The familiar fundamental equations of motion can be written with the matrices for inertia M, damping D and stiffness K and the force vector F(t) as

For (harmonic) loads F(ω) defined in the frequency domain, the system of equations shown above is transformed with the approach x(t)=U(ω) e^{iωt} resulting in

From this linear system of equations, the reaction of the system U(ω) can be determined in the context of a Response analysis. The inversion of the dynamic stiffness matrix required for this purpose leads to the so-called transfer functions H(ω): U(ω)=H(ω)F(ω).

Based on presentation (1.2), it can readily be seen that, depending on the excitation frequency, the dynamic stiffness matrix is composed of differently weighted components of

the mass, damping and stiffness matrix. Connected with this is a frequency-dependent response behavior of the dynamic structure.

How can a fatigue life analysis of a structure with these complex

dynamic properties be accomplished?

FEMFAT offers two different solution approaches for this in dependence on the excitation characteristics: for stochastic loads, the SPECTRAL module, while the area of deterministic

excitations is covered by the Harmonic tool in combination with ChannelMAX.

The concrete variants of possible vibration processes in Harmonic range from constant sinusoidal, linear and logarithmic sweeps to a tabular description of time, frequency and Amplitude and on to the processing of measured or simulated load-time signals.

An example of a fictional frame structure (see Figure 1) is provided here to illustrate the procedure: the frame is excited vertically at four points, whereby histories for the vertical

acceleration are taken into account. Instead of a time consuming transient analysis, a modal fatigue analysis is carried out with ChannelMAX and Harmonic.

Figure 1: Frame structure

In the first step, the natural frequency analysis and the response analysis for the four unit load cases (each with a constant acceleration amplitude of 1g in the z-direction) are carried

out. The transfer functions are output for the response analysis (e.g. in the form of amplitude and phase).

The ChannelMAX calculation requires in addition to the modal stresses (from the natural frequency analysis), the modal participation factors (i.e. the time history of the amplitude factors for each mode). The latter are generated with Harmonic. For this purpose, the type of the desired vibration process must be specified in the control file (here: “Signal from unit load case”). Furthermore, the corresponding files with the acceleration-time data as well as the transfer functions from the FE Solver must be specified in the control file.

Harmonic then links the information from the load-time signal and the transfer functions in the frequency domain to form a system response. This is transformed back into the time domain and output in the form of the modal participation factors which are sought being contained in a text file (RPC ASCII format).

This file of the added up modal coordinates together with the modal stresses then makes the modal fatigue life analysis possible using ChannelMAX. In this example, the damage resulting from the acceleration histories was determined in this manner (see Figure 2).

Figure 2: Damage results

**SUMMARY**

For the analysis of deterministic vibration phenomena, the classical approaches (e.g. transient analysis) are very computationally demanding. With the FEMFAT Harmonic tool, this effort can be reduced considerably by switching to the frequency domain: the structural response is described by means of the modal participation factors which can be used directly in the ChannelMAX fatigue life analysis. A further positive aspect: for a fatigue life analysis with a new signal, “only” Harmonic is required; no new FE analysis is necessary

– the dynamic behavior of the structure is characterized by the transfer functions which are already at hand.