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Qwestrum Engineering360 · Mechanical Engineering · Theory of Machines

Velocity and Acceleration Analysis

Point velocities in a linkage follow the relative-velocity equation v_B = v_A + v_B/A, and accelerations split into tangential a_t = αr and normal a_n = ω²r components. Sliding pairs add the Coriolis term 2ωv_r, as detailed by SS Rattan.
Exam tip: use ω in rad/s for dynamics formulas; N in rpm needs ω = 2πN/60. Do not confuse flywheel C_s with governor speed control.

Key formulas & points

Skim these first — then read the full notes below.

  • Instantaneous centre (I-centre) method for velocity in planar links
  • Corioliscomponentappearsinslidingpairs(a=2ωvr)Coriolis component appears in sliding pairs (a = 2\omega v_{r})
  • Velocity polygon and acceleration polygon — graphical methods

Topic details

Introduction

Velocity and acceleration analysis is the computational core of TOM and a heavy-weight GATE topic. The relative-velocity method builds a velocity polygon; the instantaneous-centre (I-centre) method offers a faster graphical alternative using the Kennedy three-centre theorem.

Scope in B.Tech and GATE syllabus

For accelerations the polygon must include both the centripetal component a_n = ω²r (always directed toward the centre) and the tangential component a_t = αr. SS Rattan emphasises drawing these to scale so the unknown link acceleration closes the polygon.

Why this topic matters in practice

The Coriolis component 2ωv_r is the classic trap: it appears whenever a point slides along a rotating link, as in the crank-and-slotted-lever quick-return mechanism. Recognising when to add it — and its direction (rotate v_r by 90° in the sense of ω) — is what separates full marks from partial credit.

Key relations & formulas

vB=vA+vB/Av_{B} = v_{A} + v_{B/A}
(relative velocity equation)
aB=aA+aB/At+aB/Ana_{B} = a_{A} + a_{B/A}^t + a_{B/A}^n
(relative acceleration)
an=v2r=ω2ra_{n} = \frac{v^{2}}{r} = \omega^{2}r
(normal/centripetal component)
at=α⋅ra_{t} = \alpha\cdot r
(tangential component)

Notation and sign conventions

Relation 1 —
vB=vA+vB/Av_{B} = v_{A} + v_{B/A}
vB=vA+vB/Av_{B} = v_{A} + v_{B/A}
(relative velocity equation)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
aB=aA+aB/At+aB/Ana_{B} = a_{A} + a_{B/A}^t + a_{B/A}^n
aB=aA+aB/At+aB/Ana_{B} = a_{A} + a_{B/A}^t + a_{B/A}^n
(relative acceleration)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
an=v2r=ω2ra_{n} = \frac{v^{2}}{r} = \omega^{2}r
an=v2r=ω2ra_{n} = \frac{v^{2}}{r} = \omega^{2}r
(normal/centripetal component)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
at=α⋅ra_{t} = \alpha\cdot r
at=α⋅ra_{t} = \alpha\cdot r
(tangential component)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The relative-velocity equation v_B = v_A + v_B/A treats the velocity of B as A's velocity plus the velocity of B relative to A, which for a rigid link is ω×r perpendicular to AB. Drawing these as vectors and closing the polygon yields unknown magnitudes.

Governing relations in practice

Acceleration of a point on a rotating rigid link has two parts: the centripetal a_n = ω²r = v²/r pointing from the point to the centre of rotation, and the tangential a_t = αr perpendicular to r. Their vector sum gives the total link-point acceleration.

Design and analysis considerations

The instantaneous-centre method exploits that at any instant a link rotates about its I-centre; velocities are then simply ω times the distance from that centre. Kennedy's theorem locates the I-centre common to three links on a straight line.

Advanced theory and extensions

When a slider moves with velocity v_r along a link rotating at ω, the Coriolis acceleration 2ωv_r must be added; forgetting it corrupts the entire acceleration polygon. Its direction is v_r rotated 90° in the direction of ω.

Assumptions and validity limits

State assumptions explicitly before using any relation for velocity and acceleration analysis — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Theory of Machines viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Theory of Machines papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to velocity and acceleration analysis.
4. Use equation 1:
vB=vA+vB/Av_{B} = v_{A} + v_{B/A}
.
5. Use equation 2:
aB=aA+aB/At+aB/Ana_{B} = a_{A} + a_{B/A}^t + a_{B/A}^n
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Velocity and Acceleration Analysis appears in linkages, cams, gear trains, and governors. In Indian mechanical curricula this topic is tested because it connects theory to kinematics and kinetics of mechanisms.
GATE and semester exams often combine velocity and acceleration analysis with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use velocity and acceleration analysis?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Omitting the Coriolis component 2ωv_r in sliding-pair (quick-return) mechanisms
• Dropping the centripetal term a_n = ω²r and drawing only tangential acceleration
• Getting the direction of a_n wrong — it always points toward the centre of rotation
• Mislabelling the velocity-polygon vectors so the relative velocity is reversed

Quick revision checklist

Before attempting velocity and acceleration analysis problems, confirm you can:
1. Instantaneous centre (I-centre) method for velocity in planar links
2.
Corioliscomponentappearsinslidingpairs(a=2ωvr)Coriolis component appears in sliding pairs (a = 2\omega v_{r})

3. Velocity polygon and acceleration polygon — graphical methods
Revise the solved examples in SS Rattan — Theory of Machines and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Centripetal acceleration of a crank pin

Problem

A crank of radius r = 0.2 m rotates at ω = 30 rad/s. Find the centripetal (normal) acceleration of the crank pin.

Solution

a_n = ω²r = 30² × 0.2 = 900 × 0.2 = 180 m/s², directed from the pin toward the crank axis.

Conceptual check — Velocity and Acceleration Analysis

Problem

In a Theory of Machines semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of velocity and acceleration analysis." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Velocity and Acceleration Analysis, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Point velocities in a linkage follow the relative-velocity equation v_B = v_A + v_B/A, and accelerations split into tangential a_t = αr and normal a_n = ω²r components. Sliding pairs add the Coriolis term 2ωv_r, as detailed by SS Rattan.
  2. 2
    State the relation v_B = v_A + v_{B/A} and name each symbol.

    Model answer

    The governing relation is vB=vA+vB/Av_{B} = v_{A} + v_{B/A}. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation a_B = a_A + a_{B/A}^t + a_{B/A}^n and name each symbol.

    Model answer

    The governing relation is aB=aA+aB/At+aB/Ana_{B} = a_{A} + a_{B/A}^t + a_{B/A}^n. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation a_n = v²/r = ω²r and name each symbol.

    Model answer

    The governing relation is an=v2r=ω2ra_{n} = \frac{v^{2}}{r} = \omega^{2}r. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation a_t = α·r and name each symbol.

    Model answer

    The governing relation is at=α⋅ra_{t} = \alpha\cdot r. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Instantaneous centre (I-centre) method for velocity in planar links

    Model answer

    Instantaneous centre (I-centre) method for velocity in planar links — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Coriolis component appears in sliding pairs (a = 2ωv_r)

    Model answer

    Corioliscomponentappearsinslidingpairs(a=2ωvr)Coriolis component appears in sliding pairs (a = 2\omega v_{r}) — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Velocity polygon and acceleration polygon — graphical methods

    Model answer

    Velocity polygon and acceleration polygon — graphical methods — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Omitting the Coriolis component 2ωv_r in sliding-pair (quick-return) mechanisms?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Dropping the centripetal term a_n = ω²r and drawing only tangential acceleration?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Getting the direction of a_n wrong — it always points toward the centre of rotation?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Mislabelling the velocity-polygon vectors so the relative velocity is reversed?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Draw velocity diagram to scale — GATE often asks Coriolis in slider-crank.
  • 2
    Avoid: Omitting the Coriolis component 2ωv_r in sliding-pair (quick-return) mechanisms
  • 3
    Avoid: Dropping the centripetal term a_n = ω²r and drawing only tangential acceleration
  • 4
    Avoid: Getting the direction of a_n wrong — it always points toward the centre of rotation

📖 Standard books (India)

  • Theory of Machines — SS Rattan

    Read: Syllabus unit

    Kinematics, cams, governors, and balancing

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