The oxymoron of damage assessment in dynamics by static approach

The evaluation of fatigue damage of structural components within dynamic systems subjected to random loads is typically addressed using multibody models wherein one or more components are modeled as flexible using the modal approach. While incorporating flexible elements allows for consideration of their influence on the overall dynamic behavior of the system, certain components are intentionally designed to function as rigid bodies. Consequently, in this case the incorporation of flexible elements within multibody models merely leads to complex and time-consuming analysis. Hence, it would be more pragmatic to realize rigid body multibody models, with components characterized solely by their inertial properties, and subsequently extract the dynamic forces applied to the components to be verified. The assessment of stress can then be carried out by exploiting the principle of superposition of effects in the time domain. In this context, the objective of the present study is to develop this methodology in the frequency domain. This approach not only capitalizes on the simplicity of rigid multibody models but also harnesses the computational capabilities of the frequency-domain method for evaluating fatigue damage in components subjected to random loading conditions. This work, therefore, provides a rapid, effective, and robust method for verifying and designing rigid mechanical components integrated into dynamic systems subjected to random loads.


Introduction
The frequency domain approach for the fatigue behavior assessment of systems and components subjected to random stationary-Guassian [1] loads in frequency domain is generally reduced to the evaluation of fatigue damage (spectral methods [2]) based on the Power Spectral Density (PSD) matrix of a specific point of the component that is typically modeled by the Finite Element Method (FEM) [3].In the analysis of structures subjected to random loads, dynamic analysis plays a fundamental role.By analyzing this step and incorporating it into a design process in which the component has not yet assumed a definition that allows for a faithful Finite Element Modeling, the authors wanted to highlight how, with knowledge of the inputs expressed in the time domain and/or the PSD matrix, it is possible to quickly obtain the necessary information for calculating stress cumulatives or rather internal forces without undertaking a dedicated dynamic analysis aimed to obtain stress PSDs.Under the assumption that the component to design and/or verify does not participate to the dynamic of the system (thus it is characterized by a stiffness that does not modify the natural frequencies of the system or does not interact with the frequency range of the excitation), the authors have demonstrated analytically and then, with a simple test case, how it is possible to obtain, starting from results expressed in the form of PSD matrices, the spectral moments [2] of individual stress characteristics or the entire stress tensor or its equivalent uniaxial synthesis for each node or element of the finite element model, using the so-called static approach [4].In the case of a rigid behavior of the component, one could consider adopting a multibody model and dynamic simulation [5,6,7] of the system to assess fatigue behavior without the need to introduce the component as flexible bodies into the multibody model.Thus, the only required information needed by the simulation would be related to the set of forces acting on the body, expressed either as time histories or as PSD matrices.These forces can be used to reconstruct the real internal forces of the real state of stress.Moreover, assuming a linear behavior of the system, through linearization and extraction of state-space matrices of the system, having as inputs the forces to the system and outputs the forces acting on individual components it is possible to obtain the state of stress of the component without the need of transient dynamic simulation.In this way, the dynamic simulation would no longer be a serial step in the design process but would run in parallel.Indeed, under the assumption of a rigid behavior of the component, if the inertial properties of the component remain approximately unchanged, the input forces to the component also unvary, and the entire detailed design phase of the single component no longer requires dynamic analyses but only progressively more complex FEM analyses.Therefore, this work represents a methodological proposal in the design of components integrated into dynamic models subjected to dynamic loads, seemingly in contrast with the dynamic nature of the problem.In fact, it suggests the use of multibody dynamic simulation for the system, a static approach to reconstruct the component's loading conditions, and frequency-based spectral methods to obtain load spectra and, consequently, the fatigue damage.This approach identifies a fast and error-free methodology if the assumption of the dynamic "rigidity" of the components is met.

The static approach
The assessment of the stress state of a component through a static approach in the frequency domain follows the classical modal approach [8].It starts with a multibody model (MBS) in which the component to be designed/analyzed is inserted as a rigid body and thus defined only by its inertia properties.From this model, the state space matrices must be extracted [9,10], where the inputs consists of the forces applied to the MBS system and the output are the forces applied to the component to be designed/verified.Thus, once the input PSD matrix of dimension [r × r × 1] defined as G x (ω) is known and thanks to the frequency response function matrix H(ω) of dimension [r × u] (where u are the input to the system) between the forces acting on the component and the forces acting to the system derivable from the state space representation [11], it is possible to calculate the PSD matrix of forces acting to the component G l (ω) as: Once the PSD matrix of forces acting to the component G l (ω) is known, it is possible to easily determine the spectral moments of the PSDs as: This calculus is performed only to the forces acting on the component and since it must not be done on each element/node of the component, the computational effort is strongly reduced.However, the spectral moments just calculated are based on the forces PSD matrix and in their actual definition are not viable for the evaluation of the stress state on the component.To this aim, a finite element (FE) model is required (it can be a simplified model made by only beam/shell element or more complex made by solid element) from which obtain the needed information finalized to the evaluation of the spectral moments of real stress state or of internal forces.These information, defined as static shape of stress or of internal forces, can thus be obtained doing as many static analysis, applying a unitary load, as many constraints on the component to design.However, it's important to emphasize that it's necessary to use the inertia relief approach [12,13], which allows not applying constraints to the system because equilibrium is achieved by applying an acceleration field to the center of gravity of the body.The correct evaluation of the absolute position of the "rigid body" may not be guaranteed, but the assessment of stress, deformation, or internal forces is assured.The output of these analysis are the static shape of stress φ σ or of internal forces φ IF with a unitary load and allows determining the real state of stress as: where t indicates the transpose.What obtained is further extendible to multiaxial state of stress.In such case it is sufficient to adopt the approach proposed by Pitoiset [14] to reduce a multiaxial state of stress into an equivalent one: where: Other methods for multiaxial synthesis can be adopted at the same manner [15,16,17].Once the spectral moments of internal forces or stress are known, it is possible to use any frequency domain approach (such as narrow-band, Dirlik, etc.) [2,18,19,20] for assessing load cumulatives and damage.

Test case
To validate the proposed static approach, until now only theoretical valid, a real model has been used as test case.In particular, the aerial platform shown in figure 1 is considered.
Two multibody models of the aerial platform were created, one with the drive chassis modeled as a rigid body and the other with the drive chassis modeled as a flexible body.The multibody models are shown in figure 2. As visible, the model includes all the real parts of the structure except for the drive chassis, which is represented by an ellipse in one case and by a finite element model imported into the multibody code in the other case.For the flexible body, 40 modes of vibration were considered, and the first 5 natural frequencies are: 242.6 Hz, 441.3 Hz, 533 Hz, 792 Hz, and 848.6 Hz.In both variants of the MBS model, the drive chassis has consistent inertial properties with the following values defined in the reference system shown in figure 2: As shown from figure 1, the drive chassis is constrained to the rest of the components in 5 points for a total of 22 reaction forces/moments.
To excite the model as it works in real-life condition (random excitation introduced by the road asperity), a four-post test-rig has been created and the MBS models are places over [21].The model is thus excited by the four-post test-rig with four uncorrelated acceleration random excitation, defined by a PSD matrix with terms only on the main diagonal (figure 2).The input PSD has a trapezoidal shape, defined in a frequency between 5-10-25 Hz and as amplitude 30-30-0 (m/s 2 ) 2 respectively.The so-realized models were linearized in the equilibrium position and from the model, the state space matrices were obtained.For the rigid model the output of the state-space representation are the input forces to the drive chassis (22 in total) while for the flexible model the output of the state-space are the modal coordinates (40 in total) with which the flexible body was created.The input of the state space form are instead, for both models, the forces acting on the entire system (4 in total).During the linearization, a modal analysis of the system was then performed.Table 1 shows a comparison between the obtained natural frequencies of both system.As visible from table 1, the first four vibrating modes obtained by the two models are identical, while differences occur from the fourth vibrating modes.This is due to the presence of the flexible body in one model.By the way, this aspect can be omitted since, as previously introduced, the maximum frequency of the excitation PSD is equal to 25 Hz and thus the fourth vibrating modes is outside the frequency range of the excitation.This allows attesting the two models are dynamically equivalent [4].According to this, it is possible to adopt the proposed static approach for the evaluation of the spectral moments and thus of the load spectra and of the fatigue damage of the analyzed rigid component to be compared with those derivable with the classic approach in frequency domain.As previously introduce, in order to evaluate the stress state by the static approach, it is necessary to have the static shape of the stress.These are obtained for each element of the model by performing 22 static analysis in FE environment with the inertia relief approach, each of them performed by imposing a unitary load/moment in each point of the model where it is constrained to the rest of the components.For what concern the classical approach in frequency domain, the same FE model has been used to perform a modal analysis considering the first forty vibrating modes extracting the modal shapes of stress for each element of the model.By adopting the results of the multibody model and of the FE analysis it was possible to evaluate the state of stress with both methods (the proposed and the reference one).To compare the accuracy of the proposed static approach, figure 3 shows a comparison between the zero-order spectral moment m 0 of the equivalent stress PSD.
As shown in figure 3, there are not visible differences between the results obtained with both methods.The results are almost identical even if small differences (lower than 2 MPa) exist.The difference is due to the flexibility of the drive chassis that, despite is practically indifferent, is real.However, a differences lower than 2 MPa is acceptable, especially in a preliminary design phase.Similar results are obtained also for the higher-order spectral moments.According to the obtained results, it is possible to attest that the proposed model is a valid tool for the design of rigid component, especially in the preliminary design phase, where a detailed FE model is not viable and only the inertia properties of the components are known.With this approach, the design phase is much faster since there is no need to upgrade the dynamic model with flexible components step by step more detailed.

Conclusion
In this work, a design/analysis method for rigid components (i.e., components whose flexibility does not significantly affect the system's dynamics) inserted into complex dynamic systems subjected to dynamic loads, specifically random, has been proposed.The proposed approach leverages the advantages of rigid body multibody modeling and the frequency domain to obtain spectral moments of internal forces or stress, which are useful for calculating load spectra or fatigue damage.The proposed approach has been validated using a real test case, comparing the results with those derived from the classical frequency domain approach.The obtained results demonstrate excellent agreement, indicating this method as a valuable tool, especially in the design phase but also during verification.

Figure 1 .Figure 2 .
Figure 1.Aerial platform used for the validation of the proposed approach

Figure 3 .
Figure 3.Comparison between 0-order spectral moment of the equivalent stress PSD: (a) Proposed approach, (b) Classical modal approach

Table 1 .
Comparison between the natural frequencies of the two MBS models.