Freno a ceppo flottante

Context[]

SetDirectory[NotebookDirectory[]]

Componenti delle parti in coseno e seno

Nxc[αα_,rr_,ff_,pp0_,σf_]= Integrate[(pp0 Cos[γ]Cos[σf] Cos[γ] )rr,{γ,-(αα),αα}]//Simplify

Nys[αα_,rr_,ff_,pp0_,σf_]= Integrate[(pp0 Sin[γ]Sin[σf] Sin[γ] )rr,{γ,-(αα),αα}]//Simplify

Tyc[αα_,rr_,ff_,pp0_,σf_]=ff*Nxc[αα,rr,ff,pp0,σf]

Txs[αα_,rr_,ff_,pp0_,σf_]=-ff*Nys[αα,rr,ff,pp0,σf]

Componenti della risultante e momento frenante (direzione σf di accostamento incognita)

Rx[αα_,ββ_,rr_,ff_,pp0_,σf_]= Integrate[(pp0 Cos[γ-σf] Cos[γ] -ff pp0 Cos[γ-σf] Sin[γ])rr,{γ,-(αα),ββ}]//Simplify

Ry[αα_,ββ_,rr_,ff_,pp0_,σf_]= Integrate[(ff pp0 Cos[γ-σf] Cos[γ] + pp0 Cos[γ-σf] Sin[γ])rr,{γ,-(αα),ββ}]//Simplify

solo azioni di attrito

Rxf[αα_,ββ_,rr_,ff_,pp0_,σf_]= Integrate[(-ff pp0 Cos[γ-σf] Sin[γ])rr,{γ,-(αα),ββ}]//Simplify

Ryf[αα_,ββ_,rr_,ff_,pp0_,σf_]= Integrate[(ff pp0 Cos[γ-σf] Cos[γ] )rr,{γ,-(αα),ββ}]//Simplify

Mo[αα_,ββ_,rr_,ff_,pp0_,σf_]=Integrate[ff pp0 Cos[γ-σf] rr^2,{γ,-(αα),ββ}]//Simplify

(*p0=0.6;*)

r=10;

Qualche caso particolare, tanto per verifica:

Mo[α ,α ,rr,f,p0, (-α+α)/2 ]

Rxf[α ,α ,rr,f,p0, (-α+α)/2 ]

Ryf[α ,α ,rr,f,p0, (-α+α)/2 ]

Integrate[Cos[x]^2,x]

Calcolo del diametro della circonferenza di Romiti:

diamRomiti0=Mo[α ,α ,rr,f,p0, (-α+α)/2 ]/Sqrt[Rxf[α ,α ,rr,f,p0, (-α+α)/2 ]^2+Ryf[α ,α ,rr,f,p0, (-α+α)/2 ]^2]//Simplify

diametro con formula semplificata:

diamRomiti[rr_,α_]=Assuming[{f>0&&p0>0&&rr>0&& 0< α <Pi/2},Simplify[diamRomiti0]]

diamRomiti[rr,αα]*Tyc[αα,rr,ff,pp0,σf]

Mo[αα ,αα ,rr,ff,pp0, σf]//TrigExpand

diamRomiti[rr,αα]*Tyc[αα,rr,ff,pp0,σf]

Solve[
         Nxc[αα,rr,ff,1,0]*y
-Nys[αα,rr,ff,1,0]*x
-Txs[αα,rr,ff,1,0]*y
-Tyc[αα,rr,ff,1,0](x-diamRomiti[rr,αα])==0,
y]//Simplify

((ff (x αα+(-2 rr+x Cos[αα]) Sin[αα]))/(αα+Cos[αα] Sin[αα]))/. {ff->0.3,rr->10,αα->40Degree,x->18}

Si tenta una soluzione analitica:

Assuming[0<αα<Pi/2&&rr>0&&ff>0&&pp0>0 &&-Pi/2< σf<Pi/2&&x>rr,
Solve[
         Nxc[αα,rr,ff,1,σf]*y
-Nys[αα,rr,ff,1,σf]*x
-Txs[αα,rr,ff,1,σf]*y
-Tyc[αα,rr,ff,1,σf](x-diamRomiti[rr,αα])==0,
σf]]//Simplify

Manipulate[(
β=α;

(* si opta per una soluzione numerica *)
solf=FindRoot[
+Nxc[α Degree,r,f,p0,σσ]*y
-Nys[α Degree,r,f,p0,σσ]*x
+Txs[α Degree,r,f,p0,σσ]*y
-Tyc[α Degree,r,f,p0,σσ](x-diamRomiti[r,α Degree])==0,
{σσ,2. Degree*Sign[-y]}];
σ=(σσ/. solf)/Degree;;



Column[{

fig=Show[{

PolarPlot[Tooltip[r +2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],
PolarPlot[Tooltip[r +0.9*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],PolarPlot[Tooltip[r +0.8*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],PolarPlot[Tooltip[r +0.7*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],
PolarPlot[Tooltip[r +0.6*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],PolarPlot[Tooltip[r +0.5*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],PolarPlot[Tooltip[r +0.4*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],
PolarPlot[Tooltip[r +0.3*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],PolarPlot[Tooltip[r +0.2*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],
PolarPlot[Tooltip[r +0.1*2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick],

Graphics[{If[(Cos[-σ Degree +α Degree]<0)||(Cos[-σ Degree -α Degree]<0) ,Red,Yellow],
Disk[{0,0},r],
Thickness[0.04315],Darker[Green],Opacity[0.68],
Line[{1.526r{x,y}/Norm[{x,y}],{x,y}}],
Brown,Thickness[0.06315],Opacity[0.8],
Line[{{3x,y},{x,y}}],
Thick,Black,Opacity[1],
Line[{{3x,y+1.7},{x,y+1.7}}],
Line[{{3x,y-1.7},{x,y-1.7}}],
AbsolutePointSize[0.5],Red,Opacity[1],
Tooltip[Point[{x,y}], "fulcro del ceppo"],
Disk[{x,y},0.5,{0,2Pi}],
Black,Thick,
Tooltip[Circle[{x,y},0.5],"fulcro (x,y) del ceppo flottante"],

Thick,Opacity[1],(*Orange,
Tooltip[Line[{{0,0},30{Cos[σ Degree],Sin[σ Degree]}}],"direzione di accostamento (p = p0)"],*)
Thin,
Darker[Green],EdgeForm[Thin],Opacity[0.68],
Disk[{0,0},1.45 r,{-α Degree,β Degree}],
White,Yellow,Opacity[1],
Disk[{0,0}, r,{-α Degree,β Degree}],
Circle[{0,0},Mo[α Degree,β Degree,r,f,p0, σ Degree]/10],
White,Yellow,
Disk[{0,0},r,{-α Degree,β Degree}],
Black,Thick,
Circle[{0,0},r],
Gray,Opacity[1],
Disk[{0,0},1,{0,2Pi}],

Black,Arrowheads[0.032175],
Circle[{0,0},1,{0,2Pi}],
AbsoluteThickness[3],
Tooltip[
Arrow[BezierCurve[4{{0.9,0.5},{0,1.65},{-0.9,0.5}}]],"vel. angolare"],


Black,
Tooltip[Line[{{0,0},1.3r{Cos[(-α+β)/2 Degree],Sin[(-α+β)/2 Degree]}}],"mezzeria"],

Black,
Tooltip[Line[{{0,0},2.5r{Cos[σ Degree],Sin[σ Degree]}}], "dir. accostamento (p=p0)"],

Thick,AbsoluteThickness[5],
Red,
Tooltip[
ar=Arrow[{{0,-Mo[α Degree,β Degree,r,f,p0, σ Degree]/Rx[α Degree,β Degree,r,f,p0, σ Degree]},{rx=Rx[α Degree,β Degree,r,f,p0, σ Degree],
-Mo[α Degree,β Degree,r,f,p0, σ Degree]/Rx[α Degree,β Degree,r,f,p0, σ Degree]+(ry=Ry[α Degree,β Degree,r,f,p0, σ Degree])}}],
"risultante"],

Opacity[1],
AbsoluteThickness[5],

Magenta,
Tooltip[arf=Arrow[{{diamRomiti[r,α Degree],0},{
(diamRomiti[r,α Degree]+Txs[α Degree,r,f,p0, σ Degree]),
(Tyc[α Degree,r,f,p0, σ Degree])}}],
"risultante T = f N delle azioni di attrito"],


Blue,
Tooltip[arp=Arrow[{{0,0},{
(Nxc[α Degree,r,f,p0, σ Degree]),
(Nys[α Degree,r,f,p0, σ Degree])}}],
"risultante N delle pressioni"],

AbsoluteThickness[2],
Brown,
Tooltip[Circle[0.5*diamRomiti[r,α Degree]*{Cos[(-α+β)/2 Degree],Sin[(-α+β)/2 Degree]},diamRomiti[r,α Degree]/2],"circ. di Romiti"],

Magenta,PointSize[0.01],

Point[diamRomiti[r,α Degree]*{Cos[(-α+β)/2 Degree],Sin[(-α+β)/2 Degree]}],
Point[{0,0}],
Yellow,
Point[diamRomiti[r,α Degree]*{Cos[(-α+β)/2 Degree],Sin[(-α+β)/2 Degree]}],
Blue,
Rotate[
Point[{0,0}],2ArcTan[f],0.5*diamRomiti[r,α Degree]*{Cos[(-α+β)/2 Degree],Sin[(-α+β)/2 Degree]}]
}],

PolarPlot[Tooltip[r +2p0  Cos[-σ Degree +t],"pressione"],{t,-α Degree,β Degree},PlotStyle->Thick]


},


Axes->True,Ticks->None,AxesLabel->{"x","y"},

PlotLabel->"Freno a tamburo flottante (M. Guiggiani, 8/3/2022, all rights reserved)",

PlotRange->{{-11,27},{-15,15}},
ImageSize->600]


}]),
Text["Parametri che NON cambiano la circonferenza di Romiti"],
{{p0,1.2},0.01,3,Appearance->"Labeled" },
{{f,0.3},0.01,0.6,Appearance->"Labeled" },
(*{{β,45},0,90,Appearance->"Labeled" },*)
{{x,18},11,25,Appearance->"Labeled" },
{{y,2.16051},-10,10,Appearance->"Labeled" },
Delimiter,
Text["parametro che cambia la circonferenza di Romiti"],
{{α,40},30,90,Appearance->"Labeled" }
(*,
{{op,1,"Circonferenza di Romiti"},{0,1}}*),
SaveDefinitions->True,
ControlPlacement->Left
]

Sovrapporre la freccetta del mouse sui vettori per vedere cosa sono.
Fare attenzione ai punti invarianti (Romiti).
x e y controllano la posizione dello snodo del ceppo.

fig

alphadoppio=

fdoppio=

Export["alphadoppio.eps",alphadoppio]

Export["fdoppio.eps",fdoppio]

romiti1=

Export["romiti1.eps",romiti1]

romiti2=

Export["romiti2.eps",romiti2]

fig

Show[{Graphics[Circle[{0,0},1,{0,2Pi}]],
PolarPlot[Tooltip[1 +0.2Cos[-30 Degree +t],"pressione"],{t,-2β Degree,2β Degree},PlotStyle->Thick]}]