Chapter 6 A Mathematical Model for Determining the Pose of a SLDV Da-Ming Chen and W.D. Zhu Abstract A laser Doppler vibrometer (LDV) combined with a pair of orthogonal scan mirrors is a scanning LDV (SLDV). A mathematical model based on the scan mirrors configuration for determining the orientation and position of a SLDV in a structure coordinate system (SCS) is presented in this work. Coordinates of a scan point in a scan mirrors coordinate system (SMCS) are derived from the mathematical model and the relation of coordinates of the same scan point in the SMCS and SCS is derived using the rigid transformation theory. A rotation matrix and a translation vector from the SMCS to SCS are obtained by applying the least squares method and singular value decomposition. An experiment to scan a 3D structure was performed with three SLDVs placed at three different locations. Rotation matrices obtained from two different sets of reference points are almost the same at the three locations, which validates that the proposed methodology is reliable. Another experiment to scan a 2D clamped plate was performed and the rotation matrices were calculated from the proposed methodology as a special case, which shows universal applicability of the proposed methodology and extends its application scope from 3D structures to 2D structures. Keywords Scanning laser Doppler vibrometer • Rigid transformation theory • Least squares method • Singular value decomposition • 3D and 2D structures 6.1 Introduction A laser Doppler vibrometer (LDV) is a non-contact measurement device that can measure the surface velocity of a vibrating structure along the laser line-of-sight direction. The LDV combined with a pair of orthogonal scan mirrors is defined as a scanning LDV (SLDV). A laser beam emitted from the LDV can be directed to any position on the structure surface by rotating the scan mirrors in a SLDV. This technology has greatly expanded application areas of the LDV, including making full-field measurement of the surface velocity of a vibrating structure. There are two different kinds of scanning modes that depend on types of input signals to two scan mirrors. One is stepped scanning, in which the laser beam is stepped from point to point, pausing at each point for long enough time to acquire sufficient vibration data of that point before moving to the next point. Another is continuous scanning where the laser beam is continuously swept over a vibrating structure surface [1]. Both scanning modes have been widely applied in vibration measurement and modal testing to analyze structure dynamic responses under different excitations. An operating deflection shape (ODS) of a clamped plate was obtained from a continuous SLDV (CSLDV) using a uniform-rate rectangular scan in [1]. Area vibration mode shapes were also obtained using the CSLDV by giving two different scan frequencies to two scan mirrors in [2]. Details on how to perform a measurement test and calculate ODSs of different structures using the CSLDV were summarized in [3, 4]. Up to date, a single SLDV is mainly applied to measure the out-of-plane vibration of a structure surface. The laser line-ofsight direction, which is the velocity measurement direction, is assumed to be aligned with the out-of-plane direction, whereas there is actually a small angle between them during scanning. This angle can be neglected when the area of the scanned structure surface is small. However, with a large scanning area and a short distance between the SLDV and measured structure this assumption can amplify the measurement error and an out-of-plane ODS of the structure may not completely agree with the corresponding measured ODS. Further, three-dimensional (3D) measurements of vibrating structures by SLDVs have been of interest, where there are three laser beams coming from three different directions, simultaneously directing to a measured point [5–7]. The orientations of three laser beams with respect to a structure coordinate system (SCS) must be D.-M. Chen • W.D. Zhu ( ) Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA e-mail: wzhu@umbc.edu © The Society for Experimental Mechanics, Inc. 2016 J. De Clerck, D.S. Epp (eds.), Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-30084-9_6 57
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