Dynamics: Kinematics of Rigid Bodies

Rigid body kinematics is the branch of dynamics that describes motion without asking what causes it. The key assumption is that the body is rigid, meaning distances between any two points on the body remain constant during motion. This idealization works well for many engineering systems, from gears and wheels to robot arms and linkages, where deformation is small compared with the overall motion.

In practice, rigid body kinematics focuses on three closely related ideas: how to describe rotation, how to relate the motion of different points on the same body (absolute and relative motion), and how to simplify planar motion using instantaneous centers. These tools let you analyze rolling, sliding, and general plane motion with clarity and speed.

What makes rigid body motion different from particle motion?

A particle has position but no size, so its motion is fully described by a single trajectory. A rigid body has size and orientation. Even if one point on the body follows a simple path, another point may follow a very different one. Kinematics must track both translation and rotation.

A rigid body’s motion can be decomposed into:

• Translation of a reference point (often the center of mass or a convenient joint)
• Rotation of the body about some axis (in planar motion, the axis is perpendicular to the plane)

This decomposition is conceptual, not necessarily a physical separation. It becomes powerful when tied to vector relationships between points.

Rotation: angular position, velocity, and acceleration

For planar motion, rotation is described by an angular position $\theta (t)$ about an axis normal to the plane. The fundamental rotational kinematic quantities are:

• Angular velocity: $\omega =\dot{\theta }$
• Angular acceleration: $\alpha =\dot{\omega }=\ddot{\theta }$

For a point at distance $r$ from the rotation axis in pure rotation about a fixed center, the speed and tangential acceleration follow familiar forms:

• Tangential velocity magnitude: $v_t=\omega r$
• Tangential acceleration magnitude: $a_t=\alpha r$
• Normal (centripetal) acceleration magnitude: $a_n={\omega }^{2}r$

The normal acceleration always points toward the center of rotation, while tangential acceleration points in the direction of increasing $\theta$ (or opposite, if $\alpha$ is negative). Keeping these directions straight is essential when building correct vector diagrams.

Rotational motion with constant angular acceleration

Many problems use constant $\alpha$, yielding the rotational analogs of constant-acceleration particle kinematics:

• $\omega ={\omega }_0+\alpha t$
• $\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$
• ${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$

These relations apply to a rigid body rotating about a fixed axis, such as a disk spinning on a shaft.

Absolute and relative motion of points on a rigid body

The most widely used kinematic relationships for rigid bodies connect the motion of two points, $A$ and $B$, on the same body. In planar motion, the rigid body constraint means the vector ${\mathbf{r}}_{B/A}$ is constant in the body, though it rotates in space.

Velocity relationship (relative motion)

For planar rigid body motion, $${\mathbf{v}}_B={\mathbf{v}}_A+\mathbf{\omega }\times {\mathbf{r}}_{B/A}$$ Here, $\mathbf{\omega }$ is the angular velocity vector (perpendicular to the plane), and ${\mathbf{r}}_{B/A}$ is the position vector from $A$ to $B$.

Interpretation matters:

• ${\mathbf{v}}_A$ is the translational contribution due to the chosen reference point’s motion.
• $\mathbf{\omega }\times {\mathbf{r}}_{B/A}$ is the rotational contribution; it is always perpendicular to ${\mathbf{r}}_{B/A}$.

This equation is the backbone of mechanisms analysis. If you know the velocity of one point (for example, a pin sliding in a slot) and the angular velocity, you can compute the velocity of any other point on the body.

Acceleration relationship

Acceleration adds one more layer because changing rotation creates tangential effects: $${\mathbf{a}}_B={\mathbf{a}}_A+\mathbf{\alpha }\times {\mathbf{r}}_{B/A}+\mathbf{\omega }\times (\mathbf{\omega }\times {\mathbf{r}}_{B/A})$$

The terms have distinct physical meanings:

• $\mathbf{\alpha }\times {\mathbf{r}}_{B/A}$ is tangential acceleration due to angular acceleration.
• $\mathbf{\omega }\times (\mathbf{\omega }\times {\mathbf{r}}_{B/A})$ is normal acceleration due to angular velocity; it points toward the instantaneous center of rotation for pure rotation.

A common mistake is to treat the acceleration of a point as purely tangential when $\alpha$ is known. Even with constant $\omega$, the normal term remains and often dominates.

General plane motion: translation plus rotation

Most rigid body motion in machines is general plane motion: the body both translates and rotates in a plane. A connecting rod in a slider-crank, a ladder slipping while rotating, or a coin that both slides and spins are all examples.

General plane motion can be analyzed using either:

• Relative motion equations (choosing a convenient point $A$ and building vector relations), or
• Instantaneous centers (turning the velocity problem into a momentary pure rotation problem)

Acceleration analysis typically leans on relative motion, because instantaneous centers are primarily a velocity tool.

Rolling, sliding, and rolling without slipping

A wheel in contact with a surface can exhibit:

• Pure rolling (rolling without slipping)
• Rolling with slipping (a combination of rolling and sliding)
• Pure sliding (rotation absent or irrelevant)

Rolling without slipping condition

For a wheel of radius $R$ rolling on a stationary surface without slipping, the contact point has zero velocity relative to the ground at the instant of contact. This yields the classic relationship between the center’s speed and angular speed: $$v_C=\omega R$$ where $v_C$ is the speed of the wheel’s center.

This condition is kinematic, not a guarantee. Whether a wheel actually rolls without slipping depends on friction and applied forces, which belongs to dynamics. Kinematics uses the no-slip constraint when it is stated or can be reasonably assumed.

Rolling with slipping

If slipping occurs, $v_C\ne \omega R$. The contact point has a nonzero relative velocity along the surface, and you must treat translation and rotation as independent unless additional constraints are provided. In practical terms, slipping invalidates many shortcuts and requires careful use of the general velocity relationships.

A useful velocity picture for rolling

For a rolling wheel, velocities along the rim can be understood by superposition:

• Translation gives every point the same velocity ${\mathbf{v}}_C$.
• Rotation adds a tangential component $\mathbf{\omega }\times \mathbf{r}$.

In pure rolling, the top point’s speed relative to the ground becomes $2v_C$, while the bottom contact point is instantaneously at rest. This is a kinematic result and often explains why rolling objects can “walk” faster at their top surface than their center moves.

Instantaneous centers of zero velocity

The instantaneous center of rotation (IC) is a powerful concept for planar rigid body kinematics. At any instant, the motion of a rigid body in a plane can be represented as a pure rotation about a point (possibly not on the body) where the velocity is zero.

How instantaneous centers simplify velocity analysis

If you locate the IC, then for any point $P$ on the body:

• The velocity of $P$ is perpendicular to the line from the IC to $P$.
• The magnitude is $v_P=\omega r_{P/IC}$.

This turns a general plane motion velocity problem into a geometry problem.

Finding the instantaneous center

A practical method:

1. Identify two points on the body with known velocity directions (not necessarily known magnitudes).
2. Through each point, draw a line perpendicular to its velocity direction.
3. The intersection of these perpendiculars is the IC (if it exists uniquely at that instant).

For a wheel rolling without slipping, the IC lies at the contact point with the ground. For a sliding ladder, the IC lies at the intersection of the perpendiculars to the velocities of two chosen points, often the endpoints.

Limitations to remember

Instantaneous centers are mainly for velocities. Using ICs to infer accelerations is generally unreliable because the IC location changes with time, and acceleration depends on time derivatives of velocity, not just the instantaneous velocity field.

Practical workflow for rigid body kinematics problems

1. Classify the motion: pure rotation, pure translation, or general plane motion.
2. Choose points strategically: pick points with known constraints (pins, sliders, contact points).
3. Use velocity relations first: apply ${\mathbf{v}}_B={\mathbf{v}}_A+\mathbf{\omega }\times {\mathbf{r}}_{B/A}$ or locate the instantaneous center.
4. Then handle acceleration: apply the full acceleration relation and separate tangential and normal components carefully.
5. Check directions and units: perpendicularity, sign conventions for $\omega$ and $\alpha$, and consistent geometry prevent most errors.

Rigid body kinematics rewards disciplined diagrams and clear reasoning