chapter 2

chapter 2 - 2-1 Chapter 2 Understanding Mechanical System...

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Unformatted text preview: 2-1 Chapter 2 Understanding Mechanical System Requirements Exit 2001 by N. Mohan Print TOC " ! 2-2 Motivation ❏ How can the ASD accelerate and decelerate the load to give desired speed profile Load ASD ωL ω L ( rad / sec ) desired speed profile 100 0 Exit 2001 by N. Mohan 1 2 3 4 5 6 7 t (sec ) TOC " ! 2-3 Systems With Linear Motion fe fL M x u= ⇒ dx ; dt a= du f − fL = e dt M fM M u= dx ; dt a= du f = M dt M x Figure on left includes load force, f L , that must be overcome x Figure on right shows only the force, f M , available to accelerate the mass, M Accelaration f −f f a= e L = M M M Exit 2001 by N. Mohan Power Input Pe (t ) = f e ⋅ u = f M ⋅ u + f L ⋅ u Kinetic energy 1 WM = Mu 2 2 TOC " ! 2-4 Rotating Systems f f 90 o β M r torque θ θ Mg x Torque = force radius [ Nm ] [N] [ m] x Example: what torque is needed to hold M motionless Exit 2001 by N. Mohan TOC " ! 2-5 ❏ Torque in an electric drive ω Motor Tem TL Load x Tem electromagnetic torque produced by motor x Tem is opposed by load torque, TL x The difference, Tem − TL = TJ , will accelerate the system x d ω Tem − TL TJ = = dt J J where J is the moment of inertia Exit 2001 by N. Mohan TOC " ! 2-6 Calculation of Moment of Inertia J of a Uniform Cylinder ! df r rdθ dM dθ ω r1 d f = dM θ dM = ρ rdθ dr & & d v dt ⇒ dT = r 2 dM d d ω = ρ ( r 3 dr dθ d ! ) ω dt dt 2π ! 0 T = ρ ( ∫ r dr ∫ dθ ∫ d ! ) 0 J solid = Exit 2001 by N. Mohan 3 d! & arc height length 0 r1 d! dr d d π ω = ( ρ ! r14 ) ω dt 2 " $# dt # % 1 π ρ ! r14 = M r12 2 2 J TOC " ! 2-7 Accelaration, Speed and Position, Power and Energy ωm Motor Tem TL Tem + Load Σ − TJ 1 α J eq ∫ ωm ∫ θ TL acceleration , ⇒ speed , dω m TJ 1 α = = (Tem −TL ) = dt J eq ( J m+ J L ) ω m (t ) = ω m (0 ) + ∫0 α (τ ) dτ t ⇒ position , θ (t ) = θ (0 ) + ∫0 ω (τ ) dτ t Pem = Tem ⋅ω m ; 1 Kinetic Energy W = J ω 2 2 Power Exit 2001 by N. Mohan PL = TL ⋅ω m TOC " ! 2-8 Frictional Torque Tf stiction 0 coloumb friction T f = Bω viscous friction stiction ω coloumb friction x x x x Exit 2001 by N. Mohan Stiction: static component Coulomb friction: dynamic component (constant magnitude) Viscous friction: speed dependent In general, friction is non-linear TOC " ! 2-9 ❏ Example: Aerodynamic drag Drag power at different speeds f L = 0.046 Cw Av 2 ; (Cw : drag coefficient) p = f L ⋅u ∴ power α speed 3 Speed Power (W) (km/h) Cw = 0.3 Cw = 0.5 50 100 2001 by N. Mohan 1.44 kW 6.9 kW 150 Exit 0.86 kW 23.3 kW 11.5 kW 38.8 kW TOC " ! 2-10 Torsional Resonances ωm ωL TL Tem Motor Tshaft Load JL d ωm dt d ωL At load end Tshaft = TL + J L dt Tshaft (θ m − θ L ) = K θ m and θ L :angular rotation at the two ends of the shaft Jm At motor end Tshaft = Tem − J m x If K → ∞ , θ m = θ L ( J M and J L can be treated as one inertial mass ) x Finite K may lead to resonances Exit 2001 by N. Mohan TOC " ! 2-11 Mechanical - Electrical Analogy • Torque • Angular Velocity • Voltage • Angular Displacement • Flux Linkage • Moment of Inertia • Capacitance • Spring Constant • 1/Inductance • Damping Coefficient • 1/Resistance • Coupling Ratio Exit • Current • Transformer ratio 2001 by N. Mohan TOC " ! 2-12 Electrical Analogy of Motor & Load ωm TL Tem Motor ωL JL Jm ωM Tem TJM TJL Tshaft JM ωm ωL 1/ K Load TL Tem TJ TL JL J eq = J M + J L Finite shaft stiffness Exit 2001 by N. Mohan Infinite shaft stiffness TOC " ! 2-13 Coupling Mechanisms ❏ Required when x a (rotary) motor is driving a load which requires linear (translational) motion x motors prefer higher rotational speed than that required by the load x the axis of rotation needs to be changed ❏ Types ! Conveyor belts (belt and pulley) ! Rack and pinion or a lead-screw type of arrangement ! Gear mechanisms Exit 2001 by N. Mohan TOC " ! 2-14 Conversion between Linear and Rotary Systems fL M u M = mass of load r ωm Motor r = pulley radius Tem du f = M + fL dt Jm u = r ωm 2 dω m T = r f =r M + rf L dt d ωm dω 2 Tem = Jm +r M + r fL " $# # dt% "##dt $## % required to accelerate motor Exit 2001 by N. Mohan J m = motor inertia due to load TOC " ! 2-15 Gears Tem Motor JM T1 r1 ωM r2 T2 TL Load ωL JL x Basic relationships: radius, speed, torque Equal speeds at gear surfaces ⇒ r1 ω M = r2 ω L Power transferred across gears ⇒ ω M T1 = ω L T2 , ⇒ r1 T ω = L = 1 r2 T2 ωM d ωM ωM d ωL & Tem − J M = TL + J L dt % ω L # dt % "## $### # "# $## T1 T2 increased, torque decreased ω L > ω M ; T2 < T1 ; r2 < r1 x Geared up: speed x Geared down: speed decreased, torque increased ω L < ω M ; T2 > T1 ; r2 > r1 Exit 2001 by N. Mohan TOC " ! 2-16 Gears (cont’d) x Equivalent Inertia Tem 2 ω L d ωm ω L = Jm + J L dt + ω TL ωm m "## $### # % J eq ⇒ J eq ω = Jm + JL L ωm 2 r = Jm + JL 1 r2 2 x Optimum gear ratio (to minimize Tem ) 2 Jm r = 1 ⋅ JL r2 opt. and (Tem )opt. = 2 J m Exit 2001 by N. Mohan ⇒ r1 = r 2 opt. Jm JL r dω L d ωm = 2J m 2 dt r1 opt. dt TOC " ! 2-17 Types of Loads Centrifugal loads Fan Constant Torque loads Hoist Exit 2001 by N. Mohan TOC " ! 2-18 Types of Loads Squared power loads Compressor Constant power loads Winder Exit 2001 by N. Mohan TOC " ! 2-19 Four-Quadrant Operation ωm (2) Tem ωm = + Tem = − p=− ωm Load Motor Power 2001 by N. Mohan ωm = + Tem = + p=+ Tem ωm = − Tem = − p=+ (3) Exit (1) ωm = − Tem = + p=− (4) TOC " ! 2-20 Dynamic Operation ❏ How the operating point changes with time ❏ Important for High Performance Drives ❏ Speed change: rapid and without any oscillations ❏ Requires good controller design Exit 2001 by N. Mohan TOC " ! 2-21 Summary " What are the MKS units for force, torque, linear velocity, angular velocity, speed, and power? " What is the relationship between force, torque, and power? " Show that torque is the fundamental variable in controlling speed and position. " What is the kinetic energy stored in a moving mass and a rotating inertia? Exit 2001 by N. Mohan TOC " ! 2-22 Summary ❏What is the mechanism for torsional resonances? ❏ What are the various types of coupling mechanisms? ❏ What is the optimum gear ratio to minimize the torque required from the drive to accelerate a load? ❏ What are the torque-speed and the power-speed profiles for various types of loads? Exit 2001 by N. Mohan TOC " ! ...
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