Screws (Part 5): The Action Screw

1. Introduction

The notion of screws is most suitable to model mechanical actions. It is a well-known principle of statics that equilibrium of a rigid body is achieved by guaranteeing that the sums of both forces and moments applied to the body vanish. The action of a force \({\bf F}_A =F_A \boldsymbol{\hat{u}}\) whose line of action passes through a point \(A\) is adequately modeled as a slider of axis \((A,\boldsymbol{\hat{u}})\): this defines a screw, called action screw, which takes the form \[ \begin{Bmatrix} {\bf F}_A \\ {\bf 0} \end{Bmatrix}_A \] It is also possible model a mechanical action on \(\Sigma\) as a system of forces whose resultant is zero, yet gives rise to a non-zero moment \(\boldsymbol{C}\) (a torque): this action is modeled as a couple \[ \begin{Bmatrix} {\bf 0} \\ \boldsymbol{C} \end{Bmatrix} \] More generally, a mechanical action caused by a material system \(\Sigma_1\) on system \(\Sigma\) can always be modeled as an action screw, characterized by a resultant force and a resultant moment about a particular point. It will be denoted as \[ \{{\cal A}_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} {\bf F}_{\Sigma_1 \to \Sigma} \\\\ {\bf M}_{A, \Sigma_1 \to \Sigma} \end{Bmatrix} \]

2. Actions At-a-Distance

Action at-a-distance do not require contact between two material systems. A typical example is given by the gravitational action caused by a celestial body. Such actions are modeled by a distribution of forces within the whole interior of a material system: they are defined in terms of a vector field \(P\in \Sigma\mapsto {\bf f}_{\Sigma_1\to \Sigma} (P)\) of volumetric forces. The global action of material system \(\Sigma_1\) on \(\Sigma\) is then modeled by the action screw \[ \{{\cal A}_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} \int_\Sigma{\bf f}_{\Sigma_1 \to \Sigma} (P) dV \\\\ \int_\Sigma{\bf r}_{AP}\times {\bf f}_{\Sigma_1 \to \Sigma} (P) dV \end{Bmatrix} \] For instance, the gravitational action screw \(\{{\cal A}^g_{\Sigma_1 \to \Sigma}\}\) due to \(\Sigma_1\) on \(\Sigma\) is obtained with \[ {\bf f}_{\Sigma_1 \to \Sigma} (P) = - \int_{\Sigma_1} G \frac {\rho(P)\rho(P_1)}{|{\bf r}_{P_1 P}|^3} {\bf r}_{P_1 P} \, dV(P_1) \] where \(G\) is the universal gravitational constant, \(\rho (P)\) is the volumetric mass density at \(P\in\Sigma\), and \(\rho(P_1)\) is the volumetric mass density at \(P_1\in\Sigma_1\). A variety of (approximate) expressions can be derived for the action screw under particular assumptions. If body \(\Sigma_1\) is assumed of spherical shape, and with a spherical mass distribution, it is well-known that \({\bf f}_{\Sigma_1\to \Sigma}\) is given by \[ {\bf f}_{\Sigma_1\to \Sigma} (P) = -M_1 G \rho(P) \frac {{\bf r}_{O_1 P}}{|{\bf r}_{O_1 P}|^3} \] where \(O_1\) is the center of \(\Sigma_1\).

3. Contact Actions

If material system \(\Sigma_1\) is in direct physical contact with \(\Sigma\), then at all point \(Q\) of the boundary \(\partial\Sigma\) of \(\Sigma\), a force per unit area due to \(\Sigma_1\) can be written as a sum of normal and tangential component: \[ {\bf f}^c _{\Sigma_1 \to \Sigma} (Q) = N (Q) \boldsymbol{\hat{n}} (Q) + \boldsymbol{T} (Q) \] where \(\boldsymbol{\hat{n}} (Q)\) is the unit outward normal vector to \(\partial\Sigma\) at \(Q\). The contact action screw due to \(\Sigma_1\) on \(\Sigma\) is represented by \[ \{{\cal A}^c_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} \int_{\partial\Sigma} {\bf f}^c_{\Sigma_1 \to \Sigma} (Q) dA \\ \\ \int_{\partial\Sigma} {\bf r}_{AQ}\times {\bf f}^c_{\Sigma_1 \to \Sigma} (Q) dA \end{Bmatrix} \] Contact action screws pause significant problems for the solution of rigid body dynamics problems, since they are generally unknown, unless empirical constitutive laws are used. With modeling simplifications and/or empirical models, rigid body dynamics problems would remains indeterminate, that is, characterized by more unknowns than available equations. For instance, in the case of point-contact between two rigid bodies \(1\) and \(2\), we usually model the corresponding action as follows (denoting as \(I\) the contact point) \[ \{{\cal A}^c_{1 \to 2}\} = \begin{Bmatrix} N_{1\to 2}\boldsymbol{\hat{n}}_{12} + \boldsymbol{T}_{1\to 2} \\ \\ {\bf 0} \end{Bmatrix}_I \] where \(N_{1\to 2}\boldsymbol{\hat{n}}_{12}\) is the normal reaction force and \(\boldsymbol{T}_{1\to 2}\) is the friction force. If the contact is dry, the two surfaces are characterized by a static friction coefficient \(\mu_s\) and a kinetic friction coefficient \(\mu_k\). Coulomb law of (dry) friction give empirical relationships between the normal and tangential component of the reaction force. Two cases must be considered:

• Case 1: Slip at \(I\). If \({\bf v}_{I\in 2/1} \neq {\bf 0}\), then the friction force is opposed to the slip velocity and its magnitude is proportional to the normal reaction: \[ \boldsymbol{T}_{1\to 2}=- \mu_k |N_{1\to 2}| \frac{{\bf v}_{I\in 2/1}}{|{\bf v}_{I\in 2/1}|} \]

• Case 2: No slip at \(I\) (\({\bf v}_{I\in 2/1} ={\bf 0}\)). Then the friction force satisfies the following bound: \[ |\boldsymbol{T}_{1\to 2}| \leq \mu_s |N_{1\to 2}| \]

4. Frictionless Joint Between Two Rigid Bodies

In most mechanical systems, rigid bodies are connected by joints. Suppose that bodies \(i\) and \(j\) of a system \(\Sigma\) are interconnected and that the kinematics of body \(j\) relative to body \(i\) is characterized by the following kinematic screw \[ \{ {\cal V}_{j/i} \} = \begin{Bmatrix} \boldsymbol{\omega}_{j/i} \\ {\bf v}_{A\in j/i} \end{Bmatrix} \] Assume the action of body \(i\) on body \(j\) gives rise to the following contact action screw \[ \{{\cal A}^c_{i \to j}\} = \begin{Bmatrix} {\bf F}^c_{i\to j} \\ \\ {\bf M}^c_{A, i\to j} \end{Bmatrix}_I \] Then the joint between body \(i\) and \(j\) is said to be frictionless if and only if the following condition is satisfied: \[ {\bf v}_{A\in j/i}\cdot {\bf F}^c_{i\to j} + \boldsymbol{\omega}_{j/i}\cdot {\bf M}^c_{A, i\to j} =0 \qquad\qquad (1) \] for all arbitrary motions allowed by the joints.

• In equation (1), point \(A\) is arbitrary: indeed it is straightforward to show that the scalar \({\bf v}_{A\in j/i}\cdot {\bf F}^c_{i\to j}+ \boldsymbol{\omega}_{j/i}\cdot {\bf M}^c_{A, i\to j}\) is an invariant, that is, is independent of the choice of \(A\).

• Equation (1) expresses that the power generated by the contact action for all possible motions of body \(j\) relative to body \(i\) is zero at all time.

• If the joint \(j/i\) has \(m\) degrees of freedom, equation (1) will yield \(m\) scalar relationships.

• Frictionless joints represent an idealized concept. However, this concept is valuable to resolve indeterminations which are inevitable in rigid body dynamics. This is achieved by addition of equations stemming from (1) to the mathematical model of the system.

• Example 1: Frictionless Pivot. If the joint between body \(1\) and body \(2\) is a pivot of axis \((O_2,\boldsymbol{\hat{z}}_1)\), then the corresponding kinematic screw is given by \[ \{ {\cal V}_{2/1} \} = \begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{z}}_1 \\ {\bf 0} \end{Bmatrix}_{O_2} \] The pivot is frictionless if \(\omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0\) according to (1). Since this must be true for all values of \(\omega_{2/1}\), this imposes the condition \(\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0\). A frictionless pivot is thus characterized by 5 components (3 components for the resultant force, and 2 components for the resultant moment about \(O_2\)). Note that point \(O_2\) can be replaced by any point on the axis of the pivot.

• Example 2: Frictionless Helical Joint. If the joint between body \(1\) and body \(2\) is helical of axis \((O_2,\boldsymbol{\hat{z}}_1)\) and pitch \(p\), then the corresponding kinematic screw is given by \[ \{ {\cal V}_{2/1} \} = \begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{z}}_1 \\ \frac{p}{2\pi} \omega_{2/1}\boldsymbol{\hat{z}}_1 \end{Bmatrix}_{O_2} \] The helical joint is frictionless if \(\frac{p}{2\pi}\omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf F}_{1\to 2}^c + \omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0\) according to (1). Since this must be true for all values of \(\omega_{2/1}\), this imposes the condition \[ \frac{p}{2\pi} \boldsymbol{\hat{z}}_1 \cdot {\bf F}_{1\to 2}^c + \boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0 \]

• Many other cases of joints can be treated in a similar manner.