Dynamics Reference

    Contact and Rolling

    Rolling motion

    A special case of rigid body motion is rolling without slipping on a stationary ground surface. This is defined by motion where the point of contact with the ground has zero velocity, so it matches the ground velocity and is not slipping.

    Rolling without slipping on stationary ground surfaces: #rko-er
    $$ \text{Contact point P has zero velocity}: \quad \vec{v}_P = 0 $$

    Rolling without slipping means by definition that the contacting points have the same velocity. For a stationary ground the velocity is zero, so the contact point on the body must also have zero velocity.

    It is helpful to think about the motion of the body in two ways:

    1. The body rotates about the moving center \( C \).
    2. The body rotates about the instantaneous center at the contact point \( P \).

    These two ways of visualizing the motion can be seen on the figure below.

    Movement:

    flat

    big flat

    rock

    inside 2-circle

    inside 3-circle

    inside rock

    outside 1-circle

    outside 2-circle

    outside rock

    Center \(C\)

    Body

    \(P\) velocity

    none

    \( \vec{v}_C \) (trans.)

    \( \vec{\omega} \times \vec{r}_{CP} \) (rot.)

    \( \vec{v}_P \) (total)

    \(P\) acceleration

    none

    \( \vec{a}_C \) (trans.)

    \( \vec{\alpha} \times \vec{r}_{CP} \) (ang.)

    \( \vec{\omega} \times (\vec{\omega} \times \vec{r}_{CP}) \) (cent.)

    \( \vec{a}_P \) (total)

    Velocity and acceleration of points on a rigid body undergoing different rolling motions.

    Rolling on a flat surface

    The most common and also the simplest form of rolling occurs on a flat surface.

    Geometry and variables for rolling without slipping on a flat surface.

    While rolling, the velocity and acceleration are directly connected to the angular velocity and angular acceleration, as shown by the next equations.

    Center velocity and acceleration while rolling on a flat surface (Tangential-Normal Basis). #rko-ef
    $$ \begin{aligned} \vec{v}_C &= r \omega \,\hat{e}_t \\ \vec{a}_C &= r \alpha \,\hat{e}_t \end{aligned} $$

    We begin by observing that the sign conventions in Figure #rko-ff mean that \( \vec\omega = -\omega\,\hat{e}_b \). Now rolling without slipping means the contact point \(A\) must instantaneously have zero velocity, so using #rkg-er gives:

    $$ \begin{aligned} \vec{v}_C &= \vec{v}_A + \vec{\omega} \times \vec{r}_{AC} \\ &= (-\omega \,\hat{e}_b) \times r \,\hat{e}_n \\ &= r \omega \,\hat{e}_t. \end{aligned} $$
    Because the surface is flat, the tangential basis vector \( \hat{e}_t \) is constant, and the radius \(r\) is also constant. Differentiating the velocity expression thus results in:
    $$ \begin{aligned} \vec{v}_C &= r \omega \,\hat{e}_t \\ \dot{\vec{v}}_C &= r \dot\omega \,\hat{e}_t \\ \vec{a}_C &= r \alpha \,\hat{e}_t. \end{aligned} $$

    Center velocity and acceleration while rolling on a flat surface (Standard Basis). #rko-ef2
    $$ \begin{aligned} \vec{v}_C &= -r \omega_z \,\hat\imath \\ \vec{a}_C &= -r \alpha_z \,\hat\imath \end{aligned} $$

    Since the body is rolling without slipping, the contact point \(A\)'s velocity is instantaneously zero. Using rigid body equations:

    $$ \begin{aligned} \vec{v}_C &= \vec{v}_A + \vec{\omega} \times \vec{r}_{AC} \\ &= \omega_z r_{AC}^\perp \\ &= -r \omega_z \,\hat\imath. \end{aligned} $$
    Differentiating to obtain acceleration results in:
    $$ \begin{aligned} \vec{v}_C &= -r \omega_z \,\hat\imath \\ \dot{\vec{v}}_C &= -r \dot\omega_z \,\hat\imath \\ \vec{a}_C &= -r \alpha_z \,\hat\imath. \end{aligned} $$

    Another way to express the connection between angular and linear velocity and acceleration is via the distance \(s\) traveled by the rolling body:

    Distance-angular relationships for rolling on a flat surface. #rko-eh
    $$ \begin{aligned} s &= r \theta \\ \dot{s} &= r \omega \\ \ddot{s} &= r \alpha \end{aligned} $$

    The contact point \(P\) and center \(C\) are offset by the constant vector \( r\,\hat{e}_n \) so \( \dot{s} = v_P = v_C = r\omega \), from #rko-ef. Because \(r\) is constant, differentiating the velocity expression gives \( \ddot{s} = r \dot\omega = r \alpha \), while integrating with zero initial displacement gives \( s = r \theta \).

    While rolling, the contact point will have zero velocity but will have a centripetal acceleration towards the rolling center:

    Contact point \(P\) velocity and acceleration while rolling on a flat surface. #rko-eo
    $$ \begin{aligned} \vec{v}_P &= 0 \\ \vec{a}_P &= \omega^2 \,\vec{r}_{PC} \end{aligned} $$

    By definition of non-slip rolling contact, the point of contact \(P\) has zero velocity. The acceleration can be computed from the center \(C\) with #rkg-e2:

    $$ \begin{aligned} \vec{a}_P &= \vec{a}_C + \vec{\alpha} \times \vec{r}_{CP} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{CP}) \\ &= \alpha r \,\hat{e}_t + (-\alpha\,\hat{e}_b) \times (-r\,\hat{e}_n) + (-\omega\,\hat{e}_b) \times \Big((-\omega\,\hat{e}_b) \times (-r\,\hat{e}_n)\Big) \\ &= \alpha r \,\hat{e}_t - \alpha r \,\hat{e}_t + \omega^2 r \,\hat{e}_n \\ &= \omega^2 \,\vec{r}_{PC}. \end{aligned} $$

    Rolling on curved surfaces

    When a circular rigid body rolls without slipping on a surface which is itself curved, the radius of curvature of the surface affects the acceleration (but not velocity) of points on the rolling body.

    Geometric quantities for rolling on a curved surface. #rko-eg
    $$ \begin{aligned} \left.\begin{aligned} R &= \rho - r \\ \vec{\omega} &= -\omega \,\hat{e}_b \\ \vec{\alpha} &= -\alpha \,\hat{e}_b \end{aligned}\right\} & \text{ when rolling on the inside of a curved surface} \\[1em] \left.\begin{aligned} R &= \rho + r \\ \vec{\omega} &= \omega \,\hat{e}_b \\ \vec{\alpha} &= \alpha \,\hat{e}_b \end{aligned}\right\} & \text{ when rolling on the outside of a curved surface} \end{aligned} $$
    These formulas are all choices of sign conventions for \( \omega \) and \( \alpha \), and definitions of \( R \). Figure #rko-fc shows the appropriate geometry. Note that \( \omega \) and \( \alpha \) are defined with positive values corresponding to motion in the tangential direction.

    Warning: Radii of curvature \( \rho \) and \( R \) may not be constant.

    The radius of curvature \( \rho \) of the surface may be varying with position as the body rolls. If \( \rho \) changes then this will also cause \( R \) to change. These two variables will only be constant if the surface is in fact perfectly circular.

    Applications

    Bearings

    This topic is in L27, slides 13-14. Pictures are from this book https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/gpjosq/alma99955068260305899

    Application for "Rolling on curved surfaces".