Assignment #1 Identification Methods for Structural Systems

Assignment #1
Identification Methods for Structural Systems
Modeling of a 3-dof system
Consider the 3-dof system which represents a shear frame model of a two-floor building with a
seismic isolation foundation. Assume that
𝑚𝑇 = 200, 𝑚2 = 𝑚3 = 20,
𝑘1 = 1 × 106 , 𝑘2 = 𝑘3 = 1 × 105 ,
𝑐1 = 500, 𝑐2 = 𝑐3 = 50
Figure 1: Three DOF system subject to base acceleration.
1) Draw the free body diagrams of the DOFs and derive the equations of motions for the system
of Fig. 1.
2) Bring these equations into the state space form and use MATLAB’s ode45 function (Runge
Kutta method) in order to numerically obtain the system response to the accelerogram of the
El Centro earthquake [ElCentro.mat file – 1st column corresponds to time and 2nd column to
the N-S component of the earthquake].
3) Form the matrices A, B, C and D of the system’s state space representation and create the
corresponding system model in MATLAB [command ss(A,B,C,D)]. Select the acceleration of
the top floor of the system (𝑥3 ) as the output of your state space model. Use the commands eig
and/or any other appropriate MATLAB function in order to extract the natural frequencies of
the system and the corresponding damping ratios from the state matrix A.
4) Use the Laplace Transform to derive the transfer function 𝐻 𝑠 for this structural system.
Which are the roots of the denominator of H(𝑠)? Are these roots stable? [roots and pzmap
MATLAB command].
5) Use the transfer function 𝐻 𝑠 and the bode or bodemag MATLAB command in order to
obtain the magnitude of the simulated system Frequency Response Function (FRF). Use this
graph and the half power method in order to calculate the natural frequencies and the damping
ratios of the system and compare with those obtained at step 3? Comment on the results.
6) Submit your m-code electronically to [email protected] along with a short report on the
steps and appropriate figures.

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