On the Uniqueness of Solutions for the Identification of Linear Structural Systems

Abstract

This work tackles the problem of global identifiability of an undamped, shear-type, N degrees of freedom linear structural system under forced excitation without any prior knowledge of its mass or stiffness distributions. Three actuator/sensor schemes are presented, which guarantee the existence of only one solution for the mass and stiffness identification problem while requiring a minimum amount of instrumentation (only 1 actuator and 1 or 2 sensors). Through a counterexample for a 3DOF system it is also shown that fewer measurements than those suggested result invariably in non-unique solutions.

Modal parameter based structural identification using input–output data: Minimal instrumentation and global identifiability issues

“…Hence, for a 3-DOF shear-type system for which the 2nd row of V is known, there may exist two possible sets of 1st and 3rd rows of V, one of which will correspond to the true system, and the other to an alternative, but also physically admissible, system. An example of such non-uniqueness was also discussed in [9] where it was shown that two 3-DOF systems, System I with {m 1 , m 2 , m 3 }¼ {0.85, 1.3, 1.1} and {k 1 , k 2 , k 3 } ¼{9, 12, 11}; and System II with {m 1 , m 2 , m 3 } ¼{2.3690, 1.3000, 0.2776,} and {k 1 , k 2 , k 3 } ¼{7.5473, 16.1429, 6.8571}, will have the same response at DOF 2 for the same applied force at DOF 2, and thus will have the same eigenvalues and the same 2nd row of V. Proof of Statement 4. The proof of this statement is intuitive.…”

“…Furthermore, while the above studies provide answers to the question of minimal instrumentation, there still exists a need to develop and validate robust methods for unique mode shape expansion and consequent parametric identification using input-output data from prescribed minimal instrumentation set-ups. As Franco et al [9] clearly state: "…satisfying global identifiability does not mean that the unique solution is found. Indeed, to find this solution it is usually necessary to solve a nonlinear optimization problem that can be quite complex…", and illustrates the complexity one will encounter using a 3-DOF system at the end of their paper.…”

“…Thus, in the structural parameter identification problem considered here, neither M nor the total mass is known, and instead the objective is to find M, along with the usual objective of the identification of K. Formulating the problem using A enables the direct estimation of the mass normalized mode shape matrix V, thereby bypassing the need of any transformation and the necessity of any a priori additional knowledge of the value of any physical parameter. Due to this reason, the formulation of the inverse problem in this paper uses the non-symmetric A matrix, instead of the usual treatment of the problem using the symmetric J matrix adopted in Gladwell [11], Udwadia et al [7,8] and Franco et al [9]. Additionally, the present formulation allows the identification of the simultaneous lower bounds of both the necessary a priori information (no information) and the necessary instrumentation (1/2 instrumented DOFs) for a unique identification of M and K for shear-type systems.…”

“…While such a specific answer will lose out on the generalization aspect, it would be expected to gain in its straightforward applicability to any system within the class. In the studies by Udwadia et al [7,8] and Franco et al [9] such attempts have been made to address this issue of identifiability in the context of shear-type systems; while Franco et al [9] address this issue for the system excited by a single actuator located at any DOF and does not require any knowledge of the mass distribution, Udwadia et al [7,8] deal with the situation where the input is base excitation, and hence requires the complete knowledge of the mass matrix M. While these studies identify certain instrumentation set-ups which guarantee unique solutions, there is still a need to investigate the issue of uniqueness for more general experimental set-ups, especially catering to situations with multiple actuators, in order to have alternative experiment designs. Furthermore, while the above studies provide answers to the question of minimal instrumentation, there still exists a need to develop and validate robust methods for unique mode shape expansion and consequent parametric identification using input-output data from prescribed minimal instrumentation set-ups.…”

“…In works dealing with incompleteness of modal information in an identification problem, usually only one of the two incompleteness issues is addressed (e.g. see [4] and the references therein, as well as [5,6] for modal incompleteness, and [7][8][9][10][11] for spatial incompleteness). These two types of incompleteness in modal information may often make the inverse identification problem ill-conditioned, and may even lead to non-unique identification results.…”