Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Professor Flint O. Thomas Department of Aerospace & Mechanical Engineering Hessert Center for Aerospace Research Mechanical Systems Selection of Material on the Subject of Projectile Motion given by Professor Thomas for EG Fall 2000

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Mechanical Systems n Today’s Lecture: “Flight Dynamics” u Goal: Develop a model of the flight of the ball. Our analysis commences at the instant the ball leaves the pouch. Trajectory will depend on the initial speed V 0 and launch angle  imparted by the launcher.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Consider “snapshots” of the ball taken at equal time intervals  t = t i+1 -t i. At any instant of time the ball will be acted on by two forces: weight and aerodynamic drag. x y yoyo t1t1 t2t2 t3t3 t0t0 titi t i+1

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u The motion of the ball will be governed by Newton’s second law: u Remember: force and acceleration are vectors which possess both magnitude and direction. Vector addition is by the parallelogram law. F FxFx FyFy u Newton’s 2nd Law may be written for each component:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Let’s apply Newton’s Second Law to the flight of the ball: u Note that the drag force D always opposes the flight direction. Also we see that:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Newton’s Law for the x-component forces: u Newton’s Law for the y-component forces: u These equations relate the instantaneous x- and y-component accelerations of the ball to the instantaneous forces acting on the ball during the flight.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Before analyzing the trajectory of the ball, let’s look at the simpler problem of its motion in a vacuum. In this case there is no air resistance and the equations governing the motion can be obtained by setting D = 0: u In this case there is no x-component acceleration so V x will be constant. The vertical acceleration is constant. The motion of the ball is a superposition of uniformly accelerated motion in y and constant speed in x.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u We will divide the flight of the ball into many small sub-flights, each of duration  t. We start with known initial conditions from launch at time t = 0: u The acceleration is the time rate of change of velocity. Over any of the short time intervals  t we have: “Recipe” for finding the speed of the ball at t+  t from the known speed at t.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u The average x and y-component speeds of the ball over the time interval  t are given by: u The new x, y position of the ball is approximated as, u To illustrate consider the following example: V 0 = 40 m/s,  35 degrees, x 0 = 0, y 0 = 5 m Ball Launched with Initial Conditions (time t = 0):

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Suppose we divide the flight into  t = 0.1 second intervals. The component speeds of the ball at the next instant of time t = 0.1 second are, = constant The average speed over this interval is,

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems The position of the ball at t=0.1 second is obtained from, The procedure is now repeated sequentially for t=0.2, 0.3, 0.4 seconds etc. Microsoft Excel is an ideal tool for performing these calculations. A sample spreadsheet is shown below. time Initial condition

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u The trajectory (and range) of the ball is shown below: u How do we know if this is correct?

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems !! Whenever possible numerical solution methods should be checked against exact analytical solutions in order to insure the accuracy of the technique. We will apply that approach here. u Recall from your calculus course that: u Acceleration is the time derivative of velocity

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Newton’s Second Law becomes: and These are called differential equations because they involve derivatives of the velocity of the ball. Note that in a vacuum the x and y motions are independent, as noted before. u In order to obtain the velocity V x (t) and V y (t) of the ball from the above equations we need to take anti-derivatives (i.e. integrate) of both sides of the equations.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems We will integrate with respect to time over the interval from t = 0 to some arbitrary time t during the flight. and So the velocity as a function of time is, Now recall: and

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u The differential equations for the trajectory of the ball are, and As before, we integrate with respect to time to obtain the trajectory of the ball x(t) and y(t). Remembering that at t = 0, x=0 and y = y 0 we get, This is the analytical solution for the trajectory of the ball. How does it compare to our numerical solution ?

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Excellent agreement!

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u With the model validated we can put it to work. Example: For a fixed initial velocity what launch angle gives maximum range? For projectile motion in a vacuum, maximum range occurs at a launch angle of 45 degrees.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Example: What combinations of V 0 and   yield a desired range of 100 meters? Solution... construct a ballistics chart: Numerous possibilities indicated…impossible for V 0 = 30 m / s.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u We do not live in a vacuum. We are at the bottom of an ocean of air! What is the effect of the air on the flight of the ball? Photograph by F. N. M. Brown, Notre Dame

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Photo by Thomas C. Corke u Visualization of flow over a cylinder: Where does the energy to put the air in motion come from?

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Conclude that the passage of the ball through air will give rise to a loss in the kinetic energy of the ball that must be accounted for in order to realistically model the trajectory. In other words, we must account for aerodynamic drag! Back to the original form of the equations of motion: These equations don’t help us unless we know how to determine D !!

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Goal : Determine the aerodynamic drag on the ball and incorporate it into our numerical simulation of the flight. u This requires some basic fluid mechanics (the branch of mechanics which deals with the dynamic behavior of gases and liquids). The objective of fluid mechanics is most often the determination of flow-induced forces. Example: Wing lift and drag:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Select a reference frame: Two ways of looking at the same problem. The drag would be identical in both cases since it is the relative motion of the air and ball that determine drag. This equivalence is one basis of wind tunnel testing.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u How is drag created? u The momentum of the “cylinder of air” is the product of its mass times its velocity. Mass of air = density X volume = Momentum of air = Mass X velocity =

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u As the cylinder of air and ball collide the momentum of the air is imparted to the ball. u Another way of writing Newton’s second law is in terms of the time rate of change of momentum.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems If we assume all the momentum of the cylinder of air is imparted to the ball during the time interval  t then the force on the ball D is given by, Momentum of air Transferred during time  t

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Is this result correct? Not necessarily.... Our calculation assumes that all of the momentum of the cylinder of air is transferred to the ball. This may not be the case! What we can say with some certainty is, Remember: = air density [ kg/m 3 ] = area of the body “seen” by the oncoming flow. [m 2 ] = fluid velocity = flight speed [m/s]

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u To make this an equality the convention is to incorporate a factor C D /2 (typically determined from experiment) such that, C D is called the “drag coefficient” u The drag coefficient C D depends on the geometry of the body and also varies in a complicated way with flow speed. u Fortunately, for flow over a sphere at the velocities that we’re likely to encounter during the launch project, to good approximation experiments show that:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Now that we have a model for the drag force we can perform a numerical simulation of the flight of the ball including the effect of drag. u We will model the drag on the ball by the relation, u As before, we will divide the flight of the ball into many small sub-flights, each of duration  t. We start with the known initial conditions from launch at time t = 0:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u The acceleration is the time rate of change of velocity. Over any of the short time intervals  t we have: These provide a “recipe” for advancing the speed of the ball in time…If I know V x (t) and V y (t) I can use these equations to find V x and V y at time t+  t later.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems The recipe... with drag given by our model, VxVx VyVy V  What about  ? !!

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems  The angle  depends on the the relative size of the x- and y-components of the velocity of the ball (as shown below) and is therefore a function of time. ‘looks like simple trig to me…”

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Now I see the procedure! 1. Using V x (t i ) and V y (t i ) compute the drag force and angle theta. 2. Use “the recipe” provided courtesy of Sir Isaac Newton to find the new velocity V x (t i  t  and V y (t i  t  a time  t later. 3. Calculate the average x- and y-component velocity for the time interval  t. 4. Use the average velocity and knowledge of the position of the ball x (t i ), y (t i ) to find the of the ball x(t i  t), y (t i  t). Has the ball hit the ground? NO YES Repeat 1-4

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u To illustrate, reconsider the example we looked at before: Ball Launched with Initial Conditions (time t = 0): V 0 = 40 m/s,    35 degrees, x 0 = 0, y 0 = 5 m With the additional information that: m = 300 grams = 0.3 kg Diameter = meters A = X m 2  As before we will divide the flight into  t = 0.1 second intervals. Initial x- and y-component velocities are:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems Compute the drag force: Compute the new x- and y-component velocities:

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems u Compute the average speed for the interval, u The position of the ball at t=0.1 second is obtained from, u The procedure is now repeated using Microsoft Excel.

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems time  D x(t i ) y(t i ) titi VxVx VyVy You will develop your own spreadsheet in the learning center. How significant is the influence of drag? Initial conditions

Introduction to Engineering Systems Copyright ©2000, University of Notre Dame Module 1: Mechanical Systems A very significant effect on range! The trajectory is no longer parabolic!