Velocity Analysis of Mechanisms

Introduction

Velocity analysis determines how fast the links and points of a mechanism move at a particular configuration. It is fundamental to the design of engines, pumps, compressors, machine tools, robotic manipulators, and transmission systems. Once velocities are known, acceleration, dynamic force, power, vibration, and wear analyses can be undertaken.

A mechanism is treated as a set of rigid links connected by kinematic pairs. In this chapter, planar mechanisms are emphasized. Motion is assumed to occur in one plane, links are rigid, joints are ideal, and deformation, clearance, and friction are neglected unless stated otherwise. The principal methods are:

• relative-velocity analysis;
• velocity polygons;
• instantaneous-centre analysis; and
• analytical vector and complex-number methods.

The same physical velocity must be obtained by each valid method. Agreement between graphical and analytical results is an important engineering check.

Learning Objectives

After studying this chapter, the reader should be able to:

1. distinguish absolute, relative, linear, and angular velocity;
2. formulate the relative-velocity equation for two points on a rigid link;
3. construct and interpret velocity polygons;
4. locate and use instantaneous centres of zero velocity;
5. derive velocity relations using vector and complex-number notation;
6. solve slider-crank and four-bar velocity problems; and
7. assess assumptions, signs, units, and limitations in practical applications.

1. Fundamental Concepts

1.1 Linear and angular velocity

The linear velocity of a point is the time rate of change of its position:

where v is velocity in m/s and r is the position vector in m. Angular velocity is the time rate of change of angular displacement:

where ω is in rad/s and θ is in radians. Since the radian is dimensionless, angular velocity has the dimensional unit s⁻¹, conventionally written rad/s.

For a point P on a link rotating about a fixed centre O:

The magnitude is

and the velocity is perpendicular to OP. Here r_P/O is the distance from O to P. This relation applies to pure rotation about a fixed point; it does not by itself describe general plane motion.

A positive angular velocity is commonly taken as counterclockwise. Reversing the sign convention is acceptable if it is used consistently.

1.2 Relative velocity of two points on a rigid link

For points A and B belonging to the same rigid link, the general planar relative-velocity equation is

where

Thus,

The relative velocity v_B/A is perpendicular to AB, and its magnitude is ω_AB·AB. The equation is valid for a rigid link undergoing any plane motion, including simultaneous translation and rotation.

A crucial result is that the component of the relative velocity along AB is zero:

Therefore, the velocities of two points on a rigid link have equal components along the line joining those points. This is often the quickest way to establish a direction in a velocity diagram.

1.3 Common constraints

For a fixed pin at O, v_O = 0. For a slider constrained to a horizontal guide, its velocity is horizontal. For a slider constrained to a vertical guide, its velocity is vertical. For a pin joint connecting two links, the velocity of the common point is the same when calculated from either link.

A rolling contact without slipping imposes equal tangential velocities at the contact point. A prismatic pair permits relative translation along the guide but prevents relative motion perpendicular to it.

2. Relative-Velocity Method

The relative-velocity method combines known velocity directions with vector addition. A systematic solution is as follows:

1. Draw the mechanism accurately and identify the input velocity.
2. Choose a positive direction for every angular velocity.
3. Write a velocity equation for each moving link.
4. Determine velocity directions from fixed pivots, sliders, and rigid-link constraints.
5. Resolve vectors into two perpendicular coordinate directions, or construct a velocity polygon.
6. Check the result using dimensions, limiting positions, and an alternative method when possible.

2.1 Four-bar mechanism

Consider a four-bar mechanism with fixed pivots O₂ and O₄, input link O₂A, coupler AB, and output link BO₄. Let the input angular velocity be ω₂. Then

Since B belongs to both the coupler and the output link,

and

Equating the two expressions gives

The unknowns are usually ω₃ and ω₄. Resolving the vector equation into x and y components produces two simultaneous linear equations.

2.2 Worked example: four-bar velocity

An illustrative four-bar has fixed pivots O₂=(0,0) m and O₄=(0.12,0) m. At the instant considered, A=(0.04,0.03) m and B=(0.10,0.08) m. The input link O₂A rotates counterclockwise at ω₂=10 rad/s. Determine the angular velocities of AB and BO₄.

Using the planar cross-product relation, for r=(x,y),

The velocity of A is

For link AB, r_B/A=(0.06,0.05) m. Therefore,

For link BO₄, r_B/O₄=(-0.02,0.08) m, so

Equating components gives

Solving, ω₃≈−2.16 rad/s and ω₄≈2.42 rad/s. The negative sign indicates that AB rotates clockwise under the selected positive counterclockwise convention. The velocity of B is

Its magnitude is approximately 0.200 m/s. The units are consistent: angular velocity in rad/s multiplied by length in m gives m/s.

3. Velocity Polygons

A velocity polygon is a graphical representation of vector equations. It is not a physical drawing of the mechanism. A convenient procedure is:

1. Select a velocity scale, such as 1 cm = 0.1 m/s.
2. Mark a pole o representing zero velocity.
3. Draw oa parallel and proportional to the known velocity of A.
4. From a, draw a line perpendicular to AB, because v_B/A is perpendicular to AB.
5. From o, draw a line perpendicular to O₄B, because v_B is perpendicular to O₄B.
6. Their intersection is b. The segments ab and ob give the relative and absolute velocities.

The direction of an unknown vector is established before its magnitude. A line drawn parallel to a velocity direction is called a velocity vector locus. The polygon closes because the vector equation is satisfied.

Graphical analysis is especially useful for visualizing motion and checking a computer calculation. Its accuracy depends on the scale, drawing quality, mechanism configuration, and ability to read small angles and lengths. For precision design, analytical or computer-based methods are preferred.

Figure: Mechanism sketch and corresponding velocity polygon; the polygon represents vector addition, not physical geometry.

4. Instantaneous Centres

The instantaneous centre (IC) of two bodies in relative plane motion is the point about which one body appears to rotate relative to the other at that instant. At the IC, the relative velocity is zero. Although the point may not be fixed in space and may not lie within either body, it provides a powerful velocity relation.

If I is the instantaneous centre of a link and points A and B lie on that link,

Consequently,

The velocities at A and B are perpendicular to IA and IB respectively. These equations apply at the instant under consideration and do not imply that the link undergoes permanent rotation about I.

4.1 Locating instantaneous centres

For a mechanism with n links, the number of possible instantaneous centres is

This includes the frame as a link. For a four-link mechanism, six centres exist. Some are directly located:

• the centre between a link and the frame is its fixed pivot;
• for a sliding pair, the instantaneous centre lies at infinity in the direction perpendicular to sliding;
• for a turning pair, the pin centre is the instantaneous centre.

The Kennedy–Aronhold theorem states that for three bodies in plane motion, their three instantaneous centres lie on one straight line. This theorem permits indirect centres to be located from two known centres. It is valid for planar rigid-body motion and is particularly useful in mechanisms with several links.

4.2 Worked example: slider velocity by an instantaneous centre

A crank OA of length 0.08 m rotates at 20 rad/s. A connecting rod AB of length 0.24 m drives a horizontal slider B. At the instant, OA makes 60° with the horizontal and AB makes 30° with the horizontal. Find the slider velocity.

The crank-pin velocity is

It is perpendicular to OA. For the connecting rod, the instantaneous centre I of AB lies at the intersection of the line through A perpendicular to v_A and the vertical line through B, because B moves horizontally. Since IA is perpendicular to v_A, IA is along OA; hence IA=0.08 m.

The vertical separation between A and B is

The horizontal distance from A to B is AB cos30°≈0.2078 m. Since I lies on the line through A along OA, its coordinates relative to A are approximately (0.04, 0.0693) m toward the crank centre, while B is (0.2078, −0.12) m relative to A. Thus

The velocity ratio gives

Therefore, v_B≈5.06 m/s, horizontally. The result is an instantaneous value; the connecting rod does not rotate about a permanently fixed point.

5. Analytical Vector and Complex-Number Methods

5.1 Vector-loop formulation

A closed mechanism can be represented by a position-vector loop. For a four-bar,

where r₂, r₃, and r₄ are link vectors and r₁ is the fixed frame vector. Differentiating with respect to time gives

Since the derivative of a rotating vector is

this becomes the velocity-loop equation. Differentiation is valid when link lengths are constant and the vectors are expressed in a common inertial reference frame.

For a slider-crank with crank radius r, connecting-rod length l, crank angle θ, and slider coordinate x, a scalar position relation is

for the configuration in which the slider lies on the positive x side and the square-root term is positive. If ω=dθ/dt is constant, differentiation gives

The sign indicates direction along the selected x-axis. This formula assumes a rigid crank and rod, a frictionless geometric constraint, and no offset between the crank axis and slider line.

5.2 Complex-number formulation

A planar vector can be represented by a complex number:

where L is length, θ is orientation, and i=√(−1). Differentiation gives

For a rigid link, dL/dt=0, so

The factor i rotates a vector by 90°, which confirms that rigid-link relative velocity is perpendicular to the link. Complex notation is compact and well suited to symbolic manipulation and computer programs. Real and imaginary parts correspond to the x and y components.

6. Choosing a Method

A sound engineering workflow often uses a graphical sketch to establish directions, an analytical calculation for numerical accuracy, and an independent check using an instantaneous centre or limiting case.

7. Practical Application: Illustrative Packaging-Machine Linkage

Background

A packaging machine uses a four-bar linkage to move a gripper between a pickup station and a conveyor. The motor drives the input crank at a nearly constant speed. The gripper is attached to the output link and must not exceed a specified linear speed near the pickup point, where fragile products are handled.

Problem

At one configuration, the input crank speed is known, but the output-link speed and gripper velocity are not. Excessive speed may cause impact, product displacement, or motor overload.

Analysis and approach

The designer first models each link as rigid and assumes pin joints have negligible clearance. The crank-pin velocity is calculated from v=ωr. The coupler equation

is combined with the output-link equation

A velocity polygon provides a rapid check of direction. A numerical vector solution is then evaluated over a complete crank revolution. Configurations near toggle positions are treated carefully because small changes in input angle can produce large changes in output velocity.

Outcome

The study identifies the maximum gripper velocity and the crank angles at which it occurs. If the speed exceeds the allowable value, the designer may reduce motor speed, alter link lengths, introduce a nonuniform drive, or redesign the coupler path. The final design should also include acceleration, force, fatigue, guarding, and emergency-stop analysis; velocity analysis alone is insufficient for machine certification.

Learning points

1. Velocity is configuration-dependent, even when input angular speed is constant.
2. A graph can reveal a problematic high-speed region before detailed simulation.
3. Mechanical redesign is often preferable to simply reducing production speed.
4. Real mechanisms require consideration of clearance, compliance, lubrication, and manufacturing tolerances.

8. Errors, Limitations, and Best Practice

Common errors include treating a relative velocity as parallel to a link rather than perpendicular to it, confusing angular velocity with linear velocity, omitting a vector sign, using degrees per second without conversion, and applying an instantaneous-centre ratio to points on different bodies.

For unit consistency, angular speed must be in rad/s when multiplied by a length in metres. If speed is given in revolutions per minute,

where N is in rev/min and ω is in rad/s. For example, 1200 rev/min equals approximately 125.66 rad/s.

Near a toggle or dead-centre position, velocity directions can change rapidly and numerical results may be sensitive to small geometric errors. Graphical methods become unreliable when the velocity polygon contains very short segments. Computer-aided mechanism analysis is valuable, but the engineer must inspect input geometry, coordinate conventions, singularities, and output units rather than accepting software results uncritically.

Modern CAD and multibody-dynamics systems can calculate velocities by numerical differentiation or analytic constraint solvers. AI-assisted design tools may suggest link dimensions or identify high-speed configurations, but their outputs require independent verification, traceable assumptions, and engineering responsibility. Safety-critical machinery must be validated against applicable machinery-safety standards and organizational design procedures.

Chapter Summary

Velocity analysis establishes the instantaneous linear and angular motion of mechanism links and points. The central relation for a rigid link is v_B=v_A+ω×r_B/A, with relative velocity perpendicular to the line joining the two points. Velocity polygons express this relation graphically, while vector and complex-number methods provide systematic numerical solutions. Instantaneous centres convert general plane motion into an equivalent instantaneous rotation and are especially useful for velocity ratios. Four-bar and slider-crank mechanisms demonstrate how fixed pivots, sliders, and rigid links determine velocity directions. Reliable analysis requires consistent signs, correct units, explicit assumptions, and checks near singular configurations. In professional design, velocity results support—but do not replace—acceleration, force, fatigue, safety, and experimental validation.

Key Terms

Mechanism: A system of rigid or nearly rigid links arranged to produce controlled relative motion.

Link: A rigid member of a mechanism.

Linear velocity: Rate of change of a point’s position, measured in m/s.

Angular velocity: Rate of change of angular displacement, measured in rad/s.

Relative velocity: Velocity of one point measured with respect to another point.

Rigid-body constraint: The condition that the distance between two points on a rigid link remains constant.

Velocity polygon: A graphical vector representation of velocity equations.

Velocity locus: A line in a velocity diagram showing the possible direction of an unknown velocity.

Instantaneous centre: The point about which a body appears to rotate at a particular instant.

Kennedy–Aronhold theorem: The theorem that three instantaneous centres for three planar bodies are collinear.

Four-bar mechanism: A mechanism containing four links, including the frame.

Slider-crank mechanism: A mechanism converting rotary motion into reciprocating motion or vice versa.

Coupler: The intermediate link connecting the input and output links.

Toggle position: A configuration in which links become aligned and mechanical advantage or velocity sensitivity may become extreme.

Velocity loop: A time-differentiated closed vector equation for a mechanism.

Complex representation: A method representing a planar vector as a complex number.

Configuration: The instantaneous geometric arrangement of a mechanism.

Prismatic pair: A joint permitting relative sliding along one direction.

Turning pair: A joint permitting relative rotation about a fixed axis.

Review Questions

Short Answer Questions

1. Why is the relative velocity of two points on a rigid link perpendicular to the line joining them?
2. Define an instantaneous centre of a moving link.
3. State the velocity relation for a point rotating about a fixed centre.
4. What does a negative angular velocity signify when counterclockwise is positive?
5. Why can a velocity polygon be used to check an analytical solution?

Long Answer Questions

1. Derive the relative-velocity equation for two points on a rigid planar link and explain each term.
2. Describe the construction of a velocity polygon for a four-bar mechanism.
3. Explain the Kennedy–Aronhold theorem and its use in locating instantaneous centres.
4. Derive the velocity-loop equations for a four-bar mechanism using vector notation.
5. Explain the assumptions and limitations of instantaneous-centre analysis.

Analytical / Application Questions

1. A link AB is 0.25 m long and rotates at 12 rad/s. Determine the relative velocity magnitude of B with respect to A.
2. A crank of radius 0.06 m rotates at 1500 rev/min. Determine the crank-pin speed in m/s.
3. Explain how you would investigate and redesign a packaging linkage whose gripper velocity becomes excessive near a toggle position.

Multiple Choice Questions with Answers

Question 1

For two points A and B on the same rigid planar link, the relative velocity v_B/A is directed:

A. Along AB

B. Perpendicular to AB

C. Toward the instantaneous centre only

D. Parallel to the absolute velocity of A

Correct Answer: B. Perpendicular to AB

Question 2

A point 0.10 m from a fixed pivot rotates at 15 rad/s. Its linear speed is:

A. 0.67 m/s

B. 1.5 m/s

C. 15 m/s

D. 150 m/s

Correct Answer: B. 1.5 m/s

Question 3

The number of instantaneous centres for a mechanism with four links is:

A. 4

B. 5

C. 6

D. 8

Correct Answer: C. 6

Question 4

If counterclockwise angular velocity is positive, an angular velocity of −3 rad/s indicates:

A. Counterclockwise rotation at 3 rad/s

B. Clockwise rotation at 3 rad/s

C. Zero rotation

D. Translation at 3 m/s

Correct Answer: B. Clockwise rotation at 3 rad/s

Question 5

For a slider constrained to move horizontally, its velocity vector is:

A. Vertical

B. Radial from the crank centre

C. Horizontal

D. Always zero

Correct Answer: C. Horizontal

Question 6

A rotating vector r=(x,y) with angular velocity ω has velocity components:

A. (ωx,ωy)

B. (−ωy,ωx)

C. (−ωx,−ωy)

D. (y/ω,x/ω)

Correct Answer: B. (−ωy,ωx)

Question 7

A crank rotates at 1200 rev/min. Its angular velocity is approximately:

A. 20 rad/s

B. 62.83 rad/s

C. 125.66 rad/s

D. 7200 rad/s

Correct Answer: C. 125.66 rad/s

Question 8

At an instantaneous centre of a link, the velocity relative to the reference body is:

A. Maximum

B. Equal to the input velocity

C. Zero

D. Always vertical

Correct Answer: C. Zero

Question 9

The complex-number expression for a rigid rotating planar vector is commonly written as:

A. v=r/ω

B. v=iωr

C. v=ω/r

D. v=r+iω

Correct Answer: B. v=iωr

Question 10

In a four-bar velocity analysis, the velocity of a common pin B must be:

A. Different for each connected link

B. Zero for every configuration

C. The same when calculated from either connected link

D. Parallel to both connected links

Correct Answer: C. The same when calculated from either connected link