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Lab 5, simulation of car trailer. EME 121, Spring 2011, UC Davis

By Nasser M. Abbasi. May 22, 2011

ManipulateBprocess@u, m, L, abRatio, IgGain, Ct, v0, r0, Θ0, ∆, f, maxt, noslipD,

88u, 79.2, "speed U km�hr"<, 10, 200, 1,

Appearance ® "Labeled", ImageSize ® Small<,

88m, 1800 � 2.2, "mass kg"<, 1, 1000, 1,


88L, 7 * 0.3048, "trailer length m"<, 1, 10, 0.1,


88abRatio, 0.85, "a�b ratio"<, 0.1, 0.9, 0.1,


88IgGain, 0.8, "Ig gain"<, 80.8 ® 0.8, 0.9 ® 0.9, 1 ® 1, 1.1 ® 1.1,

1.2 ® 1.2, 1.3 ® 1.3<, ControlType ® SetterBar, ImageSize ® Small<,

88Ct, 210 * 4.448 * 180 � Pi, "Ct N�rad"<, 30 000, 100 000,

1000, Appearance ® "Labeled", ImageSize ® Small<,

88f, 1, "spring f HHzL"<, 0.1, 3, 0.1,


88maxt, 10, "simulation time HsecL"<, 0.1, 40, 0.1,


Delimiter,

Control@88noslip, False, "Apply no-slip condition"<,

8False, True<, ControlType ® Checkbox, ImageSize ® Small<D,

Delimiter,

Text@"Initial conditions"D,

88v0, 0, "VH0L km�hr"<, 0, 10,

1, Appearance ® "Labeled", ImageSize ® Small<,

88r0, 0, "rH0L degree�sec"<, 0, 7, 1,


88Θ0, 2, "ΘH0L degree"<, 0, 7, .1,


88∆, 0, "∆sH0L mm"<, 0, 100, 1, Appearance ® "Labeled", ImageSize ® Small<,

SynchronousUpdating ® False, H*important to have this*LContinuousAction ® False,

Initialization ¦

Printed by Mathematica for Students

process@uu0_, m_, L_, abRatio_, IgGain_,

c_, vv0_, rr0_, ΘΘ0_, ∆∆0_, f_, maxt_, noslip_D := ModuleB8b = 1. � H1 + abRatioL, a, k = m * H2. * Pi * fL^2, u0 = uu0 * 1000 � 3600.,

v0 = vv0 * 1000 � 3600., Θ0 = ΘΘ0 * Pi � 180., ∆0 = ∆∆0 � 1000.,

r0 = rr0 * Pi � 180., F, eq1, eq2, eq3, eq4, t, r, v, Θ, ∆, Ig, sol, data<,

a = L - b;

F = c Hb r@tD - v@tDL � u0;

Ig = IgGain m a b;

IfBnoslip � False, :

eq1 = v'@tD �F

m+

k ∆@tDm

- r@tD u0;

eq2 = r'@tD �k ∆@tD a

Ig-

F b

Ig;

eq3 = Θ'@tD � r@tD;

eq4 = ∆'@tD � -Hv@tD + u0 Θ@tD + a r@tDL;

sol = First�DSolve@8eq1, eq2, eq3, eq4, v@0D � v0, r@0D � r0,

Θ@0D � Θ0, ∆@0D � ∆0<, 8v@tD, r@tD, Θ@tD, ∆@tD<, tD>, :

eq1 = r'@tD �-r@tD Hm b u0L + Hb + aL k ∆@tD

Ig + m b^2;

eq2 = Θ'@tD � r@tD;

eq3 = ∆'@tD � -Hb r@tD + u0 Θ@tD + a r@tDL;

sol = First�DSolve@8eq1, eq2, eq3,

r@0D � r0, Θ@0D � Θ0, ∆@0D � ∆0<, 8r@tD, Θ@tD, ∆@tD<, tD>

F;

Grid@8

8Plot@Chop�Evaluate@r@tD �. solD * 180 � Pi,

8t, 0, maxt<, PlotRange ® All, Frame ® True, FrameLabel ®

88"rHtL Hdeg�secL", None<, 8"t HsecL", "rHtL vs. time"<<,

ImagePadding ® 8840, 10<, 840, 40<<,

ImageSize ® 250, AspectRatio ® 0.8D<,

8

2 lab5.nb


8Row@8

Plot@Chop�Evaluate@Θ@tD �. solD * 180 � Pi,

8t, 0, maxt<, PlotRange ® All, Frame ® True, FrameLabel ®

88None, None<, 8"t HsecL", Row@8"Θ in deg vs. time"<D<<,

ImagePadding ® 8840, 10<, 850, 40<<, ImageSize ® 180D,

Plot@Chop�Evaluate@∆@tD �. solD, 8t, 0, maxt<,

PlotRange ® All, Frame ® True, FrameLabel ® 88None, None<,

8"t HsecL", Row@8"spring displacement HmmL vs. time"<D<<,

ImagePadding ® 8840, 10<, 850, 40<<, ImageSize ® 180D<D

<<, Frame ® All, Spacings ® 0

DF

F

lab5.nb 3


speed U km�hr 79.2

mass kg 818.182

trailer length m 2.1336

a�b ratio 0.85

Ig gain 0.8 0.9 1 1.1 1.2 1.3

Ct N�rad 53518.8

spring f HHzL 1

simulation time HsecL 10

Apply no-slip condition

Initial conditions

VH0L km�hr 0

rH0L degree�sec 0

ΘH0L degree 2

∆sH0L mm 0

0 2 4 6 8 10

-30

-20

-10

0

10

20

30

t HsecL

rHtLH

deg�

secL

rHtL vs. time

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

t HsecL

Θ in deg vs. time

0 2 4 6 8 10-0.06

-0.04

-0.02

0.00

0.02

0.04

t HsecL

spring displacement HmmL vs. time

4 lab5.nb
