Lecture 6 mechanical system modeling equivalent mass gears

Transcription

1 M Mechanical System Analysis 기계시스템해석 lecture 6,7,8 Dongjun Lee ( 이동준 ) Department of Mechanical & Aerospace Engineering Seoul National University Dongjun Lee Lecture 6 mechanical system modeling equivalent mass gears Dongjun Lee 1

2 Lecture 7 planar dynamics energy method mechanical drives spring elements Dongjun Lee natural frequency damper effect of damping Lecture 8 we are currently doing Chapter 4. Dongjun Lee 2

3 crash dummy Mechanical System Modeling suspension mass & inertia spring deformed under force; return to original shape when force removed potential energy storage 1 2 mass[kg] inertia[kgm 2 ] Dongjun Lee kinetic energy storage damper slow down motion; dissipate energy Mass Spring Damper Energetics system dynamics total energy damping dissipation energy input by Dongjun Lee 3

4 Mass/Inertia: 2D Case translation dynamics (w.r.t. CoM) rotation dynamics (w.r.t. CoM) : D Alembert often, more convenient to use or D Alembert =0 if P is fied for 2D rotation about a fied point,, Dongjun Lee rolling constraint (no sliding/bouncing) Eample α translation dynamics Dongjun Lee rotation dynamics w.r.t. G friction (unknown) sin sin to maintain rotation dynamics w.r.t. P no slip sin no need to compute! minimum friction coefficient for no slip tan producible by a P 4

5 Energy Methods energy conservation: a prelude to Lagrangian/Hamiltonian dynamics eample rolling constraints: system dynamics: 2 2 sin can think of multiple masses as one lumped mass * no need to compute friction force F: why? Dongjun Lee Mechanical Drives kinematic constraint: gear ratio speed reducer if N>1, ideal gear bo real gear bo torque amplification efficiency (friction, inertia, etc) 60 80% system dynamics / Dongjun Lee 1 5

6 E 2.3.5: Simple Robot Arm kinematic constraint 1 energies cos, motor feels lighter load system dynamics sin 1 DOF system Dongjun Lee Differential Drive Dongjun Lee 6

7 Differential Drive Dongjun Lee symbol eternal force force equation and potential energy Linear Spring Elements deflection free length >0: tension <0: compression if f=0, =0 1 2 quadratic spring constant (or stiffness) [N/m] eamples: : small deformation, linear elastic material, etc shear modulus potential Dongjun Lee * for more, see tables 4.1.1,

8 Free Length and Equilibrium, 1) w.r.t..., 2) w.r.t. (from equilibrium L) : free length : from free length : from zero 1) w.r.t. 2) w.r.t. (from equilibrium ),..,.., : static deflection = / : from equilibrium : deflection from free length w.r.t. equilibrium, we can write the system dynamics s.t. Dongjun Lee parallel connection Parallel and Serial Connections f1 f2 same deflection n spring connection serial connection f f o same force o f o n spring connection Dongjun Lee 8

9 Distributed Inertia and Equivalent Mass y : rod length : end position : initial coordinate (0, : rod coordinate (0),, 0, 0, 0 mass per unit length: infinitesimal mass moves with velocity, at linear velocity assumption, 3 Dongjun Lee equivalent mass of rod Natural Frequency and Rayleigh Method, 0, 0,0 simple harmonic oscillation Im + j j Re natural frequency (amplitude) and (phase) depend on the initial conditions if input is also sinusoid with, (resonance) Rayleigh Method: if 0, is conserved Dongjun Lee 9

10 Eample P : equilibrium 0: mass mass less rod 0?? Dongjun Lee Linear Damper symbol eternal force f linear damper equation relative velocity damping coefficient [N/(m/s)] energy is always dissipated via damper relative velocity Dongjun Lee 10

11 Eamples dashpot shock absorber hole length cylinder diameter door closer viscosity hole diameter viscous bearing pressure bearing length ale diameter viscosity lubrication layer Dongjun Lee Hydraulic Damper [Sec. 7.4] hole length cylinder diameter force equation viscosity hole diameter,, 0 continuity equation (incompressible) laminar flow resistance 1, 128, * if 2 2 (parallel) * if two holes: /2(series) 8 2 if Dongjun Lee 11

12 Door Closer? equation of motion series : deflection from door closing 1 2 orifice equation Δ 1 2 Dongjun Lee adjustable Damped Oscillations With non zero initial condition 0,0 w/ 0 Im + j j Re effect of damping 1) if 0 / keep oscillating (un damped) 2) if 2 damped oscillation (under damped) 3) if 2 sin, no oscillation (critically damped; optimal) 4) if 2 no oscillation, but slow (over damped) Dongjun Lee 12

13 Quad Car Model 1) quarter car body 2) wheel +tire +ale equation of motions are coupled: need to be simultaneously solved Dongjun Lee Net Lecture Lagrange Dynamics Dongjun Lee 13