Young's Modulus by Vibration Method - Lab Manual

To Determine Young's Modulus of the Material of a Beam by Method of Vibration

Lab Manual
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1. Aim

To determine the Young's modulus of elasticity of the material of a given beam by the method of vibration.

2. Apparatus Used

  • A metallic beam (rectangular cross-section)
  • Two knife edges to support the beam
  • A meter scale
  • A screw gauge/micrometer
  • A vernier caliper
  • A stopwatch
  • A small hammer
  • Sand/iron filings
  • Graph paper
  • A weight hanger with weights (optional)

3. Diagram

Experimental setup showing rectangular beam supported on two knife edges with measurement points for dimensions and vibration nodes representation

Fig. 1: Experimental setup showing rectangular beam supported on two knife edges with measurement points for dimensions and vibration nodes representation

4. Theory

When a beam is set into transverse vibration, it vibrates with a natural frequency that depends on its dimensions, density, and elastic properties, particularly Young's modulus.

For a uniform beam of rectangular cross-section supported at both ends, the fundamental frequency of vibration is related to Young's modulus by the following relationship:

The beam vibrates according to the wave equation, and the natural frequency of vibration depends on:

  • The length of the beam between supports (L)
  • The moment of inertia of the cross-section (I)
  • The mass per unit length (m)
  • Young's modulus (E)

For a beam with both ends free to vibrate (simply supported):

The fundamental frequency is given by:

\[f = \frac{\pi}{2L^2}\sqrt{\frac{EI}{\rho A}}\]

Where:

  • f is the frequency of vibration
  • L is the length of the beam between supports
  • E is Young's modulus
  • I is the moment of inertia of the cross-section
  • ρ is the density of the beam material
  • A is the cross-sectional area

For a rectangular beam:

  • Moment of inertia I = bd³/12
  • Area A = bd
  • Where b is the width and d is the thickness of the beam

5. Formula

For a beam with rectangular cross-section, the formula for Young's modulus is:

\[E = \frac{48\pi^2 \rho L^4 f^2}{d^2}\]

Where:

  • E = Young's modulus (N/m²)
  • ρ = Density of the material (kg/m³)
  • L = Length of the beam between supports (m)
  • f = Fundamental frequency of vibration (Hz)
  • d = Thickness/depth of the beam (m)

Alternatively, if the mass of the beam (M) is known:

\[E = \frac{48\pi^2 M L^3 f^2}{bd^3}\]

Where:

  • M = Mass of the beam (kg)
  • b = Width of the beam (m)

6. Procedure

  1. Measure the dimensions of the beam:
    • Measure the length (L) of the beam using a meter scale.
    • Measure the width (b) using vernier calipers at several positions and take the average.
    • Measure the thickness (d) using a micrometer/screw gauge at several positions and take the average.
  2. Determine the mass of the beam:
    • Weigh the beam using a physical balance to determine its mass (M).
    • Calculate the density (ρ) of the material if required, using ρ = M/(L×b×d).
  3. Set up the experimental arrangement:
    • Place the beam horizontally on two knife edges separated by a distance L.
    • Ensure the beam is placed symmetrically on the knife edges.
  4. Vibration measurement:
    • Sprinkle some sand or iron filings lightly over the beam surface.
    • Gently strike the beam at its center using a small hammer to set it into vibration.
    • Observe the pattern formed by the sand/filings. These patterns (called Chladni patterns) show the nodal lines where vibration is minimal.
    • Alternatively, observe the beam vibrations directly.
  5. Frequency determination:
    • Count the number of vibrations in a given time using the stopwatch.
    • Repeat this several times and take the average.
    • Calculate the frequency (f) by dividing the number of vibrations by the time taken.
  6. Alternative method for frequency determination:
    • Apply small weights at the center of the beam and measure the time period for different loads.
    • Plot a graph between the time period and the square root of the load.
    • Extrapolate to find the natural frequency of the unloaded beam.

7. Observation Table

A. Dimensional Measurements

Parameter Measurement 1 Measurement 2 Measurement 3 Average Value
Length (L) ... m ... m ... m ... m
Width (b) ... m ... m ... m ... m
Thickness (d) ... m ... m ... m ... m
Mass (M) ... kg ... kg ... kg ... kg

B. Frequency Determination

Trial Number of Vibrations Time Taken (s) Frequency (Hz)
1 ... ... ...
2 ... ... ...
3 ... ... ...
4 ... ... ...
5 ... ... ...
Average ...

C. Alternative Method (if used)

Load (kg) Time Period (s) √Load
0 ... 0
0.1 ... ...
0.2 ... ...
0.3 ... ...
0.4 ... ...
0.5 ... ...

8. Calculations

  1. Calculate the average dimensions:
    • Average length (L) = ... m
    • Average width (b) = ... m
    • Average thickness (d) = ... m
    • Mass (M) = ... kg
    • Density (ρ) = M/(L×b×d) = ... kg/m³
  2. Calculate the average frequency:
    • Average frequency (f) = ... Hz
  3. Calculate Young's modulus:

    Using the formula:

    \[E = \frac{48\pi^2 \rho L^4 f^2}{d^2}\]

    Substituting values:

    E = 48 × π² × (density) × (length)⁴ × (frequency)² / (thickness)²

    E = 48 × (3.14159)² × ... × ...⁴ × ...² / ...²

    E = ... N/m²

  4. Error calculation (optional):
    • Calculate the percentage error from standard value (if known)
    • Error = [(Experimental value - Standard value) / Standard value] × 100%

9. Result

The Young's modulus of elasticity of the material of the given beam determined by the method of vibration is ... × 10⁹ N/m² or ... GPa.

10. Precautions

  1. Ensure the beam has a uniform cross-section throughout its length.
  2. The supports (knife edges) should be sharp and rigid.
  3. The beam should be placed symmetrically on the supports.
  4. Measure the dimensions of the beam carefully at multiple positions.
  5. Strike the beam gently to produce small amplitude vibrations.
  6. Ensure the beam is clean and free from dirt or grease.
  7. Take multiple readings and use average values for accurate results.
  8. The beam should be free from initial stress and strain.
  9. The beam should not be too heavy or too light for the setup.
  10. Avoid any external vibrations that might interfere with the experiment.
  11. Ensure the beam is perfectly horizontal when placed on the supports.
  12. The laboratory temperature should be noted as it affects the Young's modulus.

11. Viva Voce Questions

Q: What is Young's modulus?
A: Young's modulus is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime.
Q: Why is the vibration method preferred for determining Young's modulus?
A: The vibration method is a dynamic, non-destructive technique that often yields more accurate results than static methods. It also allows for testing of materials that might be difficult to measure with traditional tensile testing.
Q: How does the frequency of vibration change with the length of the beam?
A: The frequency of vibration is inversely proportional to the square of the length. If the length doubles, the frequency decreases by a factor of four.
Q: How does the thickness of the beam affect its vibration frequency?
A: The frequency is directly proportional to the thickness. If thickness doubles, the frequency also doubles.
Q: What are Chladni patterns and how are they formed?
A: Chladni patterns are nodal patterns that form on vibrating surfaces. When sand or similar particles are placed on a vibrating surface, they move away from areas of high vibration (antinodes) and collect at areas of minimal vibration (nodes), forming characteristic patterns.
Q: How does temperature affect Young's modulus?
A: Generally, Young's modulus decreases with increasing temperature for most materials, as higher temperatures reduce the stiffness of materials.
Q: What is the difference between fundamental frequency and overtones?
A: The fundamental frequency is the lowest frequency at which an object vibrates. Overtones are higher frequencies at which the object can also vibrate, typically multiples of the fundamental frequency.
Q: How does the density of the material affect the natural frequency?
A: Higher density leads to lower natural frequency. The frequency is inversely proportional to the square root of density.
Q: What are the sources of error in this experiment?
A: Sources of error include inaccurate measurement of dimensions, imperfect beam supports, non-uniform cross-section, environmental vibrations, and timing errors when measuring frequency.
Q: Why is Young's modulus important in engineering applications?
A: Young's modulus is crucial for predicting how materials will deform under load, helping engineers design structures that are safe, functional, and efficient. It's essential in structural engineering, mechanical design, and materials science.
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