Principle of Causality

Dynamics studies the motion of material objects as a function of the physical factors affecting them. In the Newtonian formulation, those physical factors are mainly their mass and the forces acting on the objects. Whatever the forces may be, all definitions assume implicitly that there is a correlation between them and the object’s motion. This correlation is sometimes described in a simplistic way as a causality relationship: “Dynamics is the study of the motion of bodies caused by the action of forces.”

The equations of Newtonian dynamics are differential equations relating the second order derivative of the objects generalized coordinates to the coordinates and their first derivatives, all of them at a same time instant. This simultaneity of motion variables makes it difficult (and formally impossible) to distinguish “causes” and “effects,” as the former should actually precede the latter. That distinction becomes a convention.

Principle of Absolute Simultaneity

The principle of absolute simultaneityFootnote ¹ states, in short, that a sequence (order of succession) of events is the same for all observers (or independent from the reference frame). In Newton’s conception of the universe, absolute time goes hand in hand with absolute space: they constitute an immutable stage where physical events occur, they are independent external realities.

Other Assumptions

Last but not least: as in any scientific theory, Newtonian dynamics have a limited field of application (directly related to the experimental limitations of the seventeenth century!). Outside that field, it yields inaccurate results. The main limitations are:

• Low-speed dynamics: objects moving with speeds comparable to the speed of light cannot be treated successfully with this theory.

• Medium length scale dynamics: molecular dynamics and long-reach astronomy are also out of scope.

• Electromagnetic phenomena are excluded.

Galileo’s Principle of Relativity

We have just proved that all Galilean reference frames share the law governing the dynamics of a free particle.Footnote ⁴ When seeking for the law governing the dynamics of interacting particles, a logical question is: Will the Galilean reference frames also share that law? Answering that question is equivalent to establishing a principle of relativity (a principle defining the set of reference frames for which a scientific law is valid).

In Newtonian mechanics, that principle is the special principle of relativity, also known as Galileo’s principle of relativity, as it was first enunciated by Galileo in 1632. A modern formulation of that principle is: The laws governing all mechanical phenomena are the same in all reference frames where the Newton’s first law is fulfilled. A consequence of that principle is that Galilean reference frames are not distinguishable through mechanical observations.Footnote ⁵

Mach’s Empirical Propositions

Though the vector formulation of dynamics proposed by Newton is the most widespread one and its application is certainly more straightforward (as interaction forces are a more practical representation of interactions than interaction accelerations because of their symmetry, as will be seen later on), it has many drawbacks from an axiomatic point of view. It relies on the concepts mass and force, but the definition of the former is unclear and that of the latter is actually implicit.

The two axioms introduced so far (Newton’s first law and the principle of relativity) do not mention mass or force, while Newton’s second and third laws do explicitly include these concepts. Mach’s formulation relies exclusively on interaction accelerations, and for this reason it will be presented before completing Newton’s axioms.

Ernst Mach’s axiomatization is based on three empirical propositions and two definitions. It is an alternative formulation whose application is not straightforward, but it has two important advantages:

• it is based on accelerations, which are observable (and measurable) variables, and for that reason they correspond to “real” concepts;

• it allows us to define in a rigorous way mass and force.

The first empirical proposition considers two isolated interacting particles P and Q. If observed from a Galilean reference frame, their accelerations will be exclusively the consequence of their mutual interaction. If ${\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}$ and ${\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}$ are the acceleration of P under the action of Q and that of Q under the action of P, respectively, the proposition states that those accelerations will be either attractive or repulsive, in the direction of $\overline{\mathrm{QP}}$ (Fig. 1.3). Mathematically, this can be expressed as

$\frac{{\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)}=-\frac{{\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}\right)},\overline{\mathrm{QP}}\times {\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}=\overline{0}.$(1.2)

This proposition is followed by the definition of inertial mass-ratio of two particles $\left({𝘮}_{𝗤}/{𝘮}_{𝗣}\right)$:

$\frac{{𝘮}_{𝗤}}{{𝘮}_{𝗣}}\equiv \frac{\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}\right)}.$(1.3)

If we choose a particular value ${𝘮}_{𝗣}$ for a particle, the mass of the other particles may be determined straightaway.

Fig. 1.3

Equation (1.3) provides an interpretation for the inertial mass. As the product ${𝘮}_{𝗣}\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)$ is equal to ${𝘮}_{𝗣}\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)$, it follows that the particle which acquires a higher acceleration has a lower mass, and vice versa. Hence, the inertial mass is a measure of the resistance of a particle to change its state of motion.

The second empirical proposition is a principle of separation, and it guarantees that the determination of the ratio between the interaction accelerations of any pair of isolated particles (hence that of the mass-ratios) is unique through two statements:

1. A. the acceleration-ratio associated with a pair of particles is constant and independent from the interaction phenomenon

   $\frac{\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}\right)}=\text{constant}\iff \frac{{𝘮}_{𝗤}}{{𝘮}_{𝗣}}=\text{constant},$(1.4)

2. B. the acceleration-ratio is separable, that is, it can be expressed as the product of two independent acceleration-ratios.

This last idea is illustrated in Fig. 1.4 and can be formulated mathematically as:

$\frac{\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right)}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}\right)}=\frac{\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗦}\right)}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗦\to 𝗤}\right)}\cdot \frac{\left({\overline{𝗮}}_{\text{RGal}}^{𝗦\to 𝗣}\right)}{\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗦}\right)}\iff \frac{{𝘮}_{𝗤}}{{𝘮}_{𝗣}}=\frac{{𝘮}_{𝗤}}{{𝘮}_{𝗦}}\cdot \frac{{𝘮}_{𝗦}}{{𝘮}_{𝗣}}.$(1.5)

Fig. 1.4

The third empirical proposition states that the accelerations that any number of particles (Q, S, T…) induce in a particle P are independent from each other. It is actually a principle of superposition, it is illustrated in Fig. 1.5 and can be formulated mathematically as:

${\overline{𝗮}}_{\text{RGal}}^{\left(𝗤,𝗦\right)\to 𝗣}={\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}+{\overline{𝗮}}_{\text{RGal}}^{𝗦\to 𝗣}.$(1.6)

Fig. 1.5

The combination between the principle of separation and the principle of superposition for the interaction accelerations yields a principle of superposition for the mass of particles. If QP is the particle obtained when P and Q share the same location (two dimensionless particles become one single particle in that case), its mass is the addition of those of P and Q: ${𝘮}_{\mathrm{PQ}}={𝘮}_{𝗣}+{𝘮}_{𝗤}$.

Finally, Mach introduces the concept of interaction force through a definition:

${\overline{𝘍}}_{𝗤\to 𝗣}\equiv {𝘮}_{𝗣}{\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}.$(1.7)

The definition of interaction force is extremely useful: It describes the interaction between two particles through just one single magnitude $\left(\text{as},,\left({\overline{𝘍}}_{𝗤\to 𝗣}\right),=,\left({\overline{𝘍}}_{𝗣\to 𝗤}\right)\right)$. This is not possible when we use accelerations as main magnitude to describe that interaction: as the interacting particles acquire different accelerations $\left(\left({\overline{𝗮}}_{\text{RGal}}^{𝗤\to 𝗣}\right),\ne ,\left({\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}\right)\right)$, we require two related magnitudes for a complete description (Fig. 1.6).

Fig. 1.6

Newton’s Third Law

The first empirical proposition (Eq. (1.2)) and the definition of force (Eq. (1.7)) lead to:

${\overline{𝘍}}_{𝗣\to 𝗤}\equiv {𝘮}_{𝗤}{\overline{𝗮}}_{\text{RGal}}^{𝗣\to 𝗤}=-{\overline{𝘍}}_{𝗤\to 𝗣},\text{with}{\overline{𝘍}}_{𝗤\to 𝗣}\times \overline{\mathrm{QP}}=\overline{0},$(1.8)

which is Newton’s third law (principle of action and reaction). Note that the interaction forces between P and Q fulfill:

${\overline{𝘍}}_{𝗣\to 𝗤}+{\overline{𝘍}}_{𝗤\to 𝗣}=\overline{0},\overline{\mathrm{OQ}}\times {\overline{𝘍}}_{𝗣\to 𝗤}+\overline{\mathrm{OP}}\times {\overline{𝘍}}_{𝗤\to 𝗣}=\overline{0},$(1.9)

where O is any point. The cross product $\overline{\mathrm{OP}}\times {\overline{𝘍}}_{𝗤\to 𝗣}$ defines the moment (or torque) $\overline{𝘔}\left(𝗢\right)$ of force ${\overline{𝘍}}_{𝗤\to 𝗣}$ about point O. Equations (1.9) show that the resultant force and the resultant moment (about any point O) of an action–reaction pair of forces are always a zero (Fig. 1.7).

Fig. 1.7

Newton’s third law provides powerful information about the interaction force between two particles, and it is considered as the main law by many authors precisely because it introduces the symmetry in the description of the interaction mentioned earlier (description of the interaction through one single magnitude).

Newton’s Second Law

The definition of force (Eq. (1.7)) together with Mach’s third empirical proposition (superposition principle, Eq. (1.6)) lead to:

$\sum _{𝗤}{\overline{𝗙}}_{𝗤\to 𝗣}={𝘮}_{𝗣}{\overline{𝘢}}_{\text{RGal}}\left(𝗣\right),$(1.10)

which is Newton’s second law (fundamental law of dynamics).Footnote ⁶ This formulation contains a principle of superposition for the interaction forces.

Principle of Determinacy

Whether we adopt Newton’s or Mach’s axiomatics, the principle of determinacy has to be added to the other laws to complete the formulation of dynamics. It is actually known as the Newtonian principle of determinacy, though he did not postulate it as a law but commented on it as an observation. A possible formulation is: “In an isolated system (undergoing no other interactions but those between the system’s particles), the particles’ accelerations at each time instant depend exclusively on the system’s mechanical state (position and velocity of every particle) at that same time instant.”Footnote ⁷ In other words, the initial mechanical state determines univocally its future evolution. Hence, this principle states that isolated mechanical systems have no memory.

This principle may seem incompatible with everyday experiences, such as driving a vehicle, because it is without discussion that its future evolution depends on the driver and not on the present mechanical state. However, driving a vehicle (or controlling a system, in general) consists basically of varying the interactions – internal and external – acting on the system.Footnote ⁸ It may also seem incompatible with the existence of “memory” materials capable of recovering past shapes, but we must keep in mind that those are very complex systems – with interactions between deformable bodies – whose mechanical description is usually over-simplified.

Restrictions on the Position Dependency (Fig. 1.8)

• Because of the homogeneity of space in Galilean reference frames, the dependence of ${\overline{𝗙}}_{𝗤\to 𝗣}$ on the positions of P and Q cannot be through all the information contained in their position vectors $\overline{{𝗢}_{\text{RGal}}𝗣}$ and $\overline{{𝗢}_{\text{RGal}}𝗤}$, but just through their difference $\overline{\mathrm{PQ}}$.

• Isotropy does not allow a dependency on a particular direction. Thus, only the distance $\rho =\left(\overline{\mathrm{PQ}}\right)$ is acceptable in the formulation of ${\overline{𝗙}}_{𝗤\to 𝗣}$.

• Because of the principle of determinacy, the value of ${\overline{𝗙}}_{𝗤\to 𝗣}$ at any time instant may only depend on that of $\rho$ at that same time instant.

Fig. 1.8

Restrictions on the Velocity Dependency (Fig. 1.9)

• Because of Galileo’s principle of relativity, the dependence of ${\overline{𝗙}}_{𝗤\to 𝗣}$ on the velocities of P and Q cannot be through all the information contained in the vectors ${\overline{v}}_{\text{RGal}}\left(𝗣\right)$ and ${\overline{v}}_{\text{RGal}}\left(𝗤\right)$ separately, but just through their difference:

  $\Delta {\overline{v}}_{\text{RGal}}={\overline{v}}_{\text{RGal}}\left(𝗣\right)-{\overline{v}}_{\text{RGal}}\left(𝗤\right)={\left(\frac{𝘥\overline{\mathrm{QP}}}{\mathrm{dt}}\right)}_{\text{RGal}}.$

• In general, that difference will have a component parallel to the line $\overline{\mathrm{PQ}}$ and another one perpendicular to that line: $\Delta \overline{v}=\Delta {\overline{v}}_{\parallel \overline{\mathrm{QP}}}+\Delta {\overline{v}}_{\perp \overline{\mathrm{QP}}}\equiv \Delta {\overline{v}}_{\rho }+\Delta {\overline{v}}_{𝘯}$.

• Isotropy does not allow a dependency on a particular direction not associated to the existence of the two interacting particles. Isotropy restricts the type of dependence of ${\overline{𝗙}}_{𝗤\to 𝗣}$ on $\Delta \overline{v}$. The interaction force may depend on the value of $\Delta {\overline{v}}_{\rho }$ (that is, on $\dot{\rho }$, where $\dot{\rho }>0$ and $\dot{\rho }<0$ describe the separating and the approaching velocity of particles P and Q, respectively), and on just the magnitude of $\left(\Delta {\overline{v}}_{𝘯}\right)$ (though usual formulations do not depend on that component).

• Because of the principle of determinacy, the value of ${\overline{𝗙}}_{𝗤\to 𝗣}$ at any time instant may only depend on those of $\dot{\rho }$ and $\left(\Delta {\overline{v}}_{𝘯}\right)$ at that same time instant.

Summarizing: ${𝖥}_{𝗤\leftrightarrow 𝗣}=𝘧\left(\rho ,,,\dot{\rho },,,\left(\Delta {\overline{v}}_{𝘯}\right)\right)$. Note that the values $\left(\rho ,,,\dot{\rho }\right)$ are the same in all reference frames (whether Galilean or not), whereas $\left(\Delta {\overline{v}}_{𝘯}\right)$ is the same only in Galilean reference frames but may change when moving to a non-Galilean one. Hence, the formulation of any interaction depending exclusively on $\rho$ and $\dot{\rho }$ will be shared by all reference frames.

Fig. 1.9

The usual interactions in mechanical systems do not depend on $\left(\Delta {\overline{v}}_{𝘯}\right)$. Therefore, only formulations depending on $\rho$ and $\dot{\rho }$ will be considered from now on.

Gravitational Force

Newton’s law of universal gravitation states that two particles P and Q attract each other with a force directly proportional to the product of their masses and inversely proportional to ${\left(\overline{\mathrm{PQ}}\right)}^2$ (Fig. 1.11):

$\left({\overline{𝗙}}_{𝗣\leftrightarrow 𝗤}^{\text{grav}}\right)={𝘎}_0\frac{{𝘮}_{𝗣}{𝘮}_{𝗤}}{{\left(\overline{\mathrm{PQ}}\right)}^2}={𝘎}_0\frac{{𝘮}_{𝗣}{𝘮}_{𝗤}}{{\rho }^2}.$(1.11)

Fig. 1.11

Strictly speaking, ${𝘮}_{𝗣}$ and ${𝘮}_{𝗤}$ in Eq. (1.11) are the gravitational masses of the particles. They are conceptually different from the inertial masses, but there is no empirical evidence so far that they should differ, so they will be treated as one same thing. The constant parameter ${𝘎}_0$ in Eq. (1.11) is the universal constant of gravitation ${𝘎}_0=6.67·{10}^{-11}{m}^3/\left(\text{kg}·{\text{s}}^2\right)$ in the International System of Units, SI), and is independent from the particles (that is why it is a universal constant).Footnote ⁹

The vector ${\overline{𝗙}}_{𝗣\to 𝗤}^{\text{grav}}/{𝘮}_{𝗤}$ is called the gravitational field ${\overline{𝘨}}_{𝗣\to 𝗤}$ created by P at Q location. Appendix 1.A presents the formulation of the Earth’s gravitational field.

The universal law of gravitation has been widely discussed because of some very particular features. On the one hand, and as a result of the identification of inertial mass and gravitational mass, it predicts a same acceleration (relative to a Galilean reference frame) for any particle under the same gravitational field. On the other hand, Eq. (1.11) describes an instantaneous interaction. The direction and intensity of the gravitational attraction is adjusted instantaneously whatever the distance between the interacting particles might be: the gravitational field propagates with an infinite speed!Footnote ¹⁰

Interaction Force through Springs

The term spring is used to designate any element with negligible mass responsible for an interaction force depending exclusively on the distance between particles (called “spring length” from now on): $\left({\overline{𝗙}}_{𝗣\leftrightarrow 𝗤}^{\text{spring}}\right)=𝘧\left(\rho \right)$. Springs may introduce either an attraction or a repulsion between their endpoints.

The most usual case is the linear spring, where the force change associated with a change of spring length is proportional to that length change through a constant parameter called the spring constant k:

$𝘬\equiv \frac{\Delta \text{attraction}\left(\text{repulsion}\right)\text{force between the spring ends}}{\text{increase}\left(\text{decrease}\right)\text{of spring length}}.$(1.12)

Springs constants are always positive. However, the term spring may be used in a more general sense for any physical phenomenon whose net force is a function of $\rho$ (in this context, the gravitational force can be seen as a spring). In that case, we will talk of equivalent spring, and the constant may be negative.

A real spring (not an equivalent one) has a nonzero slack length ${\rho }_{\text{slack}}$ (length for which the force between the spring endpoints is zero). From that state, the spring will generate an attraction force if the length is increased $\left(\Delta \rho =\rho -{\rho }_{\text{slack}}>0\right)$ and a repulsion force if it is decreased $\left(\Delta \rho <0\right)$.

The force associated with a spring within a mechanical system may evolve from attraction to repulsion with time. When solving the system dynamics, the spring force is drawn either as an attraction or a repulsion, and its value is formulated accordingly (Fig. 1.12).

Fig. 1.12

If the spring force is formulated from a nonzero-tension state (a state where its length ${\rho }_0$ is higher or lower than ${\rho }_{\text{slack}}$, and the spring force is ${𝘍}_0$), the choice of its description as an attraction or a repulsion is made according to the nature of the initial force ${𝘍}_0$:

$\begin{array}{c}{𝘍}_0=\text{attraction}\Rightarrow {𝘍}_{\mathrm{att}}^{\text{spring}}={𝘍}_0+𝘬\Delta {\rho }_{\text{elong}}={𝘍}_0-𝘬\Delta {\rho }_{\text{short}},\\ {𝘍}_0=\text{repulsion}\Rightarrow {𝘍}_{\mathrm{rep}}^{\text{spring}}={𝘍}_0+𝘬\Delta {\rho }_{\text{short}}={𝘍}_0-𝘬\Delta {\rho }_{\text{elong}}.\end{array}$(1.13)

In the context of a particular problem, the spring length has to be expressed as a function of the generalized coordinates $\left({𝘲}_1,{𝘲}_2,\dots ,{𝘲}_{𝘯}\right)$ describing the system configuration.

► Example 1.2

The linear spring connecting the endpoints of two bars articulated at point O has a constant k and exerts an attraction force ${𝘍}_0$ between P and Q when $\theta ={\theta }_0$ (

Fig. 1.13

). For any $\theta$ value, then, the spring force is formulated as an attraction force:

${𝘍}_{\mathrm{att}}^{\text{spring}}={𝘍}_0+𝘬\Delta {\rho }_{\text{elong}}={𝘍}_0+2\mathrm{Lk}\left(sin\frac{\theta }{2}-sin\frac{{\theta }_0}{2}\right).$

Fig. 1.13

◄

► Example 1.3

The hydraulic cylinder prescribes a motion $𝙭\left(𝘵\right)$, and the angular coordinate $\theta$ evolves dynamically. When $\theta ={\theta }_0$ and $𝙭={𝙭}_0$, the spring is compressed and exerts a force ${𝘍}_0$ between its endpoints (Fig. 1.14).

Fig. 1.14

As the initial force ${𝘍}_0$ is a repulsion, the spring force for any configuration will be formulated as a repulsion force:

$\begin{array}{c}{𝘍}_{\mathrm{rep}}^{\text{spring}}={𝘍}_0+𝘬\Delta {\rho }_{\text{short}}={𝘍}_0+𝘬\left(\left(𝙭-{𝙭}_0\right)+\left(𝘓tan{\theta }_0-\frac{𝘳}{cos{\theta }_0}\right)-\left(𝘓tan\theta -\frac{𝘳}{cos\theta }\right)\right)\\ ={𝘍}_0+𝘬\left(𝙭-{𝙭}_0+\frac{𝘓sin{\theta }_0-𝘳}{cos{\theta }_0}-\frac{𝘓sin\theta -𝘳}{cos\theta }\right).\end{array}$

◄

In a nonlinear spring, the force change is not proportional to the spring length change (Fig. 1.15).

Fig. 1.15

For small length changes around a given length ${\rho }_0$, the spring force may be approximated through a Taylor series up to the linear term:

$𝘍\left(\rho \right)\approx 𝘍\left({\rho }_0\right)+{\left(\frac{\mathrm{dF}}{𝘥\rho }\right)}_{{\rho }_0}\left(\rho -{\rho }_0\right).$(1.14)

The derivative ${\left(\mathrm{dF}/𝘥\rho \right)}_{{\rho }_0}$can be understood as the constant of an equivalent linear spring for lengths close to ${\rho }_0$.

Interaction Force through Dampers

Dampers with negligible mass exert a force opposite to the approaching or separating velocity between their endpoints. The force value depends on that velocity, and sometimes also on the distance $\rho$ between those points: $\left({\overline{𝗙}}_{𝗣\leftrightarrow 𝗤}^{\text{damper}}\right)=𝘧\left(\rho ,,,\dot{\rho }\right)$.

The most usual case is the linear damper (or viscous friction damper). In those dampers, the force is strictly proportional to $\dot{\rho }$ through the damper constant, c. As in springs, the damper force may be an attraction or a repulsion. When the damper is assembled in parallel with a spring (spring–damper system), the criterion to draw and formulate that force is the same as that chosen for the spring:

$\begin{array}{c}{𝘍}_{\mathrm{att}}^{\text{damp}}=𝘤\cdot {𝙫}_{\text{separation}}\equiv 𝘤\dot{\rho }=-𝘤\cdot {𝙫}_{\text{approaching}},\\ {𝘍}_{\mathrm{rep}}^{\text{damp}}=𝘤\cdot {𝙫}_{\text{approaching}}\equiv -𝘤\dot{\rho }=-𝘤\cdot {𝙫}_{\text{separation}}.\end{array}$(1.15)

► Example 1.4

The attraction force exerted by the damper in Example 1.2 is

${𝘍}_{\mathrm{att}}^{\text{damp}}=𝘤\cdot {𝙫}_{\text{separation}}=𝘤\dot{\rho }=𝘤\frac{𝘥}{\mathrm{dt}}\left(2𝘓sin\frac{\theta }{2}\right)=𝘤𝘓\dot{\theta }cos\frac{\theta }{2}.$

◄

► Example 1.5

The repulsion force exerted by the damper in Example 1.3 is

${𝘍}_{\mathrm{rep}}^{\text{damp}}=-𝘤\cdot {𝙫}_{\text{separation}}=-𝘤\dot{\rho }=-𝘤\frac{𝘥}{\mathrm{dt}}\left(\frac{𝘓sin\theta -𝘳}{cos\theta }-𝙭\right)=-𝘤\frac{𝘓-𝘳sin\theta }{{cos}^2\theta }\dot{\theta }+𝘤\dot{𝙭}.$

Actually, the distance between the damper endpoints is $\left(\frac{𝘓sin\theta -𝘳}{cos\theta }-𝙭\right)$ plus a constant value, but that constant can be omitted because its time derivative is zero.◄

Interaction Force through Actuators

An actuator (or driver) is an element able to produce a prescribed motion or a prescribed interaction according to a predefined time law. Its functioning requires a control system and an energy source.

The most usual actuator between pairs of particles P and Q is the hydraulic cylinder. As with springs and dampers, the effect it has on P and Q is described as an action–reaction pair of forces when its mass is negligible (Fig. 1.16). The force ${𝘍}_{\mathrm{cyl}}\left(𝘵\right)$ it introduces can either be a prescribed function or an unknown whose evolution has to be adjusted in order to guarantee a prescribed motion $\rho \left(𝘵\right)$.Footnote ¹¹

Fig. 1.16

A rocket with negligible mass can be considered as an idealized actuator. It is an interesting case as it seems to be using just one force of the action–reaction pair to control the motion of a particle P (Fig. 1.17). This is not so: the reaction force is applied to the high-speed gas particles ejected by the rocket in a direction opposite to ${\overline{𝗙}}^{\text{rocket}}\left(𝘵\right)$.

Fig.1.17

Particle–Surface Interaction: Force Associated with Impenetrability

The impenetrability of a rigid body S forbids the normal approaching velocity of a particle P relative to it when they are in contact: ${\left({\overline{v}}_{𝘚}\left(𝗣\right)\right)}_{𝗇\mathrm{app}}=0$. It is a unilateral restriction, as the separating normal motion is not restricted.

The kinematic constraint ${\left({\overline{v}}_{𝘚}\left(𝗣\right)\right)}_{𝗇\mathrm{app}}=0$ translates into a repulsion normal force N on P that takes the value required to guarantee that constraint (Fig. 1.18). If that force did not exist, other interaction forces on the particle could provoke a penetration of P into the rigid body. As it is a unilateral constraint, the value of N has to be strictly positive.

Fig. 1.18

The constraint normal force can be formulated as a function of the normal deformation of objects. When that deformation is neglected (hypothesis of rigid body), that force cannot be formulated, and it becomes an unknown of the dynamical problem.

In unilateral normal forces, a zero value $\left(𝘕=0\right)$ indicates that the contact is about to be lost. When the N value required to maintain that contact is negative $\left(𝘕<0\right)$, the contact is lost and a separating normal motion is actually taking place.

If the particle P motion is restricted by two close surfaces, we say that there is a bilateral constraint (Fig. 1.19). As an adimensional particle cannot be simultaneously in contact with two surfaces, we will assume an infinitesimal mutual separation $\epsilon \to 0$ between them, and P will be either in contact with one surface or the other. In that case, the normal force on P may have either sign.

Fig. 1.19

► Example 1.6

The particle P is initially at rest relative to a curved ground (Fig. 1.20a). The normal force on P compensates the weight (otherwise P would penetrate the ground).

Fig. 1.20

If P slides on the ground with speed v (Fig. 1.20b), the value of the normal force is not the same. The normal acceleration of P relative to the ground is upward, thus N has to compensate the weight and provide that acceleration: $𝘕>\mathrm{mg}$.

When P reaches the highest position (Fig. 1.20c), its normal acceleration relative to the ground is downward, so $𝘕<\mathrm{mg}$. If the ground radius of curvature at that location is ${\Re }_0$ and the speed value is $𝙫<\sqrt{𝘨{\Re }_0}$, the normal acceleration is ${𝘢}^{𝘯}={𝙫}^2/{\Re }_0<𝘨$. As $\mathrm{mg}-𝘕=𝘮\left({𝙫}^2/{\Re }_0\right)$ and $𝘕>0$, the weight compensates the normal force and provides that acceleration.

If P goes through that position with speed ${𝙫}_0$ such that ${𝙫}_0^2/{\Re }_0=𝘨$, the normal force is zero, and that indicates that the ground contact is about to be lost. The curvature radius of the P trajectory is not imposed by the ground curvature but by the speed value and the weight (Fig. 1.20e).

If $𝙫>{𝙫}_0$, the weight will be responsible for a parabolic trajectory with ${\Re }^{\text{parab}}>{\Re }_0$,and there will be no ground contact (Fig. 1.20e).◄

Particle–Surface Interaction: Force Associated with Friction

The roughness of a rigid surface S is responsible for a tangential force (or friction force) ${\overline{𝗙}}^{\text{fric}}$ on the particle P when they are in contact.Footnote ¹² When there is sliding between P and S (that is, when ${\overline{v}}_{\text{slide}}\left(𝗣\right)={\left({\overline{v}}_{𝘚}\left(𝗣\right)\right)}_{\text{tang}}\ne \overline{0}$, so there is no kinematic restriction in that direction), that force is called dynamic or kinetic friction,Footnote ¹³ it is opposed to the sliding velocity when the roughness is isotropic, and it may be formulated as a function of that velocity.

When there is no relative sliding, the tangential force is called static friction, and it is a constraint force: its value adapts in order to keep the kinematic restriction ${\left({\overline{v}}_{𝘚}\left(𝗣\right)\right)}_{\text{tang}}=\overline{0}$. That force can be modeled as a function of the tangential deformation of objects. When that deformation is neglected (hypothesis of rigid body), that force cannot be modeled: it is an unknown of the dynamical problem.

When there is no lubricant between the particle and the surface, Coulomb’s law of friction (or dry friction model) provides the simplest description for that phenomenon. If the roughness is isotropic:

• The static friction force ${\overline{𝗙}}_{\mathrm{st}}^{\text{fric}}$ may have any direction and may take any value within the range $\left(0,{\mu }_{𝘴}𝘕\right)$ in order to guarantee the no-sliding restriction (Fig. 1.21a). The adimmensional constant coefficient ${\mu }_{𝘴}$ is called static friction coefficient. When the friction force required to maintain the kinematical constraint is $>{\mu }_{𝘴}𝘕$, the particle slides and ${\overline{𝗙}}^{\text{fric}}$ becomes a kinetic friction force.

• The kinetic friction force ${\overline{𝗙}}_{\mathrm{kin}}^{\text{fric}}$ is opposed to the sliding velocity, and its value is $\left({\overline{𝗙}}_{\mathrm{kin}}^{\text{fric}}\right)={\mu }_{𝘬}𝘕$ (Fig. 1.21b). The adimmensional coefficient ${\mu }_{𝘬}$ is the kinetic friction coefficient, and it is assumed to be constant.Footnote ¹⁴ Usually ${\mu }_{𝘬}<{\mu }_{𝘴}$.

Fig. 1.21

Friction in materials with anisotropic roughness may be modeled through direction-dependent friction coefficients. For the particular case of orthotropic surfaces (with different properties along two orthogonal directions), two different friction coefficients $\left({\mu }_{𝘴}^1,{\mu }_{𝘴}^2\right)$ describe the phenomenon. The allowed range of values for $\left({\overline{𝗙}}_{\mathrm{st}}^{\text{fric}}\right)$ is $\left(0,{\mu }_{𝘴}^{\text{mean}}𝘕\right)$, where ${\mu }_{𝘴}^{\text{mean}}=\sqrt{{\mu }_{𝘴}^1{\mu }_{𝘴}^2}$, and the direction of ${\overline{𝗙}}_{\mathrm{kin}}^{\text{fric}}$ is not any more opposed to the sliding velocity.

When one of the two friction coefficients is much lower than the other $\left({\mu }_{𝘴}^1\ll {\mu }_{𝘴}^2\right)$, a first approximation with $\left({\mu }_{𝘴}^1\approx 0,{\mu }_{𝘴}^2\equiv {\mu }_{𝘴}\right)$ yields a simpler model.

When there is lubricant between the particle and the surface $\left(\text{so},,\left({\overline{v}}_{\text{slide}}\left(𝗣\right)\right),\ne ,0\right)$, the kinetic friction force can be formulated as a viscous friction proportional to the sliding velocity: $\left({\overline{𝗙}}_{\text{visc}}^{\text{fric}}\right)={\mathrm{cv}}_{\text{slide}}$. The coefficient c may be a function of the normal constraint force.

Particle–Curve Interaction

The interaction between a rigid curve and a particle P moving along it can be visualized either as that of a small hollow sphere pierced by a thin wire (Fig. 1.22a), or a particle moving in a tube with an infinitesimal diameter $\epsilon \to 0$ (Fig. 1.22b). In both cases, we will consider that P is in contact with just one side of the wire or the tube (as was assumed earlier in particles constrained by two close surfaces), hence the normal force N on P will have any direction on the plane perpendicular to the curve (Fig. 1.22c).

Fig. 1.22

The friction force between curve and particle may be modeled through Coulomb’s friction law (Fig. 1.22d,e).

Constraints between Particles through Intermediate Elements

Two different constraint forces have been introduced so far: the normal force and the static friction force associated with a single-point contact between a particle and a surface or a curve force. Following what has been done to introduce interactions “at a distance” between particles through springs, dampers, and actuators, we will consider now the possibility of constraints “at a distance” between particles through intermediate elements with negligible mass.

Constraints between two particles P and Q can be introduced through rigid thin bars and through inextensible threads. A rigid bar between P and Q generates a bilateral constraint preventing the particles to approach or separate: $\dot{\rho }=0$ (Fig. 1.23a). That kinematical restriction translates into a pair of repulsion or attraction forces, respectively.

Fig. 1.23

An inextensible thread under tension between P and Q generates a unilateral constraint preventing just the separation velocity between them, as the thread length cannot increase (Fig. 1.23b). Hence, a tensioned thread can only transmit attraction forces between P and Q.

Analytical Characterization of Constraint Forces

The previous description of the constraint forces of a rigid body S on a particle P relies on the description of the kinematical restrictions they are supposed to guarantee. Mathematically, this can be expressed as an orthogonality between the force and the particle allowed motion:

${\overline{𝗙}}_{𝘚\to 𝗣}\cdot {\overline{v}}_{𝘚}\left(𝗣\right)=0.$(1.16)

The description of the velocity of P has to be done from the S reference frame. Eq. (1.16) is the equation for the straightforward analytical characterization of single-point constraint forces. Note that there is a one-to-one correspondence between the nonzero components in ${\overline{𝗙}}_{𝘚\to 𝗣}$ and the zero components in ${\overline{v}}_{𝘚}\left(𝗣\right)$.

When the constraint force on P comes from another particle Q through an intermediate element, the kinematic description has to be done on any reference frame R where ${\overline{v}}_{𝘙}\left(𝗤\right)=\overline{0}$.

Characterizing constraint forces and calculating them are two different things. The former consists on assessing the possibility of existence of those forces, and it does not depend on the actual motion of the constraining elements (the rigid body S or the particle Q). Calculating the constraint forces (finding out their values) requires the application of the laws of dynamics, and the result does depend on the actual motion of the particle and the constraining elements.

Relationship between Inertial Forces Associated with Non-Galilean Reference Frames with a Relative Translational Motion

As for particles, the dynamics of rigid bodies may be formulated in Galilean or non-Galilean reference frames. However, once an initial choice of reference frame has been made, one may realize that the calculation of some magnitudes (such as angular momentum, as will be seen in Chapter 4) may be simpler in a frame with a translational motion (not necessarily rectilinear and uniform) relative to the initial frame. The formulation of the theorems in the new frame is not complicated, and overall that second choice may be advantageous. As that new frame does not rotate relative to the initial one, it will be described as the “Reference frame with a Translational motion and moving along with point Q”: RTQ in short. Whenever that RTQ is not Galilean, the inertial forces on each point of the rigid body have to be taken into account.

If the first reference frame is Galilean, only transportation forces will have to be considered in the RTQ $(\text{as}{\overline{\Omega }}_{\text{RGal}}^{\mathrm{RTQ}}=\overline{0})$. As all points in the RTQ have exactly the same motion relative to Gal, the field of inertial forces acting on a system of particles will be uniform and proportional to the particle masses (Fig. 1.24).

Fig. 1.24

If the first reference frame is non-Galilean, both the transportation and the Coriolis inertial forces ${\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{tr}}$ and ${\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{Cor}}$ have to be taken into account in general even if we do not move relative to the RTQ. They are associated with ${\overline{𝗮}}_{\text{RGal}}\left(𝗢\in \text{NGal}\right),{\overline{\Omega }}_{\mathrm{tr}}\left(={\overline{\Omega }}_{\text{RGal}}^{\text{NGal}}\right)$ and ${\overline{\alpha }}_{\mathrm{tr}}\left(={\overline{\alpha }}_{\text{RGal}}^{\text{NGal}}\right)$ (Fig. 1.25a).

Fig. 1.25

If we move to the RTQ, the new inertial forces ${\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{tr}}$ and ${\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{Cor}}$ will be different. It is always possible to calculate them from ${\overline{𝗮}}_{\text{RGal}}\left(𝗤\in \mathrm{RTQ}\right),{\overline{\Omega }}_{\mathrm{tr}}^{\prime }\left(={\overline{\Omega }}_{\text{RGal}}^{\mathrm{RTQ}}\right)$ and ${\overline{\alpha }}_{\mathrm{tr}}^{\prime }\left(={\overline{\alpha }}_{\text{RGal}}^{\mathrm{RTQ}}\right)$, but they can be obtained from ${\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{tr}}$ and ${\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{Cor}}$ in a very straightforward way just by adding one term (Fig. 1.25b):

$\left({\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{Cor}}\right)=\left({\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{Cor}}\right)-{𝘮}_{𝗣}{\overline{𝗮}}_{\text{NGal}}\left(𝗤\right).$(1.20)

The additional term that has to be added when moving to the RTQ corresponds to a uniform force field associated with the transportation force on Q that would have been required if R were Galilean.

♣ Proof

The inertial forces on P associated with the NGal and the RTQ reference frames are:

$\begin{array}{c}{\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{Cor}}=-{𝘮}_{𝗣}{\overline{𝗮}}_{\text{RGal}}\left(𝗣\in \text{NGal}\right)-2{𝘮}_{𝗣}{\overline{\Omega }}_{\text{RGal}}^{\text{NGal}}\times {\overline{v}}_{\text{NGal}}\left(𝗣\right),\\ {\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{Cor}}=-{𝘮}_{𝗣}{\overline{𝗮}}_{\text{RGal}}\left(𝗣\in \mathrm{RTQ}\right)-2{𝘮}_{𝗣}{\overline{\Omega }}_{\text{RGal}}^{\mathrm{RTQ}}\times {\overline{v}}_{\mathrm{RTQ}}\left(𝗣\right).\end{array}$

Taking into account that the RTQ does not rotate relative to the NGal $(\text{so}{\overline{\Omega }}_{\text{RGal}}^{\mathrm{RTQ}}={\overline{\Omega }}_{\text{RGal}}^{\text{NGal}})$, and that ${\overline{v}}_{\text{NGal}}\left(𝗣\right)-{\overline{v}}_{\mathrm{RTQ}}\left(𝗣\right)={\overline{v}}_{\text{NGal}}\left(𝗤\right),$

$\begin{array}{c}{\overline{𝗮}}_{\text{RGal}}\left(𝗣\in \mathrm{RTQ}\right)={\overline{𝗮}}_{\text{NGal}}\left(𝗣\in \mathrm{RTQ}\right)+{\overline{𝗮}}_{\text{RGal}}\left(𝗣\in \text{NGal}\right)+2{\overline{\Omega }}_{\text{RGal}}^{\text{NGal}}\times {\overline{v}}_{\text{NGal}}\left(𝗣\in \mathrm{RTQ}\right)\\ ={\overline{𝗮}}_{\text{NGal}}\left(𝗤\right)+{\overline{𝗮}}_{\text{RGal}}\left(𝗣\in \text{NGal}\right)+2{\overline{\Omega }}_{\text{RGal}}^{\text{NGal}}\times {\overline{v}}_{\text{NGal}}\left(𝗤\right),\end{array}$

the difference between the inertial forces in those two reference frames becomes:

$\left({\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\mathrm{RTQ}\to 𝗣}^{\mathrm{Cor}}\right)-\left({\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{tr}}+{\overline{F}}_{\text{NGal}\to 𝗣}^{\mathrm{Cor}}\right)=-{𝘮}_{𝗣}{\overline{𝗮}}_{\text{NGal}}\left(𝗤\right).$

♣