Matlab homework assignments for modeling dynamical systems

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1 Physics 311 Analytical Mechanics - Matlab Exercises C1.1 Matlab homework assignments for modeling dynamical systems Physics Analytical Mechanics Professor Bruce Thompson Department of Physics Ithaca College Ithaca, NY bthompso@ithaca.edu 1 March 6 Ph311 Matlab cover.doc rev Bruce Thompson / Ithaca College

2 Physics 311 Analytical Mechanics - Matlab Exercises C1. Ph311 Matlab cover.doc rev Bruce Thompson / Ithaca College

3 Physics 311 Analytical Mechanics Matlab questions HW.1 Physics 311 Matlab Homework Learning to use Matlab for integration of equations of motion Assign. # Description 1 Matlab 1 Plotting and finding zeros The Morse Potential for a Diatomic Molecule Matlab First order ODE Linear and Quadratic Drag 3 Matlab 3 Second order ODE The Real Pendulum 4 Matlab 4 - Second order ODE with Event Sensing The Real Pendulum: Period vs. Initial Angle 5 Matlab 5 - Matlab take home exam The Morse Potential Oscillator Using Matlab to investigate a mechanical system Assign. # Description 6 Choose a Matlab project, write a description of the project. The report should include: o a one paragraph description of the system, o a diagram of the system, o the goals of the project. 7 Derive and report the equations of motion for the project in a form suitable for numerical integration. The report should include two parts. One part is the handwritten derivation of the equations of motion for the project to a form which is suitable for integration via Matlab. The second part is a progress report which includes o the previous report with revisions if necessary, o identification of the forces that are acting together with a diagram, o a summary of the derivation of the equations of motion from the forces or from Lagrange's formalism, o o identification of the significant parameters for the system, and a summary of re-writing of the equations of motion as first order equations for integration via Matlab. 8 Find preliminary solutions to the equations for the project, check them with whatever means possible and report. The report should include o the previous report, o your code for the solutions, o preliminary solutions and o how you checked them for validity. Ph311 Matlab HW.doc rev 6.1. Bruce Thompson / Ithaca College

4 Physics 311 Analytical Mechanics Matlab questions HW. 9 Investigate the solutions for the project and report. The report should include o the previous report, o one paragraph describing the goals and procedures for the investigation of the solutions, and o graphs of the solutions for various parameter values. 1 Formulate conclusions from your investigations. The report should include o the previous report and o a summary of the conclusions you have derived from analysis of the system. 11 Present a first draft of the poster pages for the project. See below for a description of the pages expected. You should make an appointment to see me individually so that we can go through these together. 1 Present the project as a poster and/or oral presentation. Be prepared to discuss and answer questions on the process and results of your project with others who have a background in physics. Matlab Project - Suggested Poster Pages By organizing your presentation into pages you have the flexibility to use them as slides for a talk, pages in a poster or stapled together as part of a report. Here are some suggestions for organizing your pages. Page Page 1 Page Page 3-7 Page 8 Page 9 Title, who, when, course name, a picture of you Description of the project. Include the physical basis for the model system and the goals for the project. Include a diagram of the system as a whole. Show the derived equations of motion and describe the physical basis for each term and the significance of each parameter. Include diagrams to make the information clear. (Variable number of pages) Show the solutions of the equations in graphical form. Summarize your conclusions about the system and if and how the goals were reached. References and credits. The typeface should be about 16. Large enough to read from a comfortable standing distance. Equations should be done with Microsoft Equation editor or a similar means. Diagrams should be done with a drawing program. Alternatively, they could be hand done and scanned to put with the text. Orientation of the page can be landscape or letter depending on the information on the page. These pages can then be easily pasted into a large poster form using Powerpoint and printed using the large format printer at the library. Ph311 Matlab HW.doc rev 6.1. Bruce Thompson / Ithaca College

5 Physics 311 Analytical Mechanics - Matlab Exercises M1.1 Matlab Exercise #1 Plotting and finding zeros of functions The Morse Potential for a Diatomic Molecule 1.1 The Morse function for a vibrating diatomic molecule is given by ( x x ) δ V( x) = V 1 e V where x is the distance between the atoms, V determines the energy scale, x is the separation when the potential is a minimum and δ determines the shape. (See Cassiday Analytical Mechanics Example.3.3) The Matlab function "MorsePotentialPlot" on the back of this page calculates the potential function for three values of the parameter delta. Since we are interested only in the shape of the graph, take V =1 and x =1. Enter this function into Matlab adding comments as you go that describe the function of each statement. Hand in a printout of the plot and the Matlab script. 1. For small vibrations about x, the Morse potential can be approximated by the ( x x ) parabola, V a = V 1. (Cassiday Example.3.4) Make a plot of the Morse δ potential and its approximation for a delta = 1.. Use the filename MorsePotentialPlotII. Estimate the range over which the approximation is reasonable. Define reasonable. Hand in a printout of the plot and the Matlab script and your estimate. 1.3 Suppose a molecule has a total vibrational energy that is 1/4 V above the minimum potential energy, -V. Assume delta = 1. Use Matlab to draw a line on your second graph at this energy and determine the minimum and maximum separation of the two molecules for both functions to 3 decimal places. One way to do this is to use the 'zoom in' function of the tool menu in the figure window to look closely at the crossing location. Check your numbers by calculating the locations by hand. Hand in a printout of the plot and the Matlab script and numerical values of the minimum and maximum separations. 1.4 Optional: Use the fzero Matlab function to automatically determine the x values and the minimum and maximum separation of the two molecules for the Morse function. Hand in a printout of the Matlab script and a plot with the values determined in this manner. 1.5 Optional: A molecule has rotational energy in addition to the vibrational potential. This can be modeled as an effective potential that consists of the Morse potential with k an additional term. The effective potential can be written: U( r) = V ( r) + where k x is a constant related to the speed of rotation and V(r) is the Morse potential given above. Plot the effective potential energy of a rotating molecule for k=, 1,, 5, 1. What does this picture say about the bound states of the hydrogen molecule at high rotation speeds? Ph311 Matlab 1.doc rev 6.1. Bruce Thompson / Ithaca College

6 Physics 311 Analytical Mechanics - Matlab Exercises M1. function MorsePotentialPlot; % calculates and plots the Morse function of the potential energy of % a vibrating diatomic molecule, see Example.3.3 and.3.4 and.3.5 % of Cassiday 6th ed. % bgt % 1/6/4 % add comments to this code!!! xmax = 5; x = linspace(,xmax); y1 = Morsef(x,.5); y = Morsef(x,1.); y3 = Morsef(x,1.5); close all; plot(x,y1,'-',x,y,'--',x,y3,':'); axis([ xmax - ]); xlabel('x/xo'), ylabel('v/vo') title('morse Potential Function - varying delta') legend('d=.5','d=1.','d=1.5') function y=morsef(x,delta); % calculates the Morse potential function % for V = 1 and x = 1 y = (1-exp(-(x-1)/delta)).^-1; Ph311 Matlab 1.doc rev 6.1. Bruce Thompson / Ithaca College

7 Physics 311 Analytical Mechanics - Matlab questions M.1 Matlab Exercise # - First order ODE Linear and Quadratic Drag dv.1 The differential equation for a falling body with linear drag is m mg c1 dt = v where c1 is the linear drag coefficient, m the mass and g the acceleration of gravity. If we let u = v t and T = where v t = mg vt and τ = then the DEQ can be written as vt τ c1 g du 1 u dt =. On the page that follows is a Matlab function, 'LinearODE', that calculates the velocity as a function of time for the case of a body starting from rest and falling vertically in a constant gravitational field with constant linear drag. Type in this function and make sure it reproduces the printout shown. Use Matlab 'help' to understand any line that you don't completely understand. I have used the case of a 1. cm ball with a density of 4. gm/cc and a drag coefficient given by 1.55E-4*D where D is the diameter of the ball. Note the value of the terminal velocity and how long it takes to get to there. Hand in the function, plot printouts and answers to the questions.. Quadratic drag gives the following DEQ: dv m mg c dt = v. Show that this can be du rewritten as 1 u dt = and define u and T. Modify the Matlab function of part 1 to produce function, called 'QuadODE', that calculates the velocity of the falling body using quadratic drag. Note the value of the terminal velocity and how long it takes to get to there. Why are these different from the linear case? Hand in the function, plot printouts and answers to the questions..3 Show that the ODE for the combined case of linear and quadratic drag acting on the du ball can be written 1 ru u dt = and determine u, T and r in terms of the fundamental constants. Now write a function called 'LQODE' that calculates the velocity for the combined linear and quadratic drags. This will require that you pass an additional parameter to the derivative function as shown in class. Note the value of the terminal velocity and how long it takes to get to there. Can you see any difference between this and the quadratic only case? What do you conclude about how a ball this size falls in the atmosphere? Hand in the function, plot printouts and answers to the questions..4 Optional: Plot both the quadratic and combined L+Q results on the same plot. You might need to use a different D, say D=.5, to see a difference between the two. Also, plot the residuals, i.e. the difference between the quadratic and combined L+Q plots. This one is tricky since the times, t, of each integration are different and you Ph311 Matlab.doc rev Bruce Thompson / Ithaca College

8 Physics 311 Analytical Mechanics - Matlab questions M. want to take the differences at the same times. One way to handle it is to resample each one using interpolation. See the 'interp1'function in Matlab or Pratap's book..5 Optional: Suppose the ball is falling vertically in the atmosphere where the quadratic y H drag coefficient varies with height due to the atmospheric density ( c =.5e where H = 8. km), calculate and plot the resulting motion and compare with the case of a constant quadratic drag. function LinearODE; % LinearODE % function to calculate and plot the velocity of a body falling in Earth's % gravity with linear drag % c.f. section.5 Cassiday % bgt 1/9/3 % constants D =.1; % size of sphere - meters R = D/; den = 4; % density of sphere - kg/m3 c1 = 1.55E-4*D; % kg/s m = den*4/3*pi*r^3; % kg g = 9.8; % m/s % characteristic velocity and time vt = m*g/c1; % m/s tau = vt/g; % s % solve ode tspan = [ 5]; % range of t/tau to consider u = ; % initial v is so v/v= [T,u]=ode45(@LDrag,tspan,u); % plot ode solution and real valued solution subplot(,1,1) plot(t,u) xlabel('t=t/tau'), ylabel('u=v/vt') title('falling Body - Linear Drag - bgt - 1/9/3') % end subplot(,1,) plot(t*tau,u*vt) xlabel('t (s)'), ylabel('v (m/s)') function uprime=ldrag(t,u); uprime = 1-u; Example plot: Ph311 Matlab.doc rev Bruce Thompson / Ithaca College

9 Physics 311 Analytical Mechanics - Matlab questions M3.1 Matlab Exercise #3 - Second order ODE The Real Pendulum 3.1 A pendulum is constructed by attaching one end of a massless rod of length l to a mass m and the other end to a pivot point. The differential equation of the motion is d θ given by ml = mg sinθ where θ is the angle of the rod with respect to the dt du vertical. Show that this equation can be written as u = sin u whereu =. dt Define u and T. Also show that it can be written as two first order equations: u 1 = uand u = sin u1. Define u1 and u. Write a function RealPendulumA' that has as a parameter the initial angle of the pendulum bob of 1 degrees. The function should use ODE45 to solve the equation of motion giving the angle and the angular velocity as functions of time and then plot them. Assume the initial angular velocity is zero and the length of the pendulum is 1 meter. The plots should be one above the other on one page (use 'subplot') and span about cycles of the motion. The units for angle should be degrees and for angular velocity, degrees/sec. Put the starting angle in degrees in the title of the graph. (Use functions 'numstr' and 'strcat'.) Hand in a printout of the function. 3. Use 'RealPendulumA' to show the motion for initial angles of 1, 9 and 17 degrees. Use your printouts to make estimates of the period of the pendulum for each starting angle and compare with the calculated period for small angles ( ω = gl). What should happen if you start the pendulum at 18 degrees? What does happen when you start your function at 18 degrees? Hand in printouts of the graphs, your estimates of the periods compared to calculations and the answers to the questions. 3.3 Show that the real pendulum with linear drag can be modeled with the equation u = bu sin u and define u, T and b. Modify function written above to produce a new function, RealPendulumB, which plots the angle and angular velocity as functions of time with the same parameters as above. You will need to pass the parameter b to the uprime function by including it in the ODE45 inputs. Look up the syntax for this using the Matlab help files. Produce plots for initial angles of 1, 9, and 17 degrees and compare them to the plots from RealPendulumA. How do the amplitude and period change? Explain how these changes make sense physically. Hand in the derivation and printouts of the function and the plots. Ph311 Matlab 3.doc rev 6..1 Bruce Thompson / Ithaca College

10 Physics 311 Analytical Mechanics - Matlab questions M3. Ph311 Matlab 3.doc rev 6..1 Bruce Thompson / Ithaca College

11 Physics 311 Analytical Mechanics - Matlab questions M4.1 Matlab Exercise #4 - Second Order ODE with Event Sensing The Real Pendulum Period vs. Initial Angle 4.1 The attached function, 'RealPendulumII', solves the ODE for a real pendulum for an array of starting angles and plots the period vs. initial angle. Write this code and be sure to understand each line as you do so. The function determines the period by telling the ODE solver to sense the event defined in the function 'events'. This function specifies the event as a zero crossing of the position variable (u(1)) in the negative direction. Returning from the ODE45 function are three additional arrays. TE contains the times that the event occurred, YE contains the two column array of the angles and angular velocities at the time of the event and IE contains a column of numbers that specify the type of the event which occurred. In our case there is just one type of event and so IE is all the same. Hand in printouts of the function and the graph. 4. As you showed in Matlab Exercise #3, the real pendulum with linear drag can be modeled with the equation u = bu sin u. Modify the function of part 1 to produce a new function, 'RealPendulumIIa', that solves the ODE for a real pendulum and a real pendulum with linear drag, calculates the periods and plots one graph showing the two curves of Period v. Initial Angle. Hand in printouts of the derivation, the function and the graph for m=1. and c=.5. Comment on the physical reasons you see the differences and similarities in the curves. Here are some tips for writing this function: a. You will need to pass the parameter b, the drag parameter, to the 'uprime' function as you did in Matlab Exercise #3. b. You need to pass the b parameter to the 'events' function in addition to the uprime function. c. I suggest you add b to the function in part one and test it before trying to do both the drag and no drag calculations at the same time. Test your function by setting b= and see if it reproduces the results of part 1. Then change it to make sure it changes the results (try b=.). d. Then add a second ODE45 call inside the loop but remember to give it different output names, e.g. [Td, ud, e. Also add a second calculation of the period array inside the loop, e.g. PeriodD(I)= f. Be sure to add comments to the code as you add operational lines to inform the reader.. (over) Ph311 Matlab 4.doc rev Bruce Thompson / Ithaca College

12 Physics 311 Analytical Mechanics - Matlab questions M Write a new function, RealPendulumIIb, which calculates and plots the amplitude of the oscillation (deg) vs. time (s) for a pendulum with drag and for a starting angle of 9 degrees. Explain what behavior do you expect to observe and why. Does your plot show that behavior? Here are some tips for writing this function: a. Change the event sensing to determine the amplitude of the oscillation, i.e. where v(t)=. b. Plot the results of the event sensing using a marker to see the individual amplitude results. c. To get a reasonable number of cycle peaks, you will probably need to extend the length of time you ask the ODE to look at. d. Be sure to get rid of unused code from the previous function and add comments so that it is easy for a reader to see what you are doing. 4.4 Optional: Write down the equation for the curve which would fit the amplitude vs. time curve of RealPendulumIIb and add code to your function that plots that curve to show that it does intersect the markers plotted previously. Hand a printout of your code and plot. Ph311 Matlab 4.doc rev Bruce Thompson / Ithaca College

13 Physics 311 Analytical Mechanics - Matlab questions M4.3 function RealPendulumII; % RealPendulumII - plots the Period of a real pendulum as a function of % starting angle % % Matlab Exercise PH 311 % Bruce G. Thompson % Ithaca College % /1/6 % constants u=; g=9.8; L=1.; deg=18/pi; %initial angular speed %gravity %pendulum length %conversion factor radians to degrees % calculated constants w=sqrt(g/l); %small angle pendulum angular frequency T=*pi/w %small angle period and print it out % initialize the initial angles in radians from almost zero to almost pi th=linspace(.1,pi-.1,5); % set the Events option to the name of the function that defines the event % that you want to report, in this case it is when the angle crosses zero in % the negative direction. The period can be calculated by subtracting % successive times of these zero crossings options=odeset('events',@events); % solve the ode for each starting angle and find the periods for I=1:length(th), % print out th to show where we are in the set of calculations fprintf('%5.f\n',th(i)*deg) % specify the initial angular speed and angle for this loop init=[th(i) u]; % use the ode solver, % TE returns the times that the zeros occur, % YE returns the angle and anglular speed at these times, % IE specifies which event occured (in this case it's all the same % event so this is irrelevant) [T,u,TE,YE,IE]=ode45(@fPendulumII,[ 1*w],init,options); % calculate the next element in the Period array % (the w converts T to real seconds) Period(I)=(TE()-TE(1))/w; end % change the starting angles to degrees for plotting thd=18/pi*th; % plot the Period (s) vs starting angle (deg) close all; plot(thd,period) axis([ 18 8]) xlabel('starting Angle (deg)') ylabel('period (s)') title('period of a Real Pendulum of length 1. m - bgt - /1/6') % end function function up=fpendulumii(t,u); % pendulum derivative functions up=[-sin(u(1)); u()]; function [value,isterminal,direction]=events(t,u) % locate the times that the angle passes through zero in the negative % direction, % value specifies the what we want to find as zero, ie the angle value = u(1); % isterminal= specifies that we don't want the ode solver to stop here isterminal=; % direction=-1 specifies that we want a zero crossing in the negative dir direction=-1; Ph311 Matlab 4.doc rev Bruce Thompson / Ithaca College

14 Physics 311 Analytical Mechanics - Matlab questions M4.4 Ph311 Matlab 4.doc rev Bruce Thompson / Ithaca College

15 Physics 311 Analytical Mechanics - Matlab questions M5.1 Matlab 5 Take home exam on mechanical analysis using Matlab Due Monday February 7 at 1AM. No late papers will be accepted. What resources you may use: The Matlab program and Help files, your textbook, any Matlab manuals, your homework, me. What you may not use: The Internet, discussions with other students, other students code found on the computer, any other outside help or documents. Part 1 - Morse Potential Oscillator The Morse Potential for a vibrating diatomic molecule (two identical atoms bonded ( ) x x δ together) is given by V( x) = V 1 e where x is the separation between the nuclei of the two atoms and V, x and δ are constants. (Note that this is slightly different than the first Matlab exercise in that this potential is zero at the minimum point, not -V as before.) This potential approximates the potential of a diatomic molecule if it is not rotating too fast. Although this is a quantum system we will be looking at the motion of the particles at the discrete energies as if they acted classically. The function has the feature that the energy quantum values (eigenvalues) of the vibrational states can be ω 1 1 calculated analytically. These are given by Eυ = ω ( υ+ ) ( υ+ ) where υ is 4V δ 1 the harmonic quantum number that has integer values such that υ mv r V mm 1 and ω =. The reduced mass, m r =, is used since there are two masses m δ m + m r 1 sharing the same spring (think: two masses, one on each end of a spring). 5.1 Show that the force function for this potential is given by ( ) V ( x x ) δ ( x x ) δ F x = e 1 e δ. 5. Show that the differential equation of motion for this potential can be written as d z z z z = = e 1 e dt and define z and T. 5.3 Show that if the total energy of the oscillator is E υ and x = x at t=, then ( ) and ( ) z = z = E υ V. (Hint: for the last, find z in terms of x and note that E υ 1 = mv where v = x ().) Ph311 Matlab 5.doc rev Bruce Thompson / Ithaca College

16 Physics 311 Analytical Mechanics - Matlab questions M5. Part The Hydrogen Molecule The H molecule has a binding energy (the energy difference between zero potential and the minimum potential) of 4.5 ev, x =.74 nm and δ =.36 nm. 5.4 Show that υ 11 and calculate the energy eigenvalues E and E. 5.5 Show that the turning points of the Morse Potential are given by ( ) x = x δ ln 1± E υ V min,max Write a Matlab function, 'MorseTurningPoints.m', that plots the minimum separation, the maximum separation and the midpoint between them as a function of Eυ for each quantum number υ. Put distance in nanometers (nm) on the y axis and energy in electron volts (ev) on the x axis. Note: since this is a discrete quantum system, plot the points as markers to emphasize that fact. What does the graph say about the size of the molecule in its various energy states? Hand in a printout of the code, the plot and the answers to the questions. 5.7 Write a Matlab function, 'MorseHoscPlot.m', that plots the separation and separation velocity as functions of time for the energy state, E 1. Do the ODE calculations using the simplified function found in part b, that is, don't pass parameters to the uprime function. Convert back to real units after doing the ODE integration. Make the scales have position in nanometers (nm), velocity in meters/second (m/s), and time in seconds (s). Use subplot to put the two graphs on the same page. The function should also find the period of the oscillation via the events Matlab option and calculate the frequency of oscillation in Hz. Hand in a printout of the code, the plot and the resulting frequency that is calculated by the program. 5.8 Use your code from part g to repeat the calculations and plots for E 7. Compare the maximum and minimum positions for the two calculations, E1 and E7. Are the oscillations symmetrical about x? Also compare the two frequencies obtained. Explain physically in terms of the potential and/or forces why the separation and velocity curves are the observed shapes for E 7. Hand in a printout of the code, the plot and the answers to the questions. 5.9 Optional: Make a plot of the potential energy function with the discrete energy levels Eυ marked by lines and the center of the separation marked with a marker. Start and stop the horizontal energy level lines at the minimum and maximum separation. Observe the effect of the shape of the higher energy levels on the average separation. Ph311 Matlab 5.doc rev Bruce Thompson / Ithaca College