Vibration of Spring-Mass System — Free and Forced Vibration Experiment Procedure

The vibration of a spring-mass system is one of the fundamental experiments in the Theory of Machines and Mechanical Vibrations laboratory. Understanding natural frequency, damping, and resonance is essential for mechanical engineers designing rotating machinery, structures, and automotive components. This guide provides the complete experiment procedure, calculations, and observation tables.

Aim of the Experiment

1. To determine the natural frequency of a spring-mass system under free vibration.
2. To study the effect of damping on free vibrations and determine the damping coefficient.
3. To study forced vibrations and observe the resonance condition.

Theory

Free Vibration — Undamped

A spring-mass system consists of a mass m attached to a spring of stiffness k. When displaced from its equilibrium position and released, the mass undergoes simple harmonic motion.

Equation of motion: mẍ + kx = 0

Natural frequency: fn = (1/2π)√(k/m) Hz
Natural circular frequency: ωn = √(k/m) rad/s
Time period: T = 2π√(m/k) seconds

Free Vibration — Damped

When damping is present (dashpot or viscous damping), the equation of motion becomes:
mẍ + cẋ + kx = 0

where c = damping coefficient (N·s/m)

Critical damping coefficient: cc = 2√(km) = 2mωn

Damping ratio: ζ = c/cc

For ζ < 1 (underdamped), the system oscillates with exponentially decaying amplitude:
x(t) = X·e^(-ζωnt)·cos(ωdt + φ)

Damped natural frequency: ωd = ωn√(1 – ζ²)

Logarithmic decrement: δ = ln(x₁/x₂) = 2πζ/√(1 – ζ²)

Forced Vibration and Resonance

When an external periodic force F(t) = F₀sin(ωt) is applied:
mẍ + cẋ + kx = F₀sin(ωt)

Amplitude at resonance (ω = ωn): X_res = F₀/(cωn)

Resonance: When the forcing frequency ω equals the natural frequency ωn, the amplitude reaches maximum. In undamped systems, amplitude theoretically becomes infinite at resonance.

Apparatus Required

• Spring-mass vibration apparatus (single degree of freedom — motorised exciter)
• Springs of different stiffness (k₁, k₂)
• Masses (disc weights) of known weight
• Dashpot assembly (for damped vibration)
• Stroboscope or optical tachometer
• Displacement sensor / dial gauge or LVDT
• Stopwatch
• Ruler

Procedure — Free Undamped Vibration

1. Measure and record the stiffness k of the spring using a load-deflection test (apply known weights and measure deflection).
2. Attach mass m₁ to the spring. Ensure the system is vertical and in equilibrium.
3. Give the mass a small displacement (5–10 mm) and release it.
4. Record the time for 10 complete oscillations using a stopwatch. Calculate the time period T and frequency fn.
5. Compare with the theoretical value: fn_th = (1/2π)√(k/m).
6. Repeat with masses m₂, m₃ (by adding disc weights). Plot fn vs. 1/√m.

Procedure — Damped Free Vibration

1. Connect the dashpot to the mass-spring system.
2. Give the system an initial displacement and release.
3. Record successive amplitudes x₁, x₂, x₃ from the displacement trace or dial gauge.
4. Calculate logarithmic decrement: δ = (1/n)ln(x₁/xₙ₊₁)
5. Calculate damping ratio: ζ = δ/√(4π² + δ²)
6. Change the dashpot oil to vary damping and repeat.

Procedure — Forced Vibration

1. Connect the motorised eccentric exciter to the spring-mass system.
2. Start the motor at the lowest speed setting.
3. Gradually increase the motor speed in steps, measuring the amplitude at each step using the dial gauge or displacement sensor.
4. Record the forcing frequency (from the stroboscope or tachometer) and corresponding amplitude.
5. Note the speed at which maximum amplitude occurs — this is the resonance condition.
6. Continue increasing speed beyond resonance and observe the decrease in amplitude.
7. Plot the frequency response curve (amplitude vs. forcing frequency ratio ω/ωn).

Observation Tables

Table 1 — Free Undamped Vibration

Table 2 — Damped Free Vibration (Logarithmic Decrement)

Calculations

Spring stiffness from load-deflection: k = W/δ = mg/x (N/m)

Natural frequency: fn = (1/2π)√(k/m)

Logarithmic decrement: δ = ln(x₁/x₂)

Damping ratio: ζ = δ/√(4π² + δ²)

Damped natural frequency: fd = fn√(1 – ζ²)

Critical damping coefficient: cc = 2mωn = 2√(km)

Actual damping coefficient: c = ζ × cc

Results

• Natural frequency of the system = _____ Hz (actual) vs _____ Hz (theoretical)
• Damping ratio ζ = _____
• The resonance condition was observed at forcing frequency = _____ Hz
• At resonance, the amplitude was _____ mm (with damping = _____ Ns/m)

Viva Questions

1. What is the difference between free vibration and forced vibration?
2. Define resonance. Why is resonance dangerous in engineering structures?
3. What is the effect of increasing damping on the resonant amplitude?
4. Define degree of freedom. Give examples of 1-DOF and 2-DOF systems.
5. What is logarithmic decrement and how is it used to find the damping ratio?
6. What is the difference between overdamped, underdamped, and critically damped systems?
7. How does adding mass affect the natural frequency of a spring-mass system?

Lab Equipment Featured in This Guide

Manufactured in-house by Scientico India — ISO 9001:2015 & CE certified, exported to 60+ countries. Request a CIF quote within 24 hours.

Building Management System Trainer | PolarX – 568View details & get quote →Recycle Loops | ThermoFlux – 7089View details & get quote →Spring Mass Vibration System | Vibrano X – 09View details & get quote →Cam Analysis Machine | FortiTestX – 24View details & get quote →Demonstration Infiltration Apparatus | FluidoSurge-X 396View details & get quote →Solar Energy Trainer with Connection to Mains | RVX-003View details & get quote →

Setting up or upgrading a lab? Send your equipment list or syllabus — get a CIF proforma invoice within 24 hours.

💬 WhatsApp a QuoteRequest a Quote