---
title: "Rayleigh-Ritz and finite difference methods"
subject: "Theory of Mechanism and Machine II"
chapter: 9
key: "semester_7/theory_of_mechanism_and_machine_ii:rayleigh-ritz-finite-difference-notes"
source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/rayleigh-ritz-finite-difference-notes
site: BCT Engineering
---

# Rayleigh-Ritz and finite difference methods

Theory of Mechanism and Machine II · Chapter 9: Vibration of Multi Degree of Freedom Systems

- Asked in: year not given, 5 marks
- Video: [Rayleigh Ritz Method in FEM( Finite Element Method) | Rayleigh Ritz Method example in FEA](https://www.youtube.com/watch?v=A3eVp1_SEXY) — Mahesh Gadwantikar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/rayleigh-ritz-finite-difference-notes
- Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-9

## Question

Write short notes on the Rayleigh-Ritz method and the finite difference method for finding natural frequencies.

## Short answer

Rayleigh-Ritz approximates mode shape as a series of admissible functions and makes Rayleigh’s quotient stationary, giving an eigenvalue problem for upper-bound frequencies. Finite difference replaces spatial derivatives with difference formulas at grid points, forming a discrete eigenvalue problem whose accuracy increases with finer discretization.

## Answer

#### Rayleigh-Ritz method
It improves Rayleigh's quotient by using a trial function with several adjustable constants.

1. Assume the mode shape as a series of admissible functions (satisfying the geometric boundary conditions):
$$
y(x) = c_1\phi_1(x) + c_2\phi_2(x) + \cdots + c_n\phi_n(x)
$$
2. Write the Rayleigh quotient $\omega^2 = \dfrac{U_{max}}{T^*_{max}}$, where $U_{max}$ is the maximum strain energy and $T^*_{max}$ is the kinetic energy expression with $\omega^2$ taken out.
3. Make $\omega^2$ stationary with respect to every constant:
$$
\frac{\partial}{\partial c_i}\left(U_{max} - \omega^2T^*_{max}\right) = 0, \qquad i = 1,2,\ldots,n
$$
This gives $\left([K] - \omega^2[M]\right)\{c\} = 0$, an eigenvalue problem of order $n$.
4. The $n$ roots approximate the lowest $n$ natural frequencies, always as upper bounds. More terms give better accuracy, and the higher modes are less accurate.

#### Finite difference method
It replaces the derivatives in the differential equation by differences between displacements at equally spaced stations.

For a beam divided into stations a distance $h$ apart:
$$
\left.\frac{d^2y}{dx^2}\right|_i \approx \frac{y_{i+1} - 2y_i + y_{i-1}}{h^2}, \qquad \left.\frac{d^4y}{dx^4}\right|_i \approx \frac{y_{i+2} - 4y_{i+1} + 6y_i - 4y_{i-1} + y_{i-2}}{h^4}
$$

Procedure:
1. Divide the system into $n$ segments and write the governing equation, e.g. $EI\dfrac{d^4y}{dx^4} = \omega^2\rho Ay$, at every interior station.
2. Apply the boundary conditions (e.g. $y = 0$ and $y'' = 0$ at a simple support) using extra fictitious stations if necessary.
3. The result is $[A]\{y\} = \omega^2[B]\{y\}$, solved as an eigenvalue problem.

Accuracy improves as $h$ decreases, but the order of the matrix grows. For a uniform beam, 4 to 6 segments already give the first frequency within a few percent.

| Point | Rayleigh-Ritz | Finite difference |
|---|---|---|
| Based on | Energy (variational) | Differential equation |
| Unknowns | Constants of trial functions | Displacements at stations |
| Result bound | Upper bound | No fixed bound |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Rayleigh-Ritz improves Rayleigh's quotient by:
   A. using one fixed trial function
   B. using a series of admissible functions with adjustable constants
   C. ignoring boundary conditions
   D. using finite differences at stations
   Answer: B. The mode shape is y = c1 phi1 + ... + cn phin, and the constants are chosen to make omega^2 stationary. (Key point: Rayleigh-Ritz trial function)

2. The constants are found by requiring:
   A. U_max = 0
   B. d/dc_i (U_max - w^2 T*_max) = 0 for each i, giving an n-order eigenvalue problem
   C. all c_i = 1
   D. T*_max = U_max for every c
   Answer: B. Making the quotient stationary gives ([K] - w^2 [M]){c} = 0. (Key point: Stationary condition)

3. The Rayleigh-Ritz frequencies are:
   A. lower bounds
   B. exact
   C. upper bounds on the true frequencies
   D. unbounded
   Answer: C. Constraining the shape to a trial family makes the system stiffer than reality, so results are upper bounds. (Key point: Bound property)

4. The finite difference method:
   A. replaces derivatives by differences of displacements at equally spaced stations
   B. uses energy only
   C. needs no boundary conditions
   D. gives exact derivatives
   Answer: A. E.g. y'' is approximated by (y_{i+1} - 2 y_i + y_{i-1})/h^2, turning the equation into an eigenvalue problem. (Key point: Finite difference idea)

5. Reducing the station spacing h in the finite difference method:
   A. lowers accuracy and shrinks the matrix
   B. improves accuracy but increases the matrix order
   C. has no effect
   D. makes the answer an upper bound
   Answer: B. Smaller h improves accuracy but adds stations, hence a larger matrix. (Key point: Accuracy trade-off)

## Related in Theory of Mechanism and Machine II

- [Natural frequencies of a two-mass system](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Atwo-dof-natural-frequencies-mode-shapes) — Theory of Mechanism and Machine II, chapter 7
- [Three-mass chain frequencies and modes](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Athree-dof-eigenvalues-mode-shapes) — Theory of Mechanism and Machine II, chapter 8
- [Rod longitudinal and shaft torsional frequencies](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Alongitudinal-rod-torsional-shaft-frequencies) — Theory of Mechanism and Machine II, chapter 10

## Related in other subjects

- [Rayleigh-Ritz method and its steps](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aritz-method-steps) — Finite Element Method, chapter 4
- [Finite difference method; explicit and implicit schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afdm-explicit-implicit) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
- [Finite difference nodal temperatures in a square plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Atwo-dimensional-conduction-nodal-temperatures) — Heat Transfer, chapter 2
- [Ritz solution of beam under uniform load](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aritz-simply-supported-beam) — Finite Element Method, chapter 4