Rotational Mechanical System

Rotational - ME375 Handouts Rotational Mechanical Systems School of Mechanical Engineering Purdue University ME375 Rotation 1 Rotational Mechanical

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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] 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 ) + 1 [Nm] = 1 [J] (Joule) z z t1 t0 t1 p(t ) dt (τ ⋅ θ ) dt t0 School of Mechanical Engineering Purdue University ME375 Rotation - 3 Basic Rotational Modeling Elements Basic • Spring – Stiffness Element • Damper – Friction Element τS θ1 θ2 τS τ S = K θ 2 − θ1 b g τ D = B θ 2 − θ1 = B ω 2 − ω1 – Analogous to Translational Damper. – Dissipates Energy. – E.g., bearings, bushings, ... d ib g – Analogous to Translational Spring. – Stores Potential Energy. – E.g., shafts School of Mechanical Engineering Purdue University ME375 Rotation - 4 2 ME375 Handouts Basic Rotational Modeling Elements • Moment of Inertia – Inertia Element Ex: J = JO + M r = Jθ = ∑τi i • Parallel Axis Theorem J = JO + M r 2 d 2 – Analogous to Mass in Translational Motion. – Stores Kinetic Energy. J = JO + M r 2 = School of Mechanical Engineering Purdue University ME375 Rotation - 5 Interconnection Laws • Newton’s Second Law Newton’ d dt aJ ω f = J θ = ∑ τ EXTi i Angular Momentum • Newton’s Third Law Newton’ – 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 K Translational ⇒ Equivalent J K B B FBD: (Use Right-Hand Rule to determine Rightdirection) School of Mechanical Engineering Purdue University ME375 Rotation - 7 Energy Distribution • EOM of a simple Mass-Spring-Damper System Mass- SpringJ θ + Bθ + K θ = τ ( t ) Contribution of Inertia 0 Contribution of the Damper 1 Contribution of the Spring 1 Total Applied Torque 3 We want to look at the energy distribution of the system. How should we start ? • Multiply the above equation by angular velocity term ω : J θ ⋅ θ + Bθ ⋅ θ + K θ ⋅ θ = τ t ⋅ ω • Integrate the second equation w.r.t. time: bg ⇐ What have we done ? z t1 t0 J θ ⋅ θ dt + z t1 t0 B ω ⋅ ω dt dt ≥ 0 + ΔKE = 1 J ω 2 2 E z z ⇐ What are we doing now ? t1 t0 K θ ⋅ θ dt = z t1 t0 τ t ⋅ ω dt ΔE bg t1 2 t0 B ω E ΔPE = 1 K θ 2 2 E Total w ork done by the applied torque τ ( t ) from tim e t 0 to t 1 School of Mechanical Engineering Purdue University ME375 Rotation - 8 4 ME375 Handouts Energy Transformers • Lever School of Mechanical Engineering Purdue University ME375 Rotation - 9 Energy Transformers • Gears N1 R1 A B A N2 R2 ω1 P ω2 θ1 θ2 B School of Mechanical Engineering Purdue University ME375 Rotation - 10 5 ME375 Handouts Gear Train θ1 J1 T N2 N1 θ2 J2 R1θ1 = R2θ 2 R1 N1 = R2 N 2 J1 T θ1 R1 F F= J2 JR θ 2 = 2 1 θ1 R2 R2 R2 J1θ1 = T − R1 F F R2 θ2 J2 R12 J1θ1 + J 2 2 θ1 = T R2 2 ⎛ ⎞ ⎛ N1 ⎞ ⎜ J1 + ⎜ J 2 ⎟θ1 = T ⎟ ⎜ ⎟ ⎝ N2 ⎠ ⎝ ⎠ ME375 Rotation - 11 J 2θ 2 = R2 F School of Mechanical Engineering Purdue University Pulley System I k 2x 2 k2 Assume pulley inertia negligible, find equations of motion. (Hint: find equivalent spring constant) x1 T T T m k1 k 1x 1 T T x2 T=m g x=2( x1+x2) School 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 Coefficient of friction μ Elemental Laws: θ f(t) R J, M K B 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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This note was uploaded on 12/17/2009 for the course IE 383 taught by Professor Leyla,o during the Spring '08 term at Purdue University-West Lafayette.

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