8  Behavior of Systems

This week we will be leveraging the concepts we have learned so far (e.g., Fourier analysis, the physical basis for the operation of resistive and capacitive sensors) to shed light on a topic that we covered earlier but without much mathematical intuition: dynamic behavior of measurement systems. We even had a homework problem designed to gain some intuition into the response of a second order system (a damped forced oscillator).

Though the behavior of a system can only be truly understood experimentally, creating models of their behavior can provide very useful approximations or at least provide a tool with which once can gain very important intuition about the behavior of the system under idealized assumptions. Each component of a measurement system (e.g., sensors, signal conditioning circuits, etc.) have their own response and together they generate the overall sensing system’s response to any input signals (forcing functions). What we’d like to do is to characterize this response by using mathematical models of the individual components and/or the overall system. In other words, we want to create mathematical models that allow us to understand how the output signal varies in response to different input signals. This is especially useful when the input signals are time-varying (dynamic), where the system’s characteristics determine how well it is able to “keep-up” (follow) the changing input signal.

• Consider, for example a bulb thermometer used to measure body temperature.
  • A step function input (dynamic)
  • A slow response (in the order of seconds to minutes)
• Consider now the vibration inside a vehicle as it travesl on a road.
  • The input signal is the normal force at the contact point of the tires to the road surface.
  • The output signal is not just a direct copy of that input force…

To study the response of the system, we could model the system in great spatial and temporal detail, taking into account the physical properties that are distributed across space and time. This model would take the general form of a partial differential equation relating inputs and outputs. And it could then be “calibrated” through experiments that provide us with pairs of input \(F(t)\) signals and their corresponding output \(y(t)\), along with the initial conditions of the experiment \(y(0)\). A very useful simplifying assumption is to suppose that the spatially distributed properties of the system can be lumped into discrete elements. This idea is called lumped parameter modeling. Since there is no spatial dependence for the behavior, the partial differential equations are transformed into ordinary differential equations of order \(n\):

\[a_n \frac{d^n y}{d t^n} + a_{n-1} \frac{d^{n-1}y}{d t^{n-1}} + \dots + a_1 \frac{dy}{dt} + a_0 y = F(t)\]

• Let’s see Example 3.1 from Figliola and Beasley (2019)

There are some special cases of the above system when \(n < 3\) and lead to very useful characterizations. Let’s see them.

8.0.1 Zeroth order systems

• Let’s now see Example 3.2 from Figliola and Beasley (2019)

8.0.2 First order systems

\[a_1 \dot{y} + a_0 y = F(t)\]

or more conveniently, by dividing both sides by \(a_0\):

\[ \tau \dot{y} + y = K F(t)\]

Where \(\tau\) is the time constant for the system and represents the time it takes for the system to get to 63.8% of the steady state value. Why?

• Step function input

If \(F(t) = AU(t)\) where \(A\) is a constant and \(U(t)\) is the unit step function, then

\[y(t) = KA + (y_0 - KA) e^{-t/\tau}\]

Figliola, Richard S., and Donald E. Beasley. 2019. Theory and Design for Mechanical Measurements. Seventh edition. Hoboken, NJ: Wiley.