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Parametric instability of a cantilevered column with end mass
Journal of Sound and Vibration (1971) 18 (1) 45-53
PARAMETRIC
INSTABILITY
OF A CANTILEVERED
COLUMN WITH END MASS K. L. HANDOOAND V. SUNDARARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 16, India (Received 26 March 1971)
Parametric instability regions in the first spatial and temporal modes of cantilevered columns having longitudinal inertia and end mass have been determined experimentally and analytically in the parametric space. Experiments have been conducted with three different columns, each having a set of five end masses. Analytical investigation has been carried out using small deflection beam theory. Under certain assumptions, the governing differential equation reduces to a modified Mathieu-Hill type equation, which gives the stability criteria. Effects of disturbances on parametric resonance have been observed experimentally and explained analytically. 1. INTRODUCTION Parametric instability differs from the more familiar instability, such as elastostatic buckling and forced vibration resonance, in various aspects: (i) it occurs over a region of parameter space and not at discrete points; (ii) it can occur in directions normal to excitation and (iii) it may occur at frequencies other than the natural frequency of the system. These particular features of parametric instability arise from the fact that the excitation function appears as a coefficient in the perturbed equilibrium of the system. Parametric instability of a column pinned at both ends, and subjected to axial time-varying load, was first considered by Beliaev [l] in 1924. He reduced the governing differential equation of the system to a standard Mathieu-Hill equation and plotted the instability regions in parametric space. Later, Mettler [2], Lubkin and Stoker [3], Bolotin [4] and many other research workers extended Beliaev’s work. Evensen and Evan-Iwanowski [5] studied the parametric instability of a pinned column taking into account the effects of concentrated and distributed masses. Stevens [6] studied the parametric vibration of a viscoelastic column and the parametric excitation of a non-homogeneous Bernoulli-Euler beam has been investigated by Francis [7]. In the present work, parametric instability of cantilevered elastic columns having longitudinal inertia and end mass has been considered experimentally and analytically. Parameters of the system have been excited by vibrating the base of the column in a known manner. Two different instability criteria have been defined. Experimentally and theoretically determined instability regions have been plotted in a two-dimensional, non-dimensionalized parametric space. 2.
EXPERIMENTAL INVESTIGATION
An experimental system capable of exciting harmonic vibrations and measuring and frequency of the excitation was designed to determine the regions of parametric of various cantilevered columns. A block diagram representing the arrangement ments has been shown in Figure 1. The excitation system was designed to provide 45
amplitude instability of instruharmonic
46
K. L. HAND00 AND V. SUNDARARAJAN
L____-__-----_--_-----------A
i
I
L__----------__-_-_(
Excitation
Meosurirq
system
Figure 1. Experimental
system
arrangement.
oscillations to the column base in the vertical direction. Experiments were conducted with three different columns of high speed steel with rectangular cross-section. Each column was excited with and without end masses. The indication of the onset of instability and the frequency of the column at which the instability occurred was obtained by recording the bending strain of the column with the help of a Sanborn strip chart recorder. 2.1.
CRITERIA OF INSTABILITY
Instability has been defined as a transverse oscillation of a column resulting from the sinusoidal excitation of the base. Small amplitude instability has been defined as a departure from the equilibrium (that is, straight) configuration, which remains bounded. Large amplitude instability has been defined as unbounded oscillations at any parameter point. 2.2.
EXPERIMENTAL PROCEDURE
Experiments were conducted with three different columns with and without end masses. The natural frequency of each column-end mass combination was determined by plotting the free transverse vibration of the column. At the onset of instability, the amplitude and frequency of the base excitation were noted and the frequency was changed till the column again became stable. The first reading gave a point on the lower boundary of the instability region and the second reading on the upper boundary. The experiment was repeated at various amplitudes till the maximum possible amplitude of the shaker element was reached. 3. ANALYTICAL INVESTIGATION The column, shown in Figure 2, is assumed to be straight and elastic. Rotatory inertia and shear effects have been neglected in the present analysis. The equation of motion of the column can be written as
where x(s,t)=f[1-(;~]“2ds+x,,cosyt.
(2)
0
Equation (I), with its time-dependent coefficients, is a typical equation describing a parametric response. Longitudinal inertia contributes to this response through its intluence on the axial loading of the column.
INSTABILITY OF A CANTILEVERED COLUMN
I
47
M
I 1
P
.s
x0
cosyt
f-7 Figure 2. Column and coordinate system. It
will now be assumed that the mode shape of the column under axial loading,
the same as in the absence of this loading, and that only the amplitude of deflection becomes modified; that is, the solution of the equation
is
with the boundary conditions Y@, t) = 0,
y”(L, t) = 0,
y”(L,t)=g$
will give the mode shape of the column governed by equation (1). Thus the non-trivial solution of (1) can be written as y(s, t) = B(t) [sin fls + sinh /Is - a(cos fls - cash /3s)], where
(4)
sin /IL+ sinh /3L cash /3L
ci = s
and /3 satisfies the frequency equation 1 + cosflLcoshbL
+ gp[sinh/3Lcos/IL
- coshjILsin/?L] = 0.
On the assumption that ay/as is small, the integrand of (2) is expanded and only the first two terms are retained. Substitution of (4) in (1) yields an ordinary second-order equation in B(t) which is non-linear both ins and t. The above equation is difficult to solve in the present form. Since this equation is valid for all s between 0 and L, and at the onset of instability all the points of the beam will start oscillating in the lateral direction, a particular value between 0 and L can now be substituted for s. If this point is taken to be s = L, the following equation is obtained:
48
K. L. HAND00 AND V. SUNDARARAJAN
where A = [(cos~L - cosh/?L) + cr(sin/3L + sinh /IL)] [(sin /i?L- sinh /3L) - a(cos /3L- cash /3L)] and
FL cosh/3L + .! cos flL sinh /3L 6
-$(fsin’/?L-
1 +cosj?Lcosh/IL-+sinh2/3L))A.
The term E@“/m represents the square of the natural frequency of the transverse oscillations of the column when no axial load is present. The termgAM/3/EZ/34 can be written as M,,g/&, where M,,g represents the equivalent axial loading and Qn the Euler load. Under the action of this equivalent loading, the natural frequency of transverse oscillation of the column will be ,233
-!y].
After the definition of a new variable, z = yt/2, equation (5) takes the form B(2) + (2f_z/y)2[l - 2/Lcos 22]+ (2k/L2) [B(z) D(z) + P(z)] D(z) = 0,
(6)
where B(z) = @/8(yt/2)2 D(yt/2) and 2P = %
E@3[1
:;&493]
*
Equation (6) is satisfied by the trivial solution D(z) = 0, in which case the straight configuration remains stable. This equation also admits non-trivial solutions for various combinations of 2&/y, p and k/L2, for which the column would become parametrically unstable. Any such combination of 2&/y, ,u and k/L2 could be called an instability point, and a plot of the totality of instability points will give the instability region in parametric space. The boundaries of the first temporal instability region for the system described by equation (6), as determined by Evensen and Evan-Iwanowsky [2] using Cunningham and Struble’s perturbational method, are 1*cL y 2= (7) 1 + k(lR,I/L)2 ’ ( 2cr,1 where [ D,,I is approximately the amplitude of parametric oscillations. These boundaries can be determined in the parametric space [CL,k/L2, (y/2~ii)~].
4. RESULTS AND DISCUSSION
Columns used in the experiments have been named CL-I, CL-2, CL-3, respectively. Details regarding their physical dimensions and mass are given in Table 1. End masses, EM1 to EM5, were used with each of these columns. The regions of instability, obtained experimentally and theoretically, for the first column CL-l with all five end masses and for columns CL-2 and CL-3 with minimum and maximum end masses have been plotted in Figures 3-8. Experimentally determined regions for each of the above column-end mass combination have two sets of boundaries. The topmost of the upper two boundaries in each
49
INSTABILITY OF A CANTILEVERED COLUMN
of these regions corresponds to the small-amplitude criteria and the other to the largeamplitude criteria. The region in between the upper two boundaries is thus the smallamplitude criteria instability region (SACI). Analytically determined boundaries of the instability regions correspond to incipient instability conditions only. TABLE 1 Column
Column
data
Length
Total mass
(cm)
km)
name
33.65 38.70 41.55
CL-1 CL-2 CL-3
77.5 83.0 153.3
4.45 x 1o-4 3.83 x 1O-4 1.18 x 1O-3
End masses [M/ml] End masses Column
name
0 0 0
CL-1 CL-2 CL-3
,JL=1,086
‘EM1
’
EM2
EM3
EM4
EMi
0.457 0.426 0.202
0.670 0.626 0.339
1.402 1.310 0.709
1.921 1.792 0.962
I
I
I
1
I
I IO- y$ =IS21 Specimen. CL-I,EM5
p,=
I.08 -
I.875
$=o.o Specimen: CL-I,
13 \N r,
EM I
__-------__x ____--------____
Stable
0.0
I.0
2.0
3.0
60
50
+0
x0 7%
Figure 3. Principal instability region in (r/2w, x0 f/g)
plane. -,
Theoretical;
x-x,
experimental.
Study of Figures 3-8 reveal the following information. Apexes of the analytically determined instability regions are at parameter point x,, r2/g = 0 and y/2~r, = 1, while those of the experimentally determined regions are at points x0 r2/g > 0 and y/2& > 1. Locations of these points depend upon the column and the end mass used. Evensen and Evan-Iwanowsky [5] have attributed the shift of the apex from y/2& axis to 4
50
K. L. HAND0
AND V. SUNDARARAJAN
viscous damping. The apex of the region corresponding to zero end mass is farthest from y/2& axis and comes nearer to it with increasing end mass. The significance of this observation is that the column remains parametrically stable up to a certain value of x0 r’/g, irrespective of
/q=I.441 I.10 -
$=o
to4
I
457
Specimen
13
I
’
: CL-I,
EM ??
Stable
? r, I.02
-
Figure 4. As Figure 3, but under different conditions. I
1
I 2.0
3.0
p,=1.348’ J,IO -
t’o8
-
0,96
-
$-=0.67
Stable
I
0,95 0.0
I,0
I
,
.%Y2&
Figure 5. As Figure 3, but under different conditions.
the frequency, and that the entire instability region shifts away from the y/263 axis. In the present analysis, this shift can be attributed to structural damping. It has been found experimentally that the fraction of critical structural damping reduces with increasing end mass for each column.
INSTABILITY
OF A CANTILEVERED
51
COLUMN
On the other hand, the apex of the instability regions corresponding to zero end mass is nearest to the y/26 = 1 line and the apexes corresponding to other end masses lie above this line. The significance of this observation is that at a parametrically unstable point the ratio of
Figure 6. As Figure 3, but under differentconditions. I ’ pL=llo2 $=I
I
’ 792
s;pecimen
: CL-2,
EM 5
B
=, 875 ‘
XM=OO Specimen
CL-2,
EM I
Ia c-u 2
o-95 00
I.0
20
3-o
40
Figure 7. As Figure 3, but under differentconditions. excitation frequency to natural frequency is not 2, as determined in the present theoretical analysis, but slightly more than 2, depending upon the end mass. If the equation obtained by the substitution of equation (4) in (1) is solved exactly, the coefficient of (1 - ~,ILCOS~Z) D(z) in equation (6) will be Q(~cG/~)~instead of only (2~3/~)~,where Q is a constant greater than one, and dependent on the end mass M, thus agreeing with the behaviour observed in the experimental analysis.
52
K. L. HAND00 AND V. SUNDARARAJAN
Slopes of the boundaries of instability regions obtained experimentally and theoretically are in good agreement. The slope of the upper boundary of the large-amplitude criteria instability region is more than that of the small-amplitude criteria instability region. Moreover, the slope of the lower instability region boundary is more than that of the upper boundaries. The area of the instability region in the parametric space increases with an increase in the end mass for each of the three columns. The small-amplitude criteria instability region shortens with increasing end mass. I
Spechen
CL-3,
I
I
4.
I.75
I
I
EM 5
#f=0.0 Specimen : CL-3,
EM
I
Stable
105 -
x-x--
13
0.97 -
0.95
I
0
I
0.5
I.0
I
I.5
I
I
I
1
1
I
20
2.5
3.0
3.5
4.0
4.5
I
50
I
55
I
6.0
xoy2&
Figure 8. As Figure 3, but under different conditions.
Column response to disturbances at parameter points in the neighbourhood of instability boundaries were studied experimentally. It was observed that in the cases where the column eventually oscillated at a steady finite amplitude (that is, in the small amplitude criteria region) further disturbances had no long-term effects on this amplitude and decayed with time. It was also observed that in the neighbourhood of the lower boundary of the instability region, disturbances had a direct bearing on the column stability and could cause a previously stable column to be dragged into an unstable configuration. This behaviour can readily be explained through equation (7). When the instability point is located between the two upper boundaries, any further disturbance will tend to increase 1I&,\ and thereby lower the upper instability boundary, which, in turn, will tend to decrease I&,,,]. Again, the upper boundary of the instability region will become raised, increasing IDool. This will go on and the column response will show “beating”. Any disturbance near the lower boundary will increase ID,,\ and thereby lower the instability boundary which will cause 1L&,l to increase without bound. Evensen and Evan-Iwanowsky [5] have also observed this behaviour and have explained it on the basis of frequency considerations. REFERENCES 1. N. M. BELIAEV 1924 Collection of Engineering Construction and Structural Mechanics. (Inzhinernye Soorzheiia i Stroitel’naia mekhanika). Leningrad: Put’. 2. E. METTLER 1940 Mitteilungen. Forschung-Anstalt GHH Konzern 3, 1-12. Biegeschwingungen eines Stables unter pulsierende axial Last.
INSTABILITYOFACANTILEVEREDCOLUMN
53
3. S. LUBKIN and J. J. STOKER 1943 Quarterly of Applied Mathematics 1, 215-238. Stability of columns and strings under periodically varying forces. 4. V. V. BOLOTIN 1964 The Dynamic Stability of Elastic Systems. New York: Holden Day. 5. H. A. EVENSEN and R. M. EVAN-IWANOWSKI 1966 Journal of Applied Mechanics 33, 141-148. Effects of longitudinal inertia upon the parametric response of elastic columns. 6. K. K. STEVENS 1966 American Institute of Aeronautics and Astronautics Journal 4, 2111-2116. On the parametric excitation of a viscoelastic column. 7. P. H. FRANCIS 1968 Journal of Mechanical Engineering Science 10, 205-212. Parametric excitation of a non-homogeneous Bernoulli-Euler beam.
APPENDIX NOMENCLATURE
B(t), D(z) time-dependent amplitude of column deflection in first spatial mode D., approximate magnitude of column oscillation in first temporal mode E Z K L M m S t X X0
Y Z Y w B
Young’s modulus column area moment of inertia dimensionless mass parameter column length end mass column mass per unit length distance along deformed or undeformed neutral axis time distance along undeformed neutral axis amplitude of base excitation lateral deflection dimensionless time frequency of base excitation frequency of free lateral vibration without considering axial load Mg frequency of free lateral vibration taking axial load Mg into account
PARAMETRIC
INSTABILITY
OF A CANTILEVERED
COLUMN WITH END MASS K. L. HANDOOAND V. SUNDARARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 16, India (Received 26 March 1971)
Parametric instability regions in the first spatial and temporal modes of cantilevered columns having longitudinal inertia and end mass have been determined experimentally and analytically in the parametric space. Experiments have been conducted with three different columns, each having a set of five end masses. Analytical investigation has been carried out using small deflection beam theory. Under certain assumptions, the governing differential equation reduces to a modified Mathieu-Hill type equation, which gives the stability criteria. Effects of disturbances on parametric resonance have been observed experimentally and explained analytically. 1. INTRODUCTION Parametric instability differs from the more familiar instability, such as elastostatic buckling and forced vibration resonance, in various aspects: (i) it occurs over a region of parameter space and not at discrete points; (ii) it can occur in directions normal to excitation and (iii) it may occur at frequencies other than the natural frequency of the system. These particular features of parametric instability arise from the fact that the excitation function appears as a coefficient in the perturbed equilibrium of the system. Parametric instability of a column pinned at both ends, and subjected to axial time-varying load, was first considered by Beliaev [l] in 1924. He reduced the governing differential equation of the system to a standard Mathieu-Hill equation and plotted the instability regions in parametric space. Later, Mettler [2], Lubkin and Stoker [3], Bolotin [4] and many other research workers extended Beliaev’s work. Evensen and Evan-Iwanowski [5] studied the parametric instability of a pinned column taking into account the effects of concentrated and distributed masses. Stevens [6] studied the parametric vibration of a viscoelastic column and the parametric excitation of a non-homogeneous Bernoulli-Euler beam has been investigated by Francis [7]. In the present work, parametric instability of cantilevered elastic columns having longitudinal inertia and end mass has been considered experimentally and analytically. Parameters of the system have been excited by vibrating the base of the column in a known manner. Two different instability criteria have been defined. Experimentally and theoretically determined instability regions have been plotted in a two-dimensional, non-dimensionalized parametric space. 2.
EXPERIMENTAL INVESTIGATION
An experimental system capable of exciting harmonic vibrations and measuring and frequency of the excitation was designed to determine the regions of parametric of various cantilevered columns. A block diagram representing the arrangement ments has been shown in Figure 1. The excitation system was designed to provide 45
amplitude instability of instruharmonic
46
K. L. HAND00 AND V. SUNDARARAJAN
L____-__-----_--_-----------A
i
I
L__----------__-_-_(
Excitation
Meosurirq
system
Figure 1. Experimental
system
arrangement.
oscillations to the column base in the vertical direction. Experiments were conducted with three different columns of high speed steel with rectangular cross-section. Each column was excited with and without end masses. The indication of the onset of instability and the frequency of the column at which the instability occurred was obtained by recording the bending strain of the column with the help of a Sanborn strip chart recorder. 2.1.
CRITERIA OF INSTABILITY
Instability has been defined as a transverse oscillation of a column resulting from the sinusoidal excitation of the base. Small amplitude instability has been defined as a departure from the equilibrium (that is, straight) configuration, which remains bounded. Large amplitude instability has been defined as unbounded oscillations at any parameter point. 2.2.
EXPERIMENTAL PROCEDURE
Experiments were conducted with three different columns with and without end masses. The natural frequency of each column-end mass combination was determined by plotting the free transverse vibration of the column. At the onset of instability, the amplitude and frequency of the base excitation were noted and the frequency was changed till the column again became stable. The first reading gave a point on the lower boundary of the instability region and the second reading on the upper boundary. The experiment was repeated at various amplitudes till the maximum possible amplitude of the shaker element was reached. 3. ANALYTICAL INVESTIGATION The column, shown in Figure 2, is assumed to be straight and elastic. Rotatory inertia and shear effects have been neglected in the present analysis. The equation of motion of the column can be written as
where x(s,t)=f[1-(;~]“2ds+x,,cosyt.
(2)
0
Equation (I), with its time-dependent coefficients, is a typical equation describing a parametric response. Longitudinal inertia contributes to this response through its intluence on the axial loading of the column.
INSTABILITY OF A CANTILEVERED COLUMN
I
47
M
I 1
P
.s
x0
cosyt
f-7 Figure 2. Column and coordinate system. It
will now be assumed that the mode shape of the column under axial loading,
the same as in the absence of this loading, and that only the amplitude of deflection becomes modified; that is, the solution of the equation
is
with the boundary conditions Y@, t) = 0,
y”(L, t) = 0,
y”(L,t)=g$
will give the mode shape of the column governed by equation (1). Thus the non-trivial solution of (1) can be written as y(s, t) = B(t) [sin fls + sinh /Is - a(cos fls - cash /3s)], where
(4)
sin /IL+ sinh /3L cash /3L
ci = s
and /3 satisfies the frequency equation 1 + cosflLcoshbL
+ gp[sinh/3Lcos/IL
- coshjILsin/?L] = 0.
On the assumption that ay/as is small, the integrand of (2) is expanded and only the first two terms are retained. Substitution of (4) in (1) yields an ordinary second-order equation in B(t) which is non-linear both ins and t. The above equation is difficult to solve in the present form. Since this equation is valid for all s between 0 and L, and at the onset of instability all the points of the beam will start oscillating in the lateral direction, a particular value between 0 and L can now be substituted for s. If this point is taken to be s = L, the following equation is obtained:
48
K. L. HAND00 AND V. SUNDARARAJAN
where A = [(cos~L - cosh/?L) + cr(sin/3L + sinh /IL)] [(sin /i?L- sinh /3L) - a(cos /3L- cash /3L)] and
FL cosh/3L + .! cos flL sinh /3L 6
-$(fsin’/?L-
1 +cosj?Lcosh/IL-+sinh2/3L))A.
The term E@“/m represents the square of the natural frequency of the transverse oscillations of the column when no axial load is present. The termgAM/3/EZ/34 can be written as M,,g/&, where M,,g represents the equivalent axial loading and Qn the Euler load. Under the action of this equivalent loading, the natural frequency of transverse oscillation of the column will be ,233
-!y].
After the definition of a new variable, z = yt/2, equation (5) takes the form B(2) + (2f_z/y)2[l - 2/Lcos 22]+ (2k/L2) [B(z) D(z) + P(z)] D(z) = 0,
(6)
where B(z) = @/8(yt/2)2 D(yt/2) and 2P = %
E@3[1
:;&493]
*
Equation (6) is satisfied by the trivial solution D(z) = 0, in which case the straight configuration remains stable. This equation also admits non-trivial solutions for various combinations of 2&/y, p and k/L2, for which the column would become parametrically unstable. Any such combination of 2&/y, ,u and k/L2 could be called an instability point, and a plot of the totality of instability points will give the instability region in parametric space. The boundaries of the first temporal instability region for the system described by equation (6), as determined by Evensen and Evan-Iwanowsky [2] using Cunningham and Struble’s perturbational method, are 1*cL y 2= (7) 1 + k(lR,I/L)2 ’ ( 2cr,1 where [ D,,I is approximately the amplitude of parametric oscillations. These boundaries can be determined in the parametric space [CL,k/L2, (y/2~ii)~].
4. RESULTS AND DISCUSSION
Columns used in the experiments have been named CL-I, CL-2, CL-3, respectively. Details regarding their physical dimensions and mass are given in Table 1. End masses, EM1 to EM5, were used with each of these columns. The regions of instability, obtained experimentally and theoretically, for the first column CL-l with all five end masses and for columns CL-2 and CL-3 with minimum and maximum end masses have been plotted in Figures 3-8. Experimentally determined regions for each of the above column-end mass combination have two sets of boundaries. The topmost of the upper two boundaries in each
49
INSTABILITY OF A CANTILEVERED COLUMN
of these regions corresponds to the small-amplitude criteria and the other to the largeamplitude criteria. The region in between the upper two boundaries is thus the smallamplitude criteria instability region (SACI). Analytically determined boundaries of the instability regions correspond to incipient instability conditions only. TABLE 1 Column
Column
data
Length
Total mass
(cm)
km)
name
33.65 38.70 41.55
CL-1 CL-2 CL-3
77.5 83.0 153.3
4.45 x 1o-4 3.83 x 1O-4 1.18 x 1O-3
End masses [M/ml] End masses Column
name
0 0 0
CL-1 CL-2 CL-3
,JL=1,086
‘EM1
’
EM2
EM3
EM4
EMi
0.457 0.426 0.202
0.670 0.626 0.339
1.402 1.310 0.709
1.921 1.792 0.962
I
I
I
1
I
I IO- y$ =IS21 Specimen. CL-I,EM5
p,=
I.08 -
I.875
$=o.o Specimen: CL-I,
13 \N r,
EM I
__-------__x ____--------____
Stable
0.0
I.0
2.0
3.0
60
50
+0
x0 7%
Figure 3. Principal instability region in (r/2w, x0 f/g)
plane. -,
Theoretical;
x-x,
experimental.
Study of Figures 3-8 reveal the following information. Apexes of the analytically determined instability regions are at parameter point x,, r2/g = 0 and y/2~r, = 1, while those of the experimentally determined regions are at points x0 r2/g > 0 and y/2& > 1. Locations of these points depend upon the column and the end mass used. Evensen and Evan-Iwanowsky [5] have attributed the shift of the apex from y/2& axis to 4
50
K. L. HAND0
AND V. SUNDARARAJAN
viscous damping. The apex of the region corresponding to zero end mass is farthest from y/2& axis and comes nearer to it with increasing end mass. The significance of this observation is that the column remains parametrically stable up to a certain value of x0 r’/g, irrespective of
/q=I.441 I.10 -
$=o
to4
I
457
Specimen
13
I
’
: CL-I,
EM ??
Stable
? r, I.02
-
Figure 4. As Figure 3, but under different conditions. I
1
I 2.0
3.0
p,=1.348’ J,IO -
t’o8
-
0,96
-
$-=0.67
Stable
I
0,95 0.0
I,0
I
,
.%Y2&
Figure 5. As Figure 3, but under different conditions.
the frequency, and that the entire instability region shifts away from the y/263 axis. In the present analysis, this shift can be attributed to structural damping. It has been found experimentally that the fraction of critical structural damping reduces with increasing end mass for each column.
INSTABILITY
OF A CANTILEVERED
51
COLUMN
On the other hand, the apex of the instability regions corresponding to zero end mass is nearest to the y/26 = 1 line and the apexes corresponding to other end masses lie above this line. The significance of this observation is that at a parametrically unstable point the ratio of
Figure 6. As Figure 3, but under differentconditions. I ’ pL=llo2 $=I
I
’ 792
s;pecimen
: CL-2,
EM 5
B
=, 875 ‘
XM=OO Specimen
CL-2,
EM I
Ia c-u 2
o-95 00
I.0
20
3-o
40
Figure 7. As Figure 3, but under differentconditions. excitation frequency to natural frequency is not 2, as determined in the present theoretical analysis, but slightly more than 2, depending upon the end mass. If the equation obtained by the substitution of equation (4) in (1) is solved exactly, the coefficient of (1 - ~,ILCOS~Z) D(z) in equation (6) will be Q(~cG/~)~instead of only (2~3/~)~,where Q is a constant greater than one, and dependent on the end mass M, thus agreeing with the behaviour observed in the experimental analysis.
52
K. L. HAND00 AND V. SUNDARARAJAN
Slopes of the boundaries of instability regions obtained experimentally and theoretically are in good agreement. The slope of the upper boundary of the large-amplitude criteria instability region is more than that of the small-amplitude criteria instability region. Moreover, the slope of the lower instability region boundary is more than that of the upper boundaries. The area of the instability region in the parametric space increases with an increase in the end mass for each of the three columns. The small-amplitude criteria instability region shortens with increasing end mass. I
Spechen
CL-3,
I
I
4.
I.75
I
I
EM 5
#f=0.0 Specimen : CL-3,
EM
I
Stable
105 -
x-x--
13
0.97 -
0.95
I
0
I
0.5
I.0
I
I.5
I
I
I
1
1
I
20
2.5
3.0
3.5
4.0
4.5
I
50
I
55
I
6.0
xoy2&
Figure 8. As Figure 3, but under different conditions.
Column response to disturbances at parameter points in the neighbourhood of instability boundaries were studied experimentally. It was observed that in the cases where the column eventually oscillated at a steady finite amplitude (that is, in the small amplitude criteria region) further disturbances had no long-term effects on this amplitude and decayed with time. It was also observed that in the neighbourhood of the lower boundary of the instability region, disturbances had a direct bearing on the column stability and could cause a previously stable column to be dragged into an unstable configuration. This behaviour can readily be explained through equation (7). When the instability point is located between the two upper boundaries, any further disturbance will tend to increase 1I&,\ and thereby lower the upper instability boundary, which, in turn, will tend to decrease I&,,,]. Again, the upper boundary of the instability region will become raised, increasing IDool. This will go on and the column response will show “beating”. Any disturbance near the lower boundary will increase ID,,\ and thereby lower the instability boundary which will cause 1L&,l to increase without bound. Evensen and Evan-Iwanowsky [5] have also observed this behaviour and have explained it on the basis of frequency considerations. REFERENCES 1. N. M. BELIAEV 1924 Collection of Engineering Construction and Structural Mechanics. (Inzhinernye Soorzheiia i Stroitel’naia mekhanika). Leningrad: Put’. 2. E. METTLER 1940 Mitteilungen. Forschung-Anstalt GHH Konzern 3, 1-12. Biegeschwingungen eines Stables unter pulsierende axial Last.
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53
3. S. LUBKIN and J. J. STOKER 1943 Quarterly of Applied Mathematics 1, 215-238. Stability of columns and strings under periodically varying forces. 4. V. V. BOLOTIN 1964 The Dynamic Stability of Elastic Systems. New York: Holden Day. 5. H. A. EVENSEN and R. M. EVAN-IWANOWSKI 1966 Journal of Applied Mechanics 33, 141-148. Effects of longitudinal inertia upon the parametric response of elastic columns. 6. K. K. STEVENS 1966 American Institute of Aeronautics and Astronautics Journal 4, 2111-2116. On the parametric excitation of a viscoelastic column. 7. P. H. FRANCIS 1968 Journal of Mechanical Engineering Science 10, 205-212. Parametric excitation of a non-homogeneous Bernoulli-Euler beam.
APPENDIX NOMENCLATURE
B(t), D(z) time-dependent amplitude of column deflection in first spatial mode D., approximate magnitude of column oscillation in first temporal mode E Z K L M m S t X X0
Y Z Y w B
Young’s modulus column area moment of inertia dimensionless mass parameter column length end mass column mass per unit length distance along deformed or undeformed neutral axis time distance along undeformed neutral axis amplitude of base excitation lateral deflection dimensionless time frequency of base excitation frequency of free lateral vibration without considering axial load Mg frequency of free lateral vibration taking axial load Mg into account