force orthogonally on a plane perpendicular to HK and take the moment of
the projection about the intersection of HK with the plane (see § 4).
Some convention as to sign is necessary; we shall reckon the moment to
be positive when the tendency of the force is right-handed as regards
the direction from H to K. Since two concurrent forces and their
resultant obviously project into two concurrent forces and their
resultant, we see that the sum of the moments of two concurrent forces
about any axis HK is equal to the moment of their resultant. Parallel
forces may be included in this statement as a limiting case. Hence, in
whatever way one system of forces is by successive steps replaced by
another, no change is made in the sum of the moments about any assigned
axis. By means of this theorem we can show that the previous reduction
of any system to a wrench is unique.
From the analogy of couples to translations which was pointed out in §
7, we may infer that a couple is sufficiently represented by a "free"
(or non-localized) vector perpendicular to its plane. The length of the
vector must be proportional to the moment of the couple, and its sense
must be such that the sum of the moments of the two forces of the couple
about it is positive. In particular, we infer that couples of the same
moment in parallel planes are equivalent; and that couples in any two
planes may be compounded by geometrical addition of the corresponding
vectors. Independent statical proofs are of course easily given. Thus,
let the plane of the paper be perpendicular to the planes of two
couples, and therefore perpendicular to the line of intersection of
these planes. By § 4, each couple can be replaced by two forces ± P
(fig. 43) perpendicular to the plane of the paper, and so that one force
of each couple is in the line of intersection (B); the arms (AB, BC)
will then be proportional to the respective moments. The two forces at B
will cancel, and we are left with a couple of moment P · AC in the plane
AC. If we draw three vectors to represent these three couples, they will
be perpendicular and proportional to the respective sides of the
triangle ABC; hence the third vector is the geometric sum of the other
two. Since, in this proof the magnitude of P is arbitrary, It follows
incidentally that couples of the same moment in parallel planes, e.g.
planes parallel to AC, are equivalent.
[Illustration: FIG. 43.]
[Illustration: FIG. 44.]
Hence a couple of moment G, whose axis has the direction (l, m, n)
relative to a right-handed system of rectangular axes, is equivalent to
three couples lG, mG, nG in the co-ordinate planes. The analytical
reduction of a three-dimensional system can now be conducted as follows.
Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action
of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H
normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two
equal and opposite forces ± X1. The force X1 at P1 with -X1 in KH forms
a couple about Oz, of moment -y1X1. Next, introduce along Ox two equal
and opposite forces ±X1. The force X1 in KH with -X1 in Ox forms a
couple about Oy, of moment z1X1. Hence the force X1 can be transferred
from P1 to O, provided we introduce couples of moments z1X1 about Oy and
-y1X1, about Oz. Dealing in the same way with the forces Y1, Z1 at P1,
we find that all three components of the force at P1 can be transferred
to O, provided we introduce three couples L1, M1, N1 about Ox, Oy, Oz
respectively, viz.
L1 = y1Z1 - z1Y1, M1 = z1X1 - x1Z1, N1 = x1Y1 - y1X1. (5)
It is seen that L1, M1, N1 are the moments of the original force at P1
about the co-ordinate axes. Summing up for all the forces of the given
system, we obtain a force R at O, whose components are
X = [Sigma](X_r), Y = [Sigma](Y_r), Z = [Sigma](Z_r), (6)
and a couple G whose components are
L = [Sigma](L_r), M = [Sigma](M_r), N = [Sigma](N_r), (7)
where r= 1, 2, 3 ... Since R² = X² + Y² + Z², G² = L² + M² + N², it is
necessary and sufficient for equilibrium that the six quantities X, Y,
Z, L, M, N, should all vanish. In words: the sum of the projections of
the forces on each of the co-ordinate axes must vanish; and, the sum of
the moments of the forces about each of these axes must vanish.
If any other point O´, whose co-ordinates are x, y, z, be chosen in
place of O, as the point to which the forces are transferred, we have to
write x1 - x, y1 - y, z1 - z for x1, y1, z1, and so on, in the preceding
process. The components of the resultant force R are unaltered, but the
new components of couple are found to be
L´ = L - yZ + zY, \
M´ = M - zX + xZ, > (8)
N´ = N - xY + yX. /
By properly choosing O´ we can make the plane of the couple
perpendicular to the resultant force. The conditions for this are L´ :
M´ : N´ = X : Y : Z, or
L - yZ + zY M - zX + xZ N - xY + yX
----------- = ----------- = ----------- (9)
X Y Z
These are the equations of the central axis. Since the moment of the
resultant couple is now
X Y Z LX + MY + NZ
G´ = --- L´ + --- M´ + --- N´ = ------------, (10)
R R R R
the pitch of the equivalent wrench is
(LX + MY + NZ)/(X² + Y² + Z²).
It appears that X² + Y² + Z² and LX + MY + NZ are absolute invariants
(cf. § 7). When the latter invariant, but not the former, vanishes, the
system reduces to a single force.
The analogy between the mathematical relations of infinitely small
displacements on the one hand and those of force-systems on the other
enables us immediately to convert any theorem in the one subject into a
theorem in the other. For example, we can assert without further proof
that any infinitely small displacement may be resolved into two
rotations, and that the axis of one of these can be chosen arbitrarily.
Again, that wrenches of arbitrary amounts about two given screws
compound into a wrench the locus of whose axis is a cylindroid.
The mathematical properties of a twist or of a wrench have been the
subject of many remarkable investigations, which are, however, of
secondary importance from a physical point of view. In the
"Null-System" of A. F. Möbius (1790-1868), a line such that the moment
of a given wrench about it is zero is called a _null-line_. The triply
infinite system of null-lines form what is called in line-geometry a
"complex." As regards the configuration of this complex, consider a
line whose shortest distance from the central axis is r, and whose
inclination to the central axis is [theta]. The moment of the
resultant force R of the wrench about this line is - Rr sin [theta],
and that of the couple G is G cos [theta]. Hence the line will be a
null-line provided
tan [theta] = k/r, (11)
where k is the pitch of the wrench. The null-lines which are at a
given distance r from a point O of the central axis will therefore
form one system of generators of a hyperboloid of revolution; and by
varying r we get a series of such hyperboloids with a common centre
and axis. By moving O along the central axis we obtain the whole
complex of null-lines. It appears also from (11) that the null-lines
whose distance from the central axis is r are tangent lines to a
system of helices of slope tan^-1 (r/k); and it is to be noticed that
these helices are left-handed if the given wrench is right-handed, and
vice versa.
Since the given wrench can be replaced by a force acting through any
assigned point P, and a couple, the locus of the null-lines through P
is a plane, viz. a plane perpendicular to the vector which represents
the couple. The complex is therefore of the type called "linear" (in
relation to the degree of this locus). The plane in question is called
the _null-plane_ of P. If the null-plane of P pass through Q, the
null-plane of Q will pass through P, since PQ is a null-line. Again,
any plane [omega] is the locus of a system of null-lines meeting in a
point, called the _null-point_ of [omega]. If a plane revolve about a
fixed straight line p in it, its null-point describes another straight
line p´, which is called the _conjugate line_ of p. We have seen that
the wrench may be replaced by two forces, one of which may act in any
arbitrary line p. It is now evident that the second force must act in
the conjugate line p´, since every line meeting p, p´ is a null-line.
Again, since the shortest distance between any two conjugate lines
cuts the central axis at right angles, the orthogonal projections of
two conjugate lines on a plane perpendicular to the central axis will
be parallel (fig. 42). This property was employed by L. Cremona to
prove the existence under certain conditions of "reciprocal figures"
in a plane (§ 5). If we take any polyhedron with plane faces, the
null-planes of its vertices with respect to a given wrench will form
another polyhedron, and the edges of the latter will be conjugate (in
the above sense) to those of the former. Projecting orthogonally on a
plane perpendicular to the central axis we obtain two reciprocal
figures.
In the analogous theory of infinitely small displacements of a solid,
a "null-line" is a line such that the lengthwise displacement of any
point on it is zero.
Since a wrench is defined by six independent quantities, it can in
general be replaced by any system of forces which involves six
adjustable elements. For instance, it can in general be replaced by
six forces acting in six given lines, e.g. in the six edges of a given
tetrahedron. An exception to the general statement occurs when the six
lines are such that they are possible lines of action of a system of
six forces in equilibrium; they are then said to be _in involution_.
The theory of forces in involution has been studied by A. Cayley, J.
J. Sylvester and others. We have seen that a rigid structure may in
general be rigidly connected with the earth by six links, and it now
appears that any system of forces acting on the structure can in
general be balanced by six determinate forces exerted by the links.
If, however, the links are in involution, these forces become infinite
or indeterminate. There is a corresponding kinematic peculiarity, in
that the connexion is now not strictly rigid, an infinitely small
relative displacement being possible. See § 9.
When parallel forces of given magnitudes act at given points, the
resultant acts through a definite point, or _centre of parallel forces_,
which is independent of the special direction of the forces. If P_r be
the force at (x_r, y_r, z_r), acting in the direction (l, m, n), the
formulae (6) and (7) reduce to
X = [Sigma](P).l, Y = [Sigma](P).m, Z = [Sigma](P).n, (12)
and
L = [Sigma](P)·(n[|y] - m[|z]), M = [Sigma](P)·(l[|z] - n[|x]), N = [Sigma](P)·(m[|x] - l[|y]), (13)
provided
[Sigma](Px) [Sigma](Py) [Sigma](Pz)
[|x] = -----------, [|y] = -----------, [|z] = -----------. (14)
[Sigma](P) [Sigma](P) [Sigma](P)
These are the same as if we had a single force [Sigma](P) acting at the
point ([|x], [|y], [|z]), which is the same for all directions (l, m,
n). We can hence derive the theory of the centre of gravity, as in § 4.
An exceptional case occurs when [Sigma](P) = 0.
If we imagine a rigid body to be acted on at given points by forces of
given magnitudes in directions (not all parallel) which are fixed in
space, then as the body is turned about the resultant wrench will
assume different configurations in the body, and will in certain
positions reduce to a single force. The investigation of such
questions forms the subject of "Astatics," which has been cultivated
by Möbius, Minding, G. Darboux and others. As it has no physical
bearing it is passed over here.
[Illustration: FIG. 45.]
§ 9. _Work._--The _work_ done by a force acting on a particle, in any
infinitely small displacement, is defined as the product of the force
into the orthogonal projection of the displacement on the direction of
the force; i.e. it is equal to F·[delta]s cos [theta], where F is the
force, [delta]s the displacement, and [theta] is the angle between the
directions of F and [delta]s. In the language of vector analysis (q.v.)
it is the "scalar product" of the vector representing the force and the
displacement. In the same way, the work done by a force acting on a
rigid body in any infinitely small displacement of the body is the
scalar product of the force into the displacement of any point on the
line of action. This product is the same whatever point on the line of
action be taken, since the lengthwise components of the displacements of
any two points A, B on a line AB are equal, to the first order of small
quantities. To see this, let A´, B´ be the displaced positions of A, B,
and let [phi] be the infinitely small angle between AB and A´B´. Then if
[alpha], [beta] be the orthogonal projections of A´, B´ on AB, we have
A[alpha] - B[beta] = AB - [alpha][beta] = AB(1 - cos [phi]) = ½AB·[phi]²,
ultimately. Since this is of the second order, the products F·A[alpha]
and F·B[beta] are ultimately equal.
[Illustration: FIG. 46.]
[Illustration: FIG. 47.]
The total work done by two concurrent forces acting on a particle, or on
a rigid body, in any infinitely small displacement, is equal to the work
of their resultant. Let AB, AC (fig. 46) represent the forces, AD their
resultant, and let AH be the direction of the displacement [delta]s of
the point A. The proposition follows at once from the fact that the sum
of orthogonal projections of [->AB], [->AC] on AH is equal to the
projection of [->AD]. It is to be noticed that AH need not be in the
same plane with AB, AC.
It follows from the preceding statements that any two systems of forces
which are statically equivalent, according to the principles of §§ 4, 8,
will (to the first order of small quantities) do the same amount of work
in any infinitely small displacement of a rigid body to which they may
be applied. It is also evident that the total work done in two or more
successive infinitely small displacements is equal to the work done in
the resultant displacement.
The work of a couple in any infinitely small rotation of a rigid body
about an axis perpendicular to the plane of the couple is equal to the
product of the moment of the couple into the angle of rotation, proper
conventions as to sign being observed. Let the couple consist of two
forces P, P (fig. 47) in the plane of the paper, and let J be the point
where this plane is met by the axis of rotation. Draw JBA perpendicular
to the lines of action, and let [epsilon] be the angle of rotation. The
work of the couple is
P·JA·[epsilon] - P·JB·[epsilon] = P·AB·[epsilon] = G[epsilon],
if G be the moment of the couple.
The analytical calculation of the work done by a system of forces in any
infinitesimal displacement is as follows. For a two-dimensional system
we have, in the notation of §§ 3, 4,
[Sigma](X[delta]x + Y[delta]y) = [Sigma]{X([lambda] - y[epsilon]) + Y([mu] + x[epsilon])}
= [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](xY - yX)[epsilon]
= X[lambda] + Y[mu] + N[epsilon]. (1)
Again, for a three-dimensional system, in the notation of §§ 7, 8,
[Sigma](X[delta]x + Y[delta]y + Z[delta]z)
= [Sigma]{(X([lambda] + [eta]z - [zeta]y) + Y([mu] + [zeta]x - [xi]x) + Z([nu] + [xi]y - [eta]x)}
= [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](Z)·[nu] + [Sigma](yZ - zY)·[xi]
+ [Sigma](zX - xZ)·[eta] + [Sigma](xY - yX)·[zeta]
= X[lambda] + Y[mu] + Z[nu] + L[xi] + M[eta] + N[zeta]. (2)
This expression gives the work done by a given wrench when the body
receives a given infinitely small twist; it must of course be an
absolute invariant for all transformations of rectangular axes. The
first three terms express the work done by the components of a force (X,
Y, Z) acting at O, and the remaining three terms express the work of a
couple (L, M, N).
[Illustration: FIG. 48.]
The work done by a wrench about a given screw, when the body twists
about a second given screw, may be calculated directly as follows. In
fig. 48 let R, G be the force and couple of the wrench,
[epsilon],[tau] the rotation and translation in the twist. Let the
axes of the wrench and the twist be inclined at an angle [theta], and
let h be the shortest distance between them. The displacement of the
point H in the figure, resolved in the direction of R, is [tau] cos
[theta] - [epsilon]h sin [theta]. The work is therefore
R([tau] cos [theta] - [epsilon]h sin [theta]) + G cos [theta]
= R[epsilon]{(p + p´) cos [theta] - h sin [theta]}, (3)
if G = pR, [tau] = p´[epsilon], i.e. p, p´ are the pitches of the two
screws. The factor (p + p´) cos[theta] - h sin[theta] is called the
_virtual coefficient_ of the two screws which define the types of the
wrench and twist, respectively.
A screw is determined by its axis and its pitch, and therefore
involves five Independent elements. These may be, for instance, the
five ratios [xi]:[eta]:[zeta]:[lambda]:[mu]:[nu] of the six quantities
which specify an infinitesimal twist about the screw. If the twist is
a pure rotation, these quantities are subject to the relation
[lambda][xi] + [mu][eta] + [nu][zeta] = 0. (4)
In the analytical investigations of line geometry, these six
quantities, supposed subject to the relation (4), are used to specify
a line, and are called the six "co-ordinates" of the line; they are of
course equivalent to only four independent quantities. If a line is a
null-line with respect to the wrench (X, Y, Z, L, M, N), the work done
in an infinitely small rotation about it is zero, and its co-ordinates
are accordingly subject to the further relation
L[xi] + M[eta] + N[zeta] + X[lambda] + Y[mu] + Z[nu] = 0, (5)
where the coefficients are constant. This is the equation of a "linear
complex" (cf. § 8).
Two screws are _reciprocal_ when a wrench about one does no work on a
body which twists about the other. The condition for this is
[lambda][xi]´ + [mu][eta]´ + [nu][zeta]´ + [lambda]´[xi] + [mu]´[eta] + [nu]´[zeta] = 0, (6)
if the screws be defined by the ratios [xi] : [eta] : [zeta] :
[lambda] : [mu] : [nu] and [xi]´ : [eta]´ : [zeta]´ : [lambda]´ :
[mu]´ : [nu]´, respectively. The theory of the screw-systems which are
reciprocal to one, two, three, four given screws respectively has been
investigated by Sir R. S. Ball.
Considering a rigid body in any given position, we may contemplate the
whole group of infinitesimal displacements which might be given to it.
If the extraneous forces are in equilibrium the total work which they
would perform in any such displacement would be zero, since they reduce
to a zero force and a zero couple. This is (in part) the celebrated
principle of _virtual velocities_, now often described as the principle
of _virtual work_, enunciated by John Bernoulli (1667-1748). The word
"virtual" is used because the displacements in question are not regarded
as actually taking place, the body being in fact at rest. The
"velocities" referred to are the velocities of the various points of the
body in any imagined motion of the body through the position in
question; they obviously bear to one another the same ratios as the
corresponding infinitesimal displacements. Conversely, we can show that
if the virtual work of the extraneous forces be zero for every
infinitesimal displacement of the body as rigid, these forces must be in
equilibrium. For by giving the body (in imagination) a displacement of
translation we learn that the sum of the resolved parts of the forces in
any assigned direction is zero, and by giving it a displacement of pure
rotation we learn that the sum of the moments about any assigned axis is
zero. The same thing follows of course from the analytical expression
(2) for the virtual work. If this vanishes for all values of [lambda],
[mu], [nu], [xi], [eta], [zeta] we must have X, Y, Z, L, M, N = 0, which
are the conditions of equilibrium.
The principle can of course be extended to any system of particles or
rigid bodies, connected together in any way, provided we take into
account the internal stresses, or reactions, between the various parts.
Each such reaction consists of two equal and opposite forces, both of
which may contribute to the equation of virtual work.
The proper significance of the principle of virtual work, and of its
converse, will appear more clearly when we come to kinetics (§ 16); for
the present it may be regarded merely as a compact and (for many
purposes) highly convenient summary of the laws of equilibrium. Its
special value lies in this, that by a suitable adjustment of the
hypothetical displacements we are often enabled to eliminate unknown
reactions. For example, in the case of a particle lying on a smooth
curve, or on a smooth surface, if it be displaced along the curve, or on
the surface, the virtual work of the normal component of the pressure
may be ignored, since it is of the second order. Again, if two bodies
are connected by a string or rod, and if the hypothetical displacements
be adjusted so that the distance between the points of attachment is
unaltered, the corresponding stress may be ignored. This is evident from
fig. 45; if AB, A´B´ represent the two positions of a string, and T be
the tension, the virtual work of the two forces ±T at A, B is T(A[alpha]
- B[beta]), which was shown to be of the second order. Again, the normal
pressure between two surfaces disappears from the equation, provided the
displacements be such that one of these surfaces merely slides
relatively to the other. It is evident, in the first place, that in any
displacement common to the two surfaces, the work of the two equal and
opposite normal pressures will cancel; moreover if, one of the surfaces
being fixed, an infinitely small displacement shifts the point of
contact from A to B, and if A´ be the new position of that point of the
sliding body which was at A, the projection of AA´ on the normal at A is
of the second order. It is to be noticed, in this case, that the
tangential reaction (if any) between the two surfaces is not eliminated.
Again, if the displacements be such that one curved surface rolls
without sliding on another, the reaction, whether normal or tangential,
at the point of contact may be ignored. For the virtual work of two
equal and opposite forces will cancel in any displacement which is
common to the two surfaces; whilst, if one surface be fixed, the
displacement of that point of the rolling surface which was in contact
with the other is of the second order. We are thus able to imagine a
great variety of mechanical systems to which the principle of virtual
work can be applied without any regard to the internal stresses,
provided the hypothetical displacements be such that none of the
connexions of the system are violated.
If the system be subject to gravity, the corresponding part of the
virtual work can be calculated from the displacement of the centre of
gravity. If W1, W2, ... be the weights of a system of particles, whose
depths below a fixed horizontal plane of reference are z1, z2, ...,
respectively, the virtual work of gravity is
W1[delta]·z1 + W2[delta]z2 + ... = [delta](W1z1 + W2z2 + ...) (7)
= (W1 + W2 + ...) [delta][|z],
where [|z] is the depth of the centre of gravity (see § 8 (14) and § 11
(6)). This expression is the same as if the whole mass were concentrated
at the centre of gravity, and displaced with this point. An important
conclusion is that in any displacement of a system of bodies in
equilibrium, such that the virtual work of all forces except gravity may
be ignored, the depth of the centre of gravity is "stationary."
The question as to stability of equilibrium belongs essentially to
kinetics; but we may state by anticipation that in cases where gravity
is the only force which does work, the equilibrium of a body or system
of bodies is stable only if the depth of the centre of gravity be a
maximum.
[Illustration: FIG. 49.]
Consider, for instance, the case of a bar resting with its ends on two
smooth inclines (fig. 18). If the bar be displaced in a vertical plane
so that its ends slide on the two inclines, the instantaneous centre
is at the point J. The displacement of G is at right angles to JG;
this shows that for equilibrium JG must be vertical. Again, the locus
of G is an arc of an ellipse whose centre is in the intersection of
the planes; since this arc is convex upwards the equilibrium is
unstable. A general criterion for the case of a rigid body movable in
two dimensions, with one degree of freedom, can be obtained as
follows. We have seen (§ 3) that the sequence of possible positions is
obtained if we imagine the "body-centrode" to roll on the
"space-centrode." For equilibrium, the altitude of the centre of
gravity G must be stationary; hence G must lie in the same vertical
line with the point of contact J of the two curves. Further, it is
known from the theory of "roulettes" that the locus of G will be
concave or convex upwards according as
cos[phi] 1 1
-------- = ----- + ------, (8)
h [rho] [rho]´
where [rho], [rho]´ are the radii of curvature of the two curves at J,
[phi] is the inclination of the common tangent at J to the horizontal,
and h is the height of G above J. The signs of [rho], [rho]´ are to be
taken positive when the curvatures are as in the standard case shown
in fig. 49. Hence for stability the upper sign must obtain in (8). The
same criterion may be arrived at in a more intuitive manner as
follows. If the body be supposed to roll (say to the right) until the
curves touch at J´, and if JJ´ = [delta]s, the angle through which the
upper figure rotates is [delta]s/[rho] + [delta]s/[rho]´, and the
horizontal displacement of G is equal to the product of this
expression into h. If this displacement be less than the horizontal
projection of JJ´, viz. [delta]s cos[phi], the vertical through the
new position of G will fall to the left of J´ and gravity will tend to
restore the body to its former position. It is here assumed that the
remaining forces acting on the body in its displaced position have
zero moment about J´; this is evidently the case, for instance, in the
problem of "rocking stones."
The principle of virtual work is specially convenient in the theory of
frames (§ 6), since the reactions at smooth joints and the stresses in
inextensible bars may be left out of account. In particular, in the case
of a frame which is just rigid, the principle enables us to find the
stress in any one bar independently of the rest. If we imagine the bar
in question to be removed, equilibrium will still persist if we
introduce two equal and opposite forces S, of suitable magnitude, at the
joints which it connected. In any infinitely small deformation of the
frame as thus modified, the virtual work of the forces S, together with
that of the original extraneous forces, must vanish; this determines S.
As a simple example, take the case of a light frame, whose bars form
the slides of a rhombus ABCD with the diagonal BD, suspended from A
and carrying a weight W at C; and let it be required to find the
stress in BD. If we remove the bar BD, and apply two equal and
opposite forces S at B and D, the equation is
W·[delta](2l cos[theta]) + 2S·[delta](l sin [theta]) = 0,
where l is the length of a side of the rhombus, and [theta] its
inclination to the vertical. Hence
S = W tan [theta] = W·BD/AC. (8)
[Illustration: FIG. 50.]
The method is specially appropriate when the frame, although just
rigid, is not "simple" in the sense of § 6, and when accordingly the
method of reciprocal figures is not immediately available. To avoid
the intricate trigonometrical calculations which would often be
necessary, graphical devices have been introduced by H. Müller-Breslau
and others. For this purpose the infinitesimal displacements of the
various joints are replaced by finite lengths proportional to them,
and therefore proportional to the velocities of the joints in some
imagined motion of the deformable frame through its actual
configuration; this is really (it may be remarked) a reversion to the
original notion of "virtual velocities." Let J be the instantaneous
centre for any bar CD (fig. 12), and let s1, s2 represent the virtual
velocities of C, D. If these lines be turned through a right angle in
the same sense, they take up positions such as CC´, DD´, where C´, D´
are on JC, JD, respectively, and C´D´ is parallel to CD. Further, if
F1 (fig. 51) be any force acting on the joint C, its virtual work will
be equal to the moment of F1 about C´; the equation of virtual work is
thus transformed into an equation of moments.
[Illustration: FIG. 12.]
[Illustration: FIG. 51.]
[Illustration: FIG. 52.]
Consider, for example, a frame whose sides form the six sides of a
hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it
is required to find the stress in CF due to a given system of
extraneous forces in equilibrium, acting on the joints. Imagine the
bar CF to be removed, and consider a deformation in which AB is fixed.
The instantaneous centre of CD will be at the intersection of AD, BC,
and if C´D´ be drawn parallel to CD, the lines CC´, DD´ may be taken
to represent the virtual velocities of C, D turned each through a
right angle. Moreover, if we draw D´E´ parallel to DE, and E´F´
parallel to EF, the lines CC´, DD´, EE´, FF´ will represent on the
same scale the virtual velocities of the points C, D, E, F,
respectively, turned each through a right angle. The equation of
virtual work is then formed by taking moments about C´, D´, E´, F´ of
the extraneous forces which act at C, D, E, F, respectively. Amongst
these forces we must include the two equal and opposite forces S which
take the place of the stress in the removed bar FC.
The above method lends itself naturally to the investigation of the
_critical forms_ of a frame whose general structure is given. We have
seen that the stresses produced by an equilibrating system of
extraneous forces in a frame which is just rigid, according to the
criterion of § 6, are in general uniquely determinate; in particular,
when there are no extraneous forces the bars are in general free from
stress. It may however happen that owing to some special relation
between the lengths of the bars the frame admits of an infinitesimal
deformation. The simplest case is that of a frame of three bars, when
the three joints A, B, C fall into a straight line; a small
displacement of the joint B at right angles to AC would involve
changes in the lengths of AB, BC which are only of the second order of
small quantities. Another example is shown in fig. 53. The graphical
method leads at once to the detection of such cases. Thus in the
hexagonal frame of fig. 52, if an infinitesimal deformation is
possible without removing the bar CF, the instantaneous centre of CF
(when AB is fixed) will be at the intersection of AF and BC, and since
CC´, FF´ represent the virtual velocities of the points C, F, turned
each through a right angle, C´F´ must be parallel to CF. Conversely,
if this condition be satisfied, an infinitesimal deformation is
possible. The result may be generalized into the statement that a
frame has a critical form whenever a frame of the same structure can
be designed with corresponding bars parallel, but without complete
geometric similarity. In the case of fig. 52 it may be shown that an
equivalent condition is that the six points A, B, C, D, E, F should
lie on a conic (M. W. Crofton). This is fulfilled when the opposite
sides of the hexagon are parallel, and (as a still more special case)
when the hexagon is regular.
[Illustration: FIG. 53.]
When a frame has a critical form it may be in a state of stress
independently of the action of extraneous forces; moreover, the
stresses due to extraneous forces are indeterminate, and may be
infinite. For suppose as before that one of the bars is removed. If
there are no extraneous forces the equation of virtual work reduces to
S·[delta]s = 0, where S is the stress in the removed bar, and [delta]s
is the change in the distance between the joints which it connected.
In a critical form we have [delta]s = 0, and the equation is satisfied
by an arbitrary value of S; a consistent system of stresses in the
remaining bars can then be found by preceding rules. Again, when
extraneous forces P act on the joints, the equation is
[Sigma](P·[delta]p) + S·[delta]s = 0,
where [delta]p is the displacement of any joint in the direction of
the corresponding force P. If [Sigma](P·[delta]p) = 0, the stresses
are merely indeterminate as before; but if [Sigma] (P·[delta]p) does
not vanish, the equation cannot be satisfied by any finite value of S,
since [delta]s = 0. This means that, if the material of the frame were
absolutely unyielding, no finite stresses in the bars would enable it
to withstand the extraneous forces. With actual materials, the frame
would yield elastically, until its configuration is no longer
"critical." The stresses in the bars would then be comparatively very
great, although finite. The use of frames which approximate to a
critical form is of course to be avoided in practice.
A brief reference must suffice to the theory of three dimensional
frames. This is important from a technical point of view, since all
structures are practically three-dimensional. We may note that a frame
of n joints which is just rigid must have 3n - 6 bars; and that the
stresses produced in such a frame by a given system of extraneous
forces in equilibrium are statically determinate, subject to the
exception of "critical forms."
§ 10. _Statics of Inextensible Chains._--The theory of bodies or
structures which are deformable in their smallest parts belongs properly
to elasticity (q.v.). The case of inextensible strings or chains is,
however, so simple that it is generally included in expositions of pure
statics.
It is assumed that the form can be sufficiently represented by a plane
curve, that the stress (tension) at any point P of the curve, between
the two portions which meet there, is in the direction of the tangent at
P, and that the forces on any linear element [delta]s must satisfy the
conditions of equilibrium laid down in § 1. It follows that the forces
on any finite portion will satisfy the conditions of equilibrium which
apply to the case of a rigid body (§ 4).
[Illustration: FIG. 54.]
We will suppose in the first instance that the curve is plane. It is
often convenient to resolve the forces on an element PQ (= [delta]s) in
the directions of the tangent and normal respectively. If T, T +
[delta]T be the tensions at P, Q, and [delta][psi] be the angle between
the directions of the curve at these points, the components of the
tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or
[delta]T, ultimately; whilst for the component along the normal at P we
have (T + [delta]T) sin [delta][psi], or T[delta][psi], or
T[delta]s/[rho], where [rho] is the radius of curvature.
Suppose, for example, that we have a light string stretched over a
smooth curve; and let R[delta]s denote the normal pressure (outwards
from the centre of curvature) on [delta]s. The two resolutions give
[delta]T = 0, T[delta][psi] = R[delta]s, or
T = const., R = T/[rho]. (1)
The tension is constant, and the pressure per unit length varies as the
curvature.
Next suppose that the curve is "rough"; and let F[delta]s be the
tangential force of friction on [delta]s. We have [delta]T ± F[delta]s =
0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be
taken according to the sense in which F acts. We assume that in
limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the
coefficient of friction. If the string be on the point of slipping in
the direction in which [psi] increases, the lower sign is to be taken;
hence [delta]T = F[delta]s = [mu]T[delta][psi], whence
T = T0 e^([mu][psi]), (2)
if T0 be the tension corresponding to [psi] = 0. This illustrates the
resistance to dragging of a rope coiled round a post; e.g. if we put
[mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn
T/T0 = 6.5. In two turns this ratio is squared, and so on.
Again, take the case of a string under gravity, in contact with a smooth
curve in a vertical plane. Let [psi] denote the inclination to the
horizontal, and w [delta]s the weight of an element [delta]s. The
tangential and normal components of w[delta]s are -s sin [psi] and
-w [delta]s cos [psi]. Hence
[delta]T = w [delta]s sin [psi], T [delta][psi] = w [delta]s cos [psi] + R[delta]s. (3)
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically
upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T =
w[delta]y. If the string be uniform, w is constant, and
T = wy + const. = w(y - y0), (4)
say; hence the tension varies as the height above some fixed level (y0).
The pressure is then given by the formula
d[psi]
R = T ------ - w cos [psi]. (5)
ds
In the case of a chain hanging freely under gravity it is usually
convenient to formulate the conditions of equilibrium of a finite
portion PQ. The forces on this reduce to three, viz. the weight of PQ
and the tensions at P, Q. Hence these three forces will be concurrent,
and their ratios will be given by a triangle of forces. In particular,
if we consider a length AP beginning at the lowest point A, then
resolving horizontally and vertically we have
T cos [psi] = T0, T sin [psi] = W, (6)
where T0 is the tension at A, and W is the weight of PA. The former
equation expresses that the horizontal tension is constant.
[Illustration: FIG. 55.]
If the chain be uniform we have W = ws, where s is the arc AP: hence ws
= T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion
of the chain whose weight would equal the horizontal tension, this
becomes
s = a tan [psi]. (7)
This is the "intrinsic" equation of the curve. If the axes of x and y be
taken horizontal and vertical (upwards), we derive
x = a log (sec [psi] + tan [psi]), y = a sec [psi]. (8)
Eliminating [psi] we obtain the Cartesian equation
x
y = a cosh --- (9)
a
of the _common catenary_, as it is called (fig. 56). The omission of the
additive arbitrary constants of integration in (8) is equivalent to a
special choice of the origin O of co-ordinates; viz. O is at a distance
a vertically below the lowest point ([psi] = 0) of the curve. The
horizontal line through O is called the _directrix_. The relations
s = a sinh x/a, y² = a² + s², T = T0 sec [psi] = wy, (10)
[Illustration: FIG. 56.]
which are involved in the preceding formulae are also noteworthy. It is
a classical problem in the calculus of variations to deduce the equation
(9) from the condition that the depth of the centre of gravity of a
chain of given length hanging between fixed points must be stationary (§
9). The length a is called the _parameter_ of the catenary; it
determines the scale of the curve, all catenaries being geometrically
similar. If weights be suspended from various points of a hanging chain,
the intervening portions will form arcs of equal catenaries, since the
horizontal tension (wa) is the same for all. Again, if a chain pass over
a perfectly smooth peg, the catenaries in which it hangs on the two
sides, though usually of different parameters, will have the same
directrix, since by (10) y is the same for both at the peg.
As an example of the use of the formulae we may determine the maximum
span for a wire of given material. The condition is that the tension
must not exceed the weight of a certain length [lambda] of the wire.
At the ends we shall have y = [lambda], or
x
[lambda] = a cosh ---, (11)
a
and the problem is to make x a maximum for variations of a.
Differentiating (11) we find that, if dx/da = 0,
x x
--- tanh --- = 1. (12)
a a
It is easily seen graphically, or from a table of hyperbolic tangents,
that the equation u tanh u = 1 has only one positive root (u = 1.200);
the span is therefore
2x = 2au = 2[lambda]/sinh u = 1.326[lambda],
and the length of wire is
2s = 2[lambda]/u = 1.667 [lambda].
The tangents at the ends meet on the directrix, and their inclination
to the horizontal is 56° 30´.
[Illustration: FIG. 57.]
The relation between the sag, the tension, and the span of a wire
(e.g. a telegraph wire) stretched nearly straight between two points
A, B at the same level is determined most simply from first
principles. If T be the tension, W the total weight, k the sag in the
middle, and [psi] the inclination to the horizontal at A or B, we have
2T[psi] = W, AB = 2[rho][psi], approximately, where [rho] is the
radius of curvature. Since 2k[rho] = (½AB)², ultimately, we have
k = (1/8)W·AB/T. (13)
The same formula applies if A, B be at different levels, provided k be
the sag, measured vertically, half way between A and B.
In relation to the theory of suspension bridges the case where the
weight of any portion of the chain varies as its horizontal projection
is of interest. The vertical through the centre of gravity of the arc AP
(see fig. 55) will then bisect its horizontal projection AN; hence if PS
be the tangent at P we shall have AS = SN. This property is
characteristic of a parabola whose axis is vertical. If we take A as
origin and AN as axis of x, the weight of AP may be denoted by wx, where
w is the weight per unit length at A. Since PNS is a triangle of forces
for the portion AP of the chain, we have wx/T0 = PN/NS, or
y = w·x²/2T0, (14)
which is the equation of the parabola in question. The result might of
course have been inferred from the theory of the parabolic funicular in
§ 2.
Finally, we may refer to the _catenary of uniform strength_, where the
cross-section of the wire (or cable) is supposed to vary as the
tension. Hence w, the weight per foot, varies as T, and we may write
T = w[lambda], where [lambda] is a constant length. Resolving along
the normal the forces on an element [delta]s, we find T[delta][psi] =
w[delta]s cos[psi], whence
ds
p = ------ = [lambda] sec [psi]. (15)
d[psi]
From this we derive
x
x = [lambda][psi], y = [lambda] log sec --------, (16)
[lambda]
where the directions of x and y are horizontal and vertical, and the
origin is taken at the lowest point. The curve (fig. 58) has two
vertical asymptotes x = ± ½[pi][lambda]; this shows that however the
thickness of a cable be adjusted there is a limit [pi][lambda] to the
horizontal span, where [lambda] depends on the tensile strength of the
material. For a uniform catenary the limit was found above to be
1.326[lambda].
[Illustration: FIG. 58.]
For investigations relating to the equilibrium of a string in three
dimensions we must refer to the textbooks. In the case of a string
stretched over a smooth surface, but in other respects free from
extraneous force, the tensions at the ends of a small element [delta]s
must be balanced by the normal reaction of the surface. It follows that
the osculating plane of the curve formed by the string must contain the
normal to the surface, i.e. the curve must be a "geodesic," and that the
normal pressure per unit length must vary as the principal curvature of
the curve.
§ 11. _Theory of Mass-Systems._--This is a purely geometrical subject.
We consider a system of points P1, P2 ..., P_n, with which are
associated certain coefficients m1, m2, ... m_n, respectively. In the
application to mechanics these coefficients are the masses of particles
situate at the respective points, and are therefore all positive. We
shall make this supposition in what follows, but it should be remarked
that hardly any difference is made in the theory if some of the
coefficients have a different sign from the rest, except in the special
case where [Sigma](m) = 0. This has a certain interest in magnetism.
In a given mass-system there exists one and only one point G such that
[Sigma](m·[->GP]) = 0. (1)
For, take any point O, and construct the vector
[Sigma](m·[->OP])
[->OG] = -----------------. (2)
[Sigma](m)
Then
[Sigma](m·[->GP]) = [Sigma]{m([->GO] + [->OP])} = [Sigma](m)·[->GO] + [Sigma](m)·[->OP] = 0. (3)
Also there cannot be a distinct point G´ such that [Sigma](m·G´P) = 0,
for we should have, by subtraction,
[Sigma]{m([->GP] + [->PG´])} = 0, or [Sigma](m)·GG´ = 0; (4)
i.e. G´ must coincide with G. The point G determined by (1) is called
the _mass-centre_ or _centre of inertia_ of the given system. It is
easily seen that, in the process of determining the mass-centre, any
group of particles may be replaced by a single particle whose mass is
equal to that of the group, situate at the mass-centre of the group.
If through P1, P2, ... P_n we draw any system of parallel planes meeting
a straight line OX in the points M1, M2 ... M_n, the collinear vectors
[->OM1], [->OM2] ... [->OM_n] may be called the "projections" of
[->OP1], [->OP2], ... [->OP_n] on OX. Let these projections be denoted
algebraically by x1, x2, ... x_n, the sign being positive or negative
according as the direction is that of OX or the reverse. Since the
projection of a vector-sum is the sum of the projections of the several
vectors, the equation (2) gives
[Sigma](mx)
[|x] = -----------, (5)
[Sigma](m)
if [|x] be the projection of [->OG]. Hence if the Cartesian co-ordinates
of P1, P2, ... P_n relative to any axes, rectangular or oblique be (x1,
y1, z1), (x2, y2, z2), ..., (x_n, y_n, z_n), the mass-centre ([|x],
[|y], [|z]) is determined by the formulae
[Sigma](mx) [Sigma](my) [Sigma](mz)
[|x] = -----------, [|y] = -----------, [|z] = -----------. (6)
[Sigma](m) [Sigma](m) [Sigma](m)
If we write x = [|x] + [xi], y = [|y] + [eta], z = [|z] + [zeta], so
that [xi], [eta], [zeta] denote co-ordinates relative to the mass-centre
G, we have from (6)
[Sigma](m[xi]) = 0, [Sigma](m[eta]) = 0, [Sigma](m[zeta]) = 0. (7)
One or two special cases may be noticed. If three masses [alpha],
[beta], [gamma] be situate at the vertices of a triangle ABC, the
mass-centre of [beta] and [gamma] is at a point A´ in BC, such that
[beta]·BA´ = [gamma]·A´C. The mass-centre (G) of [alpha], [beta],
[gamma] will then divide AA´ so that [alpha]·AG = ([beta] + [gamma])
GA´. It is easily proved that
[alpha] : [beta] : [gamma] = [Delta]BGA : [Delta]GCA : [Delta]GAB;
also, by giving suitable values (positive or negative) to the ratios
[alpha] : [beta] : [gamma] we can make G assume any assigned position
in the plane ABC. We have here the origin of the "barycentric
co-ordinates" of Möbius, now usually known as "areal" co-ordinates. If
[alpha] + [beta] + [gamma] = 0, G is at infinity; if [alpha] = [beta]
= [gamma], G is at the intersection of the median lines of the
triangle; if [alpha] : [beta] : [gamma] = a : b : c, G is at the
centre of the inscribed circle. Again, if G be the mass-centre of four
particles [alpha], [beta], [gamma], [delta] situate at the vertices of
a tetrahedron ABCD, we find
[alpha] : [beta] : [gamma] : [delta] = tet^n GBCD : tet^n GCDA : tet^n GDAB : tet^n GABC,
and by suitable determination of the ratios on the left hand we can
make G assume any assigned position in space. If [alpha] + [beta] +
[gamma] + [delta] = O, G is at infinity; if [alpha] = [beta] = [gamma]
= [delta], G bisects the lines joining the middle points of opposite
edges of the tetrahedron ABCD; if [alpha] : [beta] : [gamma] : [delta]
= [Delta]BCD : [Delta]CDA : [Delta]DAB : [Delta]ABC, G is at the
centre of the inscribed sphere.
If we have a continuous distribution of matter, instead of a system of
discrete particles, the summations in (6) are to be replaced by
integrations. Examples will be found in textbooks of the calculus and
of analytical statics. As particular cases: the mass-centre of a
uniform thin triangular plate coincides with that of three equal
particles at the corners; and that of a uniform solid tetrahedron
coincides with that of four equal particles at the vertices. Again,
the mass-centre of a uniform solid right circular cone divides the
axis in the ratio 3 : 1; that of a uniform solid hemisphere divides
the axial radius in the ratio 3 : 5.
It is easily seen from (6) that if the configuration of a system of
particles be altered by "homogeneous strain" (see ELASTICITY) the new
position of the mass-centre will be at that point of the strained
figure which corresponds to the original mass-centre.
The formula (2) shows that a system of concurrent forces represented by
m1·[->OP1], m2·[->OP2], ... m_n·[->OP_n] will have a resultant
represented hy [Sigma](m)·[->OG]. If we imagine O to recede to infinity
in any direction we learn that a system of parallel forces proportional
to m1, m2,... m_n, acting at P1, P2 ... P_n have a resultant
proportional to [Sigma](m) which acts always through a point G fixed
relatively to the given mass-system. This contains the theory of the
"centre of gravity" (§§ 4, 9). We may note also that if P1, P2, ... P_n,
and P1´, P2´, ... P_n´ represent two configurations of the series of
particles, then
[Sigma](m·[->PP´]) = Sigma(m)·[->GG´], (8)
where G, G´ are the two positions of the mass-centre. The forces
m1·[->P1P1´], m2·[->P2P2´], ... m_n·[->P_nP_n´], considered as localized
vectors, do not, however, as a rule reduce to a single resultant.
We proceed to the theory of the _plane_, _axial_ and _polar quadratic
moments_ of the system. The axial moments have alone a dynamical
significance, but the others are useful as subsidiary conceptions. If
h1, h2, ... h_n be the perpendicular distances of the particles from any
fixed plane, the sum [Sigma](mh²) is the quadratic moment with respect
to the plane. If p1, p2, ... p_n be the perpendicular distances from any
given axis, the sum [Sigma](mp²) is the quadratic moment with respect to
the axis; it is also called the _moment of inertia_ about the axis. If
r1, r2, ... r_n be the distances from a fixed point, the sum
[Sigma](mr²) is the quadratic moment with respect to that point (or
pole). If we divide any of the above quadratic moments by the total
mass [Sigma](m), the result is called the _mean square_ of the distances
of the particles from the respective plane, axis or pole. In the case of
an axial moment, the square root of the resulting mean square is called
the _radius of gyration_ of the system about the axis in question. If we
take rectangular axes through any point O, the quadratic moments with
respect to the co-ordinate planes are
I_x = [Sigma](mx²), I_y = [Sigma](my²), I_z = [Sigma](mz²); (9)
those with respect to the co-ordinate axes are
I_yz = [Sigma]{m(y² + z²)}, I_zx = [Sigma]{m(z² + x²)},
I_xy = [Sigma]{m(x² + y²)}; (10)
whilst the polar quadratic moment with respect to O is
I0 = [Sigma]{m(x² + y² + z²)}. (11)
We note that
I_yz = I_y + I_z, I_zx = I_z + I_x, I_xy = I_x + I_y, (12)
and
I0 = I_x + I_y + I_z = ½(I_yz + I_zx + I_xy). (13)
In the case of continuous distributions of matter the summations in
(9), (10), (11) are of course to be replaced by integrations. For a
uniform thin circular plate, we find, taking the origin at its centre,
and the axis of z normal to its plane, I0 = ½Ma², where M is the mass
and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = ½Ma²,
I_xy = ½Ma²; hence the value of the squared radius of gyration is for
a diameter ¼a², and for the axis of symmetry ½a². Again, for a uniform
solid sphere having its centre at the origin we find I0 = (3/5)Ma²,
I_x = I_y = I_z = (1/5)Ma², I_yz = I_zx = l_xy = (3/5)Ma²; i.e. the
square of the radius of gyration with respect to a diameter is
(2/5)a². The method of homogeneous strain can be applied to deduce the
corresponding results for an ellipsoid of semi-axes a, b, c. If the
co-ordinate axes coincide with the principal axes, we find I_x =
(1/5)Ma², I_y = (1/5)Mb², I_z = (1/5)Mc², whence I_yz = (1/5)M (b² +
c²), &c.
If [phi](x, y, z) be any homogeneous quadratic function of x, y, z, we
have
[Sigma]{m[phi](x, y, z)} = [Sigma] {m[phi]([|x] + [xi], [|y] + [eta], [|z] + [zeta])}
= [Sigma] {m[phi](x, y, z)} + [Sigma]{m[phi]([xi], [eta], [zeta])}, (14)
since the terms which are bilinear in respect to [|x], [|y], [|z], and
[xi], [eta], [zeta] vanish, in virtue of the relations (7). Thus
I_x = I[xi] + [Sigma](m)x², (15)
I_yz = I[eta][zeta] + [Sigma](m)·(y² + z²), (16)
with similar relations, and
I_O = I_G + [Sigma](m)·OG². (17)
The formula (16) expresses that the squared radius of gyration about any
axis (Ox) exceeds the squared radius of gyration about a parallel axis
through G by the square of the distance between the two axes. The
formula (17) is due to J. L. Lagrange; it may be written
[Sigma](m·OP²) [Sigma](m·GP²)
-------------- = -------------- + OG², (18)
[Sigma](m) [Sigma](m)
and expresses that the mean square of the distances of the particles
from O exceeds the mean square of the distances from G by OG². The
mass-centre is accordingly that point the mean square of whose distances
from the several particles is least. If in (18) we make O coincide with
P1, P2, ... P_n in succession, we obtain
0 + m2·P1P2² + ... + mn·P1P_n² = [Sigma](m·GP²) + [Sigma](m)·GP1², \
m1·P2P1² + 0 + ... + mn·P2P_n² = [Sigma](m·GP²) + [Sigma](m)·GP2², > (19)
... ... ... ... ... |
m1·P_nP1² + m2·P_nP2² + ... + 0 = [Sigma](m·GP²) + [Sigma](m)·GP_n². /
If we multiply these equations by m1, m2 ... m_n, respectively, and add,
we find
[Sigma][Sigma](m_r m_s·P_r P_s²) = [Sigma](m)·[Sigma](m·GP²), (20)
provided the summation [Sigma][Sigma] on the left hand be understood to
include each pair of particles once only. This theorem, also due to
Lagrange, enables us to express the mean square of the distances of the
particles from the centre of mass in terms of the masses and mutual
distances. For instance, considering four equal particles at the
vertices of a regular tetrahedron, we can infer that the radius R of the
circumscribing sphere is given by R² = (3/8)a², if a be the length of an
edge.
Another type of quadratic moment is supplied by the _deviation-moments_,
or _products of inertia_ of a distribution of matter. Thus the sum
[Sigma](m·yz) is called the "product of inertia" with respect to the
planes y = 0, z = 0. This may be expressed In terms of the product of
inertia with respect to parallel planes through G by means of the
formula (14); viz.:--
[Sigma](m·yz) = [Sigma](m·[eta][zeta]) + [Sigma](m)·yz (21)
The quadratic moments with respect to different planes through a fixed
point O are related to one another as follows. The moment with respect
to the plane
[lambda]x + [mu]y + [nu]z = 0, (22)
where [lambda], [mu], [nu] are direction-cosines, is
[Sigma]{(m([lambda]x + [mu]y + [nu]z)²} = [Sigma](mx²)·[lambda]² + [Sigma](my²)·[mu]² + [Sigma](mz²)·[nu]²
+ 2[Sigma](myz)·[mu][nu] + 2[Sigma](mzx)·[nu][lambda] + 2[Sigma](mxy)·[lambda][mu], (23)
and therefore varies as the square of the perpendicular drawn from O to
a tangent plane of a certain quadric surface, the tangent plane in
question being parallel to (22). If the co-ordinate axes coincide with
the principal axes of this quadric, we shall have
[Sigma](myz) = 0, [Sigma](mzx) = 0, [Sigma](mxy) = 0; (24)
and if we write
[Sigma](mx²) = Ma², [Sigma](my²) = Mb², [Sigma](mz²) = Mc², (25)
where M = [Sigma](m), the quadratic moment becomes M(a²[lambda]² +
b²[mu]² + c²[nu]²), or Mp², where p is the distance of the origin from
that tangent plane of the ellipsoid
x² y² z²
--- + --- + --- = 1, (26)
a² b² c²
which is parallel to (22). It appears from (24) that through any
assigned point O three rectangular axes can be drawn such that the
product of inertia with respect to each pair of co-ordinate planes
vanishes; these are called the _principal axes of inertia_ at O. The
ellipsoid (26) was first employed by J. Binet (1811), and may be called
"Binet's Ellipsoid" for the point O. Evidently the quadratic moment for
a variable plane through O will have a "stationary" value when, and only
when, the plane coincides with a principal plane of (26). It may further
be shown that if Binet's ellipsoid be referred to any system of
conjugate diameters as co-ordinate axes, its equation will be
x´² y´² z´²
--- + --- + --- = 1, (27)
a´² b´² c´²
provided
[Sigma](mx´²) = Ma´², [Sigma](my´²) Mb´², [Sigma](mz´²) = Mc´²;
also that
[Sigma](my´z´) = 0, [Sigma](mz´x´) = 0, [Sigma](mx´y´) = 0. (28)
Let us now take as co-ordinate axes the principal axes of inertia at the
mass-centre G. If a, b, c be the semi-axes of the Binet's ellipsoid of
G, the quadratic moment with respect to the plane [lambda]x + [mu]y +
[nu]z = 0 will be M(a²[lambda]² + b²[mu]² + c²[nu]²), and that with
respect to a parallel plane
[lambda]x + [mu]y + [nu]z = p (29)
will be M(a²[lambda]² + b²[mu]² + c²[nu]² + p²), by (15). This will have
a given value Mk², provided
p² = (k² - a²)[lambda]² + (k² - b²)[mu]² + (k² - c²)[nu]². (30)
Hence the planes of constant quadratic moment Mk² will envelop the
quadric
x² y² z²
------- + ------- + ------- = 1, (31)
k² - a² k² - b² k² - c²
and the quadrics corresponding to different values of k² will be
confocal. If we write
k² = a² + b² + c² + [theta],
b² + c² = [alpha]², c² + a² = [beta]², a² + b² = [gamma]² (32)
the equation (31) becomes
x² y² z²
------------------ + ----------------- + ------------------ = 1 (33)
[alpha]² + [theta] [beta]² + [theta] [gamma]² + [theta]
for different values of [theta] this represents a system of quadrics
confocal with the ellipsoid
x² y² z²
-------- + ------- + -------- = 1, (34)
[alpha]² [beta]² [gamma]²
which we shall meet with presently as the "ellipsoid of gyration" at G.
Now consider the tangent plane [omega] at any point P of a confocal, the
tangent plane [omega]´ at an adjacent point N´, and a plane [omega]´´
through P parallel to [omega]´. The distance between the planes [omega]´
and [omega]´´ will be of the second order of small quantities, and the
quadratic moments with respect to [omega]´ and [omega]´´ will therefore
be equal, to the first order. Since the quadratic moments with respect
to [omega] and [omega]´ are equal, it follows that [omega] is a plane of
stationary quadratic moment at P, and therefore a principal plane of
inertia at P. In other words, the principal axes of inertia at P arc the
normals to the three confocals of the system (33) which pass through P.
Moreover if x, y, z be the co-ordinates of P, (33) is an equation to
find the corresponding values of [theta]; and if [theta]1, [theta]2,
[theta]3 be the roots we find
[theta]1 + [theta]2 + [theta]3 = r² - [alpha]² - [beta]² -[gamma]², (35)
where r² = x² + y² + z². The squares of the radii of gyration about the
principal axes at P may be denoted by k2² + k3², k3² + k1², k1² + k2²;
hence by (32) and (35) they are r² - [theta]1, r² - [theta]2, r² -
[theta]3, respectively.
To find the relations between the moments of inertia about different
axes through any assigned point O, we take O as origin. Since the square
of the distance of a point (x, y, z) from the axis
x y z
-------- = ---- = ---- (36)
[lambda] [mu] [nu]
is x² + y² + z² - ([lambda]x + [mu]y + [nu]z)², the moment of inertia
about this axis is
I = [Sigma][m{([lambda]² + [mu]² + [nu]²)(x² + y² + z²) - ([lambda]x + [mu]y + [nu]z)²}]
= A[lambda]² + B[mu]² + C[nu]² - 2F[mu][nu] - 2G[nu][lambda] - 2H[lambda][mu], (37)
provided
A = [Sigma]{m(y² + z²)}, B = [Sigma]{m(z² + x²)}, C = [Sigma]{m(x² + y²)},
F = [Sigma](myz), G = [Sigma](mzx), H = [Sigma](mxy); (38)
i.e. A, B, C are the moments of inertia about the co-ordinate axes, and
F, G, H are the products of inertia with respect to the pairs of
co-ordinate planes. If we construct the quadric
Ax² + By² + Cz² - 2Fyz - 2Gzx - 2Hxy = M[epsilon]^4 (39)
where [epsilon] is an arbitrary linear magnitude, the intercept r which
it makes on a radius drawn in the direction [lambda], [mu], [nu] is
found by putting x, y, z = [lambda]r, [mu]r, [nu]r. Hence, by comparison
with (37),
I = M[epsilon]^4/r². (40)
The moment of inertia about any radius of the quadric (39) therefore
varies inversely as the square of the length of this radius. When
referred to its principal axes, the equation of the quadric takes the
form
Ax² + By² + Cz² = M[epsilon]^4. (41)
The directions of these axes are determined by the property (24), and
therefore coincide with those of the principal axes of inertia at O, as
already defined in connexion with the theory of plane quadratic moments.
The new A, B, C are called the _principal moments of inertia_ at O.
Since they are essentially positive the quadric is an ellipsoid; it is
called the _momental ellipsoid_ at O. Since, by (12), B + C > A, &c.,
the sum of the two lesser principal moments must exceed the greatest
principal moment. A limitation is thus imposed on the possible forms of
the momental ellipsoid; e.g. in the case of symmetry about an axis it
appears that the ratio of the polar to the equatorial diameter of the
ellipsoid cannot be less than 1/[root]2.
If we write A = M[alpha]², B = M[beta]², C = M[gamma]², the formula
(37), when referred to the principal axes at O, becomes
I = M([alpha]²[lambda]² + [beta]²[mu]² + [gamma]²[nu]²) = Mp², (42)
if p denotes the perpendicular drawn from O in the direction ([lambda],
[mu], [nu]) to a tangent plane of the ellipsoid
x² y² z²
-------- + ------- + -------- = 1 (43)
[alpha]² [beta]² [gamma]²
This is called the _ellipsoid of gyration_ at O; it was introduced into
the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal
polars with respect to a sphere having O as centre.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through
O are then principal axes, and the moment of inertia is the same for
each. The mass-system is then said to possess kinetic symmetry about O.
If all the masses lie in a plane (z = 0) we have, in the notation of
(25), c² = 0, and therefore A = Mb², B = Ma², C = M(a² + b²), so that
the equation of the momental ellipsoid takes the form
b²x² + a²y² + (a² + b²)z² = [epsilon]^4. (44)
The section of this by the plane z = 0 is similar to
x² y²
---- + ---- = 1, (45)
a² b²
which may be called the _momental ellipse_ at O. It possesses the
property that the radius of gyration about any diameter is half the
distance between the two tangents which are parallel to that diameter.
In the case of a uniform triangular plate it may be shown that the
momental ellipse at G is concentric, similar and similarly situated
to the ellipse which touches the sides of the triangle at their middle
points.
[Illustration: FIG. 59.]
[Illustration: FIG. 60.]
The graphical methods of determining the moment of inertia of a plane
system of particles with respect to any line in its plane may be
briefly noticed. It appears from § 5 (fig. 31) that the linear moment
of each particle about the line may be found by means of a funicular
polygon. If we replace the mass of each particle by its moment, as
thus found, we can in like manner obtain the quadratic moment of the
system with respect to the line. For if the line in question be the
axis of y, the first process gives us the values of mx, and the second
the value of [Sigma](mx·x) or [Sigma](mx²). The construction of a
second funicular may be dispensed with by the employment of a
planimeter, as follows. In fig. 59 p is the line with respect to which
moments are to be taken, and the masses of the respective particles
are indicated by the corresponding segments of a line in the
force-diagram, drawn parallel to p. The funicular ZABCD ...
corresponding to any pole O is constructed for a system of forces
acting parallel to p through the positions of the particles and
proportional to the respective masses; and its successive sides are
produced to meet p in the points H, K, L, M, ... As explained in § 5,
the moment of the first particle is represented on a certain scale by
HK, that of the second by KL, and so on. The quadratic moment of the
first particle will then be represented by twice the area AHK, that of
the second by twice the area BKL, and so on. The quadratic moment of
the whole system is therefore represented by twice the area AHEDCBA.
Since a quadratic moment is essentially positive, the various areas
are to taken positive in all cases. If k be the radius of gyration
about p we find
k² = 2 × area AHEDCBA × ON ÷ [alpha][beta],
where [alpha][beta] is the line in the force-diagram which represents
the sum of the masses, and ON is the distance of the pole O from this
line. If some of the particles lie on one side of p and some on the
other, the quadratic moment of each set may be found, and the results
added. This is illustrated in fig. 60, where the total quadratic
moment is represented by the sum of the shaded areas. It is seen that
for a given direction of p this moment is least when p passes through
the intersection X of the first and last sides of the funicular; i.e.
when p goes through the mass-centre of the given system; cf. equation
(15).