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In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.?
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In the mechanism shown in Figure (II.5), the crank AB is 75 mm long an...
Solution:
Given Data:
- Crank AB length, rAB = 75 mm
- Angular velocity of crank AB, ωAB = 8 rad/s
- Distance between crank pivot B and piston C, L(BC) = 300 mm
- Distance between crank pivot B and point D, L(BD) = L(CD) = L(DE)
Velocity of Piston C:
- Let ω be the angular velocity of the link CD.
- The instantaneous center of rotation of link CD is at point G, where the perpendiculars to links CD and BC intersect.
- The velocity of piston C is given by vC = vG + GC x ω, where GC is the perpendicular distance from point G to piston C.
- From the geometry of the mechanism, we have GC = BD - DC = L(BD) - L(BC) = L.
- The velocity of point G is zero since it is the instantaneous center of rotation.
- Therefore, vC = L x ω = L x (ωAB / 2) = 150 mm/s.
Acceleration of Piston C:
- The acceleration of piston C is given by aC = aG + α x GC + ω^2 x GC, where α is the angular acceleration of link CD and aG is the acceleration of point G.
- The acceleration of point G is given by aG = GC x α, where GC is the perpendicular distance from point G to piston C.
- From the geometry of the mechanism, we have GC = BD - DC = L(BD) - L(BC) = L.
- The angular acceleration of link CD is α = dω / dt = -ω^2 / L.
- Therefore, aG = GC x α = -ω^2 x L = -1200 mm/s^2.
- The acceleration of piston C is given by aC = aG + α x GC + ω^2 x GC = -ω^2 x L + α x L + ω^2 x L = α x L = (-ω^2 / L) x L = -ω^2 = -64 mm/s^2.
Velocity of Piston E:
- Let θ be the angle between links BD and DE.
- The instantaneous center of rotation of link DE is at point H, where the perpendiculars to links DE and BD intersect.
- The velocity of piston E is given by vE = vH + HE x ω, where HE is the perpendicular distance from point H to piston E.
- From the geometry of the mechanism, we have HE = DE x sin(θ) = L x sin(θ).
- The velocity of point H is zero since it is the instantaneous center of rotation.
- Therefore, vE = L x sin(θ) x ω = L x sin(θ) x (ωAB / 2).
Acceleration of Piston E:
- The acceleration of piston E is given by aE = aH + α x HE + ω^2 x HE, where α is the angular acceleration of link DE and aH is the acceleration of point H.
- The acceleration of point H is given by aH = HE x α, where HE is the perpendicular distance from point H to piston E.
- The angular acceleration of link DE is α = d^2θ / dt^2 = -ω^
Given Data:
- Crank AB length, rAB = 75 mm
- Angular velocity of crank AB, ωAB = 8 rad/s
- Distance between crank pivot B and piston C, L(BC) = 300 mm
- Distance between crank pivot B and point D, L(BD) = L(CD) = L(DE)
Velocity of Piston C:
- Let ω be the angular velocity of the link CD.
- The instantaneous center of rotation of link CD is at point G, where the perpendiculars to links CD and BC intersect.
- The velocity of piston C is given by vC = vG + GC x ω, where GC is the perpendicular distance from point G to piston C.
- From the geometry of the mechanism, we have GC = BD - DC = L(BD) - L(BC) = L.
- The velocity of point G is zero since it is the instantaneous center of rotation.
- Therefore, vC = L x ω = L x (ωAB / 2) = 150 mm/s.
Acceleration of Piston C:
- The acceleration of piston C is given by aC = aG + α x GC + ω^2 x GC, where α is the angular acceleration of link CD and aG is the acceleration of point G.
- The acceleration of point G is given by aG = GC x α, where GC is the perpendicular distance from point G to piston C.
- From the geometry of the mechanism, we have GC = BD - DC = L(BD) - L(BC) = L.
- The angular acceleration of link CD is α = dω / dt = -ω^2 / L.
- Therefore, aG = GC x α = -ω^2 x L = -1200 mm/s^2.
- The acceleration of piston C is given by aC = aG + α x GC + ω^2 x GC = -ω^2 x L + α x L + ω^2 x L = α x L = (-ω^2 / L) x L = -ω^2 = -64 mm/s^2.
Velocity of Piston E:
- Let θ be the angle between links BD and DE.
- The instantaneous center of rotation of link DE is at point H, where the perpendiculars to links DE and BD intersect.
- The velocity of piston E is given by vE = vH + HE x ω, where HE is the perpendicular distance from point H to piston E.
- From the geometry of the mechanism, we have HE = DE x sin(θ) = L x sin(θ).
- The velocity of point H is zero since it is the instantaneous center of rotation.
- Therefore, vE = L x sin(θ) x ω = L x sin(θ) x (ωAB / 2).
Acceleration of Piston E:
- The acceleration of piston E is given by aE = aH + α x HE + ω^2 x HE, where α is the angular acceleration of link DE and aH is the acceleration of point H.
- The acceleration of point H is given by aH = HE x α, where HE is the perpendicular distance from point H to piston E.
- The angular acceleration of link DE is α = d^2θ / dt^2 = -ω^
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In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.?
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In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.?.
In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the mechanism shown in Figure (II.5), the crank AB is 75 mm long and rotates clockwise at 8 rad/s. L(BC) =300 mm and L(BD) =L(CD) =L (DE). Find the velocity and acceleration of pistons C and E.?.
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