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Erschienen in:

2019 | OriginalPaper | Buchkapitel

8. Planar Kinematics of Rigid Bodies

verfasst von : Oliver M. O’Reilly

Erschienen in: Engineering Dynamics

Verlag: Springer International Publishing

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Abstract

Background material on the planar kinematics of rigid bodies is presented in this chapter. In particular, we show how to establish certain useful representations for the velocity and acceleration vectors of any material point of a rigid body. We also discuss the angular velocity vector of a rigid body. These concepts are illustrated using three important applications: mechanisms, rolling rigid bodies, and sliding rigid bodies. Finally, we discuss the linear and angular momenta of rigid bodies and the inertias that are used to relate the angular momentum of a rigid body relative to its center of mass and the angular velocity vector of the rigid body.

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Vorheriges Kapitel Dynamics of Systems of Particles

Nächstes Kapitel Kinetics of a Rigid Body

The details presented here are far more advanced than those in most undergraduate texts. This is partially because the presentation is influenced by the recent renaissance in continuum mechanics. We mention in particular the influential works by Beatty [6] and Casey [19–22], who used the fruits of this era to present enlightening treatments of rigid body mechanics. This chapter is based on the aforementioned works and Gurtin [46, Chapter 4].

The proof of this result is beyond the scope of this course. One proof may be found in Gurtin [46, Pages 49–50]. A good discussion on the relationship between this result with Euler’s theorem on the motion of a rigid body and Chasles’ theorem can be found in Beatty [6] (see also Beatty [5]). Euler’s representation of rigid body motion can be seen in Euler [33, Pages 30–32].

Details on these parametrizations can be found, for example, in Beatty [6], Greenwood [45], O’Reilly [70], Shuster [88], Synge and Griffith [95], and Whittaker [100].

Recall that the transpose of a product of two matrices $\mathsf {A}$ and $\mathsf {B}$ is $(\mathsf {AB})^T = \mathsf {B}^T\mathsf {A}^T$. Furthermore, a matrix $\mathsf {C}$ is symmetric if $\mathsf {C} = \mathsf {C}^T$ and is skew-symmetric if $\mathsf {C} = - \mathsf {C}^T$.

This basis is often referred to as a body-fixed frame or an embedded frame.

The proof of this result is beyond our scope here. A proof may be found in Casey [19]. For the special case of a fixed-axis rotation, we give an explicit demonstration of this result in Section 8.2.2.

For example,

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The study of mechanisms is an important area of mechanical engineering. Our discussion here touches but a small part of it. The interested reader is referred to the textbooks of Bottema and Roth [17] , Mabie and Ocvirk [62], and Paul [74] for further treatments and issues.

The identity that we use is $\sin (\alpha - \beta ) = \sin (\alpha )\cos (\beta ) - \sin (\beta )\cos (\alpha )$. We also recall the expression for the inverse of a matrix:

$$\begin{aligned} \left[ \begin{array}{cc} a &{} b \\ c &{} d \end{array}\right] ^{-1} = \frac{1}{ad - cb}\left[ \begin{array}{cc} d &{} - b \\ - c &{} a \end{array}\right] . \end{aligned}$$

If a body is homogeneous, then $\rho _0$ is a constant that is independent of $\mathbf{X}$. Our use of the symbol $\rho $ here should not be confused with our use of the same symbol for the radius of curvature of a space curve in Chapter 3.

The proof is beyond the scope of an undergraduate engineering dynamics course. For completeness, however, one proof is presented in Section 8.10.

Some may notice that we take the time derivative to the outside of an integral whose region of integration $\mathscr {R}$ depends on time. This is generally not possible. However, for the integral of interest it is shown in Section 8.10 that such a manipulation is justified.

The standard modern reference to this area was written by two Soviet mechanicians: Neimark and Fufaev [66]. One of the prime contributors to this area was Routh [80, 81]. Indeed, the problem of determining the motion of a sphere rolling on a surface of revolution is known as Routh’s problem. We also mention the interesting classic work on billiards (pool) by Coriolis [23] from 1835.

For further references to, and discussions of, rolling disks and sliding disks see Borisov and Mamaev [16], Cushman et al. [26], Hermans [48], and O’Reilly [69]. References [16, 69] contain discussion of the important works on these systems by Chaplygin in 1897, Appell and Korteweg in 1900, and Vierkandt in 1892.

Earlier, in Section 8.1.3, we expressed

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in terms of the fixed basis:

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-11745-0_8/69838_3_En_8_IEq257_HTML.gif

. Here, it is more convenient to express

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-11745-0_8/69838_3_En_8_IEq258_HTML.gif

in terms of the corotational basis.

These results are discussed in Casey [19, 22] and in Gurtin [46, Section 13, Chapter 4]. For further details on the transformation of the integrals, see Section 8.10 below.

These results can be easily found in texts on linear algebra; see, e.g., Bellman [9] or Strang [92]. One also uses these results for the (Cauchy) stress tensor when constructing Mohr’s circle in solid mechanics courses.

These results follow from the local forms of mass conservation, changes of variables theorem, and Reynolds’ transport theorem in continuum mechanics because a rigid body’s motion is isochoric (see, e.g., Casey [19], Gurtin [46], and Truesdell and Toupin [98] ).

For example, take a flexible ruler. Initially, suppose that it is straight. One can approximately locate its center of mass; suppose that it is at the geometric center. Now, bend the ruler into a circle. The center of mass no longer coincides with the same material point of the ruler.

Titel: Planar Kinematics of Rigid Bodies
verfasst von: Oliver M. O’Reilly
Verlag: Springer International Publishing
Buch: Engineering Dynamics
Print ISBN: 978-3-030-11744-3

Electronic ISBN: 978-3-030-11745-0

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-11745-0_8

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