Let us consider the following crank-connecting rod system:

Transcription

1 Capitolo. TEORIA DEI SISTEMI 7. The crank-connecting rod ytem The POG dynamic model Let u conider the following crank-connecting rod ytem: L R θ, τ m p b p d J m b m x The poition x(θ) of the piton can be expreed afollow: Velocity ẋ i obtained a follow: ẋ(t) = R = R x(θ) = Rcoθ+ L 2 (Rinθ d) 2 inθ (Rinθ d)coθ L2 (Rinθ d) 2 inθ (inθ β)coθ α2 (inθ β) 2 H(θ) θ ω = H(θ)ω unction H(θ) and parameter α and β are defined a follow: H(θ) = x(θ) θ, α = L R >, β = d R <. The time-varying dynamic model of the conidered rigid ytem can be obtained adding a tiffne element K between the connecting rod and the piton and then let K. Zanai Roberto - Teoria dei Sitemi A.A. 205/206

2 Capitolo 7. ESEMPI ED ESERCIZI 7.2 The POG dynamic model of the extended dynamic ytem with the additional tiffne element K i: τ J m ω b m H(θ) ẋ k ẋ The POG tate pace model of the conidered ytem i: J m K m p L ω k ẍ }{{} ẋ = H(θ) k K b m H(θ) 0 H(θ) 0 0 b p A m p ω k ẋ }{{} x + b p B τ }{{} u that i Lẋ = Ax + Bu. When the fictitiou tiffne K goe to infinity the tate pace variable are contrained a follow ẋ = H(θ)ω. Applying the following congruent tate pace tranformation x = Tω, ω k ẋ }{{} x = 0 H(θ) T one obtain the following tranformed and reduced ytem: d[j(θ) ω] dt N (θ)ω = b(θ)ω +B(θ)u () where: J(θ) = T T LT = J m +H 2 J(θ) (θ)m p, N (θ) = ṪT LT = = m p 2 H(θ)Ḣ(θ) and b(θ) = T T AT = b m +H 2 (θ)b p, B(θ) = T T B = [ H(θ) ]. ω Zanai Roberto - Teoria dei Sitemi A.A. 205/206

3 Capitolo 7. ESEMPI ED ESERCIZI 7.3 POG graphical repreentation of the reduced ytem: τ H(θ) J(θ) N (θ) b(θ) ω H(θ) ẋ The dynamic equation () can alo be written a follow: d dt (J(θ)ω) J(θ) ω = (b m +H 2 (θ)b p )ω +τ H(θ) (2) 2 The Lagrange approach Lagrange Equation: where d T T + U = Q i dt q i q i q i i =,..., N N degree of freedom of the mechanical ytem; q i generalized Lagrangian coordinate; Q i generalized force; T kinetic energy of the ytem (mae and inertia); U potential energy of the ytem (pring and gravitational force); The kinetic energy of the crank-connecting rod ytem i the following: T = 2 J m θ m pẋ 2 = 2 J m θ m p [ H(θ) θ] 2 Zanai Roberto - Teoria dei Sitemi A.A. 205/206

4 Capitolo 7. ESEMPI ED ESERCIZI 7.4 The potential energy i zero: U = 0. In thi cae the Lagrangian coordinate i q = θ. Intermediate term of the Lagrangian Equation: T θ = J m θ+m p H 2 (θ) θ d T dt θ = J m θ+mp H 2 (θ) θ+2m p H(θ)Ḣ(θ) θ T θ = m H(θ) ph(θ) θ θ = m p H(θ)Ḣ(θ) θ θ The external generalized force Q i obtained a follow: Qdθ = τ dθ b m θdθ (bp ẋ+)dx = τ dθ b m θdθ (bp H(θ) θ+)h(θ)dθ from which one obtain Q = τ b m θ bp H 2 (θ) θ H(θ). The Lagrangian equation of the ytem can be written a follow: d ( Jm θ+mp H 2 (θ) dt θ ) m p H(θ)Ḣ(θ) θ = τ b m θ bp H 2 (θ) θ H(θ). (3) Uing the following correpondence: T = J(θ) θ, θ T θ = N (θ) θ = N 2 (θ) θ, equation (3) i equivalent to equation (). b(θ) = b m +b p H 2 (θ) Zanai Roberto - Teoria dei Sitemi A.A. 205/206

5 Capitolo 7. ESEMPI ED ESERCIZI 7.5 Simulation of the crank and connecting-rod ytem The Matlab/Simulink block cheme The crank and connecting-rod ytem ha been imulated in Matlab/Simulink. The SimMechanic block cheme of the crank and connecting-rod ytem: Var_Motore Motore Elettrico Var_In_Motore Clock t Plot Data Env B Statore CS Giunto Rotore Var_Rotore Rotore CS CS3 B Po_Rotore CS Biella Giunto Biella Var_Biella CS CS3 Ata Giunto C2 Ata Var C Var_Ata CS3 CS Po_Pitone CS3 Po_Statore Po_Ata Pitone e_th Var_Pitone Giunto Pitone POG Model Giunto Pitone POG block cheme of the crank and connecting-rod ytem: Tm G_th p Tau_m K(th) e_th /J(th) N b dot_wm N(th) B(th) Wm Wm K(th) Ve Wm J(th), N(th), B(th), K(th), The two ytem have been imulated conidering a contant input torque: τ = 2 Nm. Zanai Roberto - Teoria dei Sitemi A.A. 205/206

6 Capitolo 7. ESEMPI ED ESERCIZI 7.6 Angular velocity of the motor ω and linear velocity of the final point V e : 350 Angular velocity of the motor Wm [rpm] Linear velocity of the final point 3 d_xp [m/] t [ec] The two block cheme provide the ame reult. Maximum error: Time behavior of the ytem torque: the motor torque (blue), the friction torque (magenta), the dynamic torque (red), the gravity force ( green green) and the final-point torque (black): 3.5 Torque: motor (b), dynamic (r), friction (m), gravity (g), end point (k) Torque [Nm] th [giri] Zanai Roberto - Teoria dei Sitemi A.A. 205/206

7 Capitolo 7. ESEMPI ED ESERCIZI 7.7 A three-dimenional graphical repreentation of the crank and connecting-rod ytem: The approach can be extended to more complex ytem. Example: the Trolley-older ytem: Zanai Roberto - Teoria dei Sitemi A.A. 205/206

8 Capitolo 7. ESEMPI ED ESERCIZI 7.8 The approach can be applied alo to robotic ytem. z Σ 2 θ 2 N 2 c a 2 ba θ 3 Σ 3 3 c a 3 ba N 3 c a ba N 2 Σ 0 Σ θ x POG block cheme of the ytem: G(Θ) τ K T (Θ) e J(Θ) N (Θ) B(Θ) 0 Θ K(Θ) V e Dynamic equation of the ytem: d[j(θ) Θ] dt N (Θ) = B(Θ) Θ+G(Θ) K T (Θ) e +τ J(Θ) i the inertia matrix, N 2 (Θ) i the dynamical matrix, B(Θ) i the friction matrix, H(Θ) i the Jacobian matrix, K(Θ) i the final point matrix, G(Θ) i the gravity vector, e i the final point force vector, V e i the final point velocity vector and τ i the input torque vector. Zanai Roberto - Teoria dei Sitemi A.A. 205/206