6.1 Simulation of Simple Systems

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1 LESSON 6.1: SIMULATION OF SIMPLE SYSTEMS Simulation of Simple Systems In the previous lessons of Part II we looked at many simplified, yet realistic examples of dynamic systems. We have learned how to apply Newton s second law and alternate forms of the law such as impulse and momentum. We solved the equations of motion in closed form because we were able to solve them in closed form. In this lesson we will again use the modeling tools we have learned to model systems, but instead of neglecting terms that make the situation feasible in closed form, we will include these terms and use the computer to help solve our problem. MOTIVATION: The Trebuchet is a complicated machine, and in order to completely analyze the machine before and after construction, simulation of the system s dynamic equations can be very useful in deciding design factors such as counterbalance size, rocker-arm length, etc. The modeling tools such as impulse and momentum will allow us to bound the motion variables, but the fine details of the motion must be determined through simulation. This is true for the case when the system under study has simple closed form equations which are solvable with analytical tools, and the case (e.g. the Trebuchet) when the system equations are too complex for analytical solutions. Lesson Objectives In this lesson the student will learn: 1. the details of generating a dynamic model suitable for simulation in computer languages such as C, FORTRAN, MATLAB 1, Maple 2, and Mathematica 3 2. how to implement a simulation in pseudo-code. Application Example ACME and Bungee Jumping Word of the acumen of ACME Engineering is spreading. The owner of Wild and Wacky Bungee Jumping has sought ACME s professional advice. In order to get reasonable insurance rates for the venture, complete dynamic analysis of the arrangement has to be completed. If results of the analysis fall within the realm of safety, as specified by the insurer, the project can proceed. The proposed system is shown in figure 6.1. We will write the equations of motion for this system, then we will solve the equations numerically, that is, we will simulate the system on a computer. System Extent The proposed system for the thrill ride is shown in figure 6.1. It consists of the tower, the platform, the jumper, and the bungee cord. There are several points to note about the configuration. 1. The bungee cord is like a rubber band, it only resists motion when the cord is in tension. 2. The tension of the cord will not exist until the jumper has fallen L u ft from the jump off point. 1 MATLAB is a Trademark of xxxxx. 2 Maple is a Trademark of Waterloo xxx. 3 Mathematica is a Trademark of Wolfram Research Inc.

2 274 CHAPTER 6. SIMULATION AND DESIGN y H Figure 6.1: Bungee Jumping System Spring Force F s Linear Stiffening Softening Displacement y Figure 6.2: Three Spring Types 3. The tower must be tall enough so that the jumper doesn t pancake. 4. Cord failure is unallowable. 5. There are two basic modes of motion: (a) un-pulled, and (b) pulled. These repeat until the friction in the system dissipates all motion. We will assume that bungee cords come in three varieties. Force-elongation curves are shown in the figure 6.2. The first spring type is linear, the second stiffening, and the third is softening. The cords are limited to a 60 ft stretch before failure. Choosing Coordinates The height y of the jumper is shown in figure 6.1. For this analysis we will neglect any swing motion of the jumper. As can be seen from the figure, it may not be wise to neglect the swinging motion if the tower is close to the flight path. For now we will assume that the vertical motion is the most important. Define the System The freebody diagram is shown in figure 6.3. The forces are from the cord (F c ), air drag

3 LESSON 6.1: SIMULATION OF SIMPLE SYSTEMS 275 F c F c F d F d mg mg Figure 6.3: Jumper Freebody Diagrams (F d ), and weight. The figure on the left is for the trip down and the figure on the right is for the trip up. Notice the change in sign of the air drag F d. We realize that the spring force only acts when the cord is stretched, otherwise there is no spring force. This can be characterized as: { 0 if y > H Lu F c = (6.1) f(h L u y) if y H L u To account for the sign change of the air drag F d = cv 2 shown in figure 6.3, the Sign function will be used: { 1 if v 0 Sign(v) = (6.2) 1 if v < 0 Force-Motion Relationships We can write Newton s second law for the falling vertical motion as follows: F c + F d mg = m v (6.3) where the left-hand freebody diagram of figure 6.3 was referenced. For the upward motion, the right-hand freebody diagram is valid. We have: F c F d mg = m v (6.4) Other Equations The kinematics of the system are simple. We need only take two derivatives of the displacement y to describe the velocity and acceleration in the vertical direction. To generate an equation more suitable to simulation we combine equations 6.2 and 6.3 by using Sign(v) as a switching function: F c Sign(v)cv 2 mg = m v (6.5) the other equation we need, since we wrote acceleration as the derivative of velocity, is the kinematic relationship: ẏ = v (6.6) These two equation describe the vertical motion. We do, however, have to use equation 6.1 for F c.

4 276 CHAPTER 6. SIMULATION AND DESIGN Solve and Interpret Equation 6.5 is a nonlinear equation, which is difficult, if not impossible, to solve in closed form. Therefore, it is appropriate to perform simulations of equations 6.5 and 6.6 so we can quantify the motion of the jumper. Before we proceed, it should be noted that equation 6.5 can be written completely in terms of y and its derivatives if equation 6.6 and its derivatives are put into equation 6.5. That is F c Sign(ẏ)cẏ 2 mg = mÿ (6.7) This equation is called the 2 nd order form of the equation of motion. Equations 6.5 and 6.6 are called the 1 st order form of the equations of motion. Many computer solutions to differential equations require that the equations be in first order form with only the derivative terms on the left-hand side. The initial conditions on displacement and velocity must also be provided so that a solution can be executed. Pseudo-Code There are many commercially available mathematics packages that run on all sorts of computer hardware. The authors assume that the reader can obtain the legal rights to a package and can translate the pseudo-code presented below into the syntax of the package being used. The pseudo-code shown below is very close to the notation of the package MATLAB. The authors have worked extensively with Mathematica and Maple, so sample sources of Mathematica and Maple code are also presented in the Appendix. The general structure of the simulation will be that of a main body and of a function called by the main body. For the bungee problem the simulation code will resemble the following pseudo-code. Main Program End to = 0;! Initial time. tf = 60;! Final time. yo = [100 0] ;! Initial conditions y(0)=h=100, v(0)=0. y = odesolve( deriv,to,tf,yo,t);! EOM solver, pass derivatives. plot(t,y)! Plotting y(t) and v(t) vs time. Function Program function ydot = deriv(t,y);! Function for bungee simulation. Lu = 25;! Free-length of cord in feet. H = 100;! Tower height in feet. g = 32.2;! Consistent unit parameters. m = 0 / g;! Mass. c = 0.2;! Drag coefficient. k = 4;! Spring constant.! Implement spring function. if y(1) > H - Lu y[t] < H - Lu !Free or break. f = 0; else f = k * (H - Lu - y(1)); end ydot(1) =y(2);! Equation 6.6. ydot(2) = (-m*g - sign(y(2))*c*y(2)*y(2) + f)/m;! Equation 6.5.

5 LESSON 6.1: SIMULATION OF SIMPLE SYSTEMS Displacement (ft) Velocity (ft/s) - Time (sec) Figure 6.4: Time History: Linear Spring Displacement (ft) Velocity (ft/s) - Time (sec) Figure 6.5: Time History: Softening Spring End Function This particular simulation, with a linear spring with k = 4 lb ft, results in a time history of position and velocity as shown in figure 6.4. When the spring force function is replaced with the quadratic stiffening term the result will be, with k = 0.1, very similar to the linear case. However, when k is chosen to be a larger value, large differences in the response can be observed. When the softening spring is used, with k = , the response is as shown in figure 6.5. The response is very similar to the linear case. Yet as the reader should see for themselves, the result is very sensitive to the choice of k. As will be shown in Part IV of the textbook, energy conservation equations can be used to verify the numbers that the simulation provides. By assuming the ideal case of no air drag, energy conservation will give results that are estimates for the values at specified points in the simulation. For instance, the velocity of the jumper just before the cord gets taut could be checked with energy considerations. The maximum displacement of the cord could also be checked with energy conservation. These calculations are left as an exercise for the reader in Part IV of the textbook. End Example In this example we utilized simulation to determine the time history of a system. We did not spend much time doing what if parameter studies, but the simulation can be easily adjusted to do these studies. The reader should be able to implement this simulation in their favorite mathematics package, or by writing their own simulation in a suitable programming language. A Mathematica notebook is provided in bungee.nb. In the next example we utilize Mathematica to generate a planar simulation of a projectile

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