D2. Interaction forces between particles
[math]\displaystyle{ \newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}} \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\cs}{\textrm{c}} \newcommand{\ds}{\textrm{d}} \newcommand{\ms}{\textrm{m}} \newcommand{\ts}{\textrm{t}} \newcommand{\us}{\textrm{u}} \newcommand{\vs}{\textrm{v}} \newcommand{\Fs}{\textrm{F}} \newcommand{\Rs}{\textrm{R}} \newcommand{\Ts}{\textrm{T}} \newcommand{\Ls}{\textrm{L}} \newcommand{\Ns}{\textrm{N}} \newcommand{\Bs}{\textrm{B}} \newcommand{\es}{\textrm{e}} \newcommand{\fs}{\textrm{f}} \newcommand{\is}{\textrm{i}} \newcommand{\js}{\textrm{j}} \newcommand{\rs}{\textrm{r}} \newcommand{\Os}{\textbf{O}} \newcommand{\Gs}{\textbf{G}} \newcommand{\Cbf}{\textbf{C}} \newcommand{\Or}{\Os_\Rs} \newcommand{\Qs}{\textbf{Q}} \newcommand{\Cs}{\textbf{C}} \newcommand{\Ps}{\textbf{P}} \newcommand{\Ss}{\textbf{S}} \newcommand{\deg}{^\textsf{o}} \newcommand{\xs}{\textsf{x}} \newcommand{\ys}{\textsf{y}} \newcommand{\zs}{\textsf{z}} \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}} \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}} \newcommand{\vec}[1]{\overline{#1}} \newcommand{\vecbf}[1]{\overline{\textbf{#1}}} \newcommand{\vecdot}[1]{\overline{\dot{#1}}} \newcommand{\OQvec}{\vec{\Os\Qs}} \newcommand{\QPvec}{\vec{\Qs\Ps}} \newcommand{\OPvec}{\vec{\Os\textbf{P}}} \newcommand{\OCvec}{\vec{\Os\Cs}} \newcommand{\OGvec}{\vec{\Os\Gs}} \newcommand{\abs}[1]{\left|{#1}\right|} \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}} \newcommand{\vector}[3]{ \begin{Bmatrix} {#1}\\ {#2}\\ {#3} \end{Bmatrix}} \newcommand{\vecdosd}[2]{ \begin{Bmatrix} {#1}\\ {#2} \end{Bmatrix}} \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})} \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})} \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})} \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})} \newcommand{\velo}[1]{\vvec_{\textrm{#1}}} \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}} \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}} \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})} \newcommand{\psio}{\dot{\psi}_0} \newcommand{\Pll}{\textbf{P}_\textrm{lliure}} \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)} \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)} \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)} }[/math]
Newton's second law can be used to predict the acceleration of a particle P when all the interaction forces exerted on P by all the other particles (Q) are known (all [math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}} }[/math]), or to calculate the forces needed to guarantee a predetermined movement ([math]\displaystyle{ \overline{\mathbf{O}_\Rs\mathbf{P}}(\ts) }[/math]).
This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.
D2.1 Kinematic dependence of interaction forces
According to the Principle of Determinacy, the forces of interaction between two particles P and Q at each time instant can only depend on the position and velocity of the particles at that time instant:
[math]\displaystyle{ \ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) . }[/math]
Galileo's Principle of Relativity (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on positions and velocities:
- The homogeneity of space in Galilean reference frames does not allow the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on positions [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}} }[/math] and [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}} }[/math] separately, but allows the dependence on its difference [math]\displaystyle{ \overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}) }[/math].
- The isotropy of space in Galilean reference frames does not allow the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on the velocities of de P and Q separately (as [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) }[/math]), but allows the dependence on its difference [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}} }[/math]. Vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}} }[/math] is the same in all Galilean reference frames (this is the meaning of the '[math]\displaystyle{ \forall }[/math]Gal' subscript, that means “for all Gal reference frames”). In general, [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\text {Gal }} }[/math] has a component parallel to [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math], and another one perpendicular to [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math]: [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}} }[/math]. But the space isotropy does not allow either the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on a direction different from [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math] (as seen in the presentation of Newton's third law). Therefore, we can accept the dependence on vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_\rho }[/math] but only on the value [math]\displaystyle{ \left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right| }[/math](though this dependence rarely appears).
Proof➕
Let us take any pair of Galilean reference frames RGal1 and RGal2. The velocity of particles P and Q relative to RGal2 may be expressed from that in RGal1 through a composition of velocities. If AB=RGal2 and REL=RGal1:
[math]\displaystyle{ \overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q})=\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})+\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })-\left[\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q})+\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 })\right]= }[/math]
[math]\displaystyle{ =\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q})+\left[\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 })\right] }[/math]
As both reference frames are Galilean, their relative motion is a translational one [math]\displaystyle{ \left(\bar{\Omega}_{\mathrm{RGal1}}^{\mathrm{RGal2}}=\overline{0}\right) }[/math], therefore [math]\displaystyle{ \overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P} \in \text { RGal1 })=\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q} \in \text { RGal1 }) }[/math].
Finally: [math]\displaystyle{ \overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal2 }}(\mathbf{Q})=\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {RGal1 }}(\mathbf{Q}) }[/math].
Figure D2.1 summarizes all these restrictions.
D2.2 Classification of interaction forces
If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” ([math]\displaystyle{ \rho \gt 0 }[/math]). Talking about contact interaction between particles ([math]\displaystyle{ \rho = 0 }[/math]) is not possible: if [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space.
Strictly speaking, it is only possible to talk about contact interactions between a particle [math]\displaystyle{ \mathbf{P} }[/math] and a particle [math]\displaystyle{ \mathbf{Q} }[/math] that belongs to a rigid body [math]\displaystyle{ \mathrm{S}_\mathbf{Q} }[/math], or between particles [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] that belong to rigid bodies [math]\displaystyle{ \mathrm{S}_\mathbf{P} }[/math] and [math]\displaystyle{ \mathrm{S}_\mathbf{Q} }[/math], respectively. These interactions are dealt with in unitat D3.
When [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] interact at a distance ([math]\displaystyle{ \rho \gt 0 }[/math]), we talk of direct interaction. If they are not in contact ([math]\displaystyle{ \rho \gt 0 }[/math]) but them there is an element acting as an intermediate between them, we talk of indirect interaction through the element. In dynamics, all elements with negligible mass compared to that of [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE).
Forces transmitted between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] through those elements fulfil the Action-Reaction Principle: they have the same value and are parallel to [math]\displaystyle{ \overline{\mathbf{PQ}} }[/math], though they have opposite directions (Figure D2.2).
Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.
In this course, we consider four intermediate elements between particles:
- springs: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] of any sign.
- dampers: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math]; in the absence of initial relative motion ([math]\displaystyle{ \dot{\rho}_\mathrm{inic}=0 }[/math]), these elements do not introduce any force.
- linear actuators: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: [math]\displaystyle{ \mathrm{F}_\mathrm{acc.lin}(\ts) }[/math] they are data in the problem) or forces that are suitable to control a predetermined relative motion [math]\displaystyle{ \dot{\rho}(\ts) }[/math], either an approaching or a separating motion.
- inextensible threads: they prevent the particles from separating (impedeixen [math]\displaystyle{ \dot{\rho} \gt 0 }[/math] )but not from approaching (they allow [math]\displaystyle{ \dot{\rho} \lt 0 }[/math]). Since they are intermediate elements that forbid movement, the force they introduce is called restriction or constraint force.
Figure D2.3 classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.
D2.3 Gravitational attraction
The gravitational interaction force (law of universal gravitation) was formulated by Newton. It is an attraction force, and is inversely proportional to the square of the distance between particles (Figure D2.4). It is an empirical formulation: it is based on astronomical observations accumulated over many years.
[math]\displaystyle{ \mathrm{G}_0 }[/math] is the constant of universal gravitation, and its value is [math]\displaystyle{ \mathrm{G}_0=6,67\cdot 10^{-11} \mathrm{m}^3/(\mathrm{Kg}\cdot \mathrm{s}^2) }[/math]
The distance between particles ([math]\displaystyle{ \rho }[/math]) must be expressed as a function of the coordinates that have been chosen to describe the system configuration.
✏️ Example D2.1: gravitational attraction between two satellites
- Two satellites [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math], modelled as particles of mass [math]\displaystyle{ \ms_\Ps }[/math] and [math]\displaystyle{ \ms_\mathrm{Q} }[/math] , describe circular orbits of radii [math]\displaystyle{ \rs_\Ps }[/math] and [math]\displaystyle{ \rs_\mathrm{Q} }[/math], respectively, around a planet in the same plane. The configuration of the system is described by the angles [math]\displaystyle{ \theta_\Ps }[/math] and [math]\displaystyle{ \theta_\mathrm{Q}. }[/math]
- The gravitational force they exert on each other is:
- [math]\displaystyle{ \rho=|\overline{\mathbf{P Q}}|=|\overline{\mathbf{O Q}}-\overline{\mathbf{O P}}|=\sqrt{\left(\rs_\mathrm{Q} \sin \theta_\mathrm{Q}-\rs_\Ps \sin \theta_\Ps\right)^2+\left(\rs_\mathrm{Q} \cos \theta_\mathrm{Q}-\rs_\Ps \cos \theta_\Ps\right)^2}=\sqrt{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \sin \left(\theta_\mathrm{Q}+\theta_\Ps\right)}
}[/math]
- [math]\displaystyle{ \mathrm{F}_{\Ps \leftrightarrow \mathrm{Q}}^{\text {grav }}=\mathrm{G}_0 \frac{\ms^2}{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \cos \left(\theta_\Ps-\theta_\mathrm{Q}\right)} }[/math]
D2.4 Interaction through springs
Springs introduce attractive or repulsive forces between their ends depending on their deformation. From their natural length [math]\displaystyle{ \rho_\mathrm{nat} }[/math] (for which no force is produced between the spring ends), an elongation ([math]\displaystyle{ \rho-\rho_\mathrm{nat} \gt 0 }[/math]) causes attractive forces while a shortening ([math]\displaystyle{ \rho-\rho_\mathrm{nat} \lt 0 }[/math]) causes repulsive forces.
The mathematical formulation of these forces is obtained empirically from tests that measure the force as a function of the length change. Usually, we start from a static configuration in which the spring length [math]\displaystyle{ \rho_0 }[/math] does not have to coincide with the natural one. If [math]\displaystyle{ \rho_0 \gt \rho_\mathrm{nat} }[/math] , the force [math]\displaystyle{ \mathrm{F}_0 }[/math] between the spring ends for that configuration is attractive. Otherwise, ([math]\displaystyle{ \rho_0 \lt \rho_\mathrm{nat} }[/math]), [math]\displaystyle{ \mathrm{F}_0 }[/math] is a repulsive force.
The linear springs considered in this course have a linear behaviour: The variation in force [math]\displaystyle{ \Delta\mathrm{F} }[/math] from the reference value ([math]\displaystyle{ \Delta\mathrm{F}=\mathrm{F}-\mathrm{F}_0 }[/math]) )is proportional to the variation in length [math]\displaystyle{ \Delta\rho=\rho-\rho_0 }[/math] through a constant k.
A spring that is part of a mechanical system can introduce attractive and repulsive forces throughout its operation. Despite this, these forces are drawn with a single criterion (attractive or repulsive), and are formulated so that their value can have a positive or negative sign. In this way, both attractive and repulsive forces can be reproduced with a single drawing (Figure D2.5).
✏️ Example D2.2: attraction force of a spring with linear behaviour
- The linear spring acts between particles [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] moving within two parallel guides. For [math]\displaystyle{ \mathrm{x}_1=\mathrm{x}_2 }[/math] , the spring is stretched and the force it exerts between its ends is [math]\displaystyle{ \mathrm{F}_0 }[/math].
- For [math]\displaystyle{ \mathrm{x}_1=\mathrm{x}_2 }[/math] , the spring length is L and the [math]\displaystyle{ \mathrm{F}_0 }[/math] force is attractive. For [math]\displaystyle{ \mathrm{x}_1 \neq \mathrm{x}_2 }[/math], the length increases and so does the attraction force.
- The expression of the spring force for a general configuration as an attractive force is:
[math]\displaystyle{ \mathrm{F}_\mathrm{at}^\mathrm{molla}=\mathrm{F}_0+k\Delta\rho=\mathrm{F}_0 + k [\rho(\mathrm{x}_1 \neq \mathrm{x}_2)-\rho(\mathrm{x}_1=\mathrm{x}_2)]=\mathrm{F}_0 + k[\sqrt{\mathrm{L}^2+(\mathrm{x}_1-\mathrm{x}_2)^2}-\mathrm{L}] }[/math].
D2.5 Interaction through dampers
Linear dampers introduce attractive or repulsive forces between their ends depending on their deformation rate [math]\displaystyle{ \dot{\rho} }[/math]. When the ends of the damper separate, the force is attractive; when they approach, it is repulsive. Unlike springs, dampers do not exert any force between their ends in static situations.
The force associated with linear dampers with linear behaviour is proportional to that speed [math]\displaystyle{ \dot{\rho} }[/math]:
[math]\displaystyle{ \mathrm{F}_\mathrm{at}^\mathrm{amort}=c\dot{\rho} }[/math] , [math]\displaystyle{ \mathrm{F}_\mathrm{rep}^\mathrm{amort}=-c\dot{\rho} }[/math]
In mechanical systems, dampers often appear in parallel with a spring. In that case, the force is formulated according to the criterion that has been chosen for the spring. When they are not part of a spring-damper group, the criterion is set arbitrarily.
✏️ Example D2.3: repulsive force of a damper with linear behaviour
- The repulsion force of the damper is obtained from the elongation calculated in Example D2.2 through differentiation:
- [math]\displaystyle{ \rho=\sqrt{\mathrm{L}^2+(\mathrm{x}_1-\mathrm{x}_2)^2}\equiv \sqrt{\mathrm{L}^2+\mathrm{z}^2} \Rightarrow \dot{\rho}= \frac{\mathrm{d}\rho}{\mathrm{d}\mathrm{z}}\frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\mathrm{t}}=\frac{\mathrm{z}\dot{\mathrm{z}}}{\sqrt{\mathrm{L}^2+\mathrm{z}^2}} }[/math]
- [math]\displaystyle{ \mathrm{F}_\mathrm{rep}^\mathrm{amort}=-c \frac{\mathrm{z}\dot{\mathrm{z}}}{\sqrt{\mathrm{L}^2+\mathrm{z}^2}}=-c\frac{(\mathrm{x}_1-\mathrm{x}_2)(\dot{\mathrm{x}}_1-\dot{\mathrm{x}}_2)}{\sqrt{\mathrm{L}^2+\mathrm{(\mathrm{x}_1-\mathrm{x}_2})^2}} }[/math]
D2.6 Interaction through actuators
Linear actuators appear in the vast majority of mechanical systems, and are responsible for controlling their motion. As they are elements based on phenomena that are not strictly mechanical, their formulation in the context of Newtonian dynamics is not possible. The treatment given to them is different from that of other intermediate elements. Two situations are considered:
- The force introduced between their ends is data of the problem: this means that its value over time is known: [math]\displaystyle{ \mathrm{F}_\mathrm{lin.ac}=\mathrm{F}_\mathrm{lin.ac}(\ts) }[/math]. The movement they produce, in that case, is an unknown of the problem (Figure D2.6a).
- The force introduced between their ends is the required on to guarantee a predetermined motion. In that case, that force is an unknown (Figure D2.6b).
D2.7 Constraint interactions
Constraint forces restrict relative motions between particles, between particles and surfaces, or between rigid bodies. These forces arise from small deformations of the intermediate elements connecting the particles, from local deformations of the surface or of the rigid body, respectively. This course deals with the dynamics of rigid bodies, therefore these deformations (and the associated interaction forces) cannot be formulated: they are unknowns of the dynamic problem.
Constraint forces adapt themselves to guarantee the restrictions, but always within permitted ranges. Beyond those ranges, we say that the limit condition has been exceeded, the restriction disappears and either the constraint force is replaced by a formulable force or the interaction disappears.
When a system includes constraints, it is necessary to characterize them. That means investigating the direction the associated forces can have, and the associated limit conditions. That direction is that of the kinematic restrictions to be guaranteed.
This section deals with the characterization of indirect constraints between particles through a thread, and of direct constraints between a particle and a rigid body. Constraints between rigid bodies (both direct and indirect) are covered in unit D3.
Indirect constraint through inextensible threads
Inextensible threads of negligible mass are intermediate elements that prevent particles from separating but not from approaching each other.
Let us consider two particles P and Q connected by a thread. If a force is exerted on each particle by hand in the right direction to try to separate them, the thread introduces a force in the opposite direction to prevent it: that is the constraint force transmitted between P and Q through the thread, and it is a traction force. If the forces of the hands on the particles are in the opposite direction, the thread slackens and is not able to guarantee the restriction: through the thread, one particle can pull the other but cannot push it. Therefore, the traction force cannot be negative: if after solving a problem the conclusion is that the force required to maintain the restriction through the thread has to be negative [math]\displaystyle{ \mathrm{F}_\mathrm{thread}\lt 0 }[/math], this indicates that the constraint is not acting any more (the thread has lost tension, and that is equivalent to not having thread, Figure D2.7a). It is a unilateral link.
On the other hand, the thread allows the P (or Q) motion on a spherical surface with centre Q (or P): in the directions tangent to these surfaces, the thread cannot introduce any force (Figure D2.7b). In other words: the constraint force is orthogonal to the allowed relative motion:
[math]\displaystyle{ \overline{\mathbf{F}}_{\rightarrow \mathrm{P}} \cdot \overline{\mathbf{V}}_{\mathrm{RTQ }}(\mathbf{P})=0, \overline{\mathbf{F}}_{\rightarrow \mathrm{Q}} \cdot \overline{\mathbf{V}}_{\mathrm{RTP }}(\mathbf{Q})=0 }[/math]
The limit condition for this type of constraint is set by the breaking limit of the thread: for each type of material, there is a force ([math]\displaystyle{ \mathrm{F}_{\mathrm{break}} }[/math]) at which the thread breaks. If, when solving a dynamics problem involving a thread, we detect that the force to guarantee the constraint is higher than this limit value ([math]\displaystyle{ \mathrm{F}_{\mathrm{thread}}\gt \mathrm{F}_{\mathrm{break}} }[/math]), the problem must be solved again without the thread (and that increases the number of DoF of the system under study).
Direct constraint between a particle P and a smooth rigid body S
A rigid body S in contact with a particle P is an obstacle for certain movements of P. The constraint force of S on P is the dynamic description of that obstacle.
The kinematic analysis of P to characterize the constraint force is done from S, which is the element responsible for this force. Thus, the directions of P's movements for which S constitutes an obstacle are emphasized: they are the directions for which the P velocity is zero.
Figure D2.8 presents the characterization of the contact between P and S when the rigid body is smooth. It is a unilateral constraint: the constraint force on the particle in the direction normal to the surface at the contact point can only be repulsive, since it is not capable of retaining the particle if some other interaction wants to pull it away from the rigid body. As for indirect constraints between particles through inextensible threads, there is an orthogonality condition between the constraint force and the allowed velocity of P relative to S. The existence of sliding implies that the movement of P relative to S is allowed in any direction of the plane tangent to the S at the contact point: the contact does not introduce any force component in those directions.
Direct constraint between a particle P and a rough rigid body S
When the surface of the rigid body S is rough and the particle P does not slide on it (Figure D2.9), the constraint force can have nonzero components in both tangential directions. Unlike the normal force, these components can have any sign, but their resultant cannot exceed a maximum value [math]\displaystyle{ \sqrt{\Fs_1^2 + \Fs_2^2}\leq\Fs_{\text{t, max}}^{\text{constraing}} }[/math]. In the dry friction model, that value depends on the roughness: the rougher the surface, the higher the maximum value (D2.8 Friction.
D2.8 Friction
When particle P moves relative to the rough surface of the rigid body S, the tangential force is not a constraint force but a friction force, and always opposes the speed of P relative to S:
[math]\displaystyle{ \vec{\Fs}_{\mathrm{S}\rightarrow\Ps}^\text{friction} = -|\vec{\Fs}_{\mathrm{S}\rightarrow\Ps}^\text{friction}|\frac{\vel{P}{S}}{|\vel{P}{S}|} }[/math]
There are several models for formulating the value of [math]\displaystyle{ \vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction} }[/math], depending on the characteristics of the contact between P and S.
Coulomb’s dry friction model
The surface roughness of S is described through friction coefficients. When the roughness is isotropic (equal in all directions) two friction coefficients are defined:
- static friction coefficient [math]\displaystyle{ \mu_\es }[/math]: it defines the maximum value (limit condition) of the tangential constraint force: [math]\displaystyle{ \vec{\Fs}_\text{t, max}^\text{constraint} = \mu_\es\Ns }[/math]. If a force higher than [math]\displaystyle{ \mu_\es \Ns }[/math] is needed to guarantee that P does not slide on S, sliding occurs and the friction force appears (Figure D2.10a).
- kinetic friction coefficient [math]\displaystyle{ \mu_\ds }[/math]: it defines the value of the friction force : [math]\displaystyle{ |\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}| = \mu_\ds\Ns }[/math] (Figure D2.10b).
Viscous friction model
It is a very suitable model when there is some lubrication between P and S. The friction force is formulated as a function of the relative speed between the two. If it is a linear model: [math]\displaystyle{ |\vec{\Fs}_{\Ss\rightarrow\Ps}^\text{friction}| = \cs|\vel{P}{S}| }[/math].
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