A robotic joint is modeled as a mechanical system with rotors of moments of inertia J and 2J, connected by a viscoelastic coupling (a parallel array of a torsional spring and a torsional dashpot) with torsional stiffness k and torsional damping coefficient c, as indicated in Fig. 5.23.

(a) Derive the mathematical model of the system in terms of the angles θ₁ and θ₂ shown in Fig. 5.23 if we know that the spring in the coupling is unloaded when these two coordinates are identical.

(b) Under the assumption that τ₂(t) = − τ₁(t) and that all initial conditions are zero, the torque τ (t) experienced by the coupling at its ends can be expressed in the form

Find expressions for k_e and c_e in terms of the system parameters.

(c) Under the same conditions as in item (b) above, τ₁(t) is modeled as a harmonic torque of amplitude τ₀ and frequency ω. Thus, the steady-state torque  transmitted to the coupling is also harmonic with an amplitude τ_T and a phase angle ψ. Using Bode plots, rather than lengthy calculations, find the ratio τ_T /τ₀, if ζ_e = 0.5 and ω_e = ω/5, where c_e = 2ζ_eω_eJ and k_e = ω²_e J.

Transcribed Image Text:

T(t) Fig. 5.23 Robotic joint viscoelastic coupling 8 k, c 2J 02 T(t)