Description of motion of a rigid flat body can be reduced to describing the motion of a disc in the xy plane. This disc is a cross-section of the body by the so-called steering plane. The planar motion of the disc is a combination of translational motion in the xy plane with respect to a chosen point A and rotational motion about the z' axis. This axis is perpendicular to the xy plane and passes through point A. The rotational component is determined by the angular velocity ω and angular acceleration ε. The translational motion is described using a chosen point A of the rigid body. We define displacement r_A(t), velocity V_A(t), and acceleration a_A(t).

Method of determining velocities of points on a rigid body (theorem of projections)

In a rigid body, the mutual distances between points do not change during motion, so there are certain relationships between the velocities of these points.

During any motion, the projections of the velocity vectors of any two points on the line connecting these points are equal.

The figure shows the velocity vectors of two points A and B on a rigid body. The variable positions of these points in space are described by the radius vectors r_A and r_B. The vector r, whose starting point is at point A and end point is at point B, is constant in magnitude. The theorem of projections states that V_B·cosβ = V_A·cosα, where α and β are the projections of the velocities of points A and B onto the line connecting points A and B of the considered rigid body. This relationship allows us to determine velocities in some problems concerning the kinematics of a rigid body.

Fig. 1 Relationship between velocities of points on a rigid body.

Rigid body in rotational motion

A rigid body is in rotational motion if it rotates about a fixed axis of rotation u. The paths of motion of all points on the body are circles in planes perpendicular to the axis of rotation. The exception is points lying on the axis of rotation, which remain at rest.

In the special case of rotation about the vertical axis z, the circles lie in horizontal planes, as shown in Fig. 2. The angle of rotation φ(t) can be interpreted as a vector along the z axis. Then:

\begin{aligned} \overline \omega(t)=\frac{d\varphi(t)}{dt} \overline \varepsilon(t)=\frac{d\omega(t)}{dt}=\frac{d^2\varphi(t)}{dt^2} \end{aligned}Fig. 2 Rotational motion about the vertical axis z.

We choose the origin of the xyz coordinate system at any point on the z axis. Then the following relationships hold:

\begin{aligned} \overline v(t)=\overline \omega(t)\times \overline r(t), &v(t)=\omega r' \overline a_t(t)=\overline \varepsilon(t)\times \overline r(t), &a_t(t)=\varepsilon r' \overline a_n(t)=\overline \omega(t)\times \overline V(t), &a_n(t)=\omega^2 r' \\ \text{where: } r'=\overline{O'M}=rsin\alpha \end{aligned}Method of superposition for determining instantaneous velocity

We consider the disc in an instantaneous configuration (at t>0). We are given the vectors V_A, ω, r, where r is the radius from point A to point B (figure below). The combination of translational and rotational motion of the disc leads to the following formulas:

\begin{aligned} \overline V_B=\overline V_A+\overline V_{BA} \overline V_{BA}=\overline \omega \times \overline r \text{where: } V_{BA}=\omega r \end{aligned>Note that the vector ω is perpendicular to vector r, so V_{BA}=\omega r sin90^o=\omega r.

Fig. 3

In the instantaneous configuration, we analytically sum the vectors V_A and V_{BA}, i.e., the components of vector V_B are:

\begin{aligned} V_{Bx}=V_{Ax}+V_{BAx} V_{By}=V_{Ay}+V_{BAy} \end{aligned>or plotted to scale. The modulus of vector V_B can be directly calculated as

\begin{aligned} V_B=\sqrt{V_A^2+(\omega r)^2+2V_A\cdot \omega r cos\alpha} \end{aligned>Instantaneous axis of rotation

In a given instant t>0, the disc in planar motion corresponds to a point C called the instantaneous axis of rotation, whose velocity V_C=0. The position of point C changes over time. The velocities of points on the disc are (Fig. 4):

\begin{aligned} \overline V_A=\overline \omega \times \overline r_A, V_A=\omega r_A \overline V_B=\overline \omega \times \overline r_B, V_B=\omega r_B \end{aligned>Fig. 4

If we are given vectors V_A and ω, finding the instantaneous axis of rotation C involves finding the distance between points C and A using the formula r_A=V_A/ω and drawing the radius r_A perpendicular to V_A (Fig. 5).

Fig. 5

If we are given velocity \(\overline V_A\) and direction \( \overline V_B\), the instantaneous axis of rotation C lies at the intersection of the directions \(r_A\) and \(r_B\). The value of angular velocity \(\omega\) (Fig. 6) is determined by the formula \(\omega = \frac{V_A}{r_A}\)

Fig. 6

If we are given velocities \( \overline V_A || \overline V_B\) (Fig. 7), the instantaneous axis of rotation C is found using Thales' theorem

\begin{aligned} \frac{V_A}{r_A}=\frac{V_B}{r_B} \Rightarrow V_A\cdot r_B=V_B\cdot r_A\\ \end{aligned>Where the relationship \(r_B-r_A=d\) (Fig. 7a) or \(r_A+r_B=d\) (Fig. 7b) is given.

Fig. 7

Method of superposition for determining instantaneous acceleration

Weconsider the disc in an instantaneous configuration (at t>0). We are given the vectors \( \overline a_A, \overline \omega, \overline \varepsilon, \overline r\), where \( \overline r\) is the radius from point A to point B (Fig. 8). The combination of translational and rotational motion of the disc leads to the following formulas:

\begin{aligned} \overline a_B=\overline a_A+\overline a_{BA}=\overline a_A+\overline a_{BA}^t+\overline a_{BA}^n \overline a_{BA}^t=\overline \varepsilon \times \overline r, a_{BA}^t=\varepsilon r \overline a_{BA}^n=\overline \omega \times \overline V_{BA}, \overline V_{BA}=\overline \omega \times \overline r, a_{BA}^n=\omega ^2 r \end{aligned>Fig. 8

In the instantaneous configuration, we analytically or graphically sum the vectors \( \overline a_A, \overline a_{BA}^t, \overline a_{BA}^n\). The modulus of vector \( \overline a_B\) can be directly calculated as (Fig. 8):

\begin{aligned} a_B=\sqrt{a_A^2+a_{BA}^2+2a_A\cdot a_{BA}\cdot cos\alpha} a_{BA}=r\sqrt{\varepsilon ^2 + \omega ^4} \end{aligned>