Consider the rotor with the gear train driving a robot arm through a flexible shaft, as shown in Fig. 1. An external motor (not shown) delivers the input torque T(t) to the Rotor with inertia J ₁ . The Rotor is connected to the gear wheel n ₁ of the gear train with a very short and very stiff shaft. Gear n2 is larger than gear n ₁ , and their gear ratio is n ₂ /n ₁ = N ₁ ; similarly, gear n ₄ is larger than gear n ₃ , and their gear ratio is: n ₄ /n ₃ = N ₂ . Inertia of gears n ₁ , n ₂ , n ₃ , and n ₄ can be neglected. The rotor is supported by bearings with damping coefficient b ₁ . Gear Train and Robotic Arm Assembly are connected by a flexible shaft. The elastic property of this shaft is represented by a parallel connection of a torsional spring with stiffness coefficient of k _T and torsional damper with damping coefficient of c _T . The Robotic Arm Assembly has inertia J ₂ , and is supported by bearings that contain some hydraulic fluid with viscous friction coefficient b ₂ .

a) Derive the governing equations for this system

b) Derive the transfer function ?(s)/T(s), where ?(s) and T(s) are the Laplace transforms of the rotation speed of the Robotic Arm Assembly ?(t) = , and of the applied torque T(t) respectively.

c) Obtain a state representation for the dynamic system, with T(t) as an input, and ?(t) as an output.

Transcribed Image Text:

T(t) 8,(1) 一月 J₁ Rotor n₁ Gear Train Figure 1 0₂ (1) b₂ k₁, CT Robotic Arm Assembly T(t) 8,(1) 一月 J₁ Rotor n₁ Gear Train Figure 1 0₂ (1) b₂ k₁, CT Robotic Arm Assembly

Answer rating: 100% (QA)

a Consider the rotor with gear train driving The free body diagram of the given system is Usin... View the full answer