2.2 Free Vibration of an Undamped Translational System
(2.1) F(t) = m ẍ
(2.2) M(t) = J θ̈
(2.3) m ẍ + kx = 0
(2.4a) F(t) - m ẍ = 0
(2.4b) M(t) - J θ̈ = 0
(2.5) -m ẍ δx - kx δx = 0
(2.6) d/dt (T + U) = 0
(2.7) T = ½ m ẋ²
(2.8) U = ½ k x²
(2.9) W = mg = k δst
(2.10) m ẍ + kx = 0
(2.12) m s² + k = 0
(2.13) s = ± i √(k/m)
(2.16) x(t) = A1 cos ωn t + A2 sin ωn t
(2.21) x(t) = A cos (ωn t - φ)
(2.22) A1 = A0 sin φ0, A2 = A0 cos φ0
(2.27) δst = g / ωn²
(2.28) ωn = √(g / δst)
(2.30) τn = 2π √(δst / g)
(2.34) ẋ(t) = -A ωn sin (ωn t - φ)
(2.35) sin (ωn t - φ) = -ẋ / (A ωn) = -y / A
(2.36) x² + y² = A²
2.3 Free Vibration of an Undamped Torsional System
(2.37) Mt = (G I0 / l) θ
(2.38) I0 = π d⁴ / 32
(2.40) J0 θ̈ + kt θ = 0
(2.41) ωn = √(kt / J0)
(2.42) τn = 2π √(J0 / kt)
(2.43) fn = 1/(2π) √(kt / J0)
(2.44) θ(t) = A1 cos ωn t + A2 sin ωn t
2.4 Response of First-Order Systems and Time Constant
(2.47) J v̇ + ct v = 0
(2.48) v(t) = A e^(st)
2.5 Rayleigh
s Energy Method
(2.55) T1 + U1 = T2 + U2
(2.56) T1 = U2
2.6 Free Vibration with Viscous Damping
(2.58) F = -c ẋ
(2.59) m ẍ + c ẋ + kx = 0
(2.60) x(t) = C e^(st)
(2.61) m s² + c s + k = 0
(2.62) s1,2 = (-c ± √(c² - 4mk)) / 2m
(2.64) x(t) = C1 e^(s1 t) + C2 e^(s2 t)
(2.68) s1,2 = (-ζ ± √(ζ² - 1)) ωn
(2.69) x(t) = C1 e^((-ζ + √(ζ² - 1)) ωn t) + C2 e^((-ζ - √(ζ² - 1)) ωn t)
(2.73) X = √( x0² + ((ẋ0 + ζ ωn x0)/ωd)² )
(2.74) φ0 = tan
¹ ( (x0 ωd) / (ẋ0 + ζ ωn x0) )
(2.75) φ = tan
¹ ( (ẋ0 + ζ ωn x0) / (x0 ωd) )
(2.78) x(t) = (C1 + C2 t) e^(-ωn t)
(2.79) C1 = x0, C2 = ẋ0 + ωn x0
(2.80) x(t) = [x0 + (ẋ0 + ωn x0) t] e^(-ωn t)
(2.81) x(t) = C1 e^((-ζ + √(ζ² - 1)) ωn t) + C2 e^((-ζ - √(ζ² - 1)) ωn t)
(2.83) x1/x2 = e^(ζ ωn (t2 - t1))
(2.84) x1/x2 = e^(ζ ωn td)
(2.85) δ = ln(x1/x2) = (2π ζ)/√(1 - ζ²)
(2.86) δ
2π ζ
(2.87) ζ = δ / √((2π)² + δ²)
(2.88) ζ
δ / (2π)
(2.94) ΔW = π c ωd X²
(2.95) F = -kx - c ẋ
(2.96) x(t) = X sin ωd t
(2.101) T = -ct θ̇
(2.102) J0 θ̈ + ct θ̇ + kt θ = 0
(2.105) ζ = ct / ctc = ct / (2 J0 ωn) = ct / √(4 kt J0)
2.7 Graphical Representation of Characteristic Roots and Corresponding Solutions
(2.106) m ẍ + c ẋ + kx = 0
(2.107) m s² + c s + k = 0
(2.108) s² + 2 ζ ωn s + ωn² = 0
(2.109) s1,2 = (-c ± √(c² - 4mk)) / 2m
(2.110) s1,2 = -ζ ωn ± i ωn √(1 - ζ²)
(2.112) sin θ = ζ
(2.113) θ = sin
¹ζ
2.8 Parameter Variations and Root Locus Representations
(2.114) τ = 1 / (ζ ωn)
(2.115) s1,2 = ± i √(k/m) = ± i ωn
(2.116) -σ = c/(2m) = ζ ωn, √(4mk - c²)/(2m) = ωd
(2.117) σ² + ωd² = ωn²
(2.121) s² + 16s + k = 0
(2.122) s1,2 = -8 ± √(64 - k)
2.9 Free Vibration with Coulomb Damping
(2.125) F = μN = μmg
(2.126) m ẍ + kx = -μN
(2.127) x(t) = A1 cos ωn t + A2 sin ωn t - μN/k
(2.128) m ẍ + kx = μN
(2.129) x(t) = A3 cos ωn t + A4 sin ωn t + μN/k
(2.134) r
floor((x0 - μN/k) / (2 μN/k))
(2.135) Xm = Xm-1 - 4 μN/k
(2.136) J0 θ̈ + kt θ = -T
(2.137) J0 θ̈ + kt θ = T
(2.140) r
floor((θ0 - T/kt) / (2 T/kt))
2.10 Free Vibration with Hysteretic Damping
(2.141) F = kx + c ẋ
(2.142) x(t) = X sin ω t
(2.143) F = kx + c ω √(X² - x²)
(2.144) ΔW = π ω c X²
(2.146) ΔW = π h X²
(2.147) F = (k + i ω c) x
(2.148) F = (k + i h) x
(2.149) k + i h = k (1 + i β), β = h/k
(2.151) (Xj - 0.5)/Xj+1 = √(2 + π β) - π β
(2.152) (Xj + 0.5)/Xj+1 = √(2 + π β) + π β
(2.153) Xj Xj+1
1 + π β = constant
(2.154) Xj/Xj+1 = 1 + π β
(2.157) ceq = h/ω