Mechanical Syste
Newton s Laws 1)First law : conservation of oentu no external force no oentu change linear oentu : v Jω angular oentu : dv ) Second law : F = a= dt d T T = J α = J ω dt
Three Basic Eleents in Modeling Mechanical Systes i) Inertial eleents ( kinetic energy ): asses: M oents of inertial : J ii) Spring eleents ( Potential energy ) T = k Δθ = k( θ ) 1 θ θ ΔX Torsional spring iii) Daper eleents ( energy dissipation ) Δθ T ΔX b θ θ F = bx T θ T = b θ θ b x x x
Exaples of Modeling Mechanical Systes t =, ω() = ω ex1) equation of otion : bw J b dω J = b ω dt dω dω b J + bω = + ω = dt dt J λt λt b λt let, ω( t) = ce λe + e = J b λ = J b t J ω() t = ce, t =, ω() = ω = C ω() t = ω e o b t J ω 1 e =.368ω ω() t w
Exaples of Modeling Mechanical Systes ex) Spring-Mass x(t) k t =, x() = x, x() = d x = F = kx dt k x && + kx = && x + x = k k xt () = Acos t+ Bsin t x () = A = x o k k k k k x& () = A sin t + B cos t = B = k xt () = x cos t = x cos ω n t π Period T = [sec] k/ 1 1 k Frequency f = = [ Hz] T π Natural frequency k ωn = π f = [ rad /sec]
Exaples of Modeling Mechanical Systes Spring-ass-daper x() x& () = = x x& d x = kx bx& dt b k && x + x& x + = k b b let, = ωn, = ζωn, = ζ k && + & + λ + ζωλ + ωx ω = x+ ζω + =, () = λt nx ωnx xt c e n n x ( ) λ = ζω ± ζω ω = ζω ± ζ 1ω n n n n n
Exaples of Modeling Mechanical Systes b k && x + x & x + = x() = x x& () = x& = k b b k let, = ωn, = ζωn, = ζ && x+ x& + x= Laplace Transfor ζω n ω n ( sxs ( ) sx x& ) + ζω ( sxs ( ) x ) + ω Xs ( ) = n n ( s + ζωns+ ωn) X( s) s x ζωnx = ( s + ζω n s+ ω n) X ( s ) = s x + ζω nx X() s = s x + ζω x s n + ζωns+ ωn
Exaples of Modeling Mechanical Systes s x + ζω x s x ζω x n n () = = + s + ζωns+ ωn s + ζωns+ ωn s + ζωns+ ωn X s s s + ζω s+ ω : charateristic polynoial n n + ζω s + ω = : characteistic i equation n n ( ) n n n n n s = ζω ± ζω ω = ζω ± ω ζ 1 : charateristic roots 1) ζ < 1 underdaped s = ζω ± ω ζ 1= ζω ± jω 1 ζ n n n n ( s+ ζω ) x ζω x X() s = + ( s+ ζω ) + ω (1 ζ ) ( + ζω ) + ω (1 ζ ) n n n n s n n ζω ζ n xt () = e t xcos 1 ζ ωnt+ x sin ζ ωnt 1 ζ
Exaples of Modeling Mechanical Systes ) ζ > 1 overdaped s = ζωn ± ωn ζ X() s = 1 ( s+ ζω ) x ( s + ζω 1)( 1) n ωn ζ s+ ζωn + ωn ζ ( ζωn+ ζ 1 ωn) t ( ζωn ζ 1 ωn) xt () = ke + ke 1 n t 3) ζ = 1 critically daped s = ζωn ( s+ ζωn) x X() s = s + ζω ( ) ( ζω ) t ( ζω ) xt ke kte n n () = + n 1 t
Exaples of Modeling Mechanical Systes 1) ζ < 1 Underdaped ζω ζ n x() t = e t xcos 1 ζ ωnt+ x sin ζ ωnt 1 ζ ω n : ζ : natural frequency daping ratio ω d = 1 ζ ωn ) ζ > 1 Overdaped ( ζωn+ ζ 1 ωn) t ( ζωn ζ 1 ωn) xt () = ke + ke 1 λ = ζωn ± ζ 1ω n t x () t ζ > 1 ζ = 1 3) ζ = 1 λ = ζωn xt ke kte n ζωn () = + ζω t 1 t ζ < 1 t
Dry Friction (no lubricant) Fμs = Static Friction Force Fμk = Kinetic Friction Force Fμs = μs N μs : Static Friction Coefficient Fμk = μk N k N μk k : Kinetic Friction Coefficient x & = x F μ F if F F = F μssgn( F) if F> F μs μs & F = F sgn( x& ) μ μk
ε Friction (with lubricant) bx& + G N sgn( x& ) x& > ε F & μ = F if x < ε and F ( Fs + G N) ( F + s G N)sgn( F) otherwise i.e., if x& < ε and F > ( F + s G N) b: viscous friction coefficient G: load-dependent factor N: noral force Fs: the axiu static friction ε : a sall bound for zero velocity detection
Work, Energy, and Power Mechanical work : W = F x [N ] = [Joule] = Force displaceent Energy : capacity or ability to do work. Electrical, Cheical, Mechanical, etc. Mechanical energy : Potential energy position Kinetic energy velocity
Potential Energy ex1) F = kx x1 x1 1 dw = Fdx = kx dx dw = kx dx = kx ex) M Mg E h 1 X= = gx (Potential Energy)
Kinetic Energy 1 v v : J ω : 1 Jω
Work and Energy Syste + External Work = E Energy E1 W E 1 + W = E F v1 v1 l v v E1+ W = E 1 1 v 1 + Fl= v 1 1 v 1 Fl= v
Power Power : tie rate & doing work P dw N = dt = sec [ Watt ] Ex) kg V = 7k/h = /s (in 1sec) V = 1 1 v + W = v 1 ()() 4 1 4 [ ] W = = N = kn kj W 4 1 N P = = t 1 sec = 4 1 N / s = 4 1 W = 4W 1 hp = 745.7 W P = 54 hp Power dissipated in a daper P = Fv = bx& x& = bx&
An Energy Method for Deriving Equations of Motion Conservative syste : No energy dissipation E + W = E 1 E E = W 1 Kinetic Energy T Potential Energy U Δ(T+U) = ΔW (the change in the total energy) = (the net work done on the syste by the external force) no external force ; ΔW = Δ(T+U) = T+U = constant
Exaples of Energy Method x(t) ex1) 1 1 T = x&, U = kx, T + U = C k 1 1 x& + kx = d ( T + U ) =, xx &&& + kxx & = ( x && + kx ) x & = dt x&, ( x && + kx ) = 1 1 d ( T + U ) = bx &, xx &&& + kxx & = bxx && = dt ( x && + kx + bx& ) x& = ex) x(t) x T = x&, U = kx k x&, ( x && + kx+ bx& ) =
ex3) Exaples of Energy Method Spring with ass ξ l + x x dξ s l + x k l Potential E x(t) 1 U = kx x+ kx ω n = = k 1 dξ ξ dt = s ( x& ) l + x l + x l+ x l+ x = 1 1 dt s + x& ξ l x d ξ 3 ( ) 1 1 1 = x s & ( 3 l + x ) ( l+ x) 3 3 3 Kinetic E 1 T = x 1 1 = ( 3 s ) x& T 1 1 1 1 1 1 = x& + ( s ) x& = ( + s ) x& ( + s )& x + kx = 3 3 3 ω n = k 1 + 3 s