## 1 Introduction  
In the space industry, the launch evidently dominates structural requirements. Therefore, in order to demonstrate that a structure will survive the launch, it is analyzed using the finite element method (FEM) and tested in vibration test facilities [1]. During a vibration test, accelerations are usually monitored in order to assess the loads that the structure is experiencing. Ideally, load cells are also installed at the interface of the structure to directly monitor the interface loads and compare them against the design loads. This, however, is not always possible because the use of load cells or strain gauges has many technical, operational, and financial drawbacks [2]. Consequently, the input of the vibration test, i.e., the excitation load of the structure under test, is often adjusted based on the measured accelerations rather than on loads or stresses [3]. One specific example concerns the case where loads need to be monitored at the interface of a subsystem that is part of a larger complex system such as the James Webb Space Telescope (JWST) (Figure 1). The JWST is composed of several subsystems, each of which was tested separately before integration on the JWST. Figure 1 illustrates this problem where, particularly on the bottom, are depicted all the different mechanical test campaigns in which the Near-Infrared Spectrograph (NIRSpec) has been involved. One can observe the NIRSpec optical assembly stand-alone test (OA), followed by the integrated science and instrument module test (ISIM) and the optical telescope assembly test (OTE + ISIM = OTIS). The last mechanical test prior to launch has been recently conducted, the observatory test with JWST in folded configuration. Beside these mechanical tests, many more test campaigns have been conducted. This path highlights the path followed only by NIRSpec. In the later testing phase, it is not possible to monitor the loads at the interface of the NIRSpec using load cells because there is no space for accommodating them [4]. Therefore, alternative approaches must be used. One approach is to use the coil current from the shaker, since the applied load can be correlated with the shaker current. However, this approach can only be used to estimate the load in the excitation direction [6]. Strain gauges could be used to recover strains at interfaces and thus loads. However, they require careful calibration to provide a robust indirect measurement of the interface loads. A force measurement device provides six global interface forces or moments and local load cell forces during vibration testing, allowing the measurement of the local forces in three orthogonal directions [6]. However, such devices are not available in every test facility center. Moreover, they are costly, take space that is not taken into account in the design, and often change the system's response, so they must be accounted for in all test prediction analyses [2; 5]. The mass operator is a mathematical tool used to derive loads from measured accelerations [3]. It uses measured accelerations in order to calculate the interface loads or stresses representative of the  
Figure 1: The James Webb Space Telescope (JWST) (top left) in deployed configuration and Near-Infrared Spectrograph (NIRSpec) (top right). On bottom, JWST is disassembled towards NIRSpec [5]. physical state of a structure. A simple example of a mass operator approach is the sum of weighted accelerations (SWA), which is nothing else but the application of [PERSON]'s second law \(\mathrm{F}=ma\), where a is a vector of measured accelerations, \(m\) is an equivalent mass matrix, and F is the vector of loads at chosen interfaces. Generally, the mass operator would be created before a vibration test based on finite element analyses' results. The actual computation of the mass matrix can be performed using one of several techniques. With these data, it is then possible during the vibration test to calculate interface loads based on the real-life accelerations, measured by the sensors with no additional hardware [2; 3; 5].