• Document: System Inputs, Physical Modeling, and Time & Frequency Domains
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K. Craig 2012 System Inputs, Physical Modeling, and Time & Frequency Domains There are three topics that require more discussion at this point of our study. They are: Classification of System Inputs, Physical Modeling, and Time Domain vs. Frequency Domain. Let’s develop each of these further. Classification of System Inputs The model of our physical system under investigation, both physical and mathematical, must be validated, i.e., its soundness must be confirmed, if it is to be of any use. How does an engineer validate a model of a physical system? In order to validate the physical and mathematical model of the physical system under investigation, the engineer must first cause the system to respond. This is done by introducing to both the system mathematical model and to the actual physical system an input. An input is simply some action which will cause the system to respond. The same input is introduced to both the mathematical model and to the physical system. The predicted response is obtained by solving the mathematical model, i.e., the equations describing the behavior of the physical model, with the designated input. The actual response is obtained by introducing into the actual physical system the designated input and measuring the response using instruments like multimeters, oscilloscopes, or dynamic signal analyzers. What types of system inputs are there? The diagram shows one possible classification of system inputs. The two main classifications of system inputs are initial energy storage and 1 K. Craig 2012 external driving. Let’s use as an example the common, simple, spring-mass-damper mechanical system to illustrate the difference. Without an input, the mass remains stationary in an equilibrium state with the weight of the mass balanced by the spring force. How can we put the mass into motion? One way is to simply pull the mass down and release it. The mass will then oscillate up and down and eventually come to rest. When we pull the mass down we increase the stretch of the spring, storing energy in the spring as potential energy. When we release the mass, this stored-up potential energy is then given up causing the mass to move. Another way we could put the mass into motion would be to give it an initial velocity. One could hit the mass, i.e., give it an impulsive force, and this would result in an initial velocity of the mass and resulting oscillation. In this case kinetic energy is given to the mass initially. Both of these cases are examples of a system input classified as initial energy storage. The first being an example of potential energy storage and the second being an example of kinetic energy storage. Once we release the system, no external driving action is needed to keep the system in motion. An engineer might measure the position, velocity, or acceleration of the mass as it moves and these would be called the outputs of the system. So we see initial energy storage refers to a situation where the engineer puts a system, initially in an equilibrium state, into a different state and then releases the system. The system then responds free from any external interference. We could have used as an example an electrical, electromagnetic, thermal, or fluid system. The other major classification of input is external driving. In this case, a physical quantity from the system’s environment, i.e., from outside the system boundary, is applied to the system and causes it to respond. We often choose to study the system response to an assumed ideal source, which is unaffected by the system to which it is coupled, with the view that practical situations will closely correspond to this idealized model. External inputs can be 2 K. Craig 2012 broadly classified as deterministic or random, recognizing that there is always some element of randomness and unpredictability in all real-world inputs. Deterministic input models are those whose complete time history is explicitly given, as by mathematical formula or a table of numerical values. This can be further divided into two categories. A transient input model is one having any desired shape, but existing only for a certain time interval, being constant before the beginning of the interval and after its end. An example of this type of input for the spring- mass system would be a constant force applied at some instant and then removed at some later instant. The constant force could be applied just by adding an additional mass at some initial time and then removing it at a later time. The response of the system would be observed both before, during, and after the application of the force. The second type of deterministic input is a periodic input model. This input type repeats a certain wave form over and over, ideally forever, and is further classified as either sinusoidal or non-sinusoidal. For the spring-mass system, an actuator (e.g., a motor with a rack-and-pinion gear) could apply a sinusoidal-varying or

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