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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 RightHand Rule to determine Rightdirection) School of Mechanical Engineering Purdue University ME375 Rotation  7 Energy Distribution
• EOM of a simple MassSpringDamper 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 nonnegligible, 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 UniversityWest Lafayette.
 Spring '08
 Leyla,O

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