Velocity Analysis of Mechanism free notes for Diploma / BTech.

1. What is Velocity Analysis of Mechanism ?

Velocity analysis of mechanisms is a fundamental topic in Theory of Machines, focusing on determining the velocity of different points and links in a mechanism during motion. It is the next step after displacement analysis and forms the basis for acceleration and dynamic analysis.

Definition:
Velocity analysis is the process of determining the magnitude and direction of velocities of various parts of a mechanism at a given instant.

It helps engineers:

• Understand motion behavior
• Design efficient machines
• Avoid excessive wear and vibration
• Predict performance of mechanisms

2. Importance of Velocity Analysis

Velocity analysis plays a crucial role in machine design and analysis:

• Helps in designing cams, gears, and linkages
• Used in engine mechanisms to calculate piston speed
• Important for balancing and vibration control
• Required for acceleration and force analysis

3. Types of Motion in Mechanisms

Before analyzing velocity, we must understand types of motion:

(A) Linear Motion

• Motion in a straight line
• Example: piston movement

(B) Circular Motion

• Motion along a circular path
• Example: rotating crank

(C) General Plane Motion

• Combination of translation and rotation
• Example: connecting rod

4. Basic Concepts of Velocity

(A) Absolute Velocity

Velocity of a point measured with respect to a fixed frame (ground).

(B) Relative Velocity

Velocity of one point with respect to another.$$V_{AB} = V_A – V_B$$

(C) Angular Velocity

$v = r\omega$v=rω

Where:

• $v$v = linear velocity
• $r$r = radius
• $\omega$ω = angular velocity

5. Velocity in Rigid Body

For a rigid link rotating about a fixed point:

• Velocity is perpendicular to the link
• Magnitude depends on distance from the center

If link AB rotates about A:

• Velocity of B:

$$V_B = \omega \cdot AB$$

Direction ⟂ to AB

6. Methods of Velocity Analysis

There are mainly three methods:

6.1 Relative Velocity Method

Concept:

Based on vector addition of velocities.$$\vec{V}_B = \vec{V}_A + \vec{V}_{BA}$$

Steps:

1. Identify known velocities
2. Draw velocity diagram
3. Apply vector addition
4. Solve for unknown velocities

Example: Two-Link Mechanism

• A is fixed
• B rotates around A
• C connected to B

Velocity of C:$$V_C = V_B + V_{CB}$$

6.2 Instantaneous Center Method (IC Method)

Definition:

Instantaneous center is a point where velocity is zero at that instant.

Kennedy’s Theorem:

Three bodies in relative motion have three ICs, and they lie on a straight line.

Types of IC:

1. Fixed IC
2. Permanent IC
3. Instantaneous IC

Formula:

$$V_A = \omega \cdot IA$$

Where:

• $I$I = instantaneous center

Application:

• Four-bar mechanism
• Complex linkages

IC Diagram

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6.3 Analytical Method

Uses mathematical equations instead of graphical methods.

Example:

For slider-crank mechanism:$$V_p = r\omega (\sin\theta + \frac{\sin2\theta}{2n})$$Vp​=rω(sinθ+2nsin2θ​)

Where:

• $n = \frac{l}{r}$n=rl​

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🔹 7. Velocity Analysis of Common Mechanisms

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🔹 7.1 Four-Bar Mechanism

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Procedure:

1. Known crank velocity
2. Draw perpendicular velocity at joints
3. Use velocity polygon
4. Determine unknown link velocities

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🔹 7.2 Slider-Crank Mechanism

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Key Points:

• Crank has angular velocity
• Piston has linear velocity
• Connecting rod has angular + linear motion

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7.3 Cam and Follower

Velocity depends on:

• Cam profile
• Follower type

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🔹 7.4 Gear Mechanism

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Key relation:

$$V = r\omega$$V=rω

👉 Velocity at pitch point is same for both gears

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8. Velocity Diagrams

Velocity diagrams are graphical tools used to represent velocities.

Types:

• Vector diagram
• Polygon method

Rules:

• Use proper scale
• Direction must be correct
• Close polygon indicates correct solution

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🔹 9. Coriolis Component of Velocity

Occurs when a point moves along a rotating link.$$V_c = 2 \cdot \omega \cdot v_r$$​

Direction:

• Perpendicular to sliding motion
• Depends on rotation direction

10. Velocity in Reciprocating Engine

• Maximum at mid-stroke
• Zero at dead centers

11. Applications of Velocity Analysis

• Engine design
• Robotics
• Machine tools
• Automotive systems
• Aerospace mechanisms

12. Advantages of Velocity Analysis

• Predicts motion behavior
• Improves efficiency
• Reduces wear and failure

13. Limitations

• Graphical methods less accurate
• Complex for multi-link systems

14. Important Exam Points

• Relative velocity method is most common
• IC method saves time in exams
• Coriolis component is frequently asked
• Slider-crank velocity is important

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