bruce mayer, pe registered electrical & mechanical engineer [email protected]

[email protected] • ENGR-25_Programming-1.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Prob 4.12Tutorial



Ballistic Trajectory Studied in Detail

in PHYS4A The Height, h, and

Velocity, v, as a Fcn of time, t, Launch Speed, v0, & Launch Angle, A

h

t

Agvt

tgAgtvvtv

gtAtvth

hit sin2

sin2

2sin

0

220

20

20

Athit



Parametric Description For This Problem

t

h

30°

9.81 m/s2

40 m/s

Find TIMES for Three cases

a. h ~< 15 m Or Equivalently:

h 15m

b. [h ~< 15 m] & [v ~> 36 m/s] Or By DeMorgan’s

Theorem: ~([h 15m] | [v 36 m/s])

c. [h < 5 m] I [v > 35 m/s]



1st Step → PLOT it Advice for Every Engineer and Applied

Mathematician or Physicist:

Rule-1: When in Doubt PLOT IT!

Rule-2: If you don’t KNOW when to DOUBT, then PLOT EVERYTHING



The Plot Plan The Plot Portion of the solution File% Bruce Mayer, PE * 21Feb06% ENGR25 * Problem 4-12% file = Prob4_12_Ballistic_Trajectory.m%%% INPUT PARAMETERS SECTIONA = 30*pi/180; % angle in radiansv0 = 40 % original velocity in m/Sg = 9.81 % Accel of gravity in m/sq-S%%%CALCULATION SECTION% calc landing timet_hit = 2*v0*sin(A)/g;% divide flite time into 100 equal intervalst = [0: t_hit/100: t_hit];% calc Height & Velocity Vectors as fcn of th = v0*t*sin(A) - 0.5*g*t.^2v = sqrt(v0^2 - 2*v0*g*sin(A)*t + g^2*t.^2)%% plot h & v %% MUST locate H & S Labels on plot before script continuesplot(t,h,t,v), xlabel('Time (s)'), ylabel('Height & Speed'), grid

Then the Plot

Analyses Follow

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Time (s)

Hei

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Analyze the Plots Draw HORIZONTAL or VERTICAL

Lines that Correspond to the Constraint Criteria

Where the Drawn-Lines Cross the Plotted-Curve(s) Defines the BREAK POINTS on the plots

Cast DOWN or ACROSS to determine Values for the Break-Points

See Next Slide



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Time (s)

Height

Speed

Case a.

0.98 3.1

]15 as [same 15~ :forOK ) mthmtha

Break-Pts



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Case b.

1.1 3.05

smtvmthbsmtvmthbsmtvmthb

/36 OR 15~ :forOK )3/36 AND 15 :forOK )2

/36~ AND 15~ :forOK )1

v Limits



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Time (s)

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Case c.

1.49 2.58

smtvmthc /35 OR 5 :forOK )

v Limitsv Limits



Advice on Using WHILE Loops When using Dynamically Terminated

Loops be SURE to Understand the MEANING of the• The LAST SUCEESSFUL

entry into the Loop• The First Failure Which

Terminates the Loop

Understanding First-Fail & Last-Success helps to avoid “Fence Post Errors”



Solution Game Plan Calc t_hit Plot & Analyze to determine approx.

values for the times in question• DONE

Precisely Determine time-points• For all cases

– Divide Flite-Time into 1000 intervals → time row-vector with 1001 elements

– Calc 1001 element Row-Vectors h(t) & v(t)



Solution Game Plan cont. Case-a

• Use WHILE Loops to – Count k-UP (in time) while h(k) < 15m

Save every time ta_lo = h(k)The first value to fail corresponds to the value of ta_lo

for the Left-side Break-Point

– Count m-DOWN (in time) while h(m) < 15mSave every time ta_hi = h(m)The first value to fail corresponds to the value of ta_hi

for the Right-side Break-Point



Solution Game Plan cont.

Case-b → Same TACTICS as Case-a• Use WHILE Loops to

– Count k-UP While h(k) < 15m OR v(k) > 36 m/sSave every time tb_lo = h(k) OR v(k)The Last Successful value of tb_lo is ONE index-unit LESS

than the Left Break point → add 1 to Index Find where [h<15 OR v>36] is NOT true

– Count m-DOWN While h(k) < 15m OR v(k) > 36 m/s Save every time tb_hi = h(m) OR v(m)The Last Successful value of tb_hi is ONE index-unit

MORE than the Right Break point → subtract 1 from index Find where [h<15 OR v>36] is NOT true

smtvmthsmtvmth /36 OR 15~/36~ AND 15~



Solution Game Plan cont. Case-c → Same TACTICS as Case-b

• Use WHILE Loops to – Count k-UP while h(k) < 5m OR v(k) > 35 m/s

Save every time tc_lo = h(k) OR v(k)The Last Successful value of tc_lo IS the Left-side

Break-Point as the logical matches the criteria– Count m-DOWN while

h(m) < 5m OR v(m) > 35 m/sSave every time tc_hi = h(m) OR v(k)The Last Successful value of tc_hi IS the Right-side

Break-Point as the logical matches the criteria



Solution Game Plan cont. MUST Properly LABEL the OutPut

using the Just Calculated BREAK-Pts Recall from the Analytical PLOTS

• Case-a is ONE interval (ConJoint Soln)– ta_lo → ta_hi

• Case-b is ONE interval (ConJoint Soln)– tb_lo → tb_hi

• Case-c is TWO intervals (DisJoint Soln)– 0 → tc_lo – tc_hi → t_hit



Alternate Soln → FIND Use FIND command along with a

LOGICAL test to locate the INDICES of h associated with the Break Points• LOWEST index is the Left-Break• HIGHEST Index is the Right-Break

Same Tactics for 3 Sets of BreakPts• Again, MUST label Properly• Must INcrement or DEcrement “found”

indices to match logical criteria– Need depends on Logical Expression Used



Compare: WHILE vs FIND Examine Script files

• Prob4_12_Ballistic_Trajectory_by_WHILE_1209.m• Prob4_12_Ballistic_Trajectory_by_FIND_1209.m

FIND is Definitely More COMPACT (fewer KeyStrokes)

WHILE-Counter is More INTUITIVE → Better for someone who does not think in “Array Indices”



Compare: WHILE vs FIND While vs Find; Which is Best? The “best” one is the one that

WORKS in the SHORTEST amount of YOUR TOTAL-TIME



Portion of The m-file

% Bruce Mayer, PE * 21Feb06 * Rev. 22Feb09

% ENGR25 * Problem 4-12

% file = Prob4_12_Ballistic_Trajectory.m

%

% INPUT PARAMETERS SECTION

A = 30*pi/180; % angle in radians

v0 = 40; % original velocity in m/S

g = 9.81; % Accel of gravity in m/sq-S

%

%

%CALCULATION SECTION

% calc landing time

t_hit = 2*v0*sin(A)/g;

display('The Flite Time in sec:')

disp(t_hit)

% divide flite time in 1000 equal intervals

t = [0: t_hit/1000: t_hit];

% 2X check no. of elements in vector t

Elements_in_t = length(t)

% calc Height & Velcity Vectors as fcn of t

h = v0*t*sin(A) - 0.5*g*t.^2;

v = sqrt(v0^2 - 2*v0*g*sin(A)*t + g^2*t.^2);



FOLLOWING ARE FOR PROJECTION ONTO WHITEBOARD


Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods0 0.5 1 1.5 2 2.5 3 3.5 40

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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods0 0.5 1 1.5 2 2.5 3 3.5 40

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ENGR25 P4-12 * h5 & v35



Portion of Plot m-file

% Bruce Mayer, PE * 20Feb13% ENGR25 * Problem 4-12 Constraint Plots% file = Prob4_12_Ballistic_Trajectory_Constraint_Plots_1302.m%% INPUT PARAMETERS SECTIONA = 30*pi/180; % angle in radiansv0 = 40; % original velocity in m/Sg = 9.81; % Accel of gravity in m/sq-Sdisp(' ')disp('Locate Break Points by FIND command')%%CALCULATION SECTION% calc landing timet_hit = 2*v0*sin(A)/g;display('The Flite Time in sec:')disp(t_hit)% divide flite time in 1000 equal intervalst = [0: t_hit/1000: t_hit];% 2X check no. of elements in vector tElements_in_t = length(t)% calc Height & Velcity Vectors as fcn of th = v0*t*sin(A) - 0.5*g*t.^2;v = sqrt(v0^2 - 2*v0*g*sin(A)*t + g^2*t.^2);%% plot h & v constraints: h15 & v36tln = [0 4.1]; h15 = [15, 15]; v36 = [36,36]plot(t,h,t,v, tln,h15, tln, v36, 'linewidth', 2), xlabel('Time (s)'),...

ylabel('Height & Speed'), grid, title('ENGR25 P4-12 * h15 & v36'),...

axis([0 4.1 0 40])display('showing h15 & v36 plot -- Hit Any Key to Continue')pause

bruce mayer, pe registered electrical & mechanical engineer [email protected]

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