Vibration Experiment: Free, Forced and Damped Vibration Procedure Using Vibration Test Rig

Vibration Experiment: Free, Forced and Damped Vibration Procedure Using Vibration Test Rig

The vibration experiment is a key laboratory exercise in the Dynamics and Theory of Machines syllabus for mechanical engineering programmes. It uses a vibration test rig to demonstrate the principles of free undamped vibration, free damped vibration, and forced vibration with resonance. This guide covers the complete theory of single-degree-of-freedom (SDOF) systems, experimental procedure using a standard motorised vibration test rig, observation tables, and calculation of natural frequency, damping ratio, and resonance frequency.

Objectives

• To determine the natural frequency of a spring-mass system by free vibration experiment and compare with the theoretical value.
• To determine the damping coefficient and damping ratio from a free damped vibration experiment using the logarithmic decrement method.
• To plot the frequency response curve (amplitude vs frequency) for forced vibration and identify the resonance frequency.

Theory

Free Undamped Vibration

Natural frequency: omega_n = root(k/m) rad/s, or f_n = (1/2pi) x root(k/m) Hz

Time period: T = 1/f_n = 2pi x root(m/k) seconds

Free Damped Vibration

Logarithmic decrement: delta = (1/n) x ln(x1/x(n+1)), where x1 and x(n+1) are amplitudes of successive cycles n apart.

Damping ratio: zeta = delta / root(4pi squared + delta squared)

Forced Vibration

At resonance, the forcing frequency equals the natural frequency: omega = omega_n. At resonance, amplitude is maximum and is limited only by damping.

Magnification factor: MF = X / (F/k) = 1 / root((1 – r squared)squared + (2 zeta r)squared), where r = omega / omega_n

Apparatus

Scientico CE-certified vibration test rig includes:

• Beam-and-spring assembly with adjustable mass positions
• Interchangeable spring sets (multiple stiffness values)
• Variable-speed eccentric motor (0-1500 rpm) for forced vibration excitation
• Digital tachometer for excitation frequency measurement
• Amplitude measurement scale with pointer or LVDT sensor
• Viscous damper with adjustable dashpot for damping ratio variation
• Data acquisition interface (optional computerised version)

Part A: Free Undamped Vibration Procedure

1. Set up the beam with the test mass at the specified position. Note the spring stiffness k (N/m) from the apparatus data sheet.
2. Calculate theoretical natural frequency: f_n(theory) = (1/2pi) x root(k/m).
3. Give the mass a small initial displacement (10-15 mm) and release. Start the stopwatch simultaneously.
4. Count 10 complete oscillations and record the total time t. Calculate T = t/10 (period) and f_n = 1/T.
5. Repeat three times and average the period.
6. Vary the mass (add extra weights) and repeat to show the effect of mass on natural frequency.

Part B: Free Damped Vibration Procedure

1. Engage the dashpot damper at the lowest damping setting.
2. Give the mass a measured initial displacement x0 (20-25 mm). Release and allow to oscillate freely.
3. Record the amplitude of successive peaks: x1, x2, x3, x4, x5 using the scale pointer.
4. Calculate logarithmic decrement: delta = (1/4) x ln(x1/x5).
5. Calculate damping ratio: zeta = delta / root(4pi squared + delta squared).
6. Increase dashpot setting and repeat to show effect of increased damping.

Part C: Forced Vibration and Resonance

1. Switch on the eccentric motor. Set the minimum speed.
2. At each motor speed setting, allow the system to reach steady-state (30-60 seconds). Record motor speed N (rpm) and steady-state amplitude X (mm).
3. Increase motor speed in steps of 50-100 rpm. Take 15-20 readings from well below to well above the expected resonance frequency.
4. Plot X vs N (or X vs frequency f). Identify the resonance peak (maximum amplitude).
5. Compare the resonance frequency with the natural frequency from Part A.

Observation Tables

Part A — Free Undamped Vibration: Spring stiffness k = _____ N/m | Mass m = _____ kg

Trial	Time for 10 oscillations (s)	Period T (s)	Experimental f_n (Hz)	Theoretical f_n (Hz)	% Error
1
2
3

Part B — Free Damped Vibration:

Amplitude x1 (mm)	x2 (mm)	x3 (mm)	x4 (mm)	x5 (mm)	Log decrement delta	Damping ratio zeta

Precautions

• Keep initial displacement small (below 15% of static deflection) to maintain linear SDOF behaviour.
• For forced vibration, approach resonance slowly — amplitude increases rapidly near resonance and can damage the apparatus if not controlled.
• Ensure all bolts on the beam and spring are tight before each test.
• For the damped test, ensure dashpot oil level is correct — low oil gives under-estimated damping ratio.

Related Experiments

Perform alongside the governor apparatus experiment. Browse Scientico vibration technology equipment and full Theory of Machines lab range. Request a CIF proforma invoice within 48 hours for export orders.

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