Planar Kinematics of Rigid Bodies

Introduction

Planar kinematics examines the motion of a rigid body when every point of the body remains in a plane parallel to a fixed reference plane. It is concerned only with position, velocity, and acceleration; forces and moments belong to kinetics. The subject is fundamental to the analysis and design of mechanisms such as engines, pumps, robotic manipulators, gear trains, linkages, wheels, and machine tools.

A rigid body is an idealization in which the distance between any two material points remains constant. Thus, deformation is neglected, although the body may translate, rotate, or perform both motions simultaneously. Planar rigid-body motion can be resolved into three important categories:

1. Translation, in which every line fixed in the body remains parallel to its original orientation.
2. Rotation about a fixed axis, in which the body turns about a stationary point in the plane.
3. General plane motion, which combines translation and rotation.

The central analytical tools are vector kinematics, relative velocity, relative acceleration, instantaneous centers, rotating reference frames, and kinematic constraints.

Learning Objectives

After studying this chapter, a student should be able to:

• distinguish translation, fixed-axis rotation, and general plane motion;
• calculate velocity and acceleration using relative-motion equations;
• interpret normal and tangential acceleration components;
• locate and use the instantaneous center of zero velocity;
• analyze rolling bodies, gears, and pin-jointed mechanisms;
• apply rotating-reference-frame equations;
• identify constraints and check whether a mechanism solution is physically consistent.

1. Position and Orientation of a Rigid Body

Choose a fixed Cartesian reference frame with unit vectors i and j. Let points A and B be fixed in a rigid body. The position vector of B can be written as

where r_A is the position of A and r_B/A is the vector from A to B. Because the body is rigid, the magnitude of r_B/A is constant, although its direction may change.

For a planar body, the relative position may be represented as

The coordinates x_B/A and y_B/A are measured in the fixed frame. If the line AB has length L and orientation θ,

Here L is constant and θ is the body orientation measured counterclockwise from the positive x-axis. The angular velocity and angular acceleration are

The units of ω are rad/s and those of α are rad/s². Radians are dimensionless in dimensional analysis, but they should be retained in engineering descriptions.

2. Translation

A rigid body undergoes translation if every line fixed in the body remains parallel to itself. Translation may be rectilinear or curvilinear.

For pure translation, all points have the same velocity and acceleration:

These equations apply to any two points fixed in a translating body. In rectilinear translation, all points follow parallel straight paths. In curvilinear translation, the paths may be curved, but the body does not rotate.

For a point moving along a path with speed v, acceleration can be resolved into tangential and normal components:

where

Here e_t and e_n are unit vectors tangent and normal to the path, and ρ is the local radius of curvature. The normal component points toward the center of curvature.

3. Rotation About a Fixed Axis

Consider a rigid body rotating in the xy-plane about a fixed point O. For a point P at distance r from O,

For planar motion, ω is perpendicular to the plane, so the velocity is tangent to the circular path and has magnitude

The acceleration is

The first term is tangential acceleration and the second is normal, or centripetal, acceleration. In scalar form,

Thus, the tangential component changes the speed, while the normal component changes the direction of velocity. The normal acceleration is directed toward O.

Worked Example 1: Fixed-Axis Rotation

A disk rotates counterclockwise with angular velocity 12 rad/s and angular acceleration 4 rad/s². Determine the velocity and acceleration of a point 0.25 m from the center.

The velocity magnitude is

The tangential acceleration is

The normal acceleration is

Therefore, the acceleration has a tangential component of 1.00 m/s² and an inward normal component of 36.0 m/s². Its magnitude is

The large normal component illustrates that high angular speed can produce substantial acceleration even when angular acceleration is modest.

4. General Plane Motion

General plane motion is the most important case in mechanism analysis. It consists of translation of one reference point plus rotation about that point. Differentiating the relative-position equation gives the relative velocity equation:

This equation applies when A and B are points fixed in the same rigid body and the reference frame is inertial. It does not require A to be stationary.

Differentiating again gives the relative acceleration equation:

The terms have clear meanings:

• a_A: acceleration of the reference point A;
• α × r_B/A: tangential relative acceleration;
• ω × (ω × r_B/A): normal relative acceleration.

In scalar vector-component form, if AB has length L and orientation θ,

and

The negative sign in the last term indicates that normal acceleration points from B toward A.

Worked Example 2: Relative Velocity in a Slotted Link

A link AB is 0.40 m long and is oriented at 30° above the horizontal. Point A moves horizontally to the right at 2.0 m/s. Point B is constrained to move vertically. Determine the angular velocity of the link and the velocity of B.

Let v_A = 2.0 i m/s and v_B = v_B j. The relative velocity equation is

Since B has no horizontal velocity, the i-component must vanish:

Therefore,

The vertical component is

Thus, B moves upward at approximately 3.46 m/s, assuming positive counterclockwise angular velocity.

5. Relative Acceleration and Constraint Equations

The velocity equation is often used first to determine unknown angular velocities. The acceleration equation is then used to determine angular accelerations and point accelerations. Constraints provide additional equations.

Typical constraints include:

• a slider constrained to a straight guide has velocity and acceleration only along the guide;
• a pin in a smooth slot has velocity normal to the slot equal to the slot's local velocity condition;
• a rolling wheel without slipping has zero velocity at the contact point relative to the ground;
• meshing gears have equal tangential velocities at the pitch point.

Worked Example 3: Acceleration of a Connecting Rod

A 0.50 m rod AB has angular velocity 6 rad/s counterclockwise and angular acceleration 8 rad/s² counterclockwise. At the instant considered, A has acceleration 1.5 m/s² to the right, and AB is horizontal with B to the right of A. Determine the acceleration of B.

Take r_B/A = 0.50 i. Then

The normal term is

Hence

The acceleration magnitude is

The negative horizontal component means that B accelerates leftward at this instant, despite A accelerating rightward, because the rod's normal acceleration is directed toward A.

6. Instantaneous Center of Zero Velocity

For a rigid body in general plane motion, there is, at a particular instant, a point whose velocity is zero. This point is called the instantaneous center of zero velocity, or IC. At that instant the body appears to rotate about the IC:

Consequently,

The IC may lie inside or outside the body and may move from one instant to the next. It is not necessarily a fixed material point.

The IC can be located when the velocities of two points are known. Draw a line through each point perpendicular to its velocity; the intersection is the IC. For pure translation, the IC is at infinity because all velocity vectors are parallel. For rotation about a fixed point, the fixed point is the IC.

The IC is especially useful for velocity analysis, but it should not be used indiscriminately for acceleration. The instantaneous center generally has acceleration, and treating it as a permanently fixed pivot can produce an incorrect acceleration analysis.

Rolling Without Slipping

For a wheel of radius R rolling without slipping on a stationary surface, the contact point has zero instantaneous velocity. Therefore the contact point is the IC and

where v_C is the velocity of the wheel center. If the wheel rolls without slipping on a straight stationary surface and R is constant,

and, by differentiation,

The center has no normal acceleration due to its straight-line path, although points on the wheel have normal components relative to the center.

7. Gears and Belt Drives

For two involute gears in mesh without slip at the pitch point, the tangential velocities are equal in magnitude:

where r₁ and r₂ are pitch radii. The gears rotate in opposite directions for external meshing. Thus,

where N₁ and N₂ are the numbers of teeth.

For an open belt drive without slip,

and both pulleys rotate in the same direction. For a crossed belt, the directions are opposite. These relations assume negligible belt creep, no slip, and compatible pitch dimensions.

8. Rotating Reference Frames

A rotating reference frame is useful when the geometry is most naturally described relative to a rotating member. If a vector A is observed from a rotating frame with angular velocity ω_rel relative to an inertial frame, the transport theorem is

For a particle whose position is described by an origin O' and a rotating frame,

The corresponding acceleration is

The terms are, respectively, origin acceleration, relative acceleration, Euler acceleration, centripetal acceleration, and Coriolis acceleration. The Coriolis term appears only when the particle has relative velocity in a rotating frame. These equations assume that the frame's axes rotate as a rigid set and that all vectors are expressed consistently in one coordinate system.

9. Mechanisms and Degrees of Freedom

A mechanism is a set of connected rigid bodies that produces a prescribed motion. In planar mechanisms, each free rigid body has three degrees of freedom: two translations and one rotation. Joints remove degrees of freedom. A revolute joint permits rotation but prevents relative translation; a prismatic joint permits translation along one direction but prevents rotation.

A useful planar mobility estimate is the Gruebler–Kutzbach relation:

where M is mobility, n is the number of links including the frame, j₁ is the number of one-degree-of-freedom joints, and j₂ is the number of two-degree-of-freedom joints. The relation assumes conventional planar lower and higher pairs without redundant constraints or special geometries. It is a counting formula, not a substitute for geometric analysis.

For a four-bar linkage, the position of each joint is obtained from loop-closure equations. If vectors around a closed loop are summed,

Differentiating gives a velocity loop equation, and differentiating again gives an acceleration loop equation. This procedure is systematic and is readily implemented in computer-aided mechanism software.

10. Applied Case: Crank–Slider Mechanism

Background

A crank–slider mechanism converts rotary motion into reciprocating motion. It is used in internal-combustion engines, compressors, pumps, and reciprocating saws.

Problem

Consider a crank OA of radius 0.075 m rotating at constant angular speed 1200 rpm. A connecting rod AB is 0.30 m long, and the slider B moves horizontally. At the instant OA is 60° above the positive horizontal, determine the slider velocity using relative velocity.

Analysis

Convert rotational speed:

The crank-pin velocity is perpendicular to OA:

Its components are

The rod AB has horizontal projection from A to B. Its vertical separation is fixed by the slider path. At this instant, the geometry gives

Thus φ ≈ 12.50°. The relative velocity of B with respect to A is perpendicular to AB:

Because B has no vertical velocity,

Therefore,

The negative sign indicates clockwise rotation of the connecting rod. The horizontal slider velocity is

The slider therefore moves left at approximately 7.11 m/s at the stated configuration.

Outcome and Learning Points

The example illustrates the importance of converting rpm to rad/s, resolving velocities into components, enforcing the slider constraint, and distinguishing the crank-pin velocity from the slider velocity. In practical engine analysis, acceleration is also required because it determines inertial forces, bearing loads, vibration, and balancing requirements.

11. Common Errors and Best Practices

1. Using scalar equations without directions. Always establish axes and signs before substituting magnitudes.
2. Confusing angular velocity with tangential velocity. The relation is v = rω only for a point at radius r in fixed-axis rotation.
3. Omitting normal acceleration. A point may have substantial centripetal acceleration even when angular speed is constant.
4. Treating the instantaneous center as permanently fixed. It is an instantaneous kinematic construction.
5. Mixing degrees and radians. Angular-rate formulas require consistent angular units; use radians in numerical calculations.
6. Ignoring constraints. A slider cannot have velocity or acceleration perpendicular to its guide.
7. Using a relative-motion equation for points not fixed in the same body. If a point slides in a slot, relative-motion and rotating-frame terms must be used appropriately.
8. Failing to check units and limiting cases. For example, if ω = 0, the normal acceleration term must vanish.

Computer-aided engineering packages can solve complex mechanisms, but software results remain dependent on the model, coordinate conventions, joint definitions, and numerical tolerances. Engineers should validate software output with vector equations, limiting cases, and independent measurements. In safety-critical machinery, transparent assumptions and traceable calculations are essential.

Chapter Summary

Planar rigid-body kinematics describes the motion of bodies whose points remain in parallel planes. Translation gives identical velocity and acceleration to all points. Fixed-axis rotation produces tangential acceleration rα and normal acceleration rω². General plane motion is analyzed through relative velocity and acceleration equations, combining the motion of a reference point with rotation of the body.

The instantaneous center provides an efficient velocity method for planar motion and explains rolling without slipping. Gear and belt relations follow from equal tangential velocities at contact. Rotating-reference-frame equations introduce relative, Euler, centripetal, and Coriolis acceleration terms. Mechanism analysis combines these equations with geometric and kinematic constraints. Accurate solutions require consistent signs, units, directions, assumptions, and physical checks.

Key Terms

Rigid body: Ideal body whose distance between any two points remains constant.

Planar motion: Motion in which all points move in planes parallel to one reference plane.

Translation: Motion in which body-fixed lines remain parallel to their original orientations.

Fixed-axis rotation: Rotation about a stationary axis or point.

General plane motion: Combined translation and rotation in a plane.

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

Angular acceleration: Rate of change of angular velocity, measured in rad/s².

Tangential acceleration: Acceleration component that changes speed, equal to rα in fixed-axis rotation.

Normal acceleration: Inward acceleration component that changes velocity direction, equal to rω² in circular motion.

Relative velocity: Velocity of one point of a rigid body expressed relative to another point.

Relative acceleration: Acceleration relation including tangential and normal relative terms.

Instantaneous center: Point having zero velocity at a particular instant.

Rolling without slipping: Contact condition in which the instantaneous contact point has zero relative velocity.

Pitch radius: Effective radius at which meshing gears transmit tangential velocity.

Coriolis acceleration: Acceleration 2ω × v_rel arising from relative motion in a rotating frame.

Kinematic constraint: Geometric or motion condition restricting allowable movement.

Mobility: Number of independent inputs required to specify a mechanism configuration.

Loop closure: Vector equation requiring the net displacement around a closed mechanism loop to be zero.

Review Questions

Short Answer Questions

1. What distinguishes translation from general plane motion?
2. Why is normal acceleration directed toward the center of curvature?
3. Define the instantaneous center of zero velocity.
4. State the no-slip velocity condition for a rolling wheel.
5. When does Coriolis acceleration occur?

Long Answer Questions

1. Derive the velocity and acceleration equations for two points fixed in a rigid body undergoing general plane motion.
2. Explain the physical meanings of tangential and normal acceleration in fixed-axis rotation.
3. Describe how the instantaneous center is located from two known point velocities and discuss its limitations for acceleration analysis.
4. Explain the transport theorem and identify every term in the rotating-frame acceleration equation.
5. Discuss how velocity and acceleration loop equations are used in a four-bar or crank–slider mechanism.

Analytical / Application Questions

1. A 0.20 m rotating arm has angular velocity 15 rad/s and angular acceleration 3 rad/s². Determine the tangential, normal, and total acceleration of its end.
2. A wheel of radius 0.30 m rolls without slipping while its center moves at 2.4 m/s. Determine its angular velocity and explain the direction of rotation.
3. Design an analytical procedure for finding the slider acceleration in a crank–slider mechanism, identifying the required geometry, constraints, and relative-acceleration terms.

Multiple Choice Questions with Answers

Question 1

For a rigid body undergoing pure translation, which statement is correct?

A. All points have the same velocity and acceleration.

B. All points have the same angular velocity but different linear velocities.

C. Every point has centripetal acceleration toward a fixed center.

D. The body must follow a circular path.

Correct Answer: A. All points have the same velocity and acceleration.

Question 2

A point 0.20 m from a fixed rotation axis has angular velocity 10 rad/s. Its normal acceleration is:

A. 2 m/s²

B. 20 m/s²

C. 50 m/s²

D. 100 m/s²

Correct Answer: B. 20 m/s²

Question 3

The relative velocity equation for two points fixed in the same rigid body is:

A. v_B = v_A + α × r_{B/A}

B. v_B = v_A + ω × r_{B/A}

C. v_B = a_A + ω × r_{B/A}

D. v_B = v_A + ω²r_{B/A}

Correct Answer: B. v_B = v_A + ω × r_{B/A}

Question 4

In general plane motion, the term α × r_{B/A} represents:

A. Relative tangential acceleration

B. Relative normal acceleration

C. Translation of the reference frame

D. Coriolis acceleration

Correct Answer: A. Relative tangential acceleration

Question 5

For a wheel rolling without slipping on a stationary surface, the instantaneous center is:

A. The wheel center

B. The topmost point of the wheel

C. The contact point with the surface

D. A point at infinity

Correct Answer: C. The contact point with the surface

Question 6

Two external gears mesh without slipping. If their pitch radii are r₁ and r₂, their angular velocities satisfy:

A. ω₁r₁ = ω₂r₂ with the same rotation direction

B. ω₁r₂ = ω₂r₁

C. ω₁r₁ = −ω₂r₂

D. ω₁ + ω₂ = r₁ + r₂

Correct Answer: C. ω₁r₁ = −ω₂r₂

Question 7

Which acceleration term occurs because a particle moves relative to a rotating reference frame?

A. Translational acceleration

B. Coriolis acceleration

C. Gravitational acceleration

D. Static acceleration

Correct Answer: B. Coriolis acceleration

Question 8

A rotating arm has radius 0.5 m and angular acceleration 6 rad/s². Its tangential acceleration is:

A. 0.083 m/s²

B. 3.0 m/s²

C. 6.5 m/s²

D. 12 m/s²

Correct Answer: B. 3.0 m/s²

Question 9

The no-slip condition at a gear pitch point requires equality of:

A. Angular accelerations only

B. Radial positions only

C. Tangential velocities

D. Masses of the gears

Correct Answer: C. Tangential velocities

Question 10

The planar Gruebler–Kutzbach mobility relation assumes, among other conditions:

A. All links are deformable

B. Conventional joints without redundant constraints

C. No joints are present

D. Motion is three-dimensional

Correct Answer: B. Conventional joints without redundant constraints