Purpose of this workbook:

This workbook creates polynomial coefficients for mode shapes given deflection data along the length

of a flexible beam (i.e., tower or blade). The beam need not be cantilevered. When it is not cantilevered,

the workbook calculates the mode shape of the data projected onto a line that is tangent to the

deflection at the bottom. This projected mode shape is the mode shape FAST needs. The coefficients

calculated in this workbook can be copied and pasted into FAST input files.

On the Input worksheet, mode shape data should be entered into the x and y columns. The x column

specifies relative locations along the length of the beam. The y column specifies the relative deflection

of the beam at the corresponding x location. The x and y data may be dimensional or normalized

independently, but x must increase monotonically (i.e., x must be specified from beam bottom to beam

top). Remember to delete all old x and y data before entering new data. This will prevent extraneous

old data from mistakenly being appended to new data in the situation where the new data has

less data points.

The slope of the input data (i.e. dy/dx) at the bottom of the beam needs to be specified for the

Projection Method and the Improved Direct Method. This slope applies to the unscaled data. This

slope--and the deflection at the bottom of the beam--is a very important parameter because it

determines how accurately the data is projected. It is best to enter the known slope from a program

such as BModes; however, if the slope is unknown, the cell may be set equal to the estimated slope

provided. The tangent line on the Entered Data graph of the Input worksheet gives an idea of how well

the slope was chosen. If the tangent line is not tangent to the data, the slope is not good. For a beam

cantilevered to a rigid and stationary base, the entered slope should be zero.

A scaling factor for the y data needs to be specified for the Projection Method. A small factor of y

ensures that the projection of the data forms a curve that can be represented by a function. The first

suggested factor provided is sufficiently small so that the Improved Direct Method and the Projection

Method should agree. The second suggested factor provided is based on the user specified ratio of the

maximum deflection to beam length for a deflected beam. This factor may be a good choice if a

mode shape about a deflected position is desired. See the Output section below for more information

on this factor.

Do not change any cell other than the gray input cells!

Normalized coefficients for sixth, seventh, eighth, and ninth order polynomial fits to the projected data

can be found on their corresponding worksheets. Each worksheet has columns of coefficients that can

be directly copied and pasted into FAST input files. Coefficients found by three different methods are

presented. A description of each method--and guidance on which method to choose (which depends on

the input data and application)--is provided below. The graphs show how well each method's polynomial

fits the projected data. Keep in mind that the projected data in the graphs only represents the original

data according to how well the slope was chosen.

The Direct Method does not depend on the given slope or y-scaling factor; therefore, it is a good

method to use when a reliable slope cannot be determined. For example, this may be the best method

for coarse ADAMS output. The resulting mode shape is only valid for small deflections of the beam

about the undeflected position. This method is not accurate for all mode shapes and should be avoided

when possible.

The Improved Direct Method uses the entered slope and the deflection at the bottom of the beam to

improve the fit relative to the Direct Method (when the slope and bottom deflection are known

accurately). Like the Direct Method, the Improved Direct Method does not depend on the y-scaling

factor, so, the resulting mode shape is only valid for small deflections of the beam about the

undeflected position. If an accurate slope is known--and if the deflection data at the bottom of the beam

is also accurate--this will most likely be the preferred method for calculating polynomial coefficients.

The Projection Method depends on the entered slope, the deflection at the bottom of the beam, and

the entered factor of y. This method works for any size factor as long as the projection does not fail to

produce a curve that a function can pass through. When the factor is set very small (as one of the

provided values suggests), the resulting mode shape should be identical to the mode shape derived

from the Improved Direct Method and is only valid for small deflections of the beam about the

undeflected position. When the factor is set to the provided suggested factor derived from the entered

ratio of maximum deflection to beam length, the resulting mode shape is valid for small deflections of

the beam about the deflected postion. This method is the best when trying to find mode shapes about

a deflected position because a broader range of factors can be specified. But as with the Improved

Direct Method, it is only accurate when the slope and bottom deflection are known accurately.

The standard version of FAST uses mode shapes derived from sixth order polynomials. To use data

in FAST from the higher-order worksheets--which is necessary when a higher-order polynomial fits

better--you must recompile FAST. The INTEGER(4), PARAMETER named PolyOrd in MODULE

Modes() of source file FAST_Mods.f90 determines the order of the polynomial used by FAST.

This workbook was created by Erica Bush in August 2008. Questions can be directed to bush.erica@gmail.com