Dynamics: Relative Velocity in Rigid Bodies

Understanding how different points on a machine part move relative to each other is fundamental to designing everything from engine pistons to robotic arms. The relative velocity method is a powerful analytical tool that allows engineers to determine the velocity of any point on a rigid body given the velocity of a known point and the body's angular velocity. This analysis is critical for predicting performance, identifying points of high wear, and ensuring safe operational speeds in complex mechanisms.

The Foundation: Deriving the Relative Velocity Equation

A rigid body is an idealized object whose particles remain at fixed distances from each other. Because the body is rigid, the velocity of any point B on the body can be related to the velocity of another point A on the same body. The fundamental relative velocity equation is derived from the basic kinematics of rotation and translation.

Consider a rigid body rotating with an angular velocity $\omega$ about an instantaneous axis. If you know the linear velocity of point A, ${\vec{v}}_A$, the velocity of point B is given by: $${\vec{v}}_B={\vec{v}}_A+{\vec{v}}_{B/A}$$ The term ${\vec{v}}_{B/A}$ is read as "the velocity of B relative to A." This is a purely rotational term. Since the distance between A and B is fixed (rigid body), the only motion B can have relative to A is a rotation about A. Therefore, ${\vec{v}}_{B/A}=\vec{\omega }\times {\vec{r}}_{B/A}$, where ${\vec{r}}_{B/A}$ is the position vector from A to B.

The complete relative velocity equation is thus: $${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}$$ This vector equation is the cornerstone of all planar rigid body velocity analysis. It states that the absolute velocity of point B equals the absolute velocity of point A plus a term accounting for B's rotation around A.

Strategic Reference Point Selection

The choice of reference point A is not arbitrary; a strategic choice simplifies the solution. The golden rule is: choose a point whose velocity is completely known (both magnitude and direction). Often, this is a fixed point (velocity is zero) or a point constrained to move along a known path.

For a rotating link pinned at one end, the pin's velocity is zero, making it the ideal reference point. If you need to find the velocity of the other end (point B), the equation simplifies to ${\vec{v}}_B=\vec{\omega }\times {\vec{r}}_{B/A}$. The velocity of B will be perpendicular to the link itself. In more complex linkages, you may need to apply the equation between two points where the velocity direction is known, even if the magnitude is not, to set up solvable vector equations.

Analyzing the Rotating Link

The rotating link is the simplest application. Let's analyze a link of length $L=0.5$ m, pinned at point A, rotating clockwise with an angular velocity $\omega =4$ rad/s.

1. Knowns: ${\vec{v}}_A=0$, $\omega =-4\hat{k}$ rad/s (clockwise into the page), ${\vec{r}}_{B/A}=0.5\hat{i}$ m.
2. Apply Equation: ${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}=0+(-4\hat{k})\times (0.5\hat{i})$.
3. Solve: Using the cross product $\hat{k}\times \hat{i}=\hat{j}$, we get ${\vec{v}}_B=-2\hat{j}$ m/s.

The result is a velocity of 2 m/s in the negative y-direction, which is indeed perpendicular to the link (which is along the x-axis). This validates the direct relationship: $v_B=\omega \cdot L$.

Velocity Analysis of a Slider-Crank Mechanism

The slider-crank mechanism is a classic engine component, converting rotational motion (crank) into translational motion (piston/slider). A typical problem asks: "Given the crank's angular velocity, find the piston's linear velocity."

Consider a crank (link OA) of length 0.2 m rotating at a constant 10 rad/s clockwise, connected to a connecting rod (link AB) of length 0.6 m, which is connected to a piston at B constrained to move horizontally.

1. Step 1: Find ${\vec{v}}_A$. Point O is fixed. ${\vec{v}}_A={\vec{\omega }}_{OA}\times {\vec{r}}_{A/O}$. With ${\omega }_{OA}=-10\hat{k}$ rad/s and ${\vec{r}}_{A/O}=0.2\angle 60^{\circ }$ m, you calculate ${\vec{v}}_A$ (perpendicular to OA).
2. Step 2: Apply Relative Equation between A and B. You know ${\vec{v}}_B$ is purely horizontal (direction known). You know ${\vec{v}}_A$ (magnitude and direction from Step 1). You know the direction of ${\vec{\omega }}_{AB}$ is perpendicular to the plane ($\pm \hat{k}$), but not its magnitude. The vector equation is:

$${\vec{v}}_B\hat{i}={\vec{v}}_A+({\omega }_{AB}\hat{k})\times {\vec{r}}_{B/A}$$ Here, ${\vec{r}}_{B/A}$ is known from geometry.

1. Step 3: Solve the Vector Equation. This single vector equation yields two scalar equations (i-component and j-component). The j-component equation will contain only one unknown (${\omega }_{AB}$), which you solve for. Then, substitute back into the i-component equation to solve for the unknown magnitude $v_B$.

Graphical vs. Analytical Solution Methods

Engineers have two primary tools to solve the relative velocity vector equation: graphical and analytical methods.

The Graphical Method involves drawing a velocity polygon to scale. You start by plotting the known velocity vector ${\vec{v}}_A$. From its tip, you draw a line in the direction of ${\vec{v}}_{B/A}$ (perpendicular to link AB). From the origin of the polygon, you draw a line in the known direction of ${\vec{v}}_B$ (e.g., horizontal). The intersection of these two lines defines the polygon's shape, allowing you to measure the magnitudes of ${\vec{v}}_B$ and ${\vec{v}}_{B/A}$ directly from the diagram. This method provides excellent visual intuition and a quick check but is limited by drawing accuracy.

The Analytical Method uses vector algebra or complex numbers to solve the equations exactly. As shown in the slider-crank example, you break the vector equation into scalar components. This yields a system of algebraic equations solvable with precision. For complex, multi-loop mechanisms, analytical methods implemented in software are essential. The choice often depends on the need for speed and intuition (graphical) versus precision and repeatability (analytical).

Common Pitfalls

1. Incorrect Relative Velocity Direction: The most frequent error is forgetting that ${\vec{v}}_{B/A}$ is always perpendicular to the line connecting B and A (${\vec{r}}_{B/A}$) in planar motion. If you draw this direction incorrectly in a graphical solution or misrepresent it in an equation, the entire solution fails.

Correction: Always sketch the link. The relative velocity term must rotate point B around point A. Visually confirm the perpendicular direction before writing equations.

1. Sign Errors in Angular Velocity: Consistently defining a positive rotational direction (e.g., counterclockwise as positive $\hat{k}$) is crucial. Mixing signs within a problem leads to wrong magnitudes and directions.

Correction: At the start, define and stick to a sign convention. If a link rotates clockwise, its angular velocity is negative ($-\omega \hat{k}$).

1. Using the Equation Across Different Bodies: The fundamental equation ${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}$ is only valid if points A and B are on the same rigid body. You cannot directly apply it to points on two different moving links.

Correction: Identify the connecting link. For two separate bodies connected at a pin (like point A connecting the crank and connecting rod), the velocities of point A on each body are equal. This provides the necessary connection to solve multi-body mechanisms.

1. Ignoring Vector Nature in Analytical Solutions: Writing ${\vec{v}}_B={\vec{v}}_A+\omega \cdot r$ is a scalar mistake. The cross product means the term $\omega \cdot r$ gives the magnitude of the relative velocity, but its direction must be accounted for separately in the vector components.

Correction: Always write the full vector equation and then resolve it into components (i, j) to generate solvable scalar equations.

Summary

• The relative velocity equation ${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}$ is the essential tool for analyzing interconnected rigid bodies in planar motion.
• Effective analysis hinges on strategically choosing a reference point with a fully known velocity to simplify the problem setup.
• For a rotating link, the velocity of its endpoint is always perpendicular to the link, with a magnitude proportional to the angular velocity and link length ($v=\omega L$).
• Solving a slider-crank mechanism is a two-step process: first find the velocity of the connecting pin using rotation about a fixed point, then use the relative velocity equation across the connecting rod to link the known rotational motion to the desired translational motion.
• Choose between graphical (intuitive, visual) and analytical (precise, scalable) solution methods based on the problem's requirements and your need for accuracy versus speed.