dynamics

Technical Writing: a guide to creating a technical report.

Brian J. Slaboch, Ph.D. Middle Tennessee State University

Department of Engineering Technology, Mechatronics Engineering

Abstract Abstract goes here.

1 INTRODUCTION Introduction goes here.

2 RECTILINEAR KINEMATICS Rectilinear motion is motion that occurs either forward are backward along a straight line. This type of motion is governed by four differential relationships.

2.1 DIFFERENTIAL RELATIONSHIPS The four differential relationships are given in Eq. 1, where 𝑠𝑠 is defined as the position of the particle along the line, 𝑣𝑣 is the particle’s velocity, and 𝑎𝑎 is the acceleration of the particle.

𝑣𝑣 =

𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑

𝑎𝑎 = 𝑑𝑑𝑣𝑣 𝑑𝑑𝑑𝑑

𝑎𝑎 = 𝑑𝑑2𝑠𝑠 𝑑𝑑𝑑𝑑2

𝑎𝑎 = 𝑣𝑣𝑑𝑑𝑣𝑣 𝑑𝑑𝑠𝑠

(1)

It should be noted that the fourth equation in Eq. 1 can be derived from the first two equations. That is, the fourth equation is not independent of the other three equations.

Slaboch 1

2.2 CONSTANT ACCELERATION EQUATIONS Often in dynamics, the acceleration of a particle along a straight line is constant. Integrating the equations in Eq. 1 yields the following constant acceleration equations, as given in Eq. 2.

𝑣𝑣 = 𝑣𝑣0 + 𝑎𝑎𝑐𝑐𝑑𝑑

𝑠𝑠 = 𝑠𝑠0 + 𝑣𝑣0𝑑𝑑 + 𝑎𝑎𝑐𝑐𝑑𝑑2

2

𝑣𝑣2 = 𝑣𝑣0 2 + 2𝑎𝑎𝑐𝑐(𝑠𝑠 − 𝑠𝑠0)

(2)

These equations are only applied when the acceleration of the particle is constant.

2.3 SPEED VS. VELOCITY An important distinction in dynamics is the difference between speed, velocity, average velocity, and instantaneous velocity. By definition, speed is the magnitude of velocity. Velocity has both magnitude and direction. Average velocity is given by Eq. 3 as:

𝑣𝑣𝑎𝑎𝑎𝑎𝑎𝑎 =

Δ𝑠𝑠 Δ𝑑𝑑

(3)

In the limit, as Δ𝑑𝑑 → 0, the average velocity becomes the instantaneous velocity, and this is given in Eq. 4:

𝑣𝑣 = lim

Δ𝑡𝑡→0 � Δ𝑠𝑠 Δ𝑑𝑑 � =

𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑

(4)

The differential relationship in Eq. 4 is useful for describing the velocity of a particle undergoing rectilinear motion. Note that a particle approximation is often used in dynamics (as well as statics) because it can make the math easier while still providing significant insight into the problem at hand. A particle is an object that has mass, but whose size is irrelevant to the description of the motion.

2.4 EXAMPLE APPLICATION Consider the planar robot shown in Fig. 1. Assume that the manipulator is constrained such that 𝜃𝜃 = 0∘, and therefore the end-point of the machine only moves back and forth in the horizontal direction. It is assumed that this robot is used for a manufacturing application in which part A must be adhered to part B. A controller must be developed to control the end point of the manipulator over time. It is assumed the acceleration must be controlled given the mathematical relationship in Eq. 5 given as:

�̈�𝑟 = 𝑘𝑘𝑑𝑑 m s2

(5)

where 𝑘𝑘 is an unknown constant. It is further assumed that the manipulator starts from rest at 𝑟𝑟 = 1 m. The goal is to determine a value of 𝑘𝑘 such that the manipulator ends at 𝑟𝑟 = 2 m after 3 seconds has elapsed.

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Figure 1: The planar manipulator is constrained to only move in the horizontal direction.

Assuming the machine starts at time is equal to zero, integrating Eq. 5 yields:

�̈�𝑟 =

𝑑𝑑�̇�𝑟 𝑑𝑑𝑑𝑑

= 𝑘𝑘𝑑𝑑

�𝑑𝑑�̇�𝑟 �̇�𝑟

�̇�𝑟0

= �𝑘𝑘𝑑𝑑𝑑𝑑𝑑𝑑 𝑡𝑡

0

�̇�𝑟 = 𝑘𝑘 2 𝑑𝑑2 + �̇�𝑟0

m s

(6)

Integrating Eq. 6 provides an expression for the position of the end-point over time. It is assumed that the end-point starts at 𝑟𝑟 = 0 m. This result is given in Eq. 7.

�̇�𝑟 =

𝑑𝑑𝑟𝑟 𝑑𝑑𝑑𝑑

= 𝑘𝑘 2

(𝑑𝑑2 − 𝑑𝑑0 2) + �̇�𝑟0

𝑑𝑑𝑟𝑟 = � 𝑘𝑘 2 𝑑𝑑2 + �̇�𝑟0�𝑑𝑑𝑑𝑑

�𝑑𝑑𝑟𝑟 𝑟𝑟

𝑟𝑟0

= �� 𝑘𝑘 2 𝑑𝑑2 + �̇�𝑟0�𝑑𝑑𝑑𝑑

𝑡𝑡

𝑡𝑡0

𝑟𝑟 = 𝑘𝑘 6 𝑑𝑑3 + �̇�𝑟0𝑑𝑑 + 𝑟𝑟0 m

(7)

Solving for 𝑘𝑘 in Eq. 8 gives:

𝑘𝑘 =

6(𝑟𝑟 − 𝑟𝑟0 − �̇�𝑟0𝑑𝑑) 𝑑𝑑3

(8)

Substituting in known values into Eq. 8 yields:

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𝑘𝑘 = 6(2 − 1 − 0(3))

33

Thus,

𝑘𝑘 = 2 9

m s3

.

Figure 2 shows a plot the position, velocity, and acceleration of the end-point of the machine as time varies from 𝑑𝑑 = 0 to 𝑑𝑑 = 3 seconds.

Figure 2: Position, velocity, and acceleration of the end-point of the manipulator.

From this plot, it is shown that the calculated value of 𝑘𝑘 achieves the desired result as the position starts at 𝑟𝑟 = 1 m and ends at 𝑟𝑟 = 2 m.

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