequency. frequencies
are
plotted
in
7
b/P,,)
Figure 7. Variation of non-dimensional quency (n,‘pol)for varying r.
second
natural
frequency
(p2/po1)with non-dimensional
spin fre-
158
C. H. J. FOX AND J. S. BURDESS
Figure 4 and compared with those associated with the first mode. It can be seen that as II is increased (for a given value of r) buckling of the first mode will always occur before the tuning and buckling speeds associated with the second mode are reached. Thus tuning of the second mode has no physical significance since the beam is dynamically unstable once first mode buckling has occurred. From this result it may be anticipated that the same would apply to the higher modes of vibration.
4. CONCLUSIONS
For the free motion of a rotating cantilever with an offset root, it has been shown that the fundamental natural frequency is a function of spin frequency. The form of this functional relationship however depends upon the value of the ratio r = R/L of the root offset (R) to the cantilever length (L). Three distinct regions have been identified. (i) For values of r less than approximately 0.067 the first natural frequency p1 is a monotonic increasing function of spin n and is always greater than the spin frequency. (ii) If r is increased so that it lies in the range 0.067 < r < 0.642 p1 still increases with increasing y1but it is possible to select a value of spin such that n = p, : i.e., tuning is possible. (iii) For root offsets r > 0.642, pi decreases with increasing spin. Tuning is still possible but if the spin frequency is further increased buckling will eventually occur. A similar relationship exists for the second mode. However it has been shown that the first mode will always buckle before the speeds corresponding to second mode tuning (n = pz) and buckling are reached. The speed range for stable operation is therefore limited by the behaviour of the first mode. For a fixed value of root offset R it has been shown that minimum values of tuning (ni”) and buckling (nil)) speeds (n, < nb)may be realized by correct choice of cantilever length L.
ACKNOWLEDGMENTS
The authors acknowledge with thanks the help of Professor L. Maunder in reading and commenting on the manuscript. The authors also acknowledge the financial support of the Science Research Council.
REFERENCES 1. T. H. Lru, G. W. HEMPand R. L. SIERAKOWSKI 1972 6th Southeastern Conferenceon Theoreticaland Applied Mechanics Tampa, Florida. Transverse vibrations of a rotating Rayleigh beam. 2. U.K. Patent Application 7702/78 February 1978. 3. R. E. D. BISHOPand W. G. PRICE 1978 Journal of Sound and Vibration 59,237-244. The vibration
characteristics of a beam with an axial force. 4. S. TIMOSHENKO, D. YOUNG and W. WEAVER,JR. 1974 Vibration Problems in Engineering. New York : Wiley. 5. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge University Press.