IFMASS10

229

2. MODAL SENSITIVITY ANALYSIS – BACKGROUND AND SURVEY

It is generally accepted that sensitivity analysis can be applied as a valuable tool in

structural reanalysis when sufficient modal properties are known, either through theo-

retical or experimental analysis. Modal sensitivities are the derivatives of the modal

properties of a dynamic system with respect to chosen structural variables. There are two

primary applications in the modal analysis. In the first case sensitivity data are used solely

as a qualitative indicator of the location and approximate scale of design changes to

achieve desired structural properties. The consequences of candidate design changes

would then be evaluated using more exact methods. The second strategy uses the design

sensitivities directly to predict the effect of proposed structural changes. The use of

sensitivities in this fashion relies on the Taylor’s series expansion of matrix, with the

usual implications of convergence and truncation errors. Use only of first order design

sensitivities assumes implicitly that the second (and higher) order derivatives are

negligible. The use of these second order sensitivities as suitable criteria for the accepta-

bility of first order sensitivities for predictive analysis can be interesting in some detail.

Sensitivity analysis may be applied to candidate design modifications distributed across a

number of degrees of freedom of the structure, but is limited in scale.

Modal design sensitivities are the derivatives of the eigen system of a dynamic system

with respect to those variables which are available for modification by the designer. A

typical modification would be the change in diameter of a circular section. This would

affect both the mass of the section, proportional to the square of the diameter, and its

stiffness, which depends on the second moment of area of the section. The change in

length would have a mass effect directly proportional to length, but a stiffness change

depending on the cube of length. Changing material would similarly affect mass, stiffness

and damping. Shape sensitivity analysis of physical systems under dynamic loads may be

important from different points of view to:

- understand and model the system's behaviour better with respect to shape;

- optimize the physical shapes of desired systems responses in given time interval, or

- identify shapes by utilizing the system's response measured in time.

3. PROBLEM STATEMENT

One important task in design of structures exposed to dynamic load is to increase the

eigen frequency values outside the frequency range of forced load, which can damage the

structure by the effect of resonance. Consequently, the main goal of dynamic optimiza-

tion is to increase natural frequencies and to enlarge the difference between them. Some

information should be available before setting up the finite elements (FE) model. The first

pack of information includes referent information about the structure: size, material, and

boundary conditions. Using this information it is possible to form a final model with all

elements, which is at the same time a FE model prepared to start in analysis.

An output there are

pairs of eigenvalues

and eigenvectors {Q

} which are very

important for a dynamic analysis. It was already mentioned that eigenvalues and the

oscillation shapes in some construction, which can be obtained form differential

equations

[ ]

{ }

M Q( t )

K Q( t )

⎧ ⎫

⋅

+ ⋅

⎨ ⎬

⎩ ⎭

, in the form

[ ]

(

)

{ }

K M Q

− ⋅

⋅

(1)