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Rotational_Mechanical_System

# Rotational_Mechanical_System - ME375 Handouts Rotational...

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Unformatted text preview: ME375 Handouts Rotational Mechanical Systems School of Mechanical Engineering Purdue University ME375 Rotation - 1 Rotational Mechanical Systems • • • • Variables Basic Modeling Elements Interconnection Laws Derive Equation of Motion (EOM) School of Mechanical Engineering Purdue University ME375 Rotation - 2 1 ME375 Handouts Variables • θ : angular displacement [rad] [rad] • ω : angular velocity [rad/sec] [rad/sec] • α : acceleration [rad/sec2] [rad/sec • τ : torque [Nm] • p : power [Nm/sec] [Nm/sec] • w : work ( energy ) [Nm] [Nm] 1 [Nm] = 1 [J] (Joule) d & θ =θ =ω dt d dd d2 && & ω =ω = θ = 2θ =θ =α dt dt dt dt &d p = τ ⋅ω = τ ⋅θ = w dt FI HK w ( t1 ) = w (t0 ) + = w (t0 ) + z z t1 t0 t1 t0 p( t ) d t & (τ ⋅ θ ) dt School of Mechanical Engineering Purdue University ME375 Rotation - 3 Basic Rotational Modeling Elements • Spring • Damper – Stiffness Element τS – Friction Element θ2 θ1 τS b τ S = K θ 2 − θ1 g d ib & & τ D = B θ 2 − θ1 = B ω 2 − ω1 – Analogous to Translational Spring. – Stores Potential Energy. – E.g., shafts g – Analogous to Translational Damper. – Dissipates Energy. – E.g., bearings, bushings, ... School of Mechanical Engineering Purdue University ME375 Rotation - 4 2 ME375 Handouts Basic Rotational Modeling Elements • Moment of Inertia • Parallel Axis Theorem J = JO + M r 2 – Inertia Element Ex: J = JO + M r d 2 = && Jθ = ∑τi i J = JO + M r 2 – Analogous to Mass in Translational Motion. – Stores Kinetic Energy. = School of Mechanical Engineering Purdue University ME375 Rotation - 5 Interconnection Laws • Newton’s Second Law af d && J ω = J θ = ∑ τ EXTi 2 dt 1 3 i Angular Momentum • Newton’s Third Law – Action & Reaction Torque • Angular Displacement Law School of Mechanical Engineering Purdue University ME375 Rotation - 6 3 ME375 Handouts Example Derive a model (EOM) for the following system: τ (t) J θ Translational K ⇒ Equivalent B K J B FBD: (Use Right-Hand Rule to determine Rightdirection) direction) School of Mechanical Engineering Purdue University ME375 Rotation - 7 Energy Distribution • EOM of a simple Mass-Spring-Damper System Mass-Spring&& & J θ + Bθ + K θ = τ ( t ) 0 Contribution of Inertia 1 1 Contribution of the Damper Contribution of the Spring 3 T o ta l Applied Torque We want to look at the energy distribution of the system. How should we start ? • Multiply the above equation by angular velocity term ω : ⇐ What have we done ? bg && & && & J θ ⋅ θ + Bθ ⋅ θ + K θ ⋅ θ = τ t ⋅ ω ⇐ What are we doing now ? • Integrate the second equation w.r.t. time: z t1 && & J θ ⋅ θ dt 142 4 3 t0 ΔKE = 1 J ω 2 2 E + z t1 Bω ⋅ ω dt 1 42 43 t0 z t1 2 t0 B ω dt ≥ 0 E + z t1 & K θ ⋅ θ dt 1 42 43 t0 Δ PE = 1 K θ 2 2 E School of Mechanical Engineering Purdue University = z t1 bg τ t ⋅ ω dt 1 42 43 t0 ΔE Total w ork done by the applied torque τ ( t ) from tim e t 0 to t 1 ME375 Rotation - 8 4 ME375 Handouts Energy Transformers • Lever School of Mechanical Engineering Purdue University ME375 Rotation - 9 Energy Transformers • Gears N2 N1 R2 R1 A ω2 ω1 P θ1 θ2 B A B School of Mechanical Engineering Purdue University ME375 Rotation - 10 5 ME375 Handouts Gear Train θ1 N1 J1 R1θ1 = R2θ 2 θ2 T J2 N2 J1 F= R1 θ1 T F && J1θ1 = T − R1 F J 2 && J 2 R1 && θ2 = θ1 R2 R2 R2 2 && + J R1 θ = T && J1θ1 2 21 R2 F R2 R1 N1 = R2 N 2 θ2 J2 2 ⎛ ⎞ ⎛ N1 ⎞ && ⎜ J1 + ⎜ J 2 ⎟θ1 = T ⎟ ⎜ ⎟ N2 ⎠ ⎝ ⎝ ⎠ && J 2θ 2 = R2 F School of Mechanical Engineering Purdue University ME375 Rotation - 11 Pulley System I Assume pulley inertia negligible, find equations of motion. k 2x 2 k2 (Hint: find equivalent spring constant) find equivalent spring constant) T x1 T T k 1x 1 T x2 T m k1 T=mg x=2(x1+x2) School of Mechanical Engineering Purdue University ME375 Rotation - 12 6 ME375 Handouts Pulley System II Pulley inertia non-negligible, find equations of motion. School of Mechanical Engineering Purdue University ME375 Rotation - 13 Example • Rolling without slipping x θ Elemental Laws: Coefficient of friction of friction μ R K B f(t) J, M FBD: School of Mechanical Engineering Purdue University ME375 Rotation - 14 7 ME375 Handouts Example (cont.) (cont.) School of Mechanical Engineering Purdue University ME375 Rotation - 15 Example (cont.) (cont.) Q: How would you decide whether or not the disk will slip? Q: How will the model be different if the disk rolls and slips ? School of Mechanical Engineering Purdue University ME375 Rotation - 16 8 ...
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