Analysis of pole acceleration in spatial motions by the generalization of pole changing velocity
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Abstract
It is well known in planar kinematics of rigid bodies that the acceleration of the material point coinciding with the instantaneous center of rotation (or pole) is perpendicular to the socalled pole changing velocity. In the present paper, the concept of pole changing velocity is generalized to spatial motions. Using this result, the acceleration of the material points along the instantaneous screw axis can be expressed in a straightforward way, without the tools of advanced differential geometry.
1 Introduction
Rigid body kinematics is a subject that belongs partly to geometry, partly to dynamics. Accordingly, there are approaches to this branch of science that are based on purely geometrical derivations [1, 2, 3]. Other authors—especially in books written for mechanical engineers [4, 5, 6]—put more emphasis on the timebased treatment of the important concepts, i.e., the main results are derived and formulated by using position, velocity and acceleration vectors. Geometrical kinematics has several advantages over the timebased approach in terms of mathematical elegance and rigor [7]. However, the learning (and teaching) of geometrical methods requires the application of advanced mathematical tools. Since university curricula rarely provide the necessary time to properly cover these topics, only specialists can apply geometrical kinematics to the solution of practical problems—this is why the use of timebased methods can still be justified.
The present paper focuses on the pole changing velocity that characterizes the change of the geometric position of the instantaneous center of rotation (or pole) in the case of planar motions and nonzero angular velocity. The goal of this contribution is to show that the concept of pole changing velocity can be generalized to spatial motions by using Euler’s rigid body formulas and exploiting that all motions can be interpreted as rolling or sliding of two ruled surfaces on each other. On the basis of this generalization, the acceleration of the material points along the instantaneous screw axis (ISA) can be expressed in a straightforward way, without the tools of advanced differential geometry. The obtained results may help to visualize the connection between the velocity distribution and the acceleration distribution during the spatial motion of a rigid body.
1.1 Background and literature survey
In the case of planar motion of rigid bodies, there exists a point on the (extended) body that is instantaneously at rest, provided that the angular velocity \(\varvec{\omega }\) of the body is nonzero. This point has several names in the literature. It is referred to as instant center of velocity [4, 5], instant center [8], instantaneous center of rotation [9], velocity pole [1] or simply, pole [10, 11]. For the sake of simplicity, we adopt the latter denomination. Note that certain authors [4] use the term “pole” for finite displacements, while the corresponding point for infinitesimal displacements is called “instantaneous center of velocity.”
The notion of pole was introduced by Johann Bernoulli [12] in the eighteenth century, for the characterization of planar motions. As the body moves, the geometric position of the pole changes continuously (Fig. 2). To describe this phenomenon, Poinsot [13] introduced the notions of moving and fixed polodes [1] or centrodes [9]. These curves (also referred to as body curve and space curve) describe the earlier and future geometric positions of the pole in the moving reference frame of the rigid body and in the fixed reference frame, respectively. During the motion, the moving polode rolls on the fixed polode without slip. In each instant, the actual contact point of these curves defines the pole.
In order to avoid any misunderstanding, it is worth to distinguish between two different interpretations of the pole. On the one hand, the pole can be thought of as a geometric object [2], defined by the requirement that the velocity of the material point of the body coinciding with it is zero. The term geometric pole with the notation \(P_g\) will be used throughout this paper when talking about the pole in this sense. On the other hand, the pole can be considered as a material point of the body in the examined time instant t, that has zero velocity. As follows, the pole in this “material” sense will be simply referred to as pole and denoted by P.
The difference between the geometric pole and the pole is illustrated in Fig. 1, where the moving polode is the outer contour of the ring, while the fixed polode is found on the surface of the ground. The letter A denotes a material point of the body that coincides with the geometric pole \(P_g\) (the contact point of polodes) at time t; thus, \(A \equiv P\) is the pole in the subfigure on the left. These two points are located in different places at a later time instant \(t'\), so point A is not a pole anymore in the subfigure on the right.
The geometric pole and the pole coincide in each time instant, but their velocity and acceleration are typically different. Although the velocity of the pole (as a material point) is zero, the geometric pole (that coincides with another material point in a subsequent time instant) apparently moves along the fixed polode. In the examples shown in Figs. 1 and 2, the pole is always below the center of gravity; thus, the velocity of this apparent motion—denoted by \(\mathbf {u}\)—is equal to the velocity of the center of gravity \(\mathbf {v}_C\).
There is no generally accepted name for the velocity characterizing the rate of change of the position of the geometric pole. We will use the term pole changing velocity, according to [10, 11]. This physical quantity is referred to as pole velocity in [1], IC velocity in [4], instant center’s velocity in [5] and pole transfer velocity in [14]. Other authors only paraphrase the velocity of the geometric pole without assigning a name to it, for example: “geometric velocity at which the contact changes along the centrodes” [2], “displacement velocity of the instantaneous center” [4], “evolution velocity of instant center of rotation,” “speed of change” or “velocity vector” of the instant center of rotation [15], “the velocity with which the instant center propagates along the outline of the body” [16], or “speed of progression of the rolling point along the centrode [17].”
It is well known in the planar case that the pole changing velocity \(\mathbf {u}\) is parallel to the common tangent of the fixed polode and the moving polode [1]. Moreover—since the path of the pole has a cusp at the contact point of the two polodes, as shown in Fig. 2—the acceleration \(\mathbf {a}_P\) of the pole (the material point) is just perpendicular to the pole changing velocity \(\mathbf {u}\) [2, 5, 9] and \(\mathbf {u} = \mathbf {a}_P/\omega \) [18].
It is shown in [16] that the diameter of the inflection circle can be expressed by the pole changing velocity and the magnitude of the angular velocity \(\varvec{\omega }\) of the rigid body: \(b_2 = \mathbf {u}/\omega \). Moreover, the magnitude of the acceleration of the pole can be expressed as \(\mathbf {a}_P = b_2 \omega ^2\) [9]. This latter result follows trivially from Euler’s acceleration formula if one utilizes that \(\mathbf {a}_P \perp \mathbf {a}_I\), where I denotes a point on the inflection circle, the socalled inflection pole (also shown in Fig. 3).
Most of the aforementioned results have been generalized to spatial or spherical motions. Mozzi [19] and Chasles [20] introduced the socalled screw axis and formulated the following theorem: Each Euclidean displacement in threedimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis. The spatial motion of a body can be considered as a continuous set of displacements. Applying Chasles’ theorem to infinitesimally small displacements, a welldefined screw axis—the instantaneous screw axis (ISA)—can be assigned to the rigid body at any time instant.
The Euler–Savary theorem had been generalized to the spatial case by Distelli [21, 22], and it was expressed by the pole changing velocity in the spherical case in [10, 11]. The concept of inflection circle was also generalized to spherical motions: The inflection cone (points with zero normal acceleration) and normal cone (points with zero tangential acceleration) are introduced in [2]. Based on these geometric results, the connections between the pole changing velocity and the acceleration of the points along the ISA can be established.
Despite its geometric nature, there are still several unsolved problems in kinematics. Some of the latest results are practice oriented [23], others are more inclined toward the pure theoretical extension of known concepts [24], or the goal is the determination of all possible motion types if the displacement of the body is not completely specified [25].
1.2 Formulation of the problem of interest and the scope of this study
As was mentioned in the previous section, the connections between the velocities and accelerations of spatially moving rigid bodies are established in principle. However, the goal of the books and papers cited in the previous section is the general geometric description of the motion properties. Consequently, advanced tools of differential geometry are used in them. Although this approach is elegant and powerful, the complexity of the mathematical tools may discourage potential readers from the application of the results. Moreover, as the literature review of the previous section shows, there are no generally accepted terms for the pole and the pole changing velocity. The diversity of the used terminology also hinders the orientation of engineers in this field.
Certain authors made successful attempts to derive many of the aforementioned results using timebased concepts—position, velocity and acceleration vectors—while keeping the mathematical rigor [9]. It is shown in Chapter 9.5 of the cited book that the acceleration of a chosen material point of the ISA can be decomposed into a component that is parallel with the ISA due to the translatory part of the raccording motion and another component that is related to the rolling about the ISA. It is also stated that the latter part is perpendicular to the common tangent plane of the moving and fixed axodes. However, the proof of this statement refers to the solution of a planar rolling problem, when the bodyfixed contact point passes through a cusp of its trajectory (cf. Fig. 8), implying that its acceleration is perpendicular to the tangent plane. This is certainly true, but the exact generalization to spatial motions is not given explicitly in [9].
The goal of the present paper is to find a straightforward, timebased derivation that establishes the relation between the instantaneous acceleration of points along the ISA and the apparent motion of the ISA on the fixed axode. The novelty of the proposed approach lies in the fact that the results are derived using Euler’s rigid body formulas (1), (2) and that the concept of pole changing velocity is generalized to spatial motions.
1.3 Organization of the paper
The paper is organized as follows: Sect. 2 deals with the definition and extension of the notion of pole to spatial motions. In Sect. 3, the formula of pole changing velocity is derived and the obtained result is interpreted. Further connections are established between the pole changing velocity \(\mathbf {u}\) and the pole acceleration \(\mathbf {a}_P\) in Sect. 4. It is shown in Sect. 5 as a corollary that the finitetime (continuous) motions of rigid bodies can be classified into three categories: planar motion, spherical motion and general raccording motion. Although this result (attributed to Painlevé in [9]) is well known in kinematics, the present paper provides a proof that is different from the conventional geometric approach. Section 6 illustrates the derived results via numerical examples, and the conclusions are drawn in Sect. 7.
2 Formulation of the geometric pole’s position
If \(\mathbf {v}_A \perp \varvec{\omega }\), the velocity of the material points along the ISA is zero; thus, the rigid body undergoes instantaneous rotation. In this case, the ISA is referred to as instantaneous axis of rotation (IAR). Equation (7) is valid in the planar case, too, when the points of the rigid body move in parallel planes that are perpendicular to the angular velocity \(\varvec{\omega }\). The same velocities and accelerations can be seen in these planes. Thus, the motion can be represented in a single, properly chosen plane, as shown in Fig. 3.
In principle, if the geometric pole is chosen according to (7), it moves in space during the motion of the body, forming a fixed polode (space curve). In the instant shown in Fig. 6, the reference point A just coincides with the geometric pole assigned to it, but at later instants—as the projection of A gets closer to the center point O (see points D and C and their projections)—the geometric pole also moves closer to O along the IAR. The points of the moving polode could be defined via the material points that coincide with the points of the fixed polode.
Note, however, that the polode curves defined this way depend on the choice of the reference point. If a point on the perimeter of the cone’s base (A, C or D) is chosen as a reference point, the fixed polode curves defined by (7) meander on the \(z = 0\) plane. The distance of the points of these polode curves from the center point O varies during the motion between \(b  \overline{P_{g1}P_{g2}}\) and \(b + \overline{P_{g1}P_{g2}}\), with different phases for the different reference points.
There is a more practical procedure in the case of spherical motions: In this case, it is possible to assign the geometric pole to a reference point in such a way that the distance of these points from the fixed center point O is the same. In Fig. 6, the points A, C and D are at the same distance from O, so the same geometric pole (just coinciding with A in the figure) can be assigned to all the points on the perimeter of the base of the cone. As a consequence, both the fixed and moving polodes will be circles in this special case.
Similarly, if point B is chosen to be the reference point, the distance of its projection (\(P_{g2}\)) from O does not change, so one obtains circular polodes, again.
We can conclude that there are several possibilities for the assignment of the geometric pole to the reference point; thus, the generalization of the polode curves to the spatial case is usually impractical. This is why the present paper focuses mainly on the instantaneous properties of the motion instead of the geometric objects corresponding to finitetime motion.
3 Generalization of pole changing velocity
3.1 Formal derivation of pole changing velocity
Let us assume that the angular velocity \(\varvec{\omega }\) of the rigid body, the angular acceleration \(\varvec{\alpha }\) of the rigid body and the velocity and acceleration of point A are known. The position of the geometric pole \(P_g\) can be given relative to point A by Eq. (7); consequently, \(\mathbf {r}_{P_g}(t) = \mathbf {r}_A(t) + \mathbf {r}_{AP_g}(t)\) (Figs. 5, 7).
Vector \(\mathbf {u}\) characterizes the apparent motion of the ISA; thus, it must lie in the common tangent plane of the moving and fixed axodes at \(P_g\). In the general case, the direction of the tangent plane varies along the ISA (Fig. 4).
The velocity and acceleration of point A can be expressed as \(\mathbf {v}_A = \mathbf {v}_P + \varvec{\omega }\times \mathbf {r}_{PA}\) and \(\mathbf {a}_A = \mathbf {a}_P + \varvec{\alpha }\times \mathbf {r}_{PA} +\varvec{\omega }\times (\varvec{\omega }\times \mathbf {r}_{PA})\), respectively. Note that \(\mathbf {r}_{PA}\) denotes the position vector of the material pole, so \(\mathbf {r}_{PA}(t)\) is constant and Euler’s formulas are valid in this case. Vector \(\mathbf {a}_P\) denotes the acceleration of the material point P that coincides the geometric pole \(P_g\), i.e., this is the pole acceleration.
3.2 Interpretation of the obtained result
Equation (10) shows that the pole changing velocity generally depends on the chosen reference point A, in accordance with Sect. 2 and Fig. 6.
In the case of spatial motions, the term \(\varvec{\omega }(\varvec{\alpha }\cdot \mathbf {r}_{PA})/\omega ^2\) is nonzero only if the angular acceleration \(\varvec{\alpha }\) has a component parallel to \(\mathbf {r}_{PA}\) and—consequently—perpendicular to \(\varvec{\omega }\). Since this term originates from Eq. (9), it means that the velocity of the reference point (\(\mathbf {v}_A = \varvec{\omega }\times \mathbf {r}_{PA}\)) must have a component perpendicular to \(\varvec{\alpha }\) in this case.
For the further analysis, recall that even in the most general case of raccording motion, the geometric positions of the ISA define the moving and fixed axodes. These are ruled surfaces with a common tangent plane [9]. As a consequence, the angular velocity \(\varvec{\omega }\) and the pole changing velocity \(\mathbf {u}\) are always parallel to this tangent plane.
In the case of spherical motion, one of the points of the ISA has zero velocity and acceleration. Thus, the apparent motion of the ISA can be characterized by the change of its direction. Since the angular velocity \(\varvec{\omega }\) is parallel to the ISA, the change of direction of \(\varvec{\omega }\)—described by the component of \(\varvec{\alpha }\) that is perpendicular to \(\varvec{\omega }\)—must take place parallel to the tangent plane. Thus, \(\varvec{\omega }(\varvec{\alpha }\cdot \mathbf {r}_{PA})/{\omega ^2}\) is nonzero only if \(\mathbf {r}_{PA}\) has a component parallel with the tangent plane. In this case, the velocity of the reference point (\(\mathbf {v}_A = \varvec{\omega }\times \mathbf {r}_{PA}\)) has a component perpendicular to the tangent plane.
To visualize this result, see Fig. 6, where the \(z=0\) plane is the tangent plane. If one chooses a reference point on the perimeter of the base of the cone (e.g., A, D or C), the radial position (distance from O) of the corresponding geometric pole will vary during the rolling of the cone. Clearly, the extremal positions of the geometric pole correspond to the configurations when the reference point is just on the tangent plane (point A) or at the upper position (point C). The pole changing velocities of the corresponding poles (\(P_{g1}\) and the point coinciding A) have no component parallel with \(\varvec{\omega }\), in accordance with \(\mathbf {r}_{AA} = \mathbf {0}\) and that \(\mathbf {r}_{P_{g1}A}\) is perpendicular to the contact plane. However, if point D is the reference point, the corresponding geometric pole (\(P_{g2}\)) is transferred closer to the center point O during the motion, i.e., its pole changing velocity has a component parallel to \(\varvec{\omega }\).
4 Pole acceleration
The importance of the notion of pole changing velocity lies in the fact that it is related to the acceleration of the pole \(\mathbf {a}_P\). However, Eq. (12) does not provide any information about the mutual direction of the angular velocity \(\varvec{\omega }\) and the acceleration \(\mathbf {a}_P\).
In the case of planar motion, all the accelerations are perpendicular to the angular velocity, i.e., \(\mathbf {a}_P = \mathbf {u}\times \varvec{\omega }\) is fulfilled.
5 Finitetime rotational motions

If \(\varvec{\alpha }\parallel \varvec{\omega }\), the acceleration of points and the pole changing velocity are constant along the IAR. This motion corresponds to the rolling of a cylinder, i.e., to planar motion.
 If \(\mathbf {a}_{P1} \parallel \varvec{\alpha }\times \varvec{\omega }\), there must be a point \(P_2\) along the IAR with zero acceleration. This case corresponds to the rolling of a cone, i.e., to spherical motion (Fig. 8). Recall that if the acceleration is zero at a point of the IAR, the corresponding pole changing velocity [cf. (13)] is also zero. Thus, we turn to the analysis of the pole changing velocity, again. By expressing the pole changing velocities corresponding to the points \(P_1\) and \(P_2\) in (19) by (12), one obtains^{1}If the same tangent plane is spanned by \(\mathbf {u}\) and \(\varvec{\omega }\) at all points of the IAR, the direction of \(\mathbf {u}\) is also the same along the IAR. According to (20), the magnitude of pole changing velocity varies linearly along the IAR. Thus, the geometric point with zero pole changing velocity (coinciding the material point with zero acceleration) can be found in a straightforward way, as is shown in Fig. 8. However, there are more general cases, too, when different tangent planes can be found for different points along the IAR, as illustrated in Fig. 4. For the further analysis, we expand the vector triple product in (20):$$\begin{aligned} \mathbf {u}_2 = \mathbf {u}_1 + \frac{\varvec{\omega }\times (\varvec{\alpha }\times \mathbf {r}_{P1P2})}{\omega ^2}. \end{aligned}$$(20)We search for the point \(P_2\) to which zero pole changing velocity \(\mathbf {u}_2\) is assigned. Multiplying the previous formula by \(\varvec{\omega }\),$$\begin{aligned} \mathbf {u}_2 = \mathbf {u}_1 + \varvec{\alpha }\frac{\varvec{\omega }\cdot \mathbf {r}_{P1P2}}{\omega ^2}  \mathbf {r}_{P1P2} \frac{\varvec{\alpha }\cdot \varvec{\omega }}{\omega ^2}. \end{aligned}$$(21)The last term is zero since \(\varvec{\omega }\parallel \mathbf {r}_{P1P2}\). According to the condition \(\mathbf {u}_2 = \mathbf {0}\), the lefthand side of the equation is also zero. Consequently,$$\begin{aligned} \varvec{\omega }\times \mathbf {u}_2 = \varvec{\omega }\times \mathbf {u}_1 + (\varvec{\omega }\times \varvec{\alpha }) \frac{\varvec{\omega }\cdot \mathbf {r}_{P1P2}}{\omega ^2}  (\varvec{\omega }\times \mathbf {r}_{P1P2}) \frac{\varvec{\alpha }\cdot \varvec{\omega }}{\omega ^2}. \end{aligned}$$(22)It means that to find the vector \(\mathbf {r}_{P1P2}\), the condition \(\varvec{\omega }\times \mathbf {u}_1 \parallel \varvec{\omega }\times \varvec{\alpha }\) must be fulfilled. Since \(\mathbf {u}\) and \(\varvec{\omega }\) span the tangent plane of the axodes, it means that the angular acceleration \(\varvec{\alpha }\) must be also in the tangent plane at a point along the IAR during spherical motions.$$\begin{aligned} \varvec{\omega }\times \mathbf {u}_1 = (\varvec{\omega }\times \varvec{\alpha }) \frac{\varvec{\omega }\cdot \mathbf {r}_{P1P2}}{\omega ^2}. \end{aligned}$$(23)

The aforementioned condition is not always fulfilled. In the case of the more general raccording motion (Fig. 4), the direction of the tangent plane’s normal vector varies along the IAR. Consequently, the direction of the pole changing velocity and the acceleration of material points also varies along the IAR. If a pole point has an acceleration component that is perpendicular to \(\varvec{\alpha }\times \varvec{\omega }\), there is no point along the ISA that has zero acceleration. There exists a point P that has no acceleration component parallel with \(\varvec{\alpha }\times \varvec{\omega }\). This point is referred to as striction point [9].
6 Numerical examples
6.1 Rolling of a cone
Let the radius of the base, the height and the slant height of the cone be \(r = 0.14\,\mathrm{m}\), \(h = 0.48\,\mathrm{m}\) and \(R = 0.5\,\mathrm{m}\), respectively. The apex of the cone is fixed in the origin O. The cone rolls on the xy plane without slipping. The x components of the velocity and acceleration of point B are \(v_{Bx} = 1.92\,\mathrm{m/s}\) and \(a_{Bx} = 1\,\mathrm{m/s}^2\).
We can conclude that the accelerations of the points along the instantaneous axis of rotation could be determined somewhat easier using the pole velocities than by Euler’s formula. Moreover, some information was needed about the acceleration of a contact point (e.g., \(\mathbf {a}_A\)) or the angular acceleration \(\varvec{\alpha }\) for the solution of the problem by Euler’s formula—this was the motivation of the present study.
A further advantage of the proposed solution is that while the apparent motion of the geometric pole can be described vividly, creating a mental picture about the acceleration vectors is more difficult.
6.2 Rotation with slipping
6.3 Transition between the previous cases
To determine the tangent plane, one can exploit that the pole changing velocity and the angular velocity span this plane. For the calculation of \(\mathbf {u}\), Eq. (13) can be utilized.
7 Conclusions
Since the direction of the pole changing velocity is parallel with the common tangent of the moving and fixed axodes, its direction and magnitude can be often determined easily. Thus, the obtained results can be utilized for the quick derivation of the acceleration of a chosen material point on the ISA or for the check of calculations based on other methods.
In principle, the results of the paper can be derived as special cases of more general geometrical results. Still, the author did not find these statements written explicitly in the literature. Thus, the goal of the presented calculations is to help to comprehend and visualize the spatial motions of rigid bodies.
Footnotes
Notes
Acknowledgements
Open access funding provided by Budapest University of Technology and Economics (BME). This research was supported by the Hungarian National Science Foundation Under Grant No. OTKA K 83890. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/20072013)/ERC Grant Agreement No. 340889. The author acknowledges the help of Dénes Takács by the drawing of the illustrations.
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