The movement equations that have been presented in the previous volumes of this series on non-deformable solid mechanics are the scalar expression of the fundamental principle of dynamics and the different consequences that stem from it. But in using this principle, the choice of the frame in which to apply it (depending on the motion being studied) is crucial. The step is therefore to see how this decision can be made so that mechanics users may have a proper frame for the studied motion that is suitable for its context.
The fundamental principle of dynamics is one of the general laws that govern mechanics just as the secondary principles that will be presented in section 1.2 later on. Their formulation is the result of experimental observations and measures; their validity is essentially based on the fact that they are universally used.
There exists at least one reference frame g, called Galilean, and a way to measure time (called the preferred time scale), such as, at any given time and whatever the considered physical set , the dynamic torsor of the set motion through the frame g is equal to torsor representing the efforts exerted on by all in the universe, that are outside of it, which we call . So that we state:
The application of this principle thus suggests the existence of at least one frame considered to be preferred, serving to locate a body during its motion, and of at least one preferred time scale that allows us to follow its evolution. But there are in fact an infinity of frames and ways to measure time; it is therefore important to select a frame that is suitable to the motion in question, to be able to apply this principle.
We consider that the choice of a reference frame must depend on the motion that is studied and from where we wish to observe it. The Galilean frame is the primary one in which the fundamental principle of dynamics applies. It must respect three conditions during the study of the motion of a mechanical set . First of all, it must be in a fixed position and orientation in the space where the motion is taking place; secondly, any moving body in this space, if it doesn’t experience any exterior effect, moves uniformly on a straight line; thirdly, any body which, by its presence and the action it is likely to exert onto the motion of , also obeys the condition that its own motion may be observed and studied in the same frame (the Galilean frame is then said to be closed).
In fact, if such bodies are subject to actions from bodies outside of the frame and which are not accounted for, they could have unidentified secondary effects on the body in motion; this explains the importance of the Galilean frame being closed.
The fundamental principle of dynamics only references, concerning the motion of in g, the dynamic torsor that is defined from the acceleration of any particle of M of , considered as vector density per unit of mass. This observation leads to considerations that will not be expanded upon here, that show, when a preferred frame is identified as well as a preferred time scale to create a Galilean frame, that any other frame will be considered Galilean as well if it includes:
First of all, if we are studying the observable motion of objects from our terrestrial environment, the first preferred Galilean g which naturally stands out as offering the most global view. Its origin is the center of inertia GE of the center of the solar system with three base vectors pointed towards the stars, E1, E2, E3 apparently fixed.
As the Sun represents more than 99% of the mass of the solar system, its center of inertia GH is close to that of the solar system. We therefore introduce a direct orthonormed system where
as our preferred Galilean frame.
However, most solid motion that we study takes place within the Earth’s environment under the effect of actions that emanate from it; it therefore seems reasonable to use a frame of work that would be more accessible, joined to the Earth, with its center of inertia as origin and to which three orthogonal and normed vectors – that could either be collinear to those of the Solar system or fixed in relation to our planet (along the North-South axis and two other directions orthogonal between one another in the equatorial plane) – would be attached.
Even if we admit that the North-South axis of rotation of our planet maintains a constant direction as it travels along its ellipse around the Sun, this last frame, called the terrestrial frame T, is not in translation with the solar frame; it cannot serve as a frame for the application of the fundamental principle. And if we apply the corresponding equations to it, we are led to introduce corrective terms due to the relative motion of the frame T in relation to the solar frame. These terms are often negligible, which justifies the use of this frame when the characteristics of motion allow it.
As for considerations about the time scale, they rely on the assumed uniform nature of the Earth’s rotation around its axis and reinforce the benefits of using the terrestrial frame T as our Galilean reference frame and applying the fundamental principle of dynamics. This frame is then described as pseudo Galilean.
With preferred time scale t, according to the fundamental principle, we have
With a different preferred time scale τ, we would have
and the velocity and acceleration vectors would be written
With this new time scale, the reduction elements at OS of kinetic and dynamic torsors have the following expressions:
If we examine the relation between velocities on the one hand and accelerations on the other within the two scales, we obtain
Subsequently
and lastly
The fundamental principle of dynamics is then formulated as follows with the new time scale:
When the relation between the two time scales is linear: τ = at + b, τ″ is null and τ′ = a; the fundamental principle is then written:
For this new time scale to be used as preferred, the following statement: a = 1 must be verified so that τ = t + b. We can then write
The origin of this preferred time scale is arbitrary, since in reality it is a matter of observing a motion during a corresponding time interval, and from a suitable point of origin.
These principles form the framework for general use of the fundamental principle.
In solid mechanics, the efforts exerted by over are remote actions gravitational or electromagnetic in nature, and contact actions.
Contact actions can be fluid (liquid or gas), solid (non-deformable) or elastic types.
We generally group into a first torsor written the given or calculable terms of due to gravitational or electromagnetic actions and assessable contact actions.
We group into a second torsor written the unknown link terms of .
The first secondary principle allows us to therefore state
If we now consider a partition of set such as
and an exterior partition of such as
we can express the two following secondary principles.
For a considered law 'ℓ', whether it is gravitational, electromagnetic or of contact, we can state
This principle can extend to a finite number of disjoint sets that constitute a partition of .
This principle can also extend to a finite number of disjoint sets that constitute a partition of .
Considering these principles, the fundamental principle of dynamics takes on the more general following form
where subscript symbols α′ and i′ refer to the only partitions of and that are in contact.
We consider the motion of a mechanical set in a given frame λ that is mobile in relation to the Galilean frame g and the question is to know how, in such a case, it is possible to apply the fundamental principle of dynamics.
We first examine the mobility of the frame λ in comparison to another frame μ on which we make no hypotheses for the time being.
The relative movement of the two frames is known, that is:
We also know the motion of the mechanical set within frame λ and we wish to observe its motion from frame μ. The formula for combination of accelerations applied to a particle M of
introduces the Coriolis acceleration of M in the motion of λ relative to μ
and the drive acceleration of M in this same motion
where vectors and are independent of M.
To the material particle corresponding to point M, given its elementary mass dm(M), we attribute the vector
We therefore attach to the kinetics of the material set , in the motion of the frame λ in relation to μ, the following torsor, called the Coriolis inertial torsor and written , of which the reduction elements at a given point Q are
To the material particle corresponding to point M, given its elementary mass dm(M), we also assign the vector
We also attach to the kinetics of the material set , in the motion of the frame λ relative to μ, the torsor known as drive inertial torsor and marked , of which the reduction elements at a given point Q are
In each one of the frames λ and μ, the dynamic torsor is marked
According to combination of accelerations, the reduction elements of the dynamic torsor of the motion relative to frame μ expand as follows
Thus the torsor equality that expresses the relation between the dynamic torsors of the motion of in both frames is
which is valid whatever the set or the frames λ and μ.
We consider here that the frame μ is the preferred Galilean frame g. From the previous results, we can therefore state that
and that the fundamental principle of dynamics is expressed as follows
This leads to the torsor expression of the motion of the set relative to the given frame λ
Analyzing this relation shows that the relative situation of the frame λ in relation to the considered Galilean frame g is not indifferent because, in the relation above, the outside of is considered from a Galilean frame and not from λ.
In the case where the sources of the efforts acting upon , and emanating from , coincide with the ones present in λ, and that the corrections introduced by the Coriolis and drive terms are sensibly negligible compared to the efforts, the frame λ is considered as pseudo-Galilean and the fundamental principle can be applied.
We therefore see that it is possible to express the torsor equation of the motion of a mechanical set relative to a given frame λ provided the Coriolis and inertial drive torsors are incorporated to the external efforts torsor.
However, when the frame λ moves with a uniform translation on even straight line relative to g, the rate of rotation is null and the Coriolis and drive inertial torsors will be also. The frame λ is then Galilean.
We now consider the case where the material set is a non-deformable solid (S), of mass m and center of inertia G, of origin OS.
The motion of solid (S) in λ is characterized by the velocity distribution torsor
with
The resultant of the Coriolis inertial torsor is then expressed as follows
Its moment at Os is expressed
The last integral above involves the triple vectorial product which can be expressed as follows
Subsequently
According to the definition of the inertia operator IOs (S), we can verify that
, which is the sum of the diagonal terms of the matrix representing the operator,
hence the expression of the moment at Os of the Coriolis inertial torsor of the non-deformable solid (S) in the motion of λ relative to μ
The moment at G of this torsor is equal to
The resultant of the drive inertial torsor is written
and its moment at Q has the following development
with
According to the easily verifiable property of the vectorial product: , the last integral above is marked
hence the moment at Q of the inertial drive torsor of the non-deformable solid (S) in the motion of λ relative to μ
and at its center of inertia G