Car Axis Model 

In[6]:=

X Animate[ With[{t = tt}, Show[ (* ground *) Plot[Evaluate[{h*Exp[-5*(x - 1.2)^2]*Sin[\[Omega] t]} - 0.14 /. params], {x, -0.9, 2}, Filling -> -0.5, FillingStyle -> Hue[.1, .2, .4], Prolog -> {Polygon[{Scaled[{0, 0}], Scaled[{1, 0}], Scaled[{1, 1}], Scaled[{0, 1}]}, VertexColors -> {Hue[.6, .2, 1], Hue[.6, .2, 1], Hue[.6, .7, .5], Hue[.6, .7, .5]}]}, PlotStyle -> Hue[.1, .4, .2], PlotRange -> {{-0.9, 1.9}, {-0.5, 1.5}}, Frame -> True, Axes -> False], Graphics[{ (* Axle *) {Thickness[0.03], Hue[.6, 1, .3], Line[{{xL[t] - 0.2, yL[t]}/ 2, {(xB[t]) + 0.2, (yR[t] - 0*yB[t])/2}}]}, (* tires *) {Thickness[0.09], GrayLevel[.2], Line[{{Mean[{0 - 0.22, xL[t] - 0.22}], 0}, {Mean[{0 - 0.22, xL[t] - 0.22}], yL[t]}}]}, {Thickness[0.09], GrayLevel[.2], Line[{{Mean[{xB[t] + 0.22, xR[t] + 0.22}], yB[t]}, {Mean[{xB[t] + 0.22, xR[t] + 0.22}], yR[t]}}]}, (* springs *) {Thickness[0.025], Hue[.1, 1, .9], Line[{{0 + 0.01, yL[0]/2 + 0.03}, {xL[t] + 0.01, yL[t] + 0.2}}]}, {Thickness[0.025], Hue[.1, 1, .9], Line[{{xB[t], yB[t]/2 + 0.28}, {xR[t] - 0.01, yR[t] + 0.2}}]}, (* chassis *) {Thickness[0.01], Hue[.6, 1, .3], Line[{{xL[t] - 0.03, yL[t] + 0.2}, {xR[t] + 0.03, yR[t] + 0.2}}]}, (* body *) {GrayLevel[1, .3], EdgeForm[{GrayLevel[1, .5]}], GeometricTransformation[ {Polygon[.5 + .0045 {{-185., -105.}, {-195., -100.}, {-200., \ -95.}, {-215., 0}, {-215., 45.}, {-210., 50.}, {-205., 50.}, {-200., 65.}, {-190., 75.}, {-220., 75.}, {-230., 80.}, {-230., 95.}, {-225., 100.}, {-205., 105.}, {-190., 105.}, {-185., 100.}, {-180., 90.}, {-175., 90.}, {-160., 125.}, {-145., 155.}, {-135., 165.}, {-120., 172.}, {-75., 177.}, {0., 180.}, {75., 177.}, {120., 172.}, {135., 165.}, {145., 155.}, {160., 125.}, {175., 90.}, {180., 90.}, {185., 100.}, {190., 105.}, {205., 105.}, {225., 100.}, {230., 95.}, {230., 80.}, {220., 75.}, {190., 75.}, {200., 65.}, {205., 50.}, {210., 50.}, {215., 45.}, {215., 0}, {200., -95.}, {195., -100.}, {185., -105.}}], Polygon[.5 + .0045 {{-145., 80.}, {-155., 90.}, {-145., 125.}, {-130., 155.}, {-120., 160.}, {-55., 165.}, {0., 165.}, {55., 165.}, {120., 160.}, {130., 155.}, {145., 125.}, {155., 90.}, {145., 80.}}], Polygon[.5 + .0045 {{-190., 20.}, {-185., 25.}, {-150., 25.}, {-110., 20.}, {-100., 15.}, {-100., 10.}, {-110., 0.}, {-140., -0.08875739644975056}, {-160., -10.}, \ {-175., -10.}, {-185., -5.}, {-190., 5.}}], Polygon[.5 + .0045 {{110., -0.08875739644975056}, {140., \ -0.08875739644975056}, {160., -10.}, {175., -10.}, {185., -5.}, {190., 5.}, {190., 20.}, {185., 25.}, {150., 25.}, {110., 20.}, {100., 15.}, {100., 10.}}], Polygon[.5 + .0045 {{-85., 15.}, {-75., 20.}, {-25., 25.}, {0., 25.}, {25., 25.}, {75., 20.}, {85., 15.}, {90., 5.}, {90., -5.}, {85., -30.}, {75., -60.}, {65., \ -70.}, {55., -75.}, {-55., -75.}, {-65., -70.}, {-75., -60.}, {-85., \ -30.}, {-90., -5.}, {-90., 5.}}], Polygon[.5 + .0045 {{-85., -50.}, {-90., -40.}, {-190., \ -40.}, {-205., -25.}, {-195., -50.}}], Polygon[.5 + .0045 {{85., -50.}, {195., -50.}, {205., \ -25.}, {190., -40.}, {90., -40.}}]}, RotationTransform[ArcTan[xR[t] - xL[t], yR[t] - yL[t]]]]} } /. params /. carAxis], ImageSize -> 400 ] ] , {{tt, 0, "time"}, 0, .2 Pi}, SaveDefinitions -> True, AnimationRunning -> False]

Model the motion of a car as it goes down a bumpy road.

The car is modeled as springs connected to two bars. The bottom of the left tire is the origin. The car chassis is represented by a bar of mass M. The position of the chassis is given by and .

X \!\(\* GraphicsBox[{GeometricTransformationBox[{GraphicsGroupBox[ {GrayLevel[0], {GrayLevel[0], Thickness[0.015], CapForm["Butt"], JoinForm["Round"], LineBox[{{0, 0}, {0.1, 0}, {0.14444444444444446`, -0.1}, { 0.23333333333333334`, 0.1}, {0.32222222222222224`, -0.1}, { 0.41111111111111115`, 0.1}, {0.5, -0.1}, { 0.5888888888888889, 0.1}, {0.6777777777777778, -0.1}, { 0.7666666666666667, 0.1}, {0.8555555555555555, -0.1}, { 0.9, 0}, {0.9, 0}, {1, 0}}], {GrayLevel[1], Thickness[0.0075], CapForm["Round"], LineBox[{{0, 0}, {0.1, 0}}], LineBox[{{0.14444444444444446`, -0.1}, { 0.23333333333333334`, 0.1}}], LineBox[{{0.32222222222222224`, -0.1}, { 0.41111111111111115`, 0.1}}], LineBox[{{0.5, -0.1}, {0.5888888888888889, 0.1}}], LineBox[{{0.6777777777777778, -0.1}, {0.7666666666666667, 0.1}}], LineBox[{{0.8555555555555555, -0.1}, {0.9, 0}}], LineBox[{{0.9, 0}, {1, 0}}]}}}], GraphicsGroupBox[ {GrayLevel[0], {GrayLevel[0], Thickness[0.015], CapForm["Butt"], JoinForm["Round"], LineBox[{{0, 1}, {0.1, 1}, {0.15000000000000002`, 0.9}, { 0.25, 1.1}, {0.35000000000000003`, 0.9}, { 0.45000000000000007`, 1.1}, {0.55, 0.9}, {0.65, 1.1}, { 0.7500000000000001, 0.9}, {0.8500000000000001, 1.1}, { 0.9500000000000001, 0.9}, {1., 1}, {1., 1}, {1.1, 1}}], {GrayLevel[1], Thickness[0.0075], CapForm["Round"], LineBox[{{0, 1}, {0.1, 1}}], LineBox[{{0.15000000000000002`, 0.9}, {0.25, 1.1}}], LineBox[{{0.35000000000000003`, 0.9}, {0.45000000000000007`, 1.1}}], LineBox[{{0.55, 0.9}, {0.65, 1.1}}], LineBox[{{0.7500000000000001, 0.9}, {0.8500000000000001, 1.1}}], LineBox[{{0.9500000000000001, 0.9}, {1., 1}}], LineBox[{{1., 1}, {1.1, 1}}]}}}], {GrayLevel[0], Thickness[0.1], LineBox[{{0.02, -0.1}, {-0.05, 1.1}}]}, {GrayLevel[0], Thickness[0.1], LineBox[{{1.05, -0.1}, {1.15, 1.1}}]}}, {{{0, -1}, {1, 0}}, Center}], InsetBox[ StyleBox[(0,0), FontFamily->"Verdana", FontSize->11, Background->GrayLevel[ 1.]], {-0.22570830425680358`, 0.05638961325155109}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox["", {1.3310258199581302`, 0.2685114041209091}, { Left, Baseline}, Alignment->{Left, Top}], {EdgeForm[{GrayLevel[0.], Opacity[1.], Thickness[0.1]}], EdgeForm[ None], FaceForm[GrayLevel[1]], DiskBox[{0.03504021743122654, -0.0872766004333938}, \ {0.020556065670107045`, 0.02281852572813771}]}, {EdgeForm[{GrayLevel[0.], Opacity[1.], Thickness[0.1]}], EdgeForm[ None], FaceForm[GrayLevel[1]], DiskBox[{1.0782767106181372`, -0.034868329232012885`}, \ {0.02055606567010626, 0.02281852572813764}]}, {EdgeForm[{GrayLevel[0.], Opacity[1.], Thickness[0.1]}], EdgeForm[ None], FaceForm[GrayLevel[1]], DiskBox[{1.0607911264553576`, 1.0062511477577398`}, \ {0.020556065670106927`, 0.02281852572813814}]}, {EdgeForm[{GrayLevel[0.], Opacity[1.], Thickness[0.1]}], EdgeForm[ None], FaceForm[GrayLevel[1]], DiskBox[{0.03874095566900582, 1.092984170125243}, \ {0.020556065670106705`, 0.022818525728137695`}]}, InsetBox[ StyleBox[(xL,yL), FontFamily->"Verdana", FontSize->11, Background->GrayLevel[ 1.]], {-0.28039519594520124`, 0.8886355455981194}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[(xR,yR), FontFamily->"Verdana", FontSize->11, Background->GrayLevel[ 1.]], {1.2205369669813062`, 0.7759488742792082}, { Left, Baseline}, Alignment->{Left, Top}], InsetBox[ StyleBox[(xB,yB), FontFamily->"Verdana", FontSize->11, Background->GrayLevel[ 1.]], {1.1903331252065974`, 0.13914717008851452`}, { Left, Baseline}, Alignment->{Left, Top}]}, ImagePadding->{{0., 0.}, {0., 0.}}, ImageSize->{425.671875, Automatic}, PlotRange->{{-0.6, 1.6}, {-0.4, 1.4}}, PlotRangePadding->Automatic]\)

The left tire is always on a flat surface, while the right tire periodically goes over a bump. The bump is defined as follows.

In[1]:=

X yB[t_] = h Sin[\[Omega] t];

The distance between the wheels must always remain fixed, and the car chassis must always have a fixed length.

In[2]:=

X constraints = {xL[t] xB[t] + yL[t] yB[t] == 0, (xL[t] - xR[t])^2 + (yL[t] - yR[t])^2 == L^2};

Derive the equations of motion using Newton's law.

In[3]:=

X m = (\[Epsilon]^2 M)/2; xB[t_] = Sqrt[L^2 - yB[t]^2]; Lleft = Sqrt[xL[t]^2 + yL[t]^2]; Lright = Sqrt[(xR[t] - xB[t])^2 + ( yR[t] - yB[t])^2]; eqns = ({ {m xL''[t] == ((L0 - Lleft) xL[t])/Lleft + \[Lambda]1[t] xB[t] + 2 \[Lambda]2[t] (xL[t] - xR[t])}, {m yL''[t] == ((L0 - Lleft) yL[t])/Lleft + \[Lambda]1[t] yB[t] + 2 \[Lambda]2[t] (yL[t] - yR[t]) - m}, {m xR''[t] == (L0 - Lright) (xR[t] - xB[t])/Lright - 2 \[Lambda]2[t] (xL[t] - xR[t])}, {m yR''[t] == (L0 - Lright) (yR[t] - yB[t])/Lright - 2 \[Lambda]2[t] (yL[t] - yR[t]) - m} }); ics = {xL[0] == 0, yL[0] == 1/2, yR[0] == 1/2, yL'[0] == 0, yR'[0] == 0};

Define the parameters associated with the system.

In[4]:=

X params = {L -> 1, L0 -> 1/2, \[Epsilon] -> 10^-2, M -> 10, h -> 10^-1, \[Omega] -> 10}; vars = {xL, yL, xR, yR, \[Lambda]1, \[Lambda]2};

Solve and visualize the solution.

In[5]:=

X carAxis = NDSolve[{eqns, constraints, ics} /. params, vars, {t, 0, 3}, Method -> {"IndexReduction" -> Automatic}, AccuracyGoal -> 7] ;

Out[6]=

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