78 J. Maierhofer et al. to the ratio between output and input. In the context of experimental substructuring, for example, a FRF can be acquired by measuring an acceleration on a structure (output) with a known force (input). It is common to use the admittance matrixY(s) to characterize the substructure (n), which includes the FRFs and is defined as the inverse of the dynamic stiffness matrixZ(s) as shown in Eq. (9.1). Y(s) =Z(s)−1 = K(s) +C(iω)(s) −ω 2 M(s) −1 (9.1) The dynamic stiffness matrix Z is composed of the mass matrix M, damping matrix C and stiffness matrix K. If the FRFs of the individual substructures were determined experimentally or simulatively, an analytical coupling according to the LM-FBS can be performed to obtain the assembled admittance Yassembled of the total system. uassembled = uuncoupled +,-. Yf − ucoupling + ,- . YBT(BYBT - .+ , int.flex. )−1 BYf - .+ , u =Yassembledf (9.2) Equation (9.2) can be interpreted as the assembled response of a structure, which is a combination of the uncoupled system responseuuncoupled and coupling responseucoupling, produced by the internal interface forces to keep the substructures together while be excited by the external forces. For a detailed derivation of the assembled admittance in the course of the LM-FBS procedure, see the corresponding literature [7]. 9.1.2 Motivation for an Automated FRF Measurement There are two motivating key points for automating the impact measurements, especially the impacting of the structure. The first one comes right out of the FBS method, the second one lies in possible non-linearities of the structure. The measured FRFs are in a form that is equivalent to an admittance as one gives in a force in frequency domain and get an acceleration (also in frequency domain). Looking at Eq. (9.2), it’s observed that the admittance has to be inverted for the LM-FBS-method. This inversion is somewhat problematic for the experimental side. The admittance is defined in form of an accelerance. Especially when the acceleration is very low, for example at anti resonances, very small errors in the force direction or the position of the impact cause very high errors in the FRFs. The problem here is that the force sensor is only unidirectional. Hitting the structure at a wrong angle means that the whole force is set to the assumed direction even if only one part of the force is in exactly that direction. The second effect is that a manual hammering changes the impulse position every time in a range of a few millimeters. However, for very light structures this displacement changes the measured FRF by quite an amount. When averaging multiple runs, this causes bad FRFs. This effect can be seen very well with the coherence of the FRFs which normally should be near 1. Looking at anti-resonances, the coherence often drops when using a classical manual impulse hammer. If the structure has some nonlinearities which may come e.g. from mountings, material damping or anything else, there is a different answer to different levels of excitation force. Therefore, when measuring multiple times to make some averaging, it is necessary to excite the structure always with the same peak force. Manually it seems nearly impossible to impact lightweight structures in a free-free configuration multiple times with approximately the same force. 9.2 Existing Technology The relevant literature seems to show some movement on the topic over recent years. This chapter gives an overview of the existing automated modal hammer systems and how they work. There are two fundamental categories of automated hammer systems. The linear acting and the rotating systems. Linear Actuator Hammers The device in Fig. 9.1a built by the authors in [8] uses a solenoid to generate the impact. As this hammer stands on the structure, it is only suitable for very big structures, as the added mass influences the dynamic of the structure. In [9] a milling machine is measured using a solenoid on a special frame as seen in Fig. 9.1b. The AS-1220 Automated Impact Hammer depicted in Fig. 9.1c is sold by the company Alta Solutions. The device is built for automated testing in production lines, and is mainly intended for acoustic measurements [1].
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