Figure from:System Identification via the Proper Orthogonal Decomposition

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Fig. 5.16 Percent Error of Displacement Norms for NL Experimental Beam Response — Figure 5 16 Percent Error of Displacement Norms for NL Experimental Beam Response

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Fig. 2.1 Nonlinear System Identification Process (from [3], used with author’s permission)

Fig. 3.1 Example Mass-Spring System with Equal Masses >qual to the eigenmodes. Consider the nondimensional mass-spring system shown in Fig.

This system is undamped and the mass matrix is proportional to (in fact, it is equal to) the identity matrix. The eigenmodes (scaled so that they are orthonormal) for the system are:

3.2.2 POV Recalculation from a State-Space Perspective

After ®,,(t;) has been obtained from Eq. (3.27) at every time step, the POC histories representing a response to a new initial displacement profile may be calculated at each

Fig. 4.2 Cubic Spring Force for Nonlinear Beam Model

Fig. 4.1 LTI (top), LTV (middle), and NL (bottom) Beam Models

Fig. 4.3 Time Variation of Tip Mass for LTI Beam Model

Fig. 4.4 Applied Initial Displacement Profiles to form W . Thus the dimensions of W were (25 x 500) for both models.

Fig. 4.5 Original and Recalculated POVs for Linear Beam Model

Fig 4.6 Displacement Norms for LTI Beam Model in Response to w; mass results in a mass matrix with diagonal terms that are not close in value.

Fig. 4.7 Percent Error of Displacement Norms for LTI Beam Model in Response to

Fig. 4.8 Displacement Norms for LTV Beam Model in Response to w;

Fig. 4.9 Percent Error of Displacement Norms for LTV Beam Model in Response to

Fig. 4.10 Displacement Norms for NL Beam Model in Response to w; they are dwarfed by the linearization error.

Fig. 4.11 Percent Error of Displacement Norms for NL Beam Model in Response to

Fig. 4.13 Displacement Norm for LTI Beam Model in Response to w,

Fig. 4.14 Percent Error of Displacement Norms for LTI Beam Model in Response to

Fig. 4.15 Displacement Norm for LTV Beam Model in Response to w , is attributed to projection error.

Fig. 4.16 Percent Error of Displacement Norms for LTV Beam Model in Response to Wy,

Fig. 4.17 Displacement Norm for NL Beam Model in Response to w,

Fig. 4.19 Pulse Load Applied to Beam Models models.

Fig. 4.20 Five Locations of Pulse Load A pplication

Fig 4.21 Displacement Norms for LTI Beam Response to Load 5

Fig. 4.22 Percent Error of Displacement Norms for LTI Beam Response to Load 5

Fig 4.23 Displacement Norms for LTV Beam Response to Load 5

Fig. 4.24 Percent Error of Displacement Norms for LTV Beam Response to Load 5

Fig 4.25 Displacement Norms for NL Beam Response to Load 5

Fig. 4.26 Percent Error of Displacement Norms for NL Beam Response to Load 5

Fig. 4.27 Initial Displacements in Excitation Sets a, b, and c dimensions of “W, ’W and ‘W were (25 x 500) for all beam models.

Fig. 4.28 Initial Velocities in Excitation Sets a, b, and c Next, the POD was computed for each beam’s response to the first excitation set and the

Fig. 4.31 New Load A pplied Vertically at Beam Tip

Fig. 4.32 Displacement Norms for LTI Beam in Response to W

Fig. 4.36 Displacement Norms for LTI Beam in Response to F (t)

Fig. 4.37 Percent Error of Displacement Norms for LTI Beam in Response to F (t)

Fig. 4.38 Displacement Norms for LTV Beam in Response to W, attributed to the fact that the tip mass causes high diagonalization error.

Fig. 4.42 Displacement Norms for LTV Beam in Response to F (t)

Fig. 4.43 Percent Error of Displacement Norms for LTV Beam in Response to F (t)

Fig. 4.46 Displacement Norms for NL Beam in Response to W,

Fig. 4.48 Displacement Norms for NL Beam in Response to F (¢)

Table 4.1 Truss Member Geometry and Material Properties

Fig. 4.50 Cross Section and Schematic View of a Single Truss Element [C ourtesy Armaghan Salehian] members all have hollow circular cross sections and are joined by idealized pin joints.

Fig. 4.51 Coordinates for Truss Element [C ourtesy Armaghan Salehian] rotation around each pinned end. The structure is therefore a linear time-varying system.

Table 4.2 Static Loads A pplied to Generate Initial Displacements for Truss

Fig. 4.52 Displacement Norms for Truss Model in Response to w¢

Fig. 4.53 Percent Error of Displacement Norms for Truss Model in Response to w,

Table 4.3 Impulsive Loads A pplied to Generate Initial Velocities for Truss

Fig. 4.54 Displacement Norms for Truss Model in Response to Wg

Fig. 4.56 Pulse Load A pplied to Satellite Truss M odel the left boundary).

Fig. 4.57 Displacement Norms for Truss Model in Response to Load method’s response prediction.

valid for any applied load and the extra hits do not interfere as long as they are measured.

Fig. 5.2 Application of Impulsive L oad via Shaker Strike For each load case, a Polytec OFV-303 laser vibrometer was set up to measure the

Fig. 5.3 Load Cell Output With and Without Beam Impact

Fig. 5.4 Forces Applied to Various Beam L ocations

Fig. 5.5 Two Most Dominant POMs for Linear Beam the SL and ML methods compared with the measured data.

Fig. 5.6 POVs for Experimental Linear Beam

Fig. 5.8 Percent Error of Displacement Norms for Linear Experimental Beam Response

Fig. 5.9 Rubber Band Acting as Linear Spring beam tip deflects toward the rubber band attachment point (the rubber band buckles as

Fig. 5.10 Rubber Band Providing no Stiffness

Fig. 5.13 Phase Lag for Various Impulse M agnitudes excitation magnitude, indicating that the nonlinearity is active in the beam response.

Fig. 5.14 Forced A pplied to Locations on Nonlinear Beam initial impact.

Fig. 5.15 Displacement Norms for NL Experimental Beam Response to Load at 4” Location as in the numerical example.