Traditionally fatigue damage is associated with time dependent loading, in the form of local stress or strain histories. However there often are situations where these loading time signals can not easily be determined. As examples one can think of shaker tables, or signals like the wind load on a wind mill where just the length of necessary measurements enforces to use other representations of the loads. In these cases power spectral densities define the loads.

In other cases the loads are deterministic but defined in frequency domain. Engine accesssories for example are excited by the engine and the loads are in defined small frequency bands. To simulate the fatigue contents one needs to be able to define such loads easily and for efficiency reasons it is desirable to perform the complete simulation in frequency domain.

Similarly if one wants to simulate a complete sine-sweep on a shaker table, this is by far no stationary signal so not to be calculated by a random approach, and also it is not desirable to do a complete calculation in time domain for efficiency reasons.

In the launch phase of rockets transporting expensive satellites into orbit, high sound pressures are acting on these systems such that even fatigue problems can occur. The definition and simulations of these sound loads can be handled by random acoustics methods and the fatigue problem is again based on frequency loads.

LMS has great experience for frequency based solvers and load transfers both at component and system level. With the vibration fatigue solver LMS integrates this knowledge with its leading edge knowledge in

durability assessment methods. You can benefit from easy and consistent set-up, as well as from highly efficient analysis methods. The solver combines well known methodology with technology breakthroughs like real multi-axial load and local stress behavior for more accurate insights. It also offers deterministic load in the frequency domain.

The post-processing features are especially adapted to the given applications and they are not only intended to really fast identify the critical regions but give answers to how to solve the durability problems. Due to the efficient solving and the tight integration multiple designs can be analyzed without big manual interaction to reach the design that fulfills both weight and durability requirements.

**Random Vibrations**

Fatigue is the degradation of a material due to repeated cyclic loading. For metals, this typically means the initiation of small cracks from active slip bands in grains on the free surface of a specimen, component, or structure, which eventually link to form large cracks that either break or severely degrade the performance of a component.

Get realistic loads: In the case of random vibration loads, we have some statistical information on the load. Typically at one or several points the structure is excited. From the excitation loads (forces, accelerations, displacements,…) it is assumed that they can be described as stationary random processes. These can individually be described by their power spectral densities (PSD). Often the loads are not independent. If for example the loads are the wheel forces on a vehicle the forces on the back wheels are highly correlated to the forces on the front wheels, they more or less just differ by a phase shift. This phase information is considered in the cross correlations. The whole load consisting of power spectral densities and cross powers we call cross power set.

Get the local stress cycles - For a fatigue analysis one needs to analyze the local stress cycles. First the random loads have to be transferred into local random stresses. In principle this is done by calculating the transfer functions between the loads and the local stresses. Calculating all transfer functions on a complete structure in an independent step (e.g. using a finite element package) is time and storage consuming, such that in such an approach only part of a structure could be analyzed. An efficient approach takes advantage of the modal participation factors: Once the stress mode shapes are calculated using a finite element tool, an internal solver based on a modal based forced solver is used to calculate the modal participation factors for the local stress tensor cross power set.

Once a local stress tensor cross power set is calculated one has to calculate a uni-axial stress. But one has to take into account that using a set of correlated loads, the local information also contains correlation information between the tensor components. Simple mathematics shows that using an equivalent stress approach now in any case leads to severe problems (The result is no more a stationary random process, the process of finding the equivalent stress even is dependent on the coordinate system, i.e. these are no more invariants) Therefore a critical plane approach is taken to calculate stress PSDs.

For these stress PSDs a probability distribution for the number of stress cycles with different amplitudes gets estimated. Clearly the power spectral density gives us information on a random process of the local stresses. One very useful characteristic can be calculated directly from the PSD. The so-called root mean square (rms) value is defined as the square root of the area under the PSD curve.

But what can it tell about stress cycles. Some basic analysis shows us that the moments give us some basic statistical information:

- Expected Zero-crossings,
- Peaks
- Irregularity Factor

Two of the most important statistical parameters are the number of so-called zero crossings and number of peaks in the signal. The figure above shows a interval cut out from a wide band signal.

The red points give the upward zero crossings and the blue points the upper turning points. The more upper turning points exist for which no zero crossings exist, the more irregular is the signal

Basic analysis shows that one can estimate the number of zero crossings (E(Z)) and the number of upper turning points (E(P)) from the PSD. The estimation of the stress amplitude distribution involves some knowledge about the random process.

For this different methods have been developed that try to follow "Process assumptions". For a selected class of processes, the PSD defines the damage domain response

**Deterministic Vibrations**

In the case the loads are deterministic like a sine sweep or a set of single frequency excitations; one can directly calculate the local stress amplitude distribution. These processes can also be handled in time domain by using modal transient analysis, but the analysis in frequency domain is just more efficient and easier to set up. There is no extra finite element run necessary; all needed is - as for the random approach - the modal analysis and the load definition. The load definition includes an amplitude vs. frequency and a frequency vs. duration definition.

Note that the harmonic fatigue process for sine sweeps and sine waves excitation does not include a statistical step like the random vibration process but calculates the same local stress amplitudes as a modal transient approach in time domain would do. But since for these loads the frequency based approach is much more suited it is able to perform this in a much shorter time. Especially for sine sweeps where in time domain many cycles have to be simulated both in the finite element calculation as well as in the fatigue simulation. In the frequency domain solution only a modal analysis is to be performed in a finite element solver the rest of the analysis is completely performed in the solver and is extremely fast. For a sine sweep on a bracket (approx 100 000 elements, 20 modes taken into account) simulating several hours of real time, the simulation in time domain took approx. 1 day, whereas the full simulation in frequency domain is done in approx 15 min.

**Fatigue Calculation**

The result of the analysis in the latter two chapters is the number of stress cycles of a given amplitude in a unit time interval. Knowing the total length of the signal one easily calculates the total number of cycles of a given amplitude. Using this information all types of fatigue prediction algorithms may be applied. Starting from safety factor analysis to high- and low cycle fatigue analyses may be conducted. Also stress based weld analysis may be performed the same way. But if one analyses the random and harmonic approaches in more detail one can also show that all force based tools, like the LMS seam weld approach based on element nodal force results may be applied.

**Conclusion**

The vibration fatigue solution is used whenever the load is defined in the frequency domain. It can handle:

- Random processes
- Acoustic loads
- Sine sweeps
- Set of single frequency excitations

These can be single loads or multiple loads where the phase information can be taken into account.

An important advantage of the LMS approach is that it is easy to combine different load types, e.g. the start-up phase and the cool down phase of a load signal may be applied as a time domain signal, where as the long and time consuming phase is modeled as a stationary process and handled by the random approach. The fatigue results may simply be combined.

In the cases of long running stationary processes or sine sweeps the simulation in frequency domain is much more efficient than calculating in time domain. Even events of different type may be combined.

It helps to efficiently include durability aspects early in the design cycle.