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Qwestrum Engineering360 · Mechanical Engineering · Dynamics of Machines

Vibration of Single Degree Systems

An SDOF system oscillates at natural frequency ω_n = √(k/m); damping ratio ζ = c/(2√(km)) decides whether the response is under-, critically, or over-damped. The logarithmic decrement δ = 2πζ/√(1−ζ²) extracts ζ from a decay trace, per 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.

  • Underdamped (ζ < 1): oscillatory decay; overdamped: aperiodic
  • Critical damping ζ = 1 — fastest return without oscillation
  • Logarithmicdecrementδ=ln(x1x2)=2πζ/1−ζ2Logarithmic decrement \delta = ln(\frac{x_{1}}{x_{2}}) = 2\piζ/\sqrt{1-ζ^{2}}

Topic details

Introduction

Single-degree-of-freedom vibration is the foundation of the whole vibrations course and a certain GATE question. The student writes the equation of motion mẍ + cẋ + kx = 0 (free) or = F₀sinωt (forced) from a free-body diagram, then reads off ω_n and ζ.

Scope in B.Tech and GATE syllabus

SS Rattan classifies free response by ζ: under-damped (ζ<1, decaying oscillation), critically damped (ζ=1, fastest non-oscillatory return), and over-damped (ζ>1, sluggish return). The logarithmic decrement links successive amplitude peaks to ζ, a common experimental problem.

Why this topic matters in practice

Forced vibration introduces the magnification factor and resonance at ω = ω_n; transmissibility governs vibration isolation of machines on mounts. Recognising which sub-case a problem belongs to before substituting is essential.

Key relations & formulas

ωn=km\omega_{n} = \sqrt{\frac{k}{m}}
(undamped natural frequency, rad/s)
fn=ωn(2π)=(12π)kmf_{n} = \frac{\omega_{n}}{(2\pi)} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}
(Hz)
x(t)=Acos(ωnt)+Bsin(ωnt)x(t) = A cos(\omega_{n} t) + B sin(\omega_{n} t)
(free undamped response)
ζ=c(2km)=c(2mωn)ζ = \frac{c}{(2\sqrt{km})} = \frac{c}{(2m\omega_{n})}
(damping ratio)

Notation and sign conventions

Relation 1 —
ωn=√\omega_{n} = √
ωn=km\omega_{n} = \sqrt{\frac{k}{m}}
(undamped natural frequency, rad/s)
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 —
fn=ωn/f_{n} = \omega_{n}/
fn=ωn(2π)=(12π)kmf_{n} = \frac{\omega_{n}}{(2\pi)} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}
(Hz)
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 —
xx
x(t)=Acos(ωnt)+Bsin(ωnt)x(t) = A cos(\omega_{n} t) + B sin(\omega_{n} t)
(free undamped response)
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 —
ζ=c/ζ = c/
ζ=c(2km)=c(2mωn)ζ = \frac{c}{(2\sqrt{km})} = \frac{c}{(2m\omega_{n})}
(damping ratio)
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

Newton's law on a spring-mass-damper gives mẍ + cẋ + kx = 0. Dividing by m yields ẍ + 2ζω_n·ẋ + ω_n²·x = 0, with ω_n = √(k/m) and ζ = c/(2mω_n) = c/(2√(km)).

Governing relations in practice

For ζ < 1 the motion is x = e^(−ζω_n t)(A cosω_d t + B sinω_d t), damped frequency ω_d = ω_n√(1−ζ²). The exponential envelope's decay rate gives the logarithmic decrement δ = ln(x₁/x₂) = 2πζ/√(1−ζ²).

Design and analysis considerations

Critical damping c_c = 2√(km) is the boundary; a door closer or instrument is tuned near ζ = 1 to return quickly without overshoot. Over-damped systems return slowly with no oscillation.

Advanced theory and extensions

Under forced excitation the steady amplitude peaks near ω = ω_n (resonance), limited only by damping. Machines are isolated by choosing mount stiffness so the forcing frequency is well above √2·ω_n, where transmissibility falls below one — the practical design goal.

Assumptions and validity limits

State assumptions explicitly before using any relation for vibration of single degree systems — 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 Dynamics 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 Dynamics 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 vibration of single degree systems.
4. Use equation 1:
ωn=√\omega_{n} = √
.
5. Use equation 2:
fn=ωn/f_{n} = \omega_{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

Vibration of Single Degree Systems appears in engines, flywheels, and high-speed shafts. In Indian mechanical curricula this topic is tested because it connects theory to balancing, vibration, and rotational dynamics.
GATE and semester exams often combine vibration of single degree systems with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use vibration of single degree systems?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Forgetting to convert k and m into ω_n = √(k/m) before finding frequency in Hz (÷2π)
• Using ω_n instead of the damped frequency ω_d for an under-damped period
• Mixing damping ratio ζ with damping coefficient c
• Assuming resonance is exactly at ω_n when for amplitude it is slightly below for damped systems

Quick revision checklist

Before attempting vibration of single degree systems problems, confirm you can:
1. Underdamped (ζ < 1): oscillatory decay; overdamped: aperiodic
2. Critical damping ζ = 1 — fastest return without oscillation
3.
Logarithmicdecrementδ=ln(x1x2)=2πζ/1−ζ2Logarithmic decrement \delta = ln(\frac{x_{1}}{x_{2}}) = 2\piζ/\sqrt{1-ζ^{2}}
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.

Natural frequency of an SDOF system

Problem

A mass m = 20 kg is on a spring of stiffness k = 20000 N/m. Find the undamped natural frequency in rad/s and Hz.

Solution

ω_n = √(k/m) = √(20000/20) = √1000 = 31.62 rad/s; f_n = ω_n/2π = 31.62/6.283 = 5.03 Hz.

Conceptual check — Vibration of Single Degree Systems

Problem

In a Dynamics of Machines semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of vibration of single degree systems." 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 Vibration of Single Degree Systems, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    An SDOF system oscillates at natural frequency ω_n = √(k/m); damping ratio ζ = c/(2√(km)) decides whether the response is under-, critically, or over-damped. The logarithmic decrement δ = 2πζ/√(1−ζ²) extracts ζ from a decay trace, per SS Rattan.
  2. 2
    State the relation ω_n = √ and name each symbol.

    Model answer

    The governing relation is ωn=√\omega_{n} = √. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation f_n = ω_n/ and name each symbol.

    Model answer

    The governing relation is fn=ωn/f_{n} = \omega_{n}/. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation x and name each symbol.

    Model answer

    The governing relation is xx. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation ζ = c/ and name each symbol.

    Model answer

    The governing relation is ζ=c/ζ = c/. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Underdamped (ζ < 1): oscillatory decay; overdamped: aperiodic

    Model answer

    Underdamped (ζ < 1): oscillatory decay; overdamped: aperiodic — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Critical damping ζ = 1 — fastest return without oscillation

    Model answer

    Critical damping ζ = 1 — fastest return without oscillation — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Logarithmic decrement δ = ln(x₁/x₂) = 2πζ/√(1−ζ²)

    Model answer

    Logarithmicdecrementδ=ln(x1x2)=2πζ/1−ζ2Logarithmic decrement \delta = ln(\frac{x_{1}}{x_{2}}) = 2\piζ/\sqrt{1-ζ^{2}} — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Forgetting to convert k and m into ω_n = √(k/m) before finding frequency in Hz (÷2π)?

    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: Using ω_n instead of the damped frequency ω_d for an under-damped period?

    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: Mixing damping ratio ζ with damping coefficient c?

    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: Assuming resonance is exactly at ω_n when for amplitude it is slightly below for damped systems?

    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
    SS Rattan Ch. 18 — write equation of motion from FBD first.
  • 2
    Avoid: Forgetting to convert k and m into ω_n = √(k/m) before finding frequency in Hz (÷2π)
  • 3
    Avoid: Using ω_n instead of the damped frequency ω_d for an under-damped period
  • 4
    Avoid: Mixing damping ratio ζ with damping coefficient c

📖 Standard books (India)

  • Theory of Machines — SS Rattan

    Read: Syllabus unit

    Kinematics, cams, governors, and balancing

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