Acceleration Analysis of Mechanisms free study notes for Diploma / BTech.

1. Introduction

Acceleration analysis of mechanisms deals with determining the linear and angular accelerations of various links in a machine when the velocity analysis is already known. It is a crucial step in kinematics because acceleration directly relates to forces and dynamic behavior of the system.

It helps in:

• Designing machine elements for strength
• Evaluating inertia forces
• Reducing vibration and wear
• Improving performance and reliability

2. Types of Acceleration in Mechanisms

Acceleration in mechanisms is generally divided into two main components:

(a) Tangential Acceleration (aₜ)

• Acts along the direction of motion
• Responsible for change in magnitude of velocity
• Formula: $$a_t = r \alpha$$at​=rα where
  $r$r = radius of rotation
  $\alpha$α = angular acceleration

(b) Normal (Centripetal) Acceleration (aₙ)

• Acts towards the center of rotation
• Responsible for change in direction of velocity
• Formula: $$a_n = \frac{v^2}{r} = r \omega^2$$an​=rv2​=rω2

(c) Total Acceleration

$$a = \sqrt{a_t^2 + a_n^2}$$

3. Acceleration in a Rigid Link

For a rigid link rotating about a fixed point:

• One point has known acceleration
• The acceleration of another point is determined using:
  • Tangential component (due to angular acceleration)
  • Normal component (due to angular velocity)

Vector form:$$\vec{a}_B = \vec{a}_A + \vec{a}_{BA}$$

Where:

• $\vec{a}_{BA}$aBA​ has both tangential and normal components

4. Relative Acceleration Method

This is the most widely used method.

For two points A and B on a link:$$\vec{a}_B = \vec{a}_A + \vec{a}_{BA}$$

Where:

• $\vec{a}_{BA} = \vec{a}_{t} + \vec{a}_{n}$aBA​=at​+an​

Steps:

1. Draw acceleration of known point
2. Add normal acceleration (towards center)
3. Add tangential acceleration (perpendicular to link)
4. Construct vector polygon
5. Solve graphically or analytically

5. Acceleration Diagram (Graphical Method)

Steps to draw acceleration diagram:

1. Choose a scale
2. Mark acceleration of fixed point (usually zero)
3. Draw known accelerations
4. Add normal acceleration components (towards center)
5. Add tangential acceleration components (perpendicular)
6. Close the polygon to find unknown accelerations

6. Acceleration Analysis of Common Mechanisms

(a) Four-Bar Mechanism

• Links: frame, crank, coupler, follower
• Known: angular velocity of crank
• Find: angular acceleration of other links

Procedure:

• Perform velocity analysis first
• Apply relative acceleration equations
• Construct acceleration polygon

(b) Slider-Crank Mechanism

• Used in engines and compressors

Acceleration of piston:$$a_p = r\omega^2 \left(\cos\theta + \frac{\cos 2\theta}{n}\right)$$

Where:

• $r$r = crank radius
• $\omega$ω = angular velocity
• $\theta$θ = crank angle
• $n = \frac{l}{r}$n=rl​

7. Coriolis Acceleration

6

Occurs when a point moves along a link that is rotating.

Formula:$$a_c = 2 \, \omega \, v_r$$ac​=2ωvr​

Where:

• $\omega$ω = angular velocity of link
• $v_r$vr​ = relative velocity of sliding

Direction:

• Perpendicular to the direction of relative velocity
• Determined using right-hand rule

8. Klein’s Construction (For Slider-Crank)

A graphical method to determine acceleration in slider-crank mechanism using velocity diagram.

Advantages:

• Simple and quick
• No need for complex calculations

Limitations:

• Only applicable to slider-crank mechanism

9. Analytical Method

Instead of graphical construction, equations are used:

• Use vector equations
• Resolve into horizontal and vertical components
• Solve simultaneous equations

Example:$$\vec{a}_B = \vec{a}_A + r\alpha + r\omega^2$$

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