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§29
Resonance in non-linear oscillations
91
This equation, however, does not suffice to determine the resulting ampli-
tude of the oscillations. The attainment of a finite amplitude involves non-
linear effects, and to include these in the equation of motion we must retain
also the terms non-linear in x(2):
= cos(2wo+e)t. (29.11)
The problem can be considerably simplified by virtue of the following fact.
Putting on the right-hand side of (29.11) x(2) = b cos[(wo++)+8], where
b is the required amplitude of the resonance oscillations and 8 a constant
phase difference which is of no importance in what follows, and writing the
product of cosines as a sum, we obtain a term (afb/3mwo2)
of the ordinary resonance type (with respect to the eigenfrequency wo of the
system). The problem thus reduces to that considered at the beginning of
this section, namely ordinary resonance in a non-linear system, the only
differences being that the amplitude of the external force is here represented
by afb/3wo2, and E is replaced by 1/6. Making this change in equation (29.4),
we have
Solving for b, we find the possible values of the amplitude:
b=0,
(29.12)
(29.13)
1
(29.14)
Figure 33 shows the resulting dependence of b on € for K > 0; for K < 0
the curves are the reflections (in the b-axis) of those shown. The points B
and C correspond to the values E = To the left of
B, only the value b = 0 is possible, i.e. there is no resonance, and oscillations
of frequency near wo are not excited. Between B and C there are two roots,
b = 0(BC) and (29.13) (BE). Finally, to the right of C there are three roots
(29.12)-(29.14). Not all these, however, correspond to stable oscillations.
The value b = 0 is unstable on BC, and it can also be shown that the middle
root (29.14) always gives instability. The unstable values of b are shown in
Fig. 33 by dashed lines.
Let us examine, for example, the behaviour of a system initially "at rest"
as the frequency of the external force is gradually diminished. Until the point
t This segment corresponds to the region of parametric resonance (27.12), and a com-
parison of (29.10) and (27.8) gives 1h = 2af/3mwo4. The condition 12af/3mwo3 > 4X for
which the phenomenon can exist corresponds to h > hk.
+ It should be recalled that only resonance phenomena are under consideration. If these
phenomena are absent, the system is not literally at rest, but executes small forced oscillations
of frequency y.
92
Small Oscillations
§29
C is reached, b = 0, but at C the state of the system passes discontinuously
to the branch EB. As € decreases further, the amplitude of the oscillations
decreases to zero at B. When the frequency increases again, the amplitude
increases along BE.-
b
E
E
A
B
C D
FIG. 33
The cases of resonance discussed above are the principal ones which may
occur in a non-linear oscillating system. In higher approximations, resonances
appear at other frequencies also. Strictly speaking, a resonance must occur
at every frequency y for which ny + mwo = wo with n and m integers, i.e. for
every y = pwo/q with P and q integers. As the degree of approximation
increases, however, the strength of the resonances, and the widths of the
frequency ranges in which they occur, decrease so rapidly that in practice
only the resonances at frequencies y 2 pwo/q with small P and q can be ob-
served.
PROBLEM
Determine the function b(e) for resonance at frequencies y 22 3 wo.
SOLUTION. In the first approximation, x(1) = -(f/8mwo2) cos(3wo+t) For the second
approximation x(2) we have from (29.1) the equation
= -3,8x(1)x(2)2,
where only the term which gives the required resonance has been retained on the right-hand
side. Putting x(2) = b cos[(wo+)+8] and taking the resonance term out of the product
of three cosines, we obtain on the right-hand side the expression
(3,3b2f(32mwo2) cos[(wotle)t-28].
Hence it is evident that b(e) is obtained by replacing f by 3,8b2f/32wo², and E by JE, in
(29.4):
Ab4.
The roots of this equation are
b=0,
Fig. 34 shows a graph of the function b(e) for k>0. Only the value b=0 (the e-axis) and
the branch AB corresponds to stability. The point A corresponds to EK = 3(4x2)2-A3)/4kA,
t It must be noticed, however, that all the formulae derived here are valid only when the
amplitude b (and also E) is sufficiently small. In reality, the curves BE and CF meet, and at
their point of intersection the oscillation ceases; thereafter, b = 0.
§30
Motion in a rapidly oscillating field
93
bk2 = Oscillations exist only for € > Ek, and then b > bk. Since the state
b = 0 is always stable, an initial "push" is necessary in order to excite oscillations.
The formulae given above are valid only for small E. This condition is satisfied if 1 is small
and also the amplitude of the force is such that 2/wo < A KWO.
b
B
A
€
FIG. 34
§30. Motion in a rapidly oscillating field
Let us consider the motion of a particle subject both to a time-independent
field of potential U and to a force
f=f1coswt+fasin.ou
(30.1)
which varies in time with a high frequency w (f1, f2 being functions of the
co-ordinates only). By a "high" frequency we mean one such that w > 1/T,
where T is the order of magnitude of the period of the motion which the
particle would execute in the field U alone. The magnitude of f is not assumed
small in comparison with the forces due to the field U, but we shall assume
that the oscillation (denoted below by $) of the particle as a result of this
force is small.
To simplify the calculations, let us first consider motion in one dimension
in a field depending only on the space co-ordinate X. Then the equation of
motion of the particle ist
mx = -dU/dx+f.
(30.2)
It is evident, from the nature of the field in which the particle moves, that
it will traverse a smooth path and at the same time execute small oscillations
of frequency w about that path. Accordingly, we represent the function x(t)
as a sum:
(30.3)
where (t) corresponds to these small oscillations.
The mean value of the function (t) over its period 2n/w is zero, and the
function X(t) changes only slightly in that time. Denoting this average by a
bar, we therefore have x = X(t), i.e. X(t) describes the "smooth" motion of
t The co-ordinate x need not be Cartesian, and the coefficient m is therefore not neces-
sarily the mass of the particle, nor need it be constant as has been assumed in (30.2). This
assumption, however, does not affect the final result (see the last footnote to this section).
94
Small Oscillations
§30
the particle averaged over the rapid oscillations. We shall derive an equation
which determines the function X(t).t
Substituting (30.3) in (30.2) and expanding in powers of & as far as the
first-order terms, we obtain
(30.4)
This equation involves both oscillatory and "smooth" terms, which must
evidently be separately equal. For the oscillating terms we can put simply
mg = f(X, t);
(30.5)
the other terms contain the small factor & and are therefore of a higher order
of smallness (but the derivative sur is proportional to the large quantity w2
and so is not small). Integrating equation (30.5) with the function f given by
(30.1) (regarding X as a constant), we have
& = -f/mw2.
(30.6)
Next, we average equation (30.4) with respect to time (in the sense discussed
above). Since the mean values of the first powers of f and $ are zero, the result
is
dX
which involves only the function X(t). This equation can be written
mX = dUeff/dX,
(30.7)
where the "effective potential energy" is defined ast
Ueff = U+f2/2mw2
=
(30.8)
Comparing this expression with (30.6), we easily see that the term added to
U is just the mean kinetic energy of the oscillatory motion:
Ueff= U+1mg2
(30.9)
Thus the motion of the particle averaged over the oscillations is the same
as if the constant potential U were augmented by a constant quantity pro-
portional to the squared amplitude of the variable field.
t The principle of this derivation is due to P. L. KAPITZA (1951).
++ By means of somewhat more lengthy calculations it is easy to show that formulae (30.7)
and (30.8) remain valid even if m is a function of X.
§30
Motion in a rapidly oscillating field
95
The result can easily be generalised to the case of a system with any number
of degrees of freedom, described by generalised co-ordinates qi. The effective
potential energy is then given not by (30.8), but by
Unt = Ut
= U+  ,
(30.10)
where the quantities a-1ik, which are in general functions of the co-ordinates,
are the elements of the matrix inverse to the matrix of the coefficients aik in
the kinetic energy (5.5) of the system.
PROBLEMS
PROBLEM 1. Determine the positions of stable equilibrium of a pendulum whose point of
support oscillates vertically with a high frequency y
(g/l)).
SOLUTION. From the Lagrangian derived in §5, Problem 3(c), we see that in this case the
variable force is f = -mlay2 cos yt sin (the quantity x being here represented by the angle
b). The "effective potential energy" is therefore Ueff = mgl[-cos - & st(a2y2/4gl) sin2]. The
positions of stable equilibrium correspond to the minima of this function. The vertically
downward position ( = 0) is always stable. If the condition a2y2 > 2gl holds, the vertically
upward position ( = ) is also stable.
PROBLEM 2. The same as Problem 1, but for a pendulum whose point of support oscillates
horizontally.
SOLUTION. From the Lagrangian derived in §5, Problem 3(b), we find f = mlay2 cos yt
cos and Uell = mgl[-cos 3+(a2y2/4gl) cos2]. If a2y2 < 2gl, the position = 0 is stable.
If a2y2 > 2gl, on the other hand, the stable equilibrium position is given by cos = 2gl/a22.
CHAPTER VI
MOTION OF A RIGID BODY
$31. Angular velocity
A rigid body may be defined in mechanics as a system of particles such that
the distances between the particles do not vary. This condition can, of course,
be satisfied only approximately by systems which actually exist in nature.
The majority of solid bodies, however, change so little in shape and size
under ordinary conditions that these changes may be entirely neglected in
considering the laws of motion of the body as a whole.
In what follows, we shall often simplify the derivations by regarding a
rigid body as a discrete set of particles, but this in no way invalidates the
assertion that solid bodies may usually be regarded in mechanics as continu-
ous, and their internal structure disregarded. The passage from the formulae
which involve a summation over discrete particles to those for a continuous
body is effected by simply replacing the mass of each particle by the mass
P dV contained in a volume element dV (p being the density) and the sum-
mation by an integration over the volume of the body.
To describe the motion of a rigid body, we use two systems of co-ordinates:
a "fixed" (i.e. inertial) system XYZ, and a moving system X1 = x, X2 = y,
X3 = 2 which is supposed to be rigidly fixed in the body and to participate
in its motion. The origin of the moving system may conveniently be taken
to coincide with the centre of mass of the body.
The position of the body with respect to the fixed system of co-ordinates
is completely determined if the position of the moving system is specified.
Let the origin O of the moving system have the radius vector R (Fig. 35).
The orientation of the axes of that system relative to the fixed system is given
by three independent angles, which together with the three components of
the vector R make six co-ordinates. Thus a rigid body is a mechanical system
with six degrees of freedom.
Let us consider an arbitrary infinitesimal displacement of a rigid body.
It can be represented as the sum of two parts. One of these is an infinitesimal
translation of the body, whereby the centre of mass moves to its final position,
but the orientation of the axes of the moving system of co-ordinates is un-
changed. The other is an infinitesimal rotation about the centre of mass,
whereby the remainder of the body moves to its final position.
Let r be the radius vector of an arbitrary point P in a rigid body in the
moving system, and r the radius vector of the same point in the fixed system
(Fig. 35). Then the infinitesimal displacement dr of P consists of a displace-
ment dR, equal to that of the centre of mass, and a displacement doxr
96
§31
Angular velocity
97
relative to the centre of mass resulting from a rotation through an infinitesimal
angle do (see (9.1)): dr = dR + do xr. Dividing this equation by the time
dt during which the displacement occurs, and putting
dr/dt = V,
dR/dt =
do/dt = La
(31.1)
we obtain the relation
V = V+Sxr.
(31.2)
Z
X3
P
X2
r
o
R
X1
Y
X
FIG. 35
The vector V is the velocity of the centre of mass of the body, and is also
the translational velocity of the body. The vector S is called the angular
velocity of the rotation of the body; its direction, like that of do, is along the
axis of rotation. Thus the velocity V of any point in the body relative to the
fixed system of co-ordinates can be expressed in terms of the translational
velocity of the body and its angular velocity of rotation.
It should be emphasised that, in deriving formula (31.2), no use has been
made of the fact that the origin is located at the centre of mass. The advan-
tages of this choice of origin will become evident when we come to calculate
the energy of the moving body.
Let us now assume that the system of co-ordinates fixed in the body is
such that its origin is not at the centre of mass O, but at some point O' at
a distance a from O. Let the velocity of O' be V', and the angular velocity
of the new system of co-ordinates be S'. We again consider some point P
in the body, and denote by r' its radius vector with respect to O'. Then
= r'+a, and substitution in (31.2) gives V = V+2xa+2xr'. The
definition of V' and S' shows that V = Hence it follows that
(31.3)
The second of these equations is very important. We see that the angular
velocity of rotation, at any instant, of a system of co-ordinates fixed in
the body is independent of the particular system chosen. All such systems
t
To avoid any misunderstanding, it should be noted that this way of expressing the angular
velocity is somewhat arbitrary: the vector so exists only for an infinitesimal rotation, and not
for all finite rotations.
4*
98
Motion of a Rigid Body
§32
rotate with angular velocities S which are equal in magnitude and parallel
in direction. This enables us to call S the angular velocity of the body. The
velocity of the translational motion, however, does not have this "absolute"
property.
It is seen from the first formula (31.3) that, if V and S are, at any given
instant, perpendicular for some choice of the origin O, then V' and SS are
perpendicular for any other origin O'. Formula (31.2) shows that in this case
the velocities V of all points in the body are perpendicular to S. It is then
always possible+ to choose an origin O' whose velocity V' is zero, SO that the
motion of the body at the instant considered is a pure rotation about an axis
through O'. This axis is called the instantaneous axis of rotation.t
In what follows we shall always suppose that the origin of the moving
system is taken to be at the centre of mass of the body, and so the axis of
rotation passes through the centre of mass. In general both the magnitude
and the direction of S vary during the motion.
$32. The inertia tensor
To calculate the kinetic energy of a rigid body, we may consider it as a
discrete system of particles and put T = mv2, where the summation is
taken over all the particles in the body. Here, and in what follows, we simplify
the notation by omitting the suffix which denumerates the particles.
Substitution of (31.2) gives
T = Sxx+
The velocities V and S are the same for every point in the body. In the first
term, therefore, V2 can be taken outside the summation sign, and Em is
just the mass of the body, which we denote by u. In the second term we put
EmV Sxr = Emr VxS = VxS Emr. Since we take the origin of the
moving system to be at the centre of mass, this term is zero, because Emr = 0.
Finally, in the third term we expand the squared vector product. The result
is
(32.1)
Thus the kinetic energy of a rigid body can be written as the sum of two
parts. The first term in (32.1) is the kinetic energy of the translational motion,
and is of the same form as if the whole mass of the body were concentrated
at the centre of mass. The second term is the kinetic energy of the rotation
with angular velocity S about an axis passing through the centre of mass.
It should be emphasised that this division of the kinetic energy into two parts
is possible only because the origin of the co-ordinate system fixed in the
body has been taken to be at its centre of mass.
t O' may, of course, lie outside the body.
+ In the general case where V and SC are not perpendicular, the origin may be chosen so
as to make V and S parallel, i.e. so that the motion consists (at the instant in question) of a
rotation about some axis together with a translation along that axis.
§32
The inertia tensor
99
We may rewrite the kinetic energy of rotation in tensor form, i.e. in terms
of the components Xi and O of the vectors r and L. We have
Here we have used the identity Oi = SikOk, where dik is the unit tensor,
whose components are unity for i = k and zero for i # k. In terms of the
tensor
(32.2)
we have finally the following expression for the kinetic energy of a rigid
body:
T =
(32.3)
The Lagrangian for a rigid body is obtained from (32.3) by subtracting
the potential energy:
L =
(32.4)
The potential energy is in general a function of the six variables which define
the position of the rigid body, e.g. the three co-ordinates X, Y, Z of the
centre of mass and the three angles which specify the relative orientation of
the moving and fixed co-ordinate axes.
The tensor Iik is called the inertia tensor of the body. It is symmetrical,
i.e.
Ik=Iki
(32.5)
as is evident from the definition (32.2). For clarity, we may give its com-
ponents explicitly:
TEST
(32.6)
m(x2+y2)
The components Ixx, Iyy, Izz are called the moments of inertia about the
corresponding axes.
The inertia tensor is evidently additive: the moments of inertia of a body
are the sums of those of its parts.
t In this chapter, the letters i, k, l are tensor suffixes and take the values 1, 2, 3. The
summation rule will always be used, i.e. summation signs are omitted, but summation over
the values 1, 2, 3 is implied whenever a suffix occurs twice in any expression. Such a suffix is
called a dummy suffix. For example, AiBi = A . B, Ai2 = AiA1 = A², etc. It is obvious that
dummy suffixes can be replaced by any other like suffixes, except ones which already appear
elsewhere in the expression concerned.
100
Motion of a Rigid Body
§32
If the body is regarded as continuous, the sum in the definition (32.2)
becomes an integral over the volume of the body:
(32.7)
Like any symmetrical tensor of rank two, the inertia tensor can be reduced
to diagonal form by an appropriate choice of the directions of the axes
X1, x2, X3. These directions are called the principal axes of inertia, and the
corresponding values of the diagonal components of the tensor are called the
principal moments of inertia; we shall denote them by I, I2, I3. When the
axes X1, X2, X3 are so chosen, the kinetic energy of rotation takes the very
simple form
=
(32.8)
None of the three principal moments of inertia can exceed the sum of the
other two. For instance,
m(x12+x22) = I3.
(32.9)
A body whose three principal moments of inertia are all different is called
an asymmetrical top. If two are equal (I1 = I2 # I3), we have a symmetrical
top. In this case the direction of one of the principal axes in the x1x2-plane
may be chosen arbitrarily. If all three principal moments of inertia are equal,
the body is called a spherical top, and the three axes of inertia may be chosen
arbitrarily as any three mutually perpendicular axes.
The determination of the principal axes of inertia is much simplified if
the body is symmetrical, for it is clear that the position of the centre of mass
and the directions of the principal axes must have the same symmetry as
the body. For example, if the body has a plane of symmetry, the centre of
mass must lie in that plane, which also contains two of the principal axes of
inertia, while the third is perpendicular to the plane. An obvious case of this
kind is a coplanar system of particles. Here there is a simple relation between
the three principal moments of inertia. If the plane of the system is taken as
the x1x2-plane, then X3 = 0 for every particle, and so I = mx22, I2 = 12,
I3 = (12+x2)2, whence
(32.10)
If a body has an axis of symmetry of any order, the centre of mass must lie
on that axis, which is also one of the principal axes of inertia, while the other
two are perpendicular to it. If the axis is of order higher than the second,
the body is a symmetrical top. For any principal axis perpendicular to the
axis of symmetry can be turned through an angle different from 180° about the
latter, i.e. the choice of the perpendicular axes is not unique, and this can
happen only if the body is a symmetrical top.
A particular case here is a collinear system of particles. If the line of the
system is taken as the x3-axis, then X1 = X2 = 0 for every particle, and so