c Dr. Md. Zahurul Haq (BUET) System Dynamics ME 361 (2018) 6 / 36

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1 Basic Syse Models Response of Measuring Syses, Syse Dynaics Dr. Md. Zahurul Haq Professor Deparen of Mechanical Engineering Bangladesh Universiy of Engineering & Technology (BUET) Dhaa-1, Bangladesh hp://eacher.bue.ac.bd/zahurul/ ME 361: Insruenaion and Measureen Modelling is he process of represening he behaviour of a syse by a collecion of aheaical equaions & logics. I is coprehensively uilized o sudy he response of any syse. Response of a syse is a easure of is fideliy o is purpose. Siulaion is he process of solving he odel and i is perfored using copuer(s). Equaions are used o describe he relaionship beween he inpu and oupu of a syse. Inpu = Governing Equaions = Oupu Analogy approach is widely used o sudy syse response. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 1 / 36 Mechanical Syse Eleens Mechanical Syse Eleens: (a) Spring c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 2 / 36 Mechanical Syse Eleens... (b) Dashpo/Daper T848 F x Spring = Spring Force (ension or copression), Displaceen (exension or copression), Spring consan. The bigger he value of he greaer he forces required o srech or copress he spring and so he greaer he siffness. T849 ẋ() b Daper Daper = bv = b dx d F Force opposing he oion a velociy v, b Daping coefficien. Larger he value of b he greaer he daping force a a paricular velociy. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 3 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 4 / 36

2 Mechanical Syse Eleens Mechanical Syse Eleens... (c) Mass T85 ẋ() ẍ() = a = dv d = d2 x d 2 ẍ() F Force required o cause acceleraion, a, Mass of he eleen ha is disribued hroughou soe volue. However, in any cases, i is assued o be concenraed a a poin. Spring sores energy when sreched, and he energy is released when i springs bac o is original sae. E = 1 f 2 2 Energy is sored in ass when i is oving wih a velociy, v, he energy being referred o as ineic energy. E = 1 2 v2 Dashpo dissipaes energy as hea raher han soring i, and dissipaed power, P depends on he velociy, v. P = bv 2 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 5 / 36 General Syse Modelling Sep and Haronic Inpu f () T854 { a sep funcion: = A for > { a haronic funcion: = A Òω for > A Inpu, (a) Sep inpu f () A T= 2 / (b) Haronic inpu Sep and haronic inpus are widely used o analyse syse response. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 7 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 6 / 36 General Syse Modelling Modelling of a General Measureen Syse The response of a easureen syse, i.e., oupu,, when subjeced o an inpu forcing funcion,, ay be expressed by a linear ordinary differenial equaion wih consan coefficiens of he for: d n x a n d n + a d n 1 x n 1 d n 1 + +a d 2 x 2 h order d 2 + a dx {}}{ 1 d + a x = } {{ } } 1 s order {{ } 2 nd order Inpu quaniy iposed on he syse, Oupu or he response of he syse, a s Physical syse paraeers, assued consans. Order of a syse is designaed by he order of he D.E. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 8 / 36

3 General Syse Modelling General Syse Modelling Zeroh Order Syse a x = := = 1 a o Saic sensiiviy or gain: he scaling facor beween he inpu and he oupu. For any-order syse, i always has he sae physical inerpreaion, i.e., he aoun of oupu per uni inpu when he inpu is saic and under such condiion all he derivaive ers of general equaion are zero. No equilibriu seeing force is presen. Oupu follows he inpu wihou disorion or ie lag. Syse requires no addiional dynaic consideraions. Represens ideal dynaic perforance. Exaple: Poenioeer, ideal spring ec. f () T855 A Inpu, Response, (a) Sep inpu f () A (b) Haronic inpu Zero-order insruen s response for sep and haronic inpus (for =.75). c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 9 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 1 / 36 Consider a herocouple iniially a eperaure, T is suddenly exposed o an environen a T. a 1 dx d + a x = := τ dx d + x = 1/a o saic sensiiviy, τ a 1 /a o ie-consan. a o dissipaion (elecric or heral resisance). a 1 sorage (elecric or heral capaciance). Exaple: Theroeer, capacior ec. The ie consan, τ has he diension of ie, while he saic sensiiviy has he diension of oupu divided by inpu. When τ : he effec of he derivaive ers becoes negligible and he governing equaion approaches o ha of a zero-order syse. T864 Q in = ha[t T()] = C dt() d h convecive hea ransfer coefficien, A hea ransfer surface area, ass of ercury + bulb, C specific hea of ercury + bulb. := τ dt() d Tie consan, τ C ha Saic sensiiviy, = 1. C h A = τ Insruens wih sall τ good dynaic response. + T() = T c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 11 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 12 / 36

4 Response of a 1 s Order Syse: Sep Inpu...cond Non-diensional response, M() = xo x x o = 1. ÜÔ( /τ) x = x o, F = : = ; F() = A : > τ dx d + x = F() = = (x o A) ÜÔ( /τ) + }{{}} A {{} ransien response seady sae response x( ) = A = x = Seady Sae Response Error, e = x = (x x o )e /τ Non-diensional Error, e /(x x o ) = e /τ c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 13 / 36 T856 M () %.9.8 s %.5 Iniial slope = r c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 14 / 36 /...cond. Tie Consan, τ - ie required o coplee 63.2% of he process. Rise Tie, r - ie required o achieve response fro 1% o 9% of final value. For firs order syse, r = 2.31τ.11τ = 2.2τ. Seling Tie, s - he ie for he response o reach, and say wihin 2% of is final value. For firs order syse, s = 4τ. Process is assued o be copleed when 5τ. Faser response is associaed wih shorer τ. Response of a 1 s Order Syse: Haronic Inpu If he governing equaion for firs-order syse is solved for haronic inpu and x = =, he soluion is: A = ωτ 1+(ωτ) 2 ÜÔ( /τ) + }{{} ransien response 1 Ò(ω +φ) 1+(ωτ) 2 }{{} seady sae response where, φ Ø Ò 1 ( ωτ) phase lag. Hence, ie delay,, is relaed o phase lag as: = φ ω For ωτ >> 1, response is aenuaed and ie/phase is lagged fro inpu, and for ωτ << 1, he ransien response becoes very sall and response follows he inpu wih sall aenuaion and ie/phase lag. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 15 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 16 / 36

5 Ideal response (wihou aenuaion and phase lag) is obained when he syse ie consan, τ is significanly saller han he forcing eleen period, T 2π/ω. As, he seady-sae soluion: s = A 1+(ωτ) 2 Ò(ω +φ) = G af() φ Hence, G a / 1+(ωτ) 2 seady-sae gain. The aenuaed seady-sae response is also a sine wave wih a frequency equal o he inpu signal frequency, ω, and i lags behind he inpu by phase angle, φ. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 17 / 36 Theroeer (τ = 1s), iniially a o C (ω =.25, T = 8π, G a =.37). Teperaure ( o C) T = sin(.25) = 1 s T = 8 s 25 seady-sae esabled = G a Elapsed ie, (s) f = x/g a c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 18 / 36 Teperaure ( o C) T858 2 = sin(.25) f x( = 1s) x( = 5s) x( = 5s) Elapsed ie, (s)...cond. Uni τ [s] τ/t φ [deg] [s] G a Response o haronic inpu is a sae frequency, wih a phase shif (ie lag), and reduced apliude. The larger he ie consan, he greaer he phase lag & apliude decrease (aenuaion). Effecs of ie consan on syse response. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 19 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 2 / 36

6 Transfer Funcion Transfer Funcion Transfer Funcion (TF) Transfer funcion of a linear syse, G(s), is defined as he raio of he Laplace ransfor (LT) of he oupu variable, X(s) L{}, o he LT of he inpu variable, F(s) L{}, wih all he iniial condiions are assued o be zero. Hence, G(s) X(s) F(s) Tie-doain Differenial Equaion Inpu, Oupu, L{ } L 1 { } Frequency-doain Algebraic Equaion Inpu, F(s) Oupu, X(s) The Laplace operaor, s σ+jω, is a coplex variable. For seady-sae sinusoidal inpu, σ =, and syse response can be evaluaed by seing s = jω. Apliude gain, G a (ω) G(jω) Phase lag, φ(ω) G(j ω) T861 Calculus Muliplicaion Division Exponeniaion Algebra Addiion Subracion Muliplicaion F(s) G(s) X(s) := = G a φ Transfer Funcion Transfer Funcion c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 21 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 22 / 36 Iporan Laplace Transfor Pairs Bode Diagra F(s) δ() 1 n Sep funcion, A n s n+1 A/s e a 1 s+a ω Òω s 2 +ω 2 s Ó ω s 2 +ω 2 f () sf(s) f() f () s 2 F(s) sf() f () ω n 1 ζ 2 e ζωn Òω n 1 ζ 2, ζ < 1 ω 2 n s 2 +2ζω ns+ω 2 n Bode diagra is a pair of graphs which consiss of wo plos: 1 Logarihic gain, L(ω) 2 ÐÓ 1 G a (ω) vs. ÐÓ 1(ω), and 2 Phase angle, φ(ω) vs. ÐÓ 1(ω) The verical scale of he apliude Bode diagra is in decibels (db), where a non-diensional frequency paraeer such as frequency raio, (ω/ω n ), is ofen used on he horizonal axis. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 23 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 24 / 36

7 Transfer Funcion Transfer Funcion TF of a 1 s Order Syse Bode Diagra of a 1 s Order Syse d n x d n = s n X(s), s = jω G a = G(jω) = τ dx d + x = F() F() = F(s). τsx(s)+x(s) = F(s) F(s) == τs+1 jωτ+1 = φ = G(jω) = Ø Ò 1 ( ωτ) 1+(ωτ) 2 == X(s) L 2 log 1 G a (db) (deg) db h order 1 s order h order 1 s order T862 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 25 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 26 / 36 Second Order Syse Second Order Syse T851 ω n ẋ() ẍ() Spring (a) b Daper spring consan b daping consan bẋ() (b) forcing funcion ass ẍ() f x c dx d = d2 x d 2 = d2 x d 2 + cdx d + x = f undaped naural frequency (rad/s) c c 2 criical daping coefficien ζ c/c c daping raio c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 27 / 36 Second Order Syse TF of a 2 nd Order Syse where, d a 2 x dx 2 + a d 2 1 d + a ox = := 1 d 2 x + 2 ζ dx ω 2 n d 2 ω n d + x = 1/a o saic sensiiviy, ω n ao a 2 undaped naural frequency, ζ a1 2 a oa 2 diensionless daping raio. G(s) = G(jω) = 1/ = ω2 n / 1 ω 2 s 2 +2 ζ n ωn s+1 s 2 +2ζω ns+ω 2 n [ 1/ ( ) ] 2 [ ] 1 ω +j 2ζ ω ωn ωn G a = G(jω) = φ = G(jω) = Ø Ò 1 1/ [ ( ) ] 2 2 ( ) 2 1 ω +4ζ ωn 2 ω ωn 2ζ ω [ ωn ( ) ] 2 1 ω ωn c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 28 / 36

8 Second Order Syse Second Order Syse Response of a 2 nd Order Syse: Sep Inpu x ()/A T =,.1,.4,.7, 1., = = c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 29 / 36 n...cond. T86 x () r decay envelope, exp(- )/(1-2 n ) 1/2 =.1 axiu overshoo 2% T d Second-order under-daped response specificaions. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 3 / 36 n...cond. Second Order Syse Seady sae posiion is obained afer a long period of ie. Under-daped syse (ζ < 1): response overshoos he seady-sae value iniially, & hen evenually decays o he seady-sae value. The saller he value of ζ, he larger he overshoo. The ransien response oscillaes abou he seady-value and occurs wih a period,t d, given by: T d 2π ω d : ω d ω n 1 ζ 2 Criical daping (ζ = 1): an exponenial rise occurs o approach he seady-sae value wihou any overshoo. Over-daped (ζ > 1): he syse approaches he seady-sae value wihou overshoo, bu a a slower rae. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 31 / 36 Second Order Syse Response of a 2 nd Order Syse: Haronic Inpu L 2 log 1 G a (db) (deg) T =.1,.4,.7, 1., = = = = c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 32 / 36 n =.1 = 2.

9 Second Order Syse Measuring Syse Response...cond. Measuring Syse Response Syse has a good lineariy for low daping raios and up o a frequency raio of.3 since he apliude gain is very nearly uniy (G a 1). For large values of ζ, he apliude is reduced subsanially. The phase shif characerisics are a srong funcion of frequency raio for all frequencies. As a general rule of hub, he choice of ζ =.77 is opial since i resuls in he bes cobinaion of apliude lineariy and phase lineariy over he wides range of frequencies. Response is a easure of a syse s fideliy o purpose. 1 Apliude response: A linear response o various inpu apliudes wihin range. Beyond he linear range, he syse is said o be overdriven. 2 Frequency response: is he abiliy of he syse o rea all frequencies he sae so ha he gain apliude reains he sae over he frequency range desired. 3 Phase response: is iporan for coplex wavefors. Lac of good response ay resul in severe disorion. 4 Delay, Rise ie, Slew rae: Delay or rise ie is required o respond o an inpu quaniy. Slew rae is he axiu applicable rae of change. c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 33 / 36 Measuring Syse Response Spring-ass-daper syse & analogous RLC circui c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 34 / 36 References Measuring Syse Response ẋ() ẍ() forcing funcion (N) ass (g) spring consan (N/) b daping consan (N.s/) x displaceen () ẋ dx/d velociy (/s) Spring b Daper v() v() applied volage (V) L inducance (H) C capaciance (F) R resisance (Ω) q charge (C) i dq/d curren (A) R Resisor L d2 q d 2 +R dq d + 1 C q = v() L Inducor L,R b, 1 C,v f i C Capacior 1 Haq, M.Z., "Measureen: Syse, Uncerainy and Response". In Applied Measureen Syses, 1-22, Edied by Md. Zahurul Haq, InTech, Figliola, R.S. & Beasley, D.E., Theory & Design for Mechanical Measureens, J. Wiley & Sons, Inc, Halan, J.P., Experienal Mehods for Engineers, McGraw-Hill, Inc, Becwih, T.G., Marangoni, R.D. & Lienhard, J.H., Mechanical Measureens, Addison Wesley, Inc, 26. T852 d2 x d +b dx 2 d +x = = 1 d 2 x ωn 2 d +2 ζ dx 2 ω n d +x = κ c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 35 / 36 c Dr. Md. Zahurul Haq (BUET) Syse Dynaics ME 361 (218) 36 / 36