Problem Background
Determining System Parameters
Checking the System Model
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To study system identification for a second order system, let's consider a cantilever beam. Suppose that we are interested in the vertical displacement of a point (P) at the end of the beam. If this is the case, the complicated dynamics of the beam can be simplified somewhat by thinking of the vertical motion of point P as being analogous to that of a mass-spring-damper system. The beam provides a restoring force (a spring force) to counteract a load applied vertically, and any motion of point P is opposed by a damping force. If we assume the spring force to be proportional to the displacement of P from its rest position and the damping force to be proportional to the vertical speed of the endpoint, then the system's free body diagram can be represented as follows:
In this diagram:
In our development of this problem, we will also use the variable "m" to represent the effective mass of the beam.
These variables can be used to arrive at the system equation:
To be able to accurately predict the vertical motion of point P, we need to know the approximate values of m, b, and k. These can be found mathematically using formulas that have been developed for this type of problem (and which still require knowing something about the beam's properties, such as stiffness and mass). Or, we can estimate these parameters by applying a known force to the beam and studying its response (the vertical displacement). Since we don't have an actual beam to test, a Simulink block has been created to represent how point P reacts to an applied force (download it by clicking here). This block takes as its input the force, F, applied at the end of the beam and outputs the vertical displacement, x, resulting from that force.
Let's apply a step input of 1000 N to the beam model (which could be done by placing an object weighing 1000 N on the end of the beam, for example) and analyze the response we get. The Simulink model for this should look like:
If we run this simulation for 4.0 seconds, the Scope block should display the following system response:
The system parameters (m, b, and k) can be estimated by taking a closer look at this plot. If we zoom in on the the last second of the response, we approximate the steady-state to be: xss = 0.213 m.
When the system is in its steady-state, we know that the velocity (x-dot) and the acceleration (x-double dot) of point P are zero. So, if we set both of these to zero in the system equation, we can find the effective spring constant:
Since the response to the step input oscillates before settling into the steady-state displacement, we see that the the system must be underdamped. Thus, we should be able to determine the damped natural frequency of the system using the plot. If we individually zoom in on the first two peaks in the response (see below), we estimate that they occur at about t = 0.38 sec. and t = 1.15 sec. Thus, the period of oscillation is T = 0.77 sec. You can verify this value by looking at the time between the second and third peaks in the response.
Using the period, we can determine the damped natural frequency of the system:
The damped natural frequency is related to the system parameters by:
Knowing the maximum displacement of point P (from the zoomed-in plot of the first peak in the response shown above) and the steady-state displacement (which we found earlier) allow us to determine the maximum percent overshoot of the system:
The maximum percent overshoot is related to the system parameters by:
Knowing the damped natural frequency and the maximum percent overshoot, we can find m and b. We first combine the equations for the two performance measures by getting rid of the terms containing zeta:
If we solve this equation for b, we get:
Now, plugging this expression into the equation for the damped natural frequency, we get:
If we solve this for m, the follow equation results:
Substituting in the values we know, we can solve for m:
Knowing m allows us to solve for b:
Thus, we now have estimates for the effective mass (m), effective damping coefficient (b), and effective spring constant (k) of the system.
Now that we have determined approximate values for the system parameters, let's verify that they agree with the original beam system. If we substitute the values we found into the system equation, we get:
This corresponds to the transfer function:
If we look under the "mask" of the Beam System block (right click on the block and choose Look Under Mask from the pull-down menu that appears), we see that it consists simply of the input force (F) being converted by a transfer function to the output vertical displacement (x). The transfer function that does this looks like:
Notice that the actual transfer function for the system and the one we found by using important characteristics of the system response are nearly identical. Thus, our method for identifying the system parameters appears to be valid, and the system equation given at the beginning of this section should do a good job predicting the displacement of point P when a force F is applied to the end of the beam.
Author: RDM
Updated: 6/12/00