Example Mathcad Worksheet

Estimation of Fundamental Frequency Using the Improved Rayleigh Method

A 3 tier frame is considered consisting of 3 perfectly rigid slabs of material with masses supported by weightless columns with a combined stiffness coefficient for each tier. The method of estimation used here is based on a system with a single degree of freedom and so only lateral displacement of the columns is allowed. However it is still valid as a first approximation of the fundamental frequency. The objective of this worksheet is to provide a tool for examining the effect on the predicted fundamental frequency caused by varying the stiffness coefficients and masses attributed to the system .

The basic premise of the Rayleigh method is that for a vibrating system the maximum potential energy is equal to the maximum kinetic energy. Given the shape formed by the deflected system when vibrated, known as the mode shape, the fundamental frequency can be estimated. Knowledge of the mode shape therefore appears to be a key factor which may not be readily available. In practice any plausible mode shape can be applied and reasonably accurate results obtained. A plausible mode shape is one that does not contradict any of the constraint conditions to which the system is subjected. As the solution progresses, the Improved Rayleigh Method calculates a more accurate mode shape from the forces implied by the initial mode shape.

The equation for the fundamental frequency w is:

Equation 1

Z₀ = Reference displacement occurring at the maximum amplitude of the initial mode shape.

Z = Reference displacement occurring at the maximum amplitude of the calculated mode shape.

m(x) = Mass at a point x, where there are L mass points along the length or height of the system

to be vibrated, x is therefore a value between 0 and L.

Y₀ = Initial mode shape function defined by prescribed relative displacements at mass points.

Y(x) = Calculated mode shape function determined from forces inherent in initial mode shape

and stiffness values, defined in terms of relative displacements at mass points.

The equation as given is the general equation which presumes m and Y to be written as functions of x. In cases where the value of Y₀ is not set to unity, then Y₀ will also be a function of x.

As stated Equation 1 is derived by equating the maximum potential energy to the maximum kinetic energy.

Starting with the the standard formula for kinetic energy:

Equation 2

m = mass
v = velocity

It is assumed vibration takes the form:

s = s₀.sin(wt)

Equation 3

s = displacement at point x at time t
s₀ = maximum displacement at x
w = frequency in rads/sec

The velocity v at point x can be found by differentiating equation 3 with respect to time:

v = w.s₀.cos(wt)

The maximum value of v will occur when cos(wt) is equal to 1, therefore:

v_max = w.s₀

Equation 4

But s₀ can be expressed as Y(x).Z where:

Y(x) = The shape function and is the ratio of displacement at x to the reference point displacement.

Z = The displacement at the reference point, usually at the point of maximum displacement.

So v_max can be expressed as:

v_max = w.Y(x).Z

Equation 5

Substituting Equation 5 into Equation 2 maximum total Kinetic Energy will be:

Equation 6

The maximum Potential Energy is calculated from the work done by the forces required to cause the prescribed displacements in the initial mode shape. The force F₀ applied at any point x in the initial mode shape to accomplish the prescribed displacement:

F₀ = w².m(x).s₀

Replacing s₀ with Y₀ the initial mode shape and Z₀ the initial displacement reference this becomes:

F₀ = w².m(x).Y₀.Z₀

Equation 7

The force applied initially to give the prescribed displacements assumes no stiffness resisting the motion. When the stiffness is considered a new shape function Y and reference displacement Z can be calculated and an expression for the revised displacement may be written:

r₀ = w².Y(x).Z

Equation 8

The total maximum Potential Energy PE_max is found:

Substituting F₀ and r₀ from equations 7 and 8:

Equation 9

The expression for the revised displacement can be incorporated into the equation for the total maximum Kinetic Energy. Substitute Equation 8 into 4 replacing s₀ with the definition of r₀ .

v_max = w³.Y(x).Z

The redefined v_max can then be substituted into Equation 2 from which a revised version of Equation 6 is obtained:

Equation 10

Equating KE_max to PE_max (Equations 9 and 10) and re-arranging to solve w² gives Equation 1.

Characteristics of the 3 Tier Frame

Mass of Material in Tiers

Overall Stiffness of Support Columns

Values for m₁, m₂, m₃, k₁, k₂, and k₃ in highlighted areas
can be varied to check the effect on the frequency.

Prescribed Displacements for Initial Mode Shape (millimetres)

By implication

In all cases

i.e. Y₀ = 1.0s_1, 1.0s_2, 1.0s₃

Inertial Forces From Prescribed Displacements

Forces f_x = w².m_x .s_x (s_x in metres)

Shear Force

Shear Force

Shear Force

Mode Shape Calculated from f_1-3 and k_1-3

Relative Displacement Du_x = sf_x /k_x Total Displacement r_x = Du_x + Du_x+1 + Du_x+2 +...

Let Reference Displacement for the calculated mode shape

The shape function Y can be determined from the displacements r_x and Z₁ :

Therefore:

Calculation of Fundamental Frequency

As the system consists of 3 discrete blocks of mass, the general Equation 1 can be simplified and re-arranged to solve for w:

Giving

rads/sec

Alternatively where Fr is the fundamental frequency in Hertz or cycles/sec:

Giving:

Hz

Reference: Dynamics of Structures by R. W. Clough and J. Penzien

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