MECHANICS APPLICATIONS OF SECOND-ORDER ODES 7 Mechanics applicaions of second-order ODEs Second-order linear ODEs wih consan coefficiens arise in many physical applicaions. One physical sysems whose behaviour is governed by an ODE of he form mẍ+kẋ+cx = f() () is he damped mechanical oscillaor shown in Fig.. In his case, m represens he mass of he paricle aached o he spring, k is a measure of he srengh of he damper, and c represens he spring siffness. f() is he applied exernal force. represens ime and x() is he displacemen of he mass from is res posiion. The physical inerpreaion of he individual erms in equaion () is as follows. The erm kẋ represens he force exered by he damper on he mass: for a linear damper, his force is proporional o he velociy ẋ and i resiss he moion. The erm cx represens he force exered by he spring on he mass: for a linear spring, his force is proporional o he displacemen x and i acs in he direcion opposie o he displacemen. Equaion () expresses Newon s law, which saes ha he sum of all forces acing on he paricle is equal o is mass imes is acceleraion, mẍ. Wall c k x() f() m Figure : Skech of a damped mechanical oscillaor consising of a mass m aached o a rigid wall by a linear spring of spring siffness c and a damper wih damping consan k. A ime-dependen force, f(), is applied o he mass and he displacemen of he mass from is res posiion is represened by x(). The iniial condiions are given by he iniial posiion, x( = ) = x, and he iniial velociy of he mass a ime =, dx d = v. =. The unforced case: f() = Eigenfrequencies If f() =, equaion () is reduced o is homogeneous form, mẍ+kẋ+cx =. () which describes he moion of he mass in absence of any exernal forcing. To reduce he number of parameers required o classificy he characer of he paricle s moion, we re-wrie he ODE () as ẍ+δẋ+ω x =, () where δ = k m and ω = c m. (5) Noe ha in he damped mechanical oscillaor he coefficiens m,k and c are posiive. Hence δ and ω are posiive, oo.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 8 The soluion o he corresponding characerisic equaion λ +δλ+ω =, (6) is given by or, wrien in a differen form, λ, = δ ± δ ω (7) λ, = δ ±i ω δ. (8) This allows a sraighforward idenificaion of four disinc ypes of moion: I. Purely damped moion: δ > ω In his case equaion (7) shows ha boh roos are real and he soluion is given by x() = Ae ( δ+ δ ω ) +Be ( δ δ ω ). (9) Since boh exponens are negaive, his represens a purely damped moion, i.e. he sysem does no perform any oscillaions. 5, x, = e - x = e -/ 6 8 Figure : Illusraion of a purely damped moion. The mass approaches is equilibrium posiion x = monoonically. II. Criically damped moion: δ = ω In his case he square roo in (7) vanishes and we have λ = λ = δ and he general soluion is given by x() = Ae δ +Be δ. () This represens a criically damped moion in which x() approaches zero bu can cross he value x = (a mos) once (when = A/B; i depends on he iniial condiions which deermine A and B, if his happens for > ). III. Damped oscillaion: δ < ω In his case (8) shows ha boh roos are complex. The general soluion is hen given by x() = e δ( Acos( ω δ )+Bsin( ) ω δ ). () This soluion represens a damped oscillaion wih frequency ω δ whose ampliude decays exponenially.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 9, x, = e - x = e -, x, = e - x = - e - - 6 8 6 8 Figure : Illusraion of criically damped moions. The mass approaches is equilibrium posiion, x =, wih a mos one overshoo., x, = e -/ cos() x = e -/ sin() - 6 8 Figure : Illusraion of a damped oscillaion. The mass oscillaes abou is equilibrium posiion x = and he ampliude of he oscillaions decays exponenially. IV: Undamped oscillaions δ = The soluion () is sill valid and for δ = we obain x() = Acos(ω)+Bsin(ω), () which is an undamped oscillaory moion wih eigenfrequency ω.. The inhomogeneous equaion Periodic forcing and resonance The general soluion of he inhomogeneous equaion ẍ+δẋ+ω x = F(), () where F() = f()/m is given by x() = x H ()+ (). () In (), x H () is he general soluion of he corresponding homogeneous equaion () and hus conains he wo free consans required o fulfill he iniial condiions. () (he paricular soluion) is any soluion of he inhomogeneous equaion.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES = cos() x = sin(), x, - - - 6 8 Figure 5: Illusraion of an undamped oscillaion. The mass performs harmonic oscillaions abou is equilibrium posiion x =. The mos imporan form of forcing in many engineering applicaions is he harmonic forcing in which F() has he form F() = F sin(ω) or F() = F cos(ω). I is advanageous o carry ou he calculaion wih complex numbers by wriing F() = Fe iω = F cos(ω)+if sin(ω). (5) Once he calculaion is compleed we can exrac he relevan real soluion (corresponding o he cos or sin forcing) by aking he real or imaginary pars of he complex soluion. We know from he previous secion ha we can find a soluion for exponenial forcing in he form () = Xe iω. (6) Hence we inser (6) and (5) ino () and carry ou he differeniaions. Afer cancelling he common facor e iω, his provides he following equaion for he unknown ampliude X (which is complex in general): X = F (ω Ω )+i (δω) = F (ω Ω ) i (δω) (ω Ω ) +(δω) (7) Now we can muliply ou () = Xe iω = X(cos(Ω)+isin(Ω)) and exrac he real or imaginary par o obain he relevan real soluion, i.e. and () = R(X)cos(Ω) I(X)sin(Ω) () = R(X)sin(Ω)+I(X)cos(Ω) for F() = F cos(ω) for F() = F sin(ω). Alernaively, we can re-wrie (7) in polar form and hus obain he ampliude of he response X = F (ω Ω ) +(δω) = F/ω ( (Ω/ω) ) ( ) + (δ/ω)(ω/ω) and he phase angle ( ) δω ϕ = arg(x) = arcan ω Ω. (8)
MECHANICS APPLICATIONS OF SECOND-ORDER ODES Hence, he paricular soluion can also be wrien as () = X e i(ω+ϕ). The relevan real soluions are again obained by aking he real and imaginary par of his complex soluion, i.e. () = X cos(ω+ϕ) for F() = F cos(ω) and () = X sin(ω+ϕ) for F() = F sin(ω). Comparing his o (5) shows ha he paricular soluion (i.e. he forced moion), (), has a phase difference of ϕ agains he forcing, F(). Noe ha for weak damping (small δ), he ampliude of he response, X, becomes very large when he exciaion frequency Ω is close o he eigenfrequency of he sysem, Ω ω. This is known as resonance. For vanishing damping, δ =, he response becomes unbounded when Ω ω. This is he resonance caasrophe. In pracical applicaions here is always some damping bu he ampliudes of he forced oscillaions of many physical sysems can sill become so large ha he sysem is desroyed when he exciaion frequency is sufficienly close o he eigenfrequency of he sysem. [Remember he wobbling Millenium Bridge?] Here is a plo of he (normalised) ampliude of he oscillaion as a funcion of he exciaion frequency (normalised by he sysem s eigenfrequency ω) for various values of he damping parameer δ: Normalised ampliude of he oscillaion of he harmonically forced mechanical oscillaor X / F/ω 8 6 δ/ω =.5 δ/ω =. δ/ω =. δ/ω =. Ω/ω Finally, noe ha for posiive damping (δ > ), he homogeneous soluion x H () decays rapidly wih increasing ime, i.e. x H (). For sufficienly large imes, only he forced moion () persiss. Therefore, he soluion x H () is ofen referred o as he ransien soluion.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES + x H = cos() x H = e -/ ( cos() + sin() ) x() x - 6 8 6 8 = x H = e -/ ( cos() + sin() ) ransien moion x H () x - 6 8 6 8 + = cos() x forced (periodic) moion () - 6 8 6 8 Figure 6: The displacemen of a harmonically-forced, damped mechanical oscillaor comprises he periodic (forced) soluion () and he ransien soluion x H ().