Relative Acceleration in Planar Rigid Body Motion

Hello! Welcome to your next lesson in our Dynamics module.

In the last lesson, we explored a powerful shortcut for velocity analysis: the Instantaneous Center of Zero Velocity (ICZV). We saw how it simplifies general planar motion to an instantaneous pure rotation, allowing us to use the simple scalar formula $v=\omega r$. However, we ended with a crucial warning: the ICZV method cannot be used for acceleration analysis. This is because the IC itself is generally an accelerating point.

So, how do we analyze accelerations? We return to the robust vector-based approach. Today, you will learn to analyze acceleration in planar rigid body motion by using the relative acceleration equation. This equation is a direct extension of the relative velocity equation and is the fundamental tool for acceleration analysis.

1. The Relative Acceleration Equation

If you recall the relative velocity equation, ${\vec{v}}_B={\vec{v}}_A+{\vec{v}}_{B/A}$, our new equation is derived by simply taking the time derivative of each term:

$\frac{d}{dt}({\vec{v}}_B)=\frac{d}{dt}({\vec{v}}_A)+\frac{d}{dt}({\vec{v}}_{B/A})$

This gives us the basic form: ${\vec{a}}_B={\vec{a}}_A+{\vec{a}}_{B/A}$.

However, the relative acceleration term, ${\vec{a}}_{B/A}$, is more complex than its velocity counterpart. It accounts for the acceleration of point B as seen from an observer at point A, which involves rotation. This rotation creates two distinct components:

1. Tangential Acceleration: Arises from a change in the magnitude of the angular velocity (i.e., an angular acceleration, $\alpha$).
2. Normal Acceleration: Arises from a change in the direction of the velocity vector of point B as it rotates around A. This component exists even if the angular velocity, $\omega$, is constant.

This leads to the full relative acceleration equation:

${\vec{a}}_B={\vec{a}}_A+\underset{\text{Tangential}}{\underbrace{\vec{\alpha }\times {\vec{r}}_{B/A}}}-\underset{\text{Normal}}{\underbrace{{\omega }^2{\vec{r}}_{B/A}}}$

where:

• ${\vec{a}}_A$ and ${\vec{a}}_B$ are the absolute accelerations of points A and B.
• $\vec{\alpha }$ is the angular acceleration of the rigid body. The term $\vec{\alpha }\times {\vec{r}}_{B/A}$ is the relative tangential acceleration, directed perpendicular to the position vector ${\vec{r}}_{B/A}$.
• $\omega$ is the angular velocity of the rigid body. The term $-{\omega }^2{\vec{r}}_{B/A}$ is the relative normal acceleration, directed from B towards A (along $-{\vec{r}}_{B/A}$).

The following image provides an excellent graphical breakdown of this equation.

For a guided walkthrough of this derivation and a clear explanation of each component, please watch the following video.

2. A Systematic Approach to Solving Problems

Just as with the relative velocity equation, a structured, formula-based procedure is the key to successfully applying this equation. Your background in electronics engineering will make you comfortable with this kind of systematic, component-based problem-solving, which is analogous to applying Kirchhoff's laws in circuit analysis by setting up and solving simultaneous equations.

Here is the step-by-step procedure:

1. Establish a fixed coordinate system (e.g., x-y axes).
2. Draw a kinematic diagram of the body. Indicate the vectors for all known and unknown accelerations (${\vec{a}}_A,{\vec{a}}_B$), angular velocity ($\omega$), and angular acceleration ($\alpha$). If a point moves along a constrained path (e.g., a slider in a track), its acceleration vector's direction is known.
3. Write the relative acceleration equation in vector form: ${\vec{a}}_B={\vec{a}}_A+\vec{\alpha }\times {\vec{r}}_{B/A}-{\omega }^2{\vec{r}}_{B/A}$.
4. Express all vectors in their i, j components relative to your coordinate system. Remember that $\vec{\alpha }$ and $\omega$ for planar motion are typically in the $\vec{k}$ direction (e.g., $\alpha \vec{k}$).
5. Perform the vector operations: calculate the cross product ($\vec{\alpha }\times {\vec{r}}_{B/A}$) and the scalar multiplication ($-{\omega }^2{\vec{r}}_{B/A}$).
6. Separate the vector equation into two scalar component equations (one for all the $\hat{ı}$ terms, one for all the $\hat{ȷ}$ terms).
7. Solve the resulting system of two linear algebraic equations for your two unknowns (which could be magnitudes of acceleration, or $\alpha$, etc.).

The video you just watched also outlines this procedure.

3. Worked Example: Crankshaft Mechanism

Let's apply this procedure to a classic engineering mechanism—the slider-crank—which is fundamental to piston engines you'll encounter in aerospace propulsion systems.

The following video solves a full problem for a crankshaft, finding the acceleration of the piston. Pay close attention to how the knowns and unknowns are identified and how the final equations are separated by components to find the solution.

Test your understanding!

A 10-foot rod AB slides with its ends in contact with a horizontal floor and a vertical wall. At the instant shown, end A has a velocity $v_A=2$ ft/s and an acceleration $a_A=3$ ft/s², both directed to the right. The rod is at a $45^{\circ }$ angle with the floor.

From a previous velocity analysis (or by using the ICZV), we know the angular velocity at this instant is $\omega =0.283$ rad/s (clockwise).

Find the angular acceleration ($\alpha$) of the rod and the acceleration of end B ($a_B$).

Show answer

1. Coordinate System & Kinematics: Let's set up a standard x-y system with the origin at the corner.

   • ${\vec{a}}_A=3\hat{ı}$ ft/s²
   • End B is constrained to move vertically, so ${\vec{a}}_B=a_B\hat{ȷ}$. We assume it accelerates up, so $a_B$ is positive. If we get a negative result, it's accelerating down.
   • ${\vec{r}}_{B/A}$ is the vector from A to B: ${\vec{r}}_{B/A}=10\cos (45^{\circ })\hat{ı}+10\sin (45^{\circ })\hat{ȷ}=7.07\hat{ı}+7.07\hat{ȷ}$ ft.
   • The angular velocity is clockwise, so $\vec{\omega }=-0.283\hat{k}$ rad/s.
   • The angular acceleration $\alpha$ is unknown. Let's assume it's counter-clockwise, so $\vec{\alpha }=\alpha \hat{k}$.

2. Relative Acceleration Equation:
   ${\vec{a}}_B={\vec{a}}_A+\vec{\alpha }\times {\vec{r}}_{B/A}-{\omega }^2{\vec{r}}_{B/A}$

3. Substitute Components:
   $a_B\hat{ȷ}=3\hat{ı}+(\alpha \hat{k})\times (7.07\hat{ı}+7.07\hat{ȷ})-(0.283{)}^2(7.07\hat{ı}+7.07\hat{ȷ})$

4. Perform Vector Operations:

   • Cross product: $\alpha \hat{k}\times 7.07\hat{ı}=7.07\alpha \hat{ȷ}$ and $\alpha \hat{k}\times 7.07\hat{ȷ}=-7.07\alpha \hat{ı}$.
   • Scalar multiplication: $(0.283{)}^2\approx 0.08$. So, $-0.08(7.07\hat{ı}+7.07\hat{ȷ})=-0.566\hat{ı}-0.566\hat{ȷ}$.

5. Combine and Separate Equations:
   $a_B\hat{ȷ}=3\hat{ı}+(-7.07\alpha \hat{ı}+7.07\alpha \hat{ȷ})-(0.566\hat{ı}+0.566\hat{ȷ})$
   Group the $\hat{ı}$ and $\hat{ȷ}$ terms:
   $0\hat{ı}+a_B\hat{ȷ}=(3-7.07\alpha -0.566)\hat{ı}+(7.07\alpha -0.566)\hat{ȷ}$

6. Solve the Component Equations:

   • $\hat{ı}$ component: $0=3-0.566-7.07\alpha ⟹7.07\alpha =2.434⟹\alpha =0.344$ rad/s².
   • $\hat{ȷ}$ component: $a_B=7.07\alpha -0.566$.

7. Final Answer:

   • Since $\alpha$ is positive, our assumption was correct. $\alpha =0.344$ rad/s² (counter-clockwise).
   • Substitute $\alpha$ into the j-equation: $a_B=7.07(0.344)-0.566=2.43-0.566=1.864$ ft/s².
   • Since $a_B$ is positive, our assumption was correct. ${\vec{a}}_B=1.864\hat{ȷ}$ ft/s² (upwards).

4. Special Case: Rolling Without Slipping

A common scenario in dynamics is a wheel or disk rolling without slipping. This case has specific kinematic conditions you should know. For a disk rolling on a flat, stationary surface:

• Velocity: The contact point is the ICZV ($v_C=0$), and the velocity of the center is $v_G=\omega r$.
• Acceleration: The acceleration of the center is purely tangential, so $a_G=\alpha r$. The acceleration of the contact point C is not zero. It has an upward normal acceleration, $a_C={\omega }^{2}r$. This again highlights why the ICZV can't be used for acceleration.

Let's see an example.

Conclusion

Today we have developed the primary tool for acceleration analysis in rigid body kinematics. By mastering its application, you can analyze a vast range of mechanical systems, from simple links to complex engines.

Key Takeaways:

• General planar acceleration is analyzed using the relative acceleration equation: ${\vec{a}}_B={\vec{a}}_A+\vec{\alpha }\times {\vec{r}}_{B/A}-{\omega }^2{\vec{r}}_{B/A}$.
• The relative acceleration term, ${\vec{a}}_{B/A}$, has two components:
  • A tangential component ($\vec{\alpha }\times {\vec{r}}_{B/A}$) due to angular acceleration.
  • A normal component ($-{\omega }^2{\vec{r}}_{B/A}$) due to rotation, which always points towards the center of rotation.
• Problem-solving involves a systematic vector-based approach: define coordinate systems, write the equation in component form, and solve the resulting scalar equations.

Next Lesson Preview:

In our next lesson, we will focus exclusively on the kinematics of bodies that are rolling without slipping. We'll formalize the velocity and acceleration relationships for this important special case, building directly on the example you saw today. This is a critical topic for analyzing anything with wheels, from landing gear on an aircraft to ground vehicles.