OSE801 Engineering System Identification Spring 2010 Lecture 3: Identification Models Instructors: K. C. Park (Division of Ocean Systems Engineering) Y. J. Park (Division of Mechanical Engineering)

Discrete form of equations of motion for structures Traditionally, the governing equations for vibrating structures are given in terms of the familiar second-order differential equations in time. Solution algorithms for the transient response analysis of structures have thus been developed by exploiting the second-order characteristics of structural dynamics equations. Hence, the bulk of vibration theories are presented based on the second-order structural dynamics equations. Consequently, modal testing practice and its link with vibration theories rely extensively on the second-order structural models.

Equations of motion for structures - cont’d Beginning about the early 80s there has been an increasing tendency by the structural dynamics specialists to employ concurrently both the second-order as well as equivalent first-order state space models. This was motivated by two demands. First, active vibration control theories are almost exclusively based on a first-order state space form of differential equations. Second, realistic damping models are difficult to express in terms of normal modes and proportional damping theory is too restrictive for practical considerations. A major difficulty in the use of a first-order state space form of structural dynamics equations is in the physical interpretation of the resulting complex eigenvectors and eigenvalues. This is not to say that there exists a theoretically inherent difficulty. Rather, it is in the inadequate familiarization by most structural engineers with the first-order state space systems.

Equations of motion for structures - cont’d There is a third motivation that necessitates the use of a first-order state space model in structural system identification. For systems theory on which modern system realization theories are based has been developed for state space models whose origin is from electrical engineering. For these reasons we will adopt both the second-order and equivalent state space form of structural dynamics models for use in system identification. To this end, we first present a discrete second-order matrix equation for structures. Various state space forms and their characterization will then be treated.

Equations of motion for structures - cont’d Once the mathematical models are derived, we will then study their mathematical and physical properties so that they can be utilized in system identification. These include mass normalization, relations between the second-order eigenvectors and the state space eigenvectors, phase properties, poles and zeroes and their relations to eigenvalues and eigenvectors. proportional vs. non-proportional damping, stiffness vs. flexibility, Sturm sequence properties, among others.

MODEL EQUATIONS FOR STRUCTURAL SYSTEM IDENTIFICATION where q is the n-displacement vector; u is a m-input force vector; y is a l-sensor output vector, either displacement, velocity, acceleration or their combinations; is the input-state influence matrix, and are state-output influence matrices for displacement, velocity, and acceleration, respectively. The reason that we introduce the input and output Boolean influence matrices is that both the number of excitations m and of output measurements l are in general much smaller than the number of discrete internal variable n.

MODAL FORM OF STRUCTURAL EQUATIONS Transformation from physical to modal form:

TRANSFER FUNCTION OR INPUT/OUTPUT RELATION The transfer function is invariant with respect to the choice of the coordinate systems employed!

GENERAL STATE SPACE REPRESENTATIONS where x is a (N ×1) internal state vector, u(t) is a (m ×1) applied force vector; y(t) is a (l ×1) output (or physical sensor measurements such as displacement, velocity or acceleration) vector; A is a (N × N) matrix that is called a state transition matrix; B is a (N × m) input (force or actuator application) location matrix; C is a (l × N) output (sensor measurement) location matrix; and, D is a (l × m) matrix that represents any direct input/output feed-through. Observe that the size of the internal variable vector N is twice the size of the second-order structural dynamics equation, namely, N = 2n.

Transfer function for state space models Question: Would the transfer function, , obtained by the second-order model be the same as that, , of the state space model? The answer is: Proof: Suppose we have employed a general transformed expression of x: Then the input/output under the new internal variable z becomes:

State Space Model with Generalized Moment Choice of

State Space Model with Generalized Moment - cont’d Choice of

INPUT/OUTPUT INVARIANCE PROPERTY Begins with the transfer function obtained by the State space model: Ends with the transfer function obtained by the Second-order structural equations of motion!

State Space Model with Normal Modes

System-Identified State Space Model Physical Model Modal Model

Summary of Today’s Lecture: In preparing for the experiment, we need to choose an analytical form of model equations so that, when the system parameters are identified, we can construct the corresponding equations of motion. In particular ,the recovery of physical coordinate model from a mathematical identified model requires a careful one-to-one correspondence comparison. For this alone, a judicious selection of model equations play an integral part of the system identification. However, regardless of the internal system models chosen, the input/output relation is invariant under different choices of State variables or the order of equations one chooses to use.