2 Added points of intersection
source | link
p = 1.6;
α = 0.001;
r = 0.6;
η = 0.04;
ω = 1;
eqn = R ω p Sin[ω τ] + R ω p α - 
     9/4 r p R^3 ω - η p R == 0 // Rationalize // Simplify

(* 3*R*(13 + 450*R^2) == 1000*R*Sin[τ] *)

The points of intersection are

pts = {τ, R} /. SortBy[
   Solve[#, {τ, R}][[1]] & /@
    ({Reduce[
          {eqn, eqn /. τ -> 20 - τ, 0 < τ < 20, 
           0 < R < 2}, {τ, R}] // Simplify // ToRules} /. 
      Rule -> Equal),
   #[[1, -1]] &]

enter image description here

Which are approximately

pts // N

(* {{0.575222, 0.611629}, {6.85841, 0.611629}, {13.1416, 0.611629}, {19.4248, 
  0.611629}} *)

Plotting

ContourPlot[
 Evaluate@{eqn, eqn /. τ -> 20 - τ},
 {τ, 0, 20}, {R, 0, 2},
 ContourStyle -> {Blue, Green},
 BaseStyle -> Thick,
 PlotLegends -> Placed["Expressions", {.5, .75}]]],
 Epilog -> {Red, AbsolutePointSize[5], Point[pts]}]

enter image description hereenter image description here

p = 1.6;
α = 0.001;
r = 0.6;
η = 0.04;
ω = 1;
eqn = R ω p Sin[ω τ] + R ω p α - 
     9/4 r p R^3 ω - η p R == 0 // Rationalize // Simplify

(* 3*R*(13 + 450*R^2) == 1000*R*Sin[τ] *)

ContourPlot[
 Evaluate@{eqn, eqn /. τ -> 20 - τ},
 {τ, 0, 20}, {R, 0, 2},
 ContourStyle -> {Blue, Green},
 BaseStyle -> Thick,
 PlotLegends -> Placed["Expressions", {.5, .75}]]

enter image description here

p = 1.6;
α = 0.001;
r = 0.6;
η = 0.04;
ω = 1;
eqn = R ω p Sin[ω τ] + R ω p α - 
     9/4 r p R^3 ω - η p R == 0 // Rationalize // Simplify

(* 3*R*(13 + 450*R^2) == 1000*R*Sin[τ] *)

The points of intersection are

pts = {τ, R} /. SortBy[
   Solve[#, {τ, R}][[1]] & /@
    ({Reduce[
          {eqn, eqn /. τ -> 20 - τ, 0 < τ < 20, 
           0 < R < 2}, {τ, R}] // Simplify // ToRules} /. 
      Rule -> Equal),
   #[[1, -1]] &]

enter image description here

Which are approximately

pts // N

(* {{0.575222, 0.611629}, {6.85841, 0.611629}, {13.1416, 0.611629}, {19.4248, 
  0.611629}} *)

Plotting

ContourPlot[
 Evaluate@{eqn, eqn /. τ -> 20 - τ},
 {τ, 0, 20}, {R, 0, 2},
 ContourStyle -> {Blue, Green},
 BaseStyle -> Thick,
 PlotLegends -> Placed["Expressions", {.5, .75}],
 Epilog -> {Red, AbsolutePointSize[5], Point[pts]}]

enter image description here

1
source | link

p = 1.6;
α = 0.001;
r = 0.6;
η = 0.04;
ω = 1;
eqn = R ω p Sin[ω τ] + R ω p α - 
     9/4 r p R^3 ω - η p R == 0 // Rationalize // Simplify

(* 3*R*(13 + 450*R^2) == 1000*R*Sin[τ] *)

ContourPlot[
 Evaluate@{eqn, eqn /. τ -> 20 - τ},
 {τ, 0, 20}, {R, 0, 2},
 ContourStyle -> {Blue, Green},
 BaseStyle -> Thick,
 PlotLegends -> Placed["Expressions", {.5, .75}]]

enter image description here