What is Rayleigh’s method?

Short Answer:

Rayleigh’s method is an approximate energy-based technique used to determine the natural frequency of a vibrating system. It works by comparing the system’s maximum potential energy and kinetic energy during vibration. When these two energies are equal, the system’s natural frequency can be estimated.

This method is especially useful for complex structures like beams, plates, and continuous systems where solving differential equations is difficult. Rayleigh’s method gives accurate results for the fundamental (lowest) natural frequency and helps engineers analyze vibration behavior efficiently.

Detailed Explanation :

Rayleigh’s Method

Rayleigh’s method is one of the most widely used approximate techniques in vibration analysis to calculate the natural frequency of vibrating systems. It is based on the principle of conservation of energy, which states that, during vibration, the total energy of a conservative system remains constant. This means that the maximum potential energy and kinetic energy are equal for any given mode of vibration.

The method was introduced by Lord Rayleigh (John William Strutt) in the 19th century and remains a cornerstone of vibration theory due to its simplicity and accuracy for many practical systems.

Rayleigh’s method is particularly effective for systems where the motion is distributed, such as beams, plates, and continuous shafts, and where solving the full differential equations of motion would be complex or time-consuming.

Basic Principle

The basic principle behind Rayleigh’s method is the energy balance between potential and kinetic energy in a vibrating system.

During vibration:

  • At maximum displacement, the potential energy is maximum, and kinetic energy is zero.
  • At mean position, the kinetic energy is maximum, and potential energy is zero.

Thus, the total energy of the system remains constant and can be expressed as:

From this condition, the natural frequency () of the system can be estimated.

Mathematical Expression

For a vibrating system, let

  •  = Maximum Kinetic Energy
  •  = Maximum Potential Energy

Then, at equilibrium:

For a continuous system with distributed mass, the expression can be written as:

Where:

  •  = Natural frequency (rad/s)
  •  = Stiffness per unit length
  •  = Mass per unit length
  •  = Mode shape or assumed deflection function
  •  = Coordinate along the system length

This equation gives the approximate value of the natural frequency if the assumed shape  is close to the actual mode shape.

For a simple system (lumped mass and spring):

which is the same result obtained from Newton’s law of motion, but derived here using energy balance principles.

Steps in Rayleigh’s Method

  1. Identify the System:
    Select the system to be analyzed (beam, shaft, spring-mass system, etc.).
  2. Assume the Mode Shape:
    Assume a reasonable deflection shape  that satisfies the boundary conditions of the system. The accuracy of the result depends on the correctness of this assumed shape.
  3. Determine Maximum Potential Energy (V):
    Calculate the strain or potential energy stored in the system due to deformation.
    For a spring-mass system,

For a beam,

where  is Young’s modulus and  is the moment of inertia.

  1. Determine Maximum Kinetic Energy (T):
    Calculate the kinetic energy of the system due to its velocity.

where  is density and  is cross-sectional area.

  1. Apply Energy Balance:
    Equate the maximum potential energy to the maximum kinetic energy:
  1. Solve for Natural Frequency:
    Substitute all values into the formula and calculate  or .

Application to a Simple Example

Consider a simple cantilever beam with a concentrated mass  at its free end.

  1. Assumed mode shape:  (satisfying boundary conditions at the fixed end).
  2. Potential energy:
  1. Kinetic energy:
  1. Equate V and T:

Solving this gives an approximate expression for the natural frequency of the beam.

Although this approach is approximate, the results are close to the exact solution if the assumed deflection shape is chosen well.

Advantages of Rayleigh’s Method

  1. Simple and direct — avoids solving differential equations.
  2. Provides accurate estimates of fundamental natural frequency.
  3. Useful for systems with complex geometry or boundary conditions.
  4. Applicable to both lumped and continuous systems.
  5. Provides insight into how mass and stiffness distribution affect vibration.

Limitations of Rayleigh’s Method

  1. Only gives the first (fundamental) natural frequency accurately.
  2. Accuracy depends on the assumed mode shape.
  3. Cannot easily handle systems with damping or external forces.
  4. Higher-order natural frequencies require more advanced techniques like Rayleigh–Ritz method.

Applications of Rayleigh’s Method

  • Vibration analysis of beams, plates, and rotating shafts.
  • Estimation of natural frequencies in mechanical and civil structures.
  • Preliminary design of machine components to avoid resonance.
  • Aeroelastic and structural vibration studies in aerospace engineering.
  • Analysis of bridges and tall structures under wind or seismic loads.
Conclusion

Rayleigh’s method is an approximate energy-based technique used to estimate the natural frequency of vibrating systems. It is based on the principle that the maximum kinetic energy equals the maximum potential energy during vibration. The method is simple, efficient, and provides accurate results for the fundamental frequency of systems like beams, plates, and mechanical structures. Although approximate, it remains one of the most useful tools in vibration analysis, especially for initial design and estimation purposes.