Hw2-S05_Solution

# Hw2-S05_Solution - HW 2-SOLUTION ME 304 Modeling Transfer...

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HW 2-SOLUTION ME 304 Modeling, Transfer Functions & Block Diagrams CONTROL SYSTEMS (01) March 23, 2005 PROBLEM 1: a) Compatibility eqn’s o xR = −θ (1) 2i = θ (2) Elemental equations damper: . 2 b fb ( V x =− ) x . spring: s fk = k J, R o , R i Rack Pinion b Disk θ x 2 m x(t) (output) V(t) (input) Torque balance on pinion and disk: f b .. 2 io Rb(V x ) RF J −= θ x x (3) F (contact force between gear and pinion) Force balance on gear: - Fk m .. F (4) = m f s From (3) => 2 i o Rb(V x ) J F R −− = . θ Insert last equation in (4) => . .. 2 i o kx m x R θ = Use compatibility equations to eliminate x 2 and θ in the last equation Insert x 2 from (2), . .. 2 .. ii oo o RbV bR J kx mx RRR θ θ −+ + = Insert θ from (1), . .. 2 .. 22 ooo bR x Jx kx −− − = Re-arrange last equation, 2 .. . o RR J (m )x b x kx b V R ⎛⎞ ++ + = ⎜⎟ ⎝⎠ Prof. Dr. Y. Samim Ünlüsoy, Res. Asst. Kemal Çal ı ş kan 1

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HW 2-SOLUTION ME 304 Modeling, Transfer Functions & Block Diagrams CONTROL SYSTEMS (01) March 23, 2005 As a check, note that the terms on the left hand side represent resistances to motion, while the term on the right hand side is the forcing function causing the motion. It is a negative term since V is defined to cause motion of the rack in the negative x direction. b) Take Laplace transform of differential equation of motion; 2 2 ii 22 oo o bR R b J (m )s X(s) sX(s) kX(s) V(s) RR R +++ = Transfer function => i 2 2 i 2 -R b X(s) bR J V(s) R(m ) s s Rk = c) Block Diagram: + - i o Rb R 1 m 1 s 1 s k i o R R 2 o J R - . x .. mx .. x . 2 x .. o J R θ x kx + V(t) - PROBLEM 2: T in θ 3 θ 1 I 1 Wheel b b b I 4 R 2 R 1 I 2 θ 2 I 3 b θ out Rigid Shaft k a) Prof. Dr. Y. Samim Ünlüsoy, Res. Asst. Kemal Çal ı ş kan 2
HW 2-SOLUTION ME 304 Modeling, Transfer Functions & Block Diagrams CONTROL SYSTEMS (01) March 23, 2005 Elastic Shaft (massless) : => 11 Tk 2 ) ( =

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Hw2-S05_Solution - HW 2-SOLUTION ME 304 Modeling Transfer...

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