Functional Relationships

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The equation for the natural frequency of a spring-mas system is deduced using the Rayleigh method.  This method seeks to find exponents that make the basic dimensions match. 

By User:Lzyvzl at en.wikipedia [Public domain], from Wikimedia Commons

The form of equations can often be deduced from an analysis of the dimensions of the variables or parameters that affect the solution to a problem. The process can best be illustrated by an example.

The form of equations can often be deduced from an analysis of the dimensions of the variables or parameters that affect the solution to a problem.  The process can best be illustrated by an example.

The natural frequency of a spring-mass system is observed to be related to the stiffness, k, of the spring and the mass, m.  What is the functional relationship between the spring stiffness and mass?  

In equation form, the frequency is some function , F, of k and m

A functional relationship that uses exponents for each variable can be written as

where the coefficient C and exponents a and b are unknowns.  The use of this functional relationship is known as Rayleigh’s Method.  The exponents a and b can be determined through dimensional analysis while the coefficient C must be determined through experiment or some other method.

The basic dimensions of the various terms are substituted into the equation 

The powers of each basic dimension must be equal for the equation to be dimensionally consistent.  Therefore,

Solving the two equations, gives a = ½ and b=-1/2.  The functional relationship can thus be written as

The true relationship can be determined analytically to be

A comparison shows that the coefficient C is equal to 1/2π.  The correct functional relationship was determined purely from making sure that the dimensions were correct.

The number of basic dimensions and number of exponents must be equal for this method to work.  In the example, there are two basic dimension, M and t, and two exponents, a and b.  This results in two equations with two unknowns, which can be solved to find the unknown exponents.

 

Learning Objectives:

Learn how to determine the functional relationship between parameters using Rayleigh’s method.

Learn how to determine when the Rayleigh method will or will not work.