# . What are the four differential relationships and why do we use them in dynamics?

ENGR 2120: Dynamics Project #1a

Project Assignment For your project, you will gain a fundamental understanding of the dynamics of a planar, two link robotic manipulator. The purpose of this project is to apply what you learn in dynamics to a practical system. The manipulator is shown in the figure below.

Figure 1: Two-link robotic manipulator

The manipulator has a revolute joint (i.e., rotational joint) at its base and a prismatic joint (i.e., sliding joint) that connects two links of the robot. The revolute joint is fixed to ground. There is a variable distance, ??, from the base of the robot to the end-point of the manipulator. In addition, there is a variable angle, ??, from the horizontal to the centerline of link 1.

Your project will be written as a technical report, and for each assignment you will perform analysis and write that part of your technical report. For week 1, please complete the following tasks:

1. Download the technical writing template from D2L. 2. Replace my name with your name. 3. Read through the technical writing template. 4. Rename section 2 “Rectilinear Kinematics” 5. Within section 2, please write a description of the following. This should be in paragraph form, and

the purpose of writing this is to help you become more familiar with the topics that we cover in class. Feel free to add any appropriate sections or subsections to your report.

Project Assignment For your project, you will gain a fundamental understanding of the dynamics of a planar, two link robotic manipulator. The purpose of this project is to apply what you learn in dynamics to a practical system. The manipulator is shown in the figure below.

Figure 1: Two-link robotic manipulator

The manipulator has a revolute joint (i.e., rotational joint) at its base and a prismatic joint (i.e., sliding joint) that connects two links of the robot. The revolute joint is fixed to ground. There is a variable distance, ??, from the base of the robot to the end-point of the manipulator. In addition, there is a variable angle, ??, from the horizontal to the centerline of link 1.

Your project will be written as a technical report, and for each assignment you will perform analysis and write that part of your technical report. For week 1, please complete the following tasks:

1. Download the technical writing template from D2L. 2. Replace my name with your name. 3. Read through the technical writing template. 4. Rename section 2 “Rectilinear Kinematics” 5. Within section 2, please write a description of the following. This should be in paragraph form, and

the purpose of writing this is to help you become more familiar with the topics that we cover in class. Feel free to add any appropriate sections or subsections to your report.

a. What is rectilinear kinematics? b. What are the four differential relationships and why do we use them in dynamics? c. How do you know when to use constant acceleration equations as opposed to the general

differential relationships? d. What is the difference between speed and velocity? e. What is the difference between average and instantaneous velocity? f. Why do we use a particle abstraction in dynamics (and often in statics)?

6. Imagine that you are an engineer working in a manufacturing environment! Assume that you have created a machine that is used to adhere one part to another. Figure 2 shows Part A being adhered to Part B. You will assume that you have constrained the manipulator such that ?? = 0∘ for all time. Additionally, you will control the acceleration of the end-point of the manipulator, and this expression is given by �̈�? = ???? m

s2 , where ?? is a constant.

Figure 2: The two-link robotic manipulator at the position ?? = 0∘. Part A must be adhered to part B.

The machine will operate during three second time intervals. Your job as a robot designer is to determine a value of ?? such that over a three second time interval the end-point will have an ending position equal to ?? = 2 m. You may assume the manipulator starts at ?? = 1 m when ?? = 0 s. Assume that the manipulator starts from rest and remains at ?? = 0∘ throughout the motion. The derivation for your solution should be shown in your technical report and should be in variable form. That is, you should only substitute numbers once you have an expression for ?? in variable form.

Please create plots of the position, velocity, and acceleration of the end-point of the machine as time varies from ?? = 0 seconds to ?? = 3 seconds with your value of ??. Include this plot and the mathematical analysis in your technical report. The plot must be computer generated and include appropriate labels. You should plot the position, velocity, and acceleration on the same plot. Including a figure of the manipulator at ?? = 0∘ in your technical report would also be appropriate. Please note that the plot should be a continuous time plot.

differential relationships? d. What is the difference between speed and velocity? e. What is the difference between average and instantaneous velocity? f. Why do we use a particle abstraction in dynamics (and often in statics)?

6. Imagine that you are an engineer working in a manufacturing environment! Assume that you have created a machine that is used to adhere one part to another. Figure 2 shows Part A being adhered to Part B. You will assume that you have constrained the manipulator such that ?? = 0∘ for all time. Additionally, you will control the acceleration of the end-point of the manipulator, and this expression is given by �̈�? = ???? m

s2 , where ?? is a constant.

Figure 2: The two-link robotic manipulator at the position ?? = 0∘. Part A must be adhered to part B.

The machine will operate during three second time intervals. Your job as a robot designer is to determine a value of ?? such that over a three second time interval the end-point will have an ending position equal to ?? = 2 m. You may assume the manipulator starts at ?? = 1 m when ?? = 0 s. Assume that the manipulator starts from rest and remains at ?? = 0∘ throughout the motion. The derivation for your solution should be shown in your technical report and should be in variable form. That is, you should only substitute numbers once you have an expression for ?? in variable form.

Please create plots of the position, velocity, and acceleration of the end-point of the machine as time varies from ?? = 0 seconds to ?? = 3 seconds with your value of ??. Include this plot and the mathematical analysis in your technical report. The plot must be computer generated and include appropriate labels. You should plot the position, velocity, and acceleration on the same plot. Including a figure of the manipulator at ?? = 0∘ in your technical report would also be appropriate. Please note that the plot should be a continuous time plot.