Work in Progress: Connections Between First-Order and Second-Order Dynamic Systems Lessons in Limit Behavior

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1 Paper ID #3580 Work i Progress: Coectios etwee First-Order ad Secod-Order Dyamic Systems Lessos i Limit ehavior Dr. Vicet C. Pratil, Milwaukee School of Egieerig Dr. Vicet Pratil eared his S, MS, ad PhD degrees i Mechaical Ad Aerospace Egieerig at Corell Uiversity. He has worked as a seior memer of techical staff i the Applied Mechaics ad Materials Modelig Directorates at Sadia Natioal Laoratories i Livermore, Califoria where he was a co-recipiet of the R&D00 Award for developmet of Microstructure-Property Model Software i 000. He has pulished 3 peer-reviewed joural ad coferece papers i the areas of fiite elemet aalysis, crystal plasticity, respose of dry graular materials, fluid power hydraulics, heat treatmet distortio, ad teachig methods for udergraduate mechaics curricula. He has ee a faculty memer at the Milwaukee School of Egieerig sice 000 i the Mechaical Egieerig Departmet where he has taught 0 courses ragig from udergraduate mechaics, dyamics, modelig, simulatio, fiite elemet aalysis ad umerical methods to capstoe desig. I additio to teachig udergraduate egieerig core curriculum courses, Dr. Pratil is curretly co-authorig a ook o Fiite Elemet Simulatio of Case Studies for Udergraduates. Dr. Mark L. Nagurka, Marquette Uiversity MARK NAGURKA, Ph.D. is a Associate Professor of Mechaical ad iomedical Egieerig at Marquette Uiversity. He received his.s. ad M.S. i Mechaical Egieerig ad Applied Mechaics from the U.of Pesylvaia ad a Ph.D. i Mechaical Egieerig from M.I.T. He taught at Caregie Mello efore joiig Marquette Uiversity. His professioal iterests are i the desig of mechaical ad electromechaical systems ad i egieerig educatio. He is a Fellow of the America Society of Mechaical Egieers (ASME). c America Society for Egieerig Educatio, 08

2 Work-i-Progress: Coectios etwee First-Order ad Secod-Order Dyamic Systems - Lessos i Limit ehavior Astract There exists a uique relatioship etwee the atural frequecy ad dampig ratio of a lumped-parameter secod-order dyamic system ad the time costats of equivalet first-order systems. These first-order systems result i the limit of vaishig stiffess or iertia, with the system the capale of storig oly a sigle type of eergy. To emphasize the correspodece of first-order-like ehavior with storage of primarily oe type of system eergy, a pair of two degree-of-freedom systems, oe iertia-domiated ad oe stiffess-domiated, are preseted. Although the goverig ordiary differetial equatios are secod-order, these systems are overdamped oly. I studyig limitig ehaviors, the paper raises the questio of what it meas for a system that ca ever e uderdamped to possess a atural frequecy. The paper shows that expressios for atural frequecy ad dampig ratio ca e explicitly writte i terms of pairs of time costats that arise aturally from the limitig process. The aalysis is preseted i a way that is ameale to udergraduate egieerig studets i courses i system dyamics. Itroductio I udergraduate system dyamics courses, studets are itroduced to lumped-parameter, idealized models of mechaical, electrical, fluid, ad thermal systems that are govered y aalogous ordiary differetial equatios. For simple models, these equatios are either first-order or secod-order i time. For example, i a lumped-parameter model of a mass ad viscous damper i series the positio of the mass is govered y a first-order, liear, ordiary differetial equatio. From this first-order equatio, the time-costat is defied i terms of the mass ad dampig. I a lumped-parameter mass-sprig-damper model of a mechaically traslatig system the positio of the mass is govered y a secod-order, liear, ordiary differetial equatio. From the secod-order equatio, the atural frequecy ad dampig ratio are defied i terms of the mass, stiffess, ad dampig. I the stadard presetatio, there is limited, if ay, coectio made etwee the properties of first-order systems (the time-costat) ad secod-order systems (the atural frequecy ad dampig ratio). First-order ad secod-order systems are geerally taught as separate cases. This paper shows that first-order systems are limitig cases of secod-order systems. The aalysis uderlyig the coectios etwee the properties of these systems is ameale to udergraduate egieerig studets. The idea provides a alterate ad itriguig way to approach core cocepts of system dyamics.

3 Uderdamped systems exhiit oscillatory ehavior, ad expressios for their atural frequecy as typically preseted have uamiguous meaig. However, as systems ecome overdamped, their ehavior is o-oscillatory ad ca e characterized y time costats of firstorder-like systems. This ecomes the case i the limit as secod-order systems ecome domiated y their stiffess or iertia. These limits, derived elow, represet overdamped ehavior for which the meaig of atural frequecy is questioale. The meaig of atural frequecy for secod-order systems that do ot oscillate, e.g., overdamped systems, is ot typically addressed i egieerig textooks [-5]. It is uclear whether there is agreemet i the physics ad mechaics commuities o a precise ad uamiguous defiitio of a system s atural frequecy. I the physics commuity, the topic has ee raised, specifically i terms of what a uamiguous defiitio of atural frequecy would offer i clarifyig how to preset overdamped system ehavior to udergraduates [6]. Some asic cocept questios ca e posed. Do all dyamic systems govered y secodorder ordiary differetial equatios exhiit a atural frequecy or does the term apply oly for udamped ad uderdamped systems that experiece oscillatory ehavior? Does it make sese that a mathematical defiitio of atural frequecy exists without kowig if the system will oscillate? Ca expressios for udamped atural frequecy iclude dissipative terms? This paper presets a series of example prolems i which these questios arise. Models ad ehavior Mass-Sprig-Damper Secod-Order System Cosider the classical lumped-parameter liear model of a mass-sprig-damper system, depicted i Figure, with mass m, viscous dampig, liear stiffess k, ad applied force F(t). Figure. A secod-order mass-sprig-damper system. From Newto s secod law, the equatio of motio i terms of displacemet x(t) is which ca e writte i ormalized form, i terms of the udamped atural frequecy, mx( t) x( t) kx( t) F( t), () x( t) x( t) x( t) F( t), () m k, (3) m

4 ad dampig ratio,. (4) km The mathematical expressio for the atural frequecy is writte solely i terms of parameters related to eergy storage, i.e., kietic eergy (related to m) ad potetial eergy (related to k). Focusig o the atural or free (uforced) respose, the solutio of Eq.() i the asece of exteral excitatio or forcig is give y exp exp x t C t C t, (5) where C ad C are costats determied from the iitial coditios, ad are the roots of the characteristic equatio, ad exp e is the atural expoetial fuctio. For a uderdamped system for which 0, the roots are complex cojugates, where i ad the free respose ca e expressed as,, i, (6) exp cos si x t t A dt dt, (7) where is the damped atural frequecy ad A ad are costats determied d from the iitial coditios. The uderdamped respose is oscillatory with a expoetially decayig evelope. For a overdamped system for which, the roots are distict ad real frequecies,, (8), givig the free respose, x t A t t exp exp (9) or, i terms of the mass, stiffess, ad dampig, 4 4 exp km x t A t exp km t. (0) m m Toward fidig a more geeral defiitio of atural frequecy, the ext two susectios of the paper explore the limitig ehavior of secod-order systems as iertia or stiffess properties domiate.

5 Iertia-Domiated Secod-Order System As the stiffess of the mass-sprig-damper system of Figure ecomes smaller, the system ecomes iertia-domiated. As the stiffess teds toward zero, the respose ecomes overdamped ad the solutio is give y Eqs.(9) or (0). I the limit as k 0, the expoet i the first term of Eq.(0) teds towards zero (ad oly a costat term survives) ad the expoet i the secod term simplifies, t t lim xt Aexp0t exp t A exp A exp k0 m m /, () m where. Takig the limit of Eq.() with acceleratio ormalized, k lim x( t) x( t) x( t) x( t) x( t) v( t) v( t) k0 m m, () m where v( t) x( t) is the velocity, or takig the limit of Eq.(), Although y defiitio. (3) lim x( t) x( t) x( t) x( t) x( t) v( t) v( t) k0 m, (4) it is oly i the limit of vaishig stiffess that lim m lim. (5) k0 k0 I summary, the secod-order mass-sprig-damper system with vaishig stiffess ecomes a first-order system with time costat. I the limit as the stiffess teds toward zero, the iverse of two times the product of the secod-order system parameters, ad, i.e.,, reduces to the first-order system parameter,. This coectio etwee parameters of first ad secod-order systems may ot e ituitively ovious to a udergraduate studet ad it is ot geerally preseted i textooks. Stiffess-Domiated Secod-Order System As the mass of the mass-sprig-damper system of Figure ecomes smaller, the system ecomes stiffess-domiated. As the mass teds toward zero, the respose ecomes overdamped ad the solutio is give y Eqs.(9) or (0). I the limit as m 0, the secod term of Eq.(0) teds expoetially towards zero ad oly the first term survives,

6 4km lim x t lim A exp t, (6) m0 m0 m where the expoet teds toward 0 0 ecessitatig applicatio of L Hôpital s rule, d 4km 4km lim xt lim Aexp t lim Aexp dm t m0 m0 m m0 d m dm 4km 4kt lim exp t t A t Aexp Aexp m0 / k, (7) where. k Takig the limit of Eq.() with positio ormalized, or takig the limit of Eq.(), m lim x( t) x( t) x( t) x( t) x( t) x( t) x( t) m0 k k, (8) k lim x( t) x( t) x( t) x( t) x( t) m 0. (9) Although y defiitio it is oly i the limit of vaishig mass that k, (0) m0 m0 lim lim. () k I summary, the secod-order mass-sprig-damper system with vaishig mass ecomes a first-order system with time costat. I the limit as the mass teds toward zero the term, which is a fuctio of the two secod-order system parameters, ad, reduces to the first-order system parameter,. Agai, this coectio etwee parameters of first ad secod-order systems may ot e ituitively ovious to a udergraduate studet ad it is geerally ot preseted i textooks.

7 Geeral Secod-Order System Expressed i Terms of Time Costats The goverig differetial equatio, Eq.(), ca e expressed i terms of first-order time costats defied aove. With ad m, Eq.() ca e writte as k x( t) x( t) x( t) F( t). () k Comparig to the form of Eq.(), the atural frequecy is ad the dampig ratio is (3). (4) Eqs.(3) ad (4) hold for oth uder- ad overdamped secod order systems ad are itriguig. The products ad ratios of time costats are directly related to the two characteristic properties of secod-order systems. I particular, the atural frequecy is the iverse of the square root of the products, i.e., the iverse of the geometric mea of the time costats. The dampig ratio is the square root of the ratio of the time costats divided y two. The authors have ot see these equatios i system dyamics textooks. For the iertia-domiated case, the limit of vaishig stiffess k 0 correspods to. Rewritig Eq.() as ad takig the limit gives x( t) x( t) x( t) F( t), (5) x( t) x( t) F( t), (6) which is a first-order differetial equatio i terms of velocity, v, v( t) v( t) F( t). (7) Aalogously, for the stiffess-domiated case, the limit of vaishig iertia m 0 correspods to the limit of vaishig. Takig the limit 0, Eq.() ecomes x( t) x( t) F( t). (8) k

8 Eqs.(6) ad (8) are the equatios of the first-order systems that are the limitig cases of the secod-order system descried y Eq.(). Secod-Order Systems with Sigle Type of Eergy Storage Two example prolems are preseted ext i which atural frequecy expressios cotai dissipative terms. They follow the form of Eq.(3) except the dissipatio terms do ot cacel. oth prolems ivolve two degree-of-freedom systems that iclude two types of system elemets: oe type stores (kietic or potetial) eergy ad oe type dissipates eergy. It is kow that sigle degree-of-freedom systems that store oly oe type of eergy are govered y firstorder differetial equatios. These systems caot oscillate. Secod-Order System Storig Kietic Eergy A two degree-of-freedom mechaical system that stores oly kietic eergy is the series mass-damper system show i Figure. Figure. A secod-order dual mass-damper system. This system with masses m ad m, viscous dampers ad, ad applied force F(t) ca e viewed as two cascaded first-order systems. The equatio of motio for velocity v(t) is v( t) v( t) v( t) F( t) F( t). (9) m m m m m mm m Comparig to the form of Eq.(), the atural frequecy is. (30) mm Although the dimesios of this expressio are those of frequecy, the meaig of atural frequecy for this system is ot ovious. The system is icapale of oscillatig; without sprigs there ca e o trasfer of kietic eergy to potetial eergy. Although the equatio of motio is secod order ad a mathematical expressio for atural frequecy exists, the masses ca ever oscillate. The existece of a secod-order goverig differetial equatio does ot mea oscillatory ehavior is possile as implied y the word frequecy. I additio, Eq.(30) cotais dissipatio parameters, ad. If represets the udamped atural frequecy, the would e zero. It is possile to fid a expressio for the dampig ratio i terms of system parameters. Agai, comparig Eq.(9) to the form of Eq.(),

9 ad solvig for the dampig ratio ζ, m m m, (3) m m m m m m. (3) m Itroducig the o-dimesioal property ratios ad, the dampig ratio ζ ca m e expressed as or. (33) From the coditio 0 for fixed, the miimum value of ζ occurs at (34) ad is give y. (35) mi Sice 0 the mi, provig the respose caot e uderdamped. From the coditio 0 for fixed, the miimum value of ζ occurs at (36) ad is give y. (37) mi m Sice 0 the mi, agai provig the respose caot e uderdamped. m A three-dimesioal plot of dampig ratio vs. mass ratio ad dampig coefficiet ratio is give i Figure 3. It cofirms that values less tha oe do ot exist. The symmetry of the plot reflects the iterchageaility of ad i Eq.(33).

10 Figure 3. Dampig ratio vs. property ratios (mass ratio ad dampig coefficiet ratio) i a secod-order dual mass-damper system. m Itroducig time costats ad writte as m, Eq.(30) for the atural frequecy ca e which matches Eq.(3), ad from Eq.(9),, (38) or givig expressios for the dampig ratio, or (39), (40) which may e recast i time costat ratio form as ( ) ( ). (4) or Agai, ca e writte explicitly as a ratio of time costats. Compared to Eq.(4), the dampig ratio expressios of Eq.(4) are parameter ratio weighted sums of (square roots of) time costat ratios. The miimum values for ζ are give y Eqs.(35) ad (37).

11 Secod-Order System Storig Electrostatic Potetial Eergy A secod-order system that stores oly electrostatic eergy is the dual series RC electrical circuit, show i Figure 5, with resistaces R ad R, capacitaces C ad C, ad voltage source, es(t). Figure 5. A secod-order dual RC circuit The differetial equatio for the voltage across capacitor C, e(t), is e( t) e( t) e( t) es( t). (4) RC RC RC RC RC RC RC Comparig to the form of Eq.(), the atural frequecy is. (43) R C R C Agai, a expressio for atural frequecy exists for a system that is icapale of a oscillatory respose. It cotais dissipative parameters, R ad R. I this case, if the resistaces vaish, Eq.(43) predicts ifiite atural frequecy. It is also possile to fid a expressio for the dampig ratio, R C R C R C R C R C R C. (44) Itroducig the o-dimesioal property ratios R R C C ad, Eq.(44) ca e writte as or. (45) From the coditio 0 for fixed, the miimum value of ζ occurs at (46) ad is give y

12 mi. (47) R Sice 0 the mi, provig the respose caot e uderdamped. R A three-dimesioal plot of the dampig ratio as a fuctio of capacitace ratio ad resistace ratio is show i Figure 4. It cofirms that there are o values less tha oe. (I cotrast to Figure 3, the plot is ot symmetric; ad are ot iterchageale i Eq.(45).) I summary, this sectio of the paper has preseted two example prolems that are modeled as secod-order systems that are always overdamped. I these examples the expressios for atural frequecy deped explicitly o dissipatio terms. I the followig sectio a geeral secodorder example with two types of eergy storage is cosidered. The expressio for atural frequecy i this prolem depeds explicitly o dissipatio terms i ratio form. Figure 4. Dampig ratio vs. property ratios (resistace ratio ad capacitace ratio) i a secod-order dual RC circuit Secod-Order System with Two Types of Eergy Storage A physically motivated model is a motor that turs a flywheel via a log, elastic shaft supported y a pair of earigs, as show i Figure 5. The motor shaft rotates at agular velocity ΩM(t) ad the flywheel at agular velocity ΩF(t). The motor is modeled as a torque source, T(t), the log shaft as a torsioal sprig with stiffess, K, the earigs as viscous torsioal dampers with dampig ad, ad the flywheel as a iertia, J. This system cotais two types of eergy storage (potetial eergy i the shaft ad kietic eergy i the flywheel) ad may exhiit oscillatory ehavior.

13 Figure 5. Motor-Shaft-Flywheel System The equatio of motio for the flywheel agular velocity, ΩF(t), is : K K K F ( t) F ( t) F ( t) T ( t) J J J Comparig to the form of Eq.(), the atural frequecy is. (48) K J. (49) It is atypical i most udergraduate system dyamics courses to ecouter expressios for atural frequecy that deped o dissipative system parameters, eve appearig i ratio form. Some textooks preset a defiitio of atural frequecy as the frequecy at which uforced systems oscillate i the asece of dissipatio. This example shows that the atural frequecy is a fuctio of a ratio of dampig terms. I the previous examples, the atural frequecy expressios were fuctios of dissipatio terms (, i Eq.(30), R, R i Eq.(43)). It is also possile to fid a expressio for the dampig ratio. Agai, comparig to the form of Eq.(), J K, (50) ad solvig for the dampig ratio ζ, where. JK Itroducig the time costats ad K JK (5) J, the differetial equatio ca e writte, This differetial equatio is idetical to oe goverig lood flow rate i a 4-elemet Widkessel model for arterial lood flow [7].

14 F ( t) F ( t) F ( t) T ( t) givig the atural frequecy, (5), (53) ad, (54) which ca e solved for the dampig ratio,. (55) Whe expressed i terms of time costats, the limits of stiffess-domiated ad iertiadomiated systems are straightforward ad give rise to first-order differetial equatios. Eq.(5) ca e writte equivaletly as, F ( t) F ( t) F ( t) T ( t). (56) I the stiffess-domiated regime ( 0 ), from Eq.(56), F( t) F( t) T ( t). (57) I the iertia-domiated regime ( ), from Eq.(5), where ( t) ( t). F F F( t) F( t) 0, (58)

15 Discussio Limit ehavior The paper has show that there is a cotiuous limit of secod-order ehavior that exhiits first-order ehavior whe the dampig ratio is sufficietly large. I the cotiuous limit of uouded dampig ratio, there exists a uique relatioship etwee the atural frequecy ad dampig ratio of the secod-order system ad the time costat of a equivalet first-order system. For mechaical systems, a uouded dampig ratio correspods to either vaishig stiffess or vaishig iertia, with the resultig system capale of storig oly a sigle type of eergy, either kietic or potetial eergy. The aalysis is ameale to udergraduate egieerig studets i courses i system dyamics. Oe author (VP) has offered the exercise of limitig ehavior of iertia-domiated ad stiffess-domiated secod-order systems (for extra credit) to studets for over a decade. The goal is for studets to fid the first-order-like time costats preseted here. We have ot see this presetatio i textooks that cover dyamics, system dyamics, ad modelig [-5], although it appears, i part, i a thorough discussio y Rowell [8] i course otes at MIT. Meaig of Natural Frequecy Despite the ofte-accepted uderstadig that all secod-order systems possess a atural frequecy, oly secod-order systems with two idepedet ad differet eergy storage elemets ca exhiit resposes with a atural frequecy. A example is the mass-sprig-damper system for which kietic eergy is stored y the mass ad potetial eergy is stored y the sprig. Secod-order systems cotaiig oly a sigle type of eergy storage are persistetly overdamped. Although govered y secod-order equatios that have mathematical expressios for atural frequecy, o oscillatory ehavior is possile. What is the meaig of atural frequecy i this case? Karopp ad Fisher [] ote that for this case the motio ca e viewed as harmoic motio i which the dampig is so heavy that the motio ever completes a half cycle. ut, if there is o retur motio, the it is ot cyclic ad there is o frequecy. The explaatio of the atural frequecy as the frequecy at which a uforced secod-order system oscillates i the asece of dissipatio is ot cosistet with the systems preseted here. I the examples dissipative terms appear i expressios for atural frequecy. There remais the questio of a uamiguous defiitio of atural frequecy that specifies what, if ay, explicit role dissipative parameters may play. Clarity of the defiitio of atural frequecy seems warrated. We put this questio of a appropriate defiitio to the pedagogical commuity ad welcome your iput. We ivite coferece participats ad readers of this mauscript to sed the authors their iterpretatios ad defiitios. We propose collectig these ad presetig them at ASEE 09. The authors ackowledge that the exercise is challegig ad requires some mathematical rigor. A former MSOE udergraduate mechaical egieerig studet, Elise Stroach, who is curretly pursuig doctoral studies at MIT, successfully solved for the limits. Most studets do ot take o the challege.

16 Coclusios The paper explores limitig ehavior of lumped-parameter, secod-order systems ad shows how expressios for secod-order parameters reduce to first-order time costats i the limits. Expressios for a atural frequecy appear to exist for ay secod-order system, whether or ot there are two types of eergy storage elemets for which oscillatory ehavior ca occur. Examples i the paper show that they ca cotai dissipative properties. A geeralized, uamiguous defiitio of atural frequecy ameale to udergraduates is sought with iput from readers ad the scholarly commuity at large. Refereces [] Close, C.M. ad Frederick, D.K., Modelig ad Aalysis of Dyamic Systems, Wiley, 00. [] Esfadiari, R. ad Lu,., Modelig ad Aalysis of Dyamic Systems, d ed., CRC Press, 04. [3] Kelly, S.G., System Dyamics ad Respose, Thomso Caada, 007. [4] Ogata, K., System Dyamics, 4 th ed., Pearso, 003. [5] Palm, W., System Dyamics, 3 rd ed., McGraw-Hill, 03. [6] The Physics Forum, [7] Kerer, D.R., Solvig Widkessel Models with MLA, 0. [8] Rowell, D. Review of First- ad Secod-Order System Respose, MIT, 004. [9] earoya, H., Nagurka, M.L. ad Ha, S., Mechaical Viratio: Aalysis, Ucertaities, ad Cotrol, 4th ed., CRC Press, 08. [0] Rao, S., Mechaical Viratios, 6 th ed., Pearso, 06. [] Karopp,.H. ad Fisher, F.E., O the Viratios of Overdamped Systems, The Frakli Istitute, Vol. 37, No. 4, 990, pp