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ME302: Mechatronics

Unit 6: System Dynamics   Any mechanical, electrical, fluidic, or thermal system has certain unique characteristics in how it responds to external excitation. Prior to attempting to control a physical system, it is important to understand its dynamic response to an external excitation. This unit will introduce you to the concept of system dynamics and its importance to mechatronic system design. Critical to the understanding and modeling the dynamics of a system is the differential equation that relates the input of the system to the output of the system. Differential equations are the ideal tool for capturing the dynamics of a system and its response to external inputs.

It is critical that prior to starting this unit, you review what you have studied on differential equations, especially those used to describe mechanical systems (mainly contained in ME202: Mechanics II). If necessary, review ME202 before beginning this unit.

Unit 6 Time Advisory
This unit should take approximately 5.5 hours to complete.

☐    Subunit 6.1: 3 hours

☐    Subunit 6.2: 1.25 hours

☐    Subunit 6.3: 1.25 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define the concept of a system. - Identify real-life examples of systems as well as their inputs and outputs. - Use ordinary differential equations to capture the dynamics of simple mechanical systems. - Identify examples of first-order mechanical systems, and solve problems using the Laplace transform method. - Identify examples of second-order mechanical systems, and solve problems using the Laplace transform method.

6.1 System Dynamics   Understanding the dynamics of a system is very important in achieving successful control. You will see from the video below how failure to understand system dynamics can lead to disastrous outcomes, as can be seen in the collapse of the Tacoma Narrows bridge.

  • Web Media: Archive.org: “Tacoma Narrows Bridge Collapse” Link: Archive.org: “Tacoma Narrows Bridge Collapse” (Flash)

    Instructions: Please watch this video.

    Terms of Use: This resource is in the public domain. The original version can be found here.

6.1.1 The Concept of a System   The mechatronic system that we are trying to control is usually referred to as the plant (e.g., mechanical system, electrical system, etc.). The plant that we are attempting to control is a system. In the case of the video below, the plant is mass vertically suspended on a spring. A system is a set of interconnected components that interacts with its environment. We are interested in the inputs and outputs to and from the system. Below, you will watch a video on system modeling by Kevin Craig.

  • Web Media: Rensselear Polytechnic Institute and Marquette University: Kevin Craig’s Multidisciplinary Mechatronic Innovations: “Intro to Modeling” Link: Rensselear Polytechnic Institute and Marquette University: Kevin Craig’s Multidisciplinary Mechatronic Innovations“Intro to Modeling” (QuickTime)

    Instructions: Please click the link above, and watch the video by Dr. Kevin Craig that provides an introduction to the concept of system modeling. In this video, Dr. Craig looks at a mass suspended on a spring and attempts to find the dynamic model that represents it.

    Watching this video and pausing to take notes should take approximately 1 hour and 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

6.1.2 The Use of Ordinary Differential Equations to Describe the Dynamic Behavior of a System   In any mechatronic system, we are interested in a specific output variable that we are attempting to control. We will usually control the output variable by setting the input variable to the desired value. Hence, the dynamic relationship between the input variable and the output variable is very important. Most systems have multiple inputs and multiple outputs, and their analysis become more involved. In the material discussed in this unit, we will restrict the analysis to single input single output (SISO) systems. The ideal tool to capture the dynamic relationship between the input variable and the output variable is the differential equation.

  • Web Media: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 2 –Modeling Systems” Link: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 2 –Modeling Systems” (YouTube)

    Instructions: Please click on the link above and then watch the video to learn about modeling systems.

    Watching this video and pausing to take notes should take approximately 1 hour and 15 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

6.2 First-Order Systems   The simplest form of a dynamic model for a system is the first order system. A first order system is typically represented by a time constant. When we try to charge a capacitor in a resistor-capacitor circuit, the response follows a first order system response. A tank full of water that has a small hole at its lower end will also follow a first order system response. The following video will discuss the response of a first order system.

  • Web Media: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 3 - Time Response” Link: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 3 - Time Response” (YouTube)

    Instructions: Please click on the link above and then watch the video. Note how different types of signal can be injected into first order systems (impulse, step, and ramp). Also, note how the Laplace transform and the inverse Laplace transform are used to find the time domain response.

    Watching this video and pausing to take notes should take approximately 1 hour and 15 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

6.3 Second-Order Systems   Few of the dynamic systems found in practice are not first order systems. They are usually second order system or higher. Second order systems are obviously more complicated, and their analysis is more involved. The following video discusses the response of second order systems to a step input. You will notice that the response of second order system to a step input could either be over-damped, critically damped, or under-damped.

  • Web Media: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 4 – Second Order Time Response” Link: YouTube: Dr. Ben Drew’s “Control Systems Engineering – Lecture 4 – Second Order Time Response” (YouTube)

    Instructions: Please click on the link above and then watch the video. Note how the response of a second order system depends on the damping ratio, zeta. Where the damping ratio is more than 1, the response to a step input will be under-damped and no overshoot will result. Where the damping ratio is less than 1, the system is under-damped and you will notice an overshoot and decaying sinusoidal response.

    Watching this video and pausing to take notes should take approximately 1 hour and 15 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.