Base force and moment based finite element model correlation method

Highlights

• A method to predict the FEM capability to represent the dynamic characteristics.

• Method is well suited for base excitation problems such as a spacecraft.

• Correlation is demonstrated using flightworthy spacecraft structure.

• Correlation can be done as part of base excited qualification test.

Abstract

The accuracy of the finite element model is an important aspect, especially in the aerospace domain, where design margins are barely minimum. Hence, standard test-analysis model correlation methods apply to assess the quality of the analytical model. However, it is observed that these standard correlation techniques are inadequate to assure the model capability to predict the forced response characteristics of complex structures. The base force assurance criterion is found to be a good tool to overcome this inadequacy. During the dynamic test of the structure, the force measurement device can be used to measure the transmitted force to the base. Thus measured base forces, and the moments computed from the measured forces are vector correlated with the analytical force and moments to predict qualitatively the analytical model capability to represent the design critical peak acceleration responses and the global modes. The procedure is demonstrated using a set of experiments conducted on a real spacecraft under the sine excitations in two mutually perpendicular directions. The sensitivity of the base force assurance criterion to the inaccuracies that may be present in the finite element model and/or in the experimental results is evaluated using simplified Monte Carlo simulations.

Introduction

Structural analyses are widely utilised to predict the behaviour of a complex system under both dynamic and static loading conditions. This helps to design the structure optimally, verify the design margins prior to the qualification test, and to plan the tests for design qualification. A Finite Element Model (FEM) created using the commercially available software is generally employed for the analyses of complex structures. The accuracy of the analytical results is critical, especially with aerospace structures such as spacecraft to minimize the launch cost. Inaccuracy in the analytical model also leads to improper load estimates that may lead to hardware failures. Conventionally, the accuracy of analytical results is assured by comparing the mode shapes extracted from the modal test with those obtained from the analysis using vector correlation methods such as the modal assurance criterion (MAC) (Allemang, 2003, Allemang and Brown, 1982), the coordinate modal assurance criterion (Lieven et al., 1988) and the normalized cross-orthogonality (NCO) check (Ewins, 2000). The NCO check requires a compatible mass or stiffness matrix that suits to the mode shapes to perform the correlation and it can be done using the system equivalent reduction expansion process (SEREP) (Mercer, 2016, O'callahan et al., 1989) whereas the MAC is performed with two equal size mode shapes. The SEREP uses generalised inverse (Penrose, 1955) to reduce the analytical mass or stiffness matrices to the test location. The frequencies obtained from the test and analysis are directly compared(Siano, 2015). It is accepted that the analytical model passes the quality check if it possesses certain threshold values of MAC or NCO along with the frequency variations within a certain range as specified by the different space agencies (Loads analysis of spacecraft and payloads, 1996, Modal survey assessment, 2008). However, one has to note that even if the analytical model passes all the standard criteria, considerable errors can arise in the predictions of structural responses such as peak acceleration under the forced vibration (Sairajan et al., 2015). The acceleration responses and the stiffness requirements are very important to check and verify the design adequacy of the structural elements, load analysis, and in the design of those subsystems to be attached to the main structure.

Model correlation can also be performed by comparing the frequency response functions obtained from the test and the analysis. Two such correlation criteria are the frequency response assurance criterion (FRAC) (Nefske and Sung, 1996) and the frequency domain assurance criterion (FDAC)(Pascual et al., 1997). In FRAC, the frequency response functions obtained from the test and the analysis are compared against each frequency, whereas the FDAC uses the MAC approach for the response function correlation. In a complex structure, there could be hundreds of response locations that need to be monitored during the test, and hence correlation will be a tedious task. Besides this, the responses highly depend on the locations of measurements, and hence very careful evaluation is needed for the successful correlation. The advancement of dynamic force measurement device (FMD) that can be used during the shaker test or the based fixed shaker driven test, has led to the development of the base force assurance criterion (BFAC) (Sairajan and Aglietti, 2014) to assess the quality of finite element models to predict the dynamic force response characteristics. The BFAC overall shows how well the FEM can predict the forced response characteristics primarily during the major modes of excitation under the base excited conditions. In the case of spacecraft, coupled load analyses (CLA) are performed for the critical load cases that may be encountered during the launch, to determine the critical responses, interface forces, and moments after integrating the reduced model of spacecraft with the launch vehicle. The CLA will be performed by an external launch vehicle team (Lim, 2014) and it is a time consuming and expensive process. The BFAC can also be used to decide whether CLA needs to be performed again after incorporating certain design changes in the spacecraft and FEM by correlating the base forces from the original FEM and the revised FEM. It was also shown that BFAC is more effective than the MAC or NCO check in the prediction of loss factor for visco-elastic systems (Sairajan et al., 2014).

Base force assurance criterion requires interface forces at the boundaries obtained from the mathematical model and the dynamic test. However, complex mathematical models of structures are created based on many assumptions such as rigid boundary conditions and uniform behavior of identical substructures. Besides, there are modeling limitations in the real loading conditions (Abdullah, 2017) and simplified modeling techniques are adopted to represent the joints to limit the overall degrees of freedom (DOF) of the model. The experimental results are also affected by the noise in the measurements, sensitivity of the instruments, non-perfect control system, and signal delays (Marcello et al., 2017). All these lead to inaccuracies in the results. Such inaccuracies are unavoidable in any real system (Remedia et al., 2015). An exact error analysis of a simple beam with a single design variable was performed by Olhoff and Rasmussen (Olhoff and Rasmussen, 1991). Generally, for complex systems, probabilistic methods are adopted to find the sensitivity of inaccuracies (Capiez-Lernout et al., 2006, Chowdhury et al., 2011, Remedia et al., 2015, De Lellis et al., 2020). A probabilistic study to assess the effect of errors in the experimental mode shapes on the test analysis orthogonality check of a spacecraft was performed by Bergman et al. (Bergman et al., 2010).

This work extends an earlier reported work (Sairajan et al., 2015) and here the experimentally determined moments are also included in the computation of BFAC for a flight-worthy spacecraft under the base excitation in two directions. To calculate the base moments using the experimental force data, it is required to separately acquire the three mutually perpendicular components of the forces using tri-axial force gauges. For computing the BFAC, there is no need to perform any additional test like modal testing for MAC or NCO, as the required measurements will be available from the base fixed sine excitation test, done while qualifying aerospace structures such as a spacecraft. Monte Carlo simulations are used to assess the robustness of BFAC to the inaccuracies that may be presented in the FEM and/or in the experimental results using a simplified error model. The procedure is then demonstrated using a mini-satellite model and its experimental results.