Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

310

()( )

(

)

OP OP

′′ ′′

−⋅= − =×

JJGJJJG

ωaa vv

(5.2.10)

We may thus state that the product of the modulus of the difference of the projections of

the accelerations of two points of a rigid solid on the straight line which links them by

the distance between these points is equal to the square of the difference of the

velocities of the respective points.

2.2 Particular cases of motion of the rigid solid

In what follows, we consider some particular cases of motion of the rigid solid, i.e.:

the motion of translation, the motion of rotation and the motion of rototranslation. As

well, we emphasize the motion of the local frames of Frenet and Darboux.

2.2.1 Motion of translation

We say that a rigid solid is subjected to a motion of translation if an arbitrary straight

bar of it remains parallel to itself at any moment of the motion. In particular, the axes

Ox , 1, 2, 3i = , of the movable frame of reference R must enjoy this property; as a

matter of fact, it is sufficient that the three axes do remain parallel to themselves (hence,

the moving frame of reference must have a displacement parallel to itself), in order that

any other straight line do enjoy the same property (because any straight line of the rigid

solid is rigidly connected to the frame

R ). The position vector r remains constant in

the relation (5.2.1) (it has a displacement parallel to itself), in this case; the trajectory

of the point

is obtained from the trajectory

of the point

by a translation of

vector

′′

=−

rr r (Fig.5.12). If the point

attains the point

on the curve

, then

the point

attains the point

on the curve

, while the vector

11 1

OP =

JJJG

equipollent to the vector

r .

Figure 5.12. Motion of translation of a rigid solid.

Taking into account Poisson’s formula (A.238), it results that

i0, 1, 2, 3j = , if

and only if

=ω 0 ; in this case, the distribution laws of the velocities and accelerations

are given by

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311

′

vv,

′

aa.

(5.2.11)

Hence, at a given moment, all the points of a rigid solid in a motion of translation have

the same velocity and the same acceleration; the respective vectors may be modelled by

free vectors. If the velocity

′

v is given, then the equation of the trajectory becomes

() ( ) ( )d

tt ττ

′′ ′

∫

rr v

(5.2.12)

vers const

′

JJJJG

v , then the motion of translation is rectilinear; otherwise, it is

curvilinear. If

const

′

v , then the motion of translation is uniform.

2.2.2 Motion of rotation

We say that a rigid solid is subjected to a motion of rotation if two points of it remain

fixed during the motion; the straight line which links the two points (called axis of

rotation) is fixed too, in this case. We assume (without losing from the generality of the

study) that

′

≡

and we choose as axis of rotation

Ox Ox

′

≡

. The rigid solid

remains with only one degree of freedom, which is the angle

()tθθ

(5.2.13)

between the axes

′′

and Ox

1, 2α

(Fig.5.13). Noting that the unit vectors of

the movable frame may be expressed with the aid of the unit vectors of the fixed frame

of reference in the form

Figure 5.13. Motion of rotation of a rigid solid.

′′

iii

112

cos sinθθ,

′

−+iii

212

sin cosθθ,

′

the formula (A.2.36) leads to

0ωω , ==

() () ()tttωωθ



(5.2.13')

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312

hence, the vector

ω is a sliding vector situated along the axis of rotation

(

= i

() ()ttωω ), being the angular velocity vector. Thus, we have put in evidence the

mechanical significance of the vector

, formally introduced in the formula (5.2.3).

The velocity of a point of the rigid solid is thus given by

=×= ×

JJG



ωvr r

(5.2.14)

and can be considered to be the moment of the vector

with respect to the point P .

Multiplying scalarly this relation by the vector

r and by the vector

, respectively,

and noting that one obtains null mixed products, one gets

⋅= =



()0

⋅

=⋅=



ir ir

wherefrom

constr = and

(

)

,const

) ir . It follows that the trajectory of a point

is the intersection of a sphere of centre

and r radius and a cone, the vertex of

which is at

too; hence, these trajectories are circles situated in planes normal to the

axis of rotation, having the centres in points

′

on the very same axis. Because

OP OP P P OP

′′ ′

== + = +

JJJG JJJJG JJJJG

JJG

, where P

′

is the projection of the point

on the

axis of rotation, one can write the relation (5.2.14) also in the form

×vr

v ω

; (5.2.14')

comparing this formula with the formula (5.1.41), which corresponds to the circular

motion of a particle, we see that the velocity vector is tangent to the trajectory in the

normal plane to the axis of rotation. Noting that the vectors

r and v are polar vectors,

it results that

ω is an axial vector; besides, the above considerations justify the

denomination given to this vector. Taking into account the considerations in Chap. 3,

Subsec. 1.2.3, we may attach to the vector

an antisymmetric tensor of second order,

given by the formula (3.1.62'); we write

ijk k

ωω=∈ ,

ijk jk

ωω=∈ ,

(5.2.15)

and the relations (5.2.14), (5.2.14') become

ijjij

ijk k

vxxωω

∈=,

vxω

− ,

vxω

, (5.2.14'')

so that all the points of the rigid solid have the same angular velocity. We notice that

the points on the axis of rotation are the only ones of vanishing velocity, while the

velocity vectors of the points of a straight line parallel to this axis are equipollent; as

well, the velocities are parallel and their moduli have a linear variation for the points of

a normal to the axis of rotation. Taking into account (A.2.31'), (A.2.31''), the relations

curl 2

, div 0v

(5.2.16)

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313

hold, the field of velocities being thus solenoidal.

Rivals formula (5.2.6''') leads to

ωω=×− =×−



arr rrωω

a ωω=+



(5.2.17)

corresponding to the formula (5.1.41'), which gives the distribution of accelerations in

the circular motion of particle; we have

ij i

ijk k

axxωω=∈ −



1, 2, 3i

121

axxωω=− −



21 2

ax xωω=−



(5.2.17')

so that the accelerations are contained in a plane normal to the axis of rotation and

directed towards the interior of the circular trajectory. The determinant of the

homogeneous system which gives the points of null acceleration is

0ωω+≠



and

allows to affirm that only the points of the axis of rotation enjoy this property (it is one

of the cases mentioned in Subsec.2.1.1, the angular acceleration vector



εω being a

sliding vector too, the support of which is the same axis of rotation. As in the case of

the velocities, the accelerations are equipollent vectors for the points of a straight line

parallel to the axis of rotation; as well, for the points of a straight line normal to the axis

of rotation, the velocities are parallel, their modulus having a linear variation. If



ω ,

then the motion of rotation is uniform. If the vectors

and



have the same direction,

then the motion is accelerated; otherwise, it is decelerated.

2.2.3 Helical motion. Motion of rototranslation

We say that a rigid solid has a helical motion if two of its points remain on a fixed

straight line during the motion; we may also say that a straight line rigidly linked to the

rigid solid (axis of rotation and sliding) maintains its support fixed (or slides along a

fixed support). We choose the axis of rotation and sliding as axis

′

and the point O

on this axis (in general,

′

), so that the axis

Ox coincides with it. This motion is

defined by the scalar functions (Fig.5.14)

Figure 5.14. Helical motion of a rigid solid.

()

xxt

′′

, ()tθθ

(5.2.18)

MECHANICAL SYSTEMS, CLASSICAL MODELS

314

which specifies the position of the rigid solid at a given moment (the rigid solid has two

degrees of freedom). The distribution of the velocities is given by

′′

=+×

vv r

222

vv rω

′′

vxω

′

=−

vxω

′

vv v

′

′′

(5.2.19)

where

() ()ttωθ=



has the significance given at the previous subsection. The formulae

(5.2.19) show that the modulus of the velocity is minimal (equal to

′

v ) for the points

of the axis of rotation and sliding; in fact, the velocity

()

′

vv is a function of

time, so that one may have points of null velocity only – eventually – for some

particular cases. As in the case of a motion of rotation, the velocities of the points of a

straight line parallel to the axis of rotation and sliding are equipollent vectors. In a plane

normal to the axis of rotation and sliding we obtain a component of the velocity which

has a behaviour analogous to that in case of a motion of rotation; the component of the

velocity along the direction of the same axis behaves as in the case of a motion of

translation.

The distribution of the accelerations is given by

′′

=+×−



aa r r

ω ,

()

2422

aa rωω

′′

=++



(5.2.20)

leading to the same components

′

as those given by the formula (5.2.17') and to

the component

′′

= . As in the case of the velocities, the modulus of the

acceleration is minimal (equal to

′

) for the points of the axis of rotation and sliding;

but there cannot be points of null acceleration, excepting – eventually – for some

particular moments (case mentioned in Subsec. 2.1.1). We notice that

() () () ()ttt tωθ== =





iiεω is the angular acceleration, the same for any point of

the rigid solid. The accelerations are equipollent vectors for the points of a straight line

parallel to the axis of rotation and sliding. The component of the acceleration which is

contained in a plane normal to the axis of rotation and sliding has a behaviour

analogous to that in case of a motion of rotation; while the component of the

acceleration along the same axis has a behaviour analogous to that of a motion of

translation.

From the above considerations, it results that the helical motion can be obtained,

from the point of view of the distribution of the velocities and accelerations, by the

composition of a motion of rotation with a motion of translation along the axis of

rotation. We notice that the vectors

()

′

v and ()t

have the same constant direction

(

()

′

×=v0

The considered motion is called helical, the trajectory of a point of the rigid solid

being situated on a circular cylinder. If the first of the scalar functions (5.2.18) which

defines the motion verifies a relation of the form

xkθ

′

, k being a constant the

dimension of which is a length, then the trajectories are helices and we have to do with

a screw motion; the rigid solid has – in this case – only one degree of freedom. We

obtain

vkω

′

akω

′



, so that

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315

vrkω=+,

()

24 2 2 2

ar rkωω=++



(5.2.21)

The pitch of the helix is

2pkπ= , so that

v ω

′

a ω

′



(5.2.22)

We notice that the helical motion is a particular case of a motion of rototranslation,

i.e., the case in which the vectors

and

′

v have the same support. In the general

case,

′

×≠

v0ω , ω and

′

v having fixed directions, and we obtain an arbitrary

motion of rototranslation (called also motion of finite rototranslation), which is not a

helical motion; this case is basic for the study of the general motion of the rigid solid.

The relations (5.2.16) remain still valid in the case of a motion of rototranslation.

2.2.4 Motion of Frenet’s and Darboux’s trihedra

If we know the trajectory of a point of the rigid solid, then we may choose as

movable frame of reference Frenet’s trihedron. Taking into account Poisson’s formulae

(A.2.38) and Frenet’s formulae (4.1.10)-(4.1.10''), and noting that

ddd

tst

ττ

ddd

tst

ddd

tst

ββ

we may write

=×=

τν

=×=− −

′

ντ

=×=

′

ωβ

ν ,

(5.2.23)

which leads to interesting kinematic interpretations for the radii of curvature and

torsion. In this case, the angular velocity vector is given by

τν

ωωω

++ωτν

−

′

, 0

(5.2.24)

and we may write

vω

ρρ

′

(5.2.24')

In the case of a torsionless motion, hence in the case in which the trajectory of a point

of the rigid solid is a plane curve, we have

MECHANICAL SYSTEMS, CLASSICAL MODELS

316

v ρω

. (5.2.24'')

Analogously, choosing as a movable frame of reference Darboux’s trihedron

(assuming that we know a surface on which moves a point of the rigid solid, for

instance a cylinder in the case of a motion of rototranslation) and using the formulae

(A.2.38) and (4.1.25), as well as observations analogous to those above, we obtain

t ρρ

=×= +

ωτ

t ρρ

=×=− −

′

ωτ,

ρρ

=×=− +

′

ωτ

;

(5.2.25)

we are thus led to interesting kinematic interpretations for the radii of normal curvature,

as well as of geodesic curvature and torsion. We may express the angular velocity

vector in the form

gnτ

ωωω=++

nωτ

=−

′

=−

(5.2.26)

wherefrom

222

111

gg n

vω

ρρ ρ

=++

′

(5.2.26')

2.3 General motion of the rigid solid

We consider, in the following, the general motion of the rigid solid as an

instantaneous helical motion, introducing the fixed and the movable axoids; according

to the form of the axoids (conical or cylindrical), we obtain the motion of a rigid solid

with a fixed point or the plane-parallel motion of the rigid solid, respectively.

2.3.1 Instantaneous helical motion. Static-kinematic analogy

In the general case of motion of a rigid solid we use the results given in Subsecs

2.1.1 and 2.1.2 concerning the distribution of velocities and accelerations. Using the

formula (5.2.3), which gives the velocity

′

v of a point P of the rigid solid with respect

to a fixed frame of reference, and the formula (5.2.14), we observe that

{

}

,{}

′

τvωω,

{

}

,{}

′

τvωω, with

PO O O

PO OP

′′ ′ ′

=+×=+×=+×

JJG JJJG

vv v v rωω ω.

(5.2.27)

We obtain thus a torsor of the angular velocities,

playing the rôle of the resultant

vector (an axial, sliding vector), while

′

v is the moment resultant vector (a polar,

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317

bounded vector). This observation stays at the basis of the static-kinematic analogy,

which allows the kinematic study of the motion of the rigid solid with the aid of the

static methods. Analysing the distribution of the velocities, we see that – at a given

moment – the general motion of a rigid solid may be identified with an instantaneous

motion of rototranslation, characterized by an instantaneous torsor of the angular

velocities.

Hence, the general motion of a rigid solid is characterized by the vectors

ω and

′

(the components of the torsor {}

of the angular velocities with respect to the pole

O , rigidly linked to the rigid solid); in general, these vectors are not collinear. There

exist points (which can belong to the rigid solid or not) for which

′

& v

, the general

motion of the rigid solid becoming thus an instantaneous helical motion. Indeed, taking

into account the scalar of the torsor

′

⋅

=⋅vv

ω ,

(5.2.28)

which is an invariant, it follows that

′

, where we have emphasized the

components parallel to the vector

of these vectors. The formula (5.2.27) becomes

⊥⊥

′

+×vv r

(5.2.29)

where the components of the velocities normal to the vector

have been introduced.

Using the method of determination of the central axis (emphasized in Chap. 2, Subsec.

2.2.5), we put the condition

⊥

′

v0, wherefrom

′

;

(5.2.30)

we obtain thus the instantaneous axis (the locus of the searched points) with respect to

which we have an instantaneous helical motion. This axis is parallel to the vector ω ; a

vector equipollent to the latter one, the support of which is the instantaneous axis, is

called Chasles’ vector. We may write the equation (5.2.30) also in the form (a vector

product by

eliminates the scalar λ )

′

⋅

′

−× =

ω ;

(5.2.30')

in components one has

()()

12332 23113

vx x vx xωω ωω

ωω

′′

−+ = −+

()()

31221 112233

OOOO

vx x v v vωω ω ω ω

′′′′

=−+= ++

(5.2.30'')

We notice that the instantaneous axis, defined by the equations (5.2.30)-(5.2.30''), is the

locus of the points for which, at a given moment, the modulus of the velocity is

minimal. The above considerations allow us to state

MECHANICAL SYSTEMS, CLASSICAL MODELS

318

Theorem 5.2.3 (Chasles). The general motion of a rigid solid at a given moment may

be identified with an instantaneous helical motion about an instantaneous axis of

rotation and sliding.

We can say that this helical motion is tangent to the general motion of the rigid solid

at a given moment.

From the above analysis it follows that if

′

v0, then the rigid solid is in

rest. If

0ω ,

′

≠

v0, then a motion of translation takes place; if ≠ 0ω ,

′

v0,

then the motion of the rigid solid can be identified with a motion of rotation about an

instantaneous axis of rotation passing through the fixed point

(in particular, a finite

rotation), hence with a rigid solid with a fixed point. If

≠

′

≠

v0, then two cases

may occur: if the torsor’s scalar vanishes (

′

⋅

), then the motion of the rigid

solid is an instantaneous motion of rotation about an instantaneous axis of rotation

(which does not pass through a fixed point); if the scalar does not vanish (

′

⋅≠

vω ),

then a general motion of the rigid solid takes place.

In the general motion of the rigid solid, the distribution of accelerations is that

emphasized in Subsec. 2.1.2.

2.3.2 The fixed and the movable axoids

We notice that the instantaneous axis of rotation and sliding varies both with respect

to the fixed and the movable frames of reference, because

and

′

v are functions of

time. The locus of the instantaneous axes of rotation and sliding with respect to the

frame

′

R is a ruled surface

, called the fixed axoid, while the locus of the same

axis with respect to the frame

R is a ruled surface

too, called the movable axoid

(Fig.5.15); these two surfaces play an important rôle in the general motion of a rigid

solid, as it was emphasized by Poncelet.

Figure 5.15. Fixed and movable axoids of a rigid solid.

Let

P be a point of the rigid solid on the instantaneous axis of rotation and sliding,

rigidly linked to this axis, for which a relation of the form (5.2.1) takes place.

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319

Differentiating with respect to the frame

′

R and using the formula (A.2.37), we

obtain

ttt

′

∂

++×

∂

ω ;

(5.2.31)

d/d

′′

=vr is the velocity of the point P in the motion of the instantaneous axis of

rotation and sliding with respect to the frame

′

R , while /

∂∂vr is the velocity

of the same point in the motion of this axis with respect to the frame

R. Noting that

d/d

′′

=vr and

′

+×=vr

ω , where λ is a scalar, because the translation

velocity of the point

P and the angular velocity

are collinear vectors for the points

of the instantaneous axis of rotation and sliding, we get the relation

′

−

=vv

(5.2.31')

If the velocities

v and

′

v would be equal, then the axoids would have a motion of

rolling without sliding; because of the vector difference

, occurs also a sliding of the

movable axoid on the fixed one along the direction of the vector

, with the velocity

λω . Projecting the relation (5.2.31) on a direction normal to the vector ω, we obtain

⊥

′

vv,

(5.2.32)

the components normal to the instantaneous axis of rotation and sliding of the velocities

of a point

P of this axis with respect to the fixed and movable frames of reference,

respectively, being equal. We may thus state

Theorem 5.2.4. The general motion of a rigid solid takes place so that the movable

axoid be tangent to the fixed axoid along the instantaneous axis of rotation and sliding

(the common generatrix of the two axoids), its motion being a rolling about this axis,

over the fixed axoid, together with a sliding along the same instantaneous axis.

We notice that the two axoids must be both developable or warped surfaces.

The motion of the movable axoid characterizes thus completely the general motion

of the rigid solid.

2.3.3 Motion of the rigid solid with a fixed point

In the case of a rigid solid with a fixed point both origins may coincide with this

point, without losing anything of the generality (

′

≡

); we have thus

′

vv0,

′

aa0, the vectors

and



being arbitrary. The rigid solid

remains with three degrees of freedom and its position at a given moment may be

specified with the aid of Euler’s angles

,,ψθϕ, emphasized in Chap. 3, Subsec. 2.2.3.

The distribution of the velocities is given by the relation

×vr

, (5.2.33)