The Inerter Concept and Its Application. Malcolm C. Smith Department of Engineering University of Cambridge U.K.

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1 Department of Engineering University of Cambridge U.K. Society of Instrument and Control Engineers (SICE) Annual Conference Fukui, Japan 4 August 2003 Plenary Lecture 1

2 Motivating Example Vehicle Suspension Performance objectives 1. Control vehicle body in the face of variable loads. 2. Minimise roll, pitch (dive and squat). 3. Improve ride quality (comfort). 4. Improve tyre grip (road holding). SICE Annual Conference, Fukui, Japan, 4 August

3 Types of suspensions 1. Passive. 2. Semi-active. 3. Self-levelling. 4. Fully active. Constraints 1. Suspension deflection hard limit. 2. Actuator constraints (e.g. bandwidth). 3. Difficulty of measurement (e.g. absolute ride-height). A Challenging set of Problems for the Designer SICE Annual Conference, Fukui, Japan, 4 August

4 Quarter-car Vehicle Model load disturbances F s m s suspension force u z s Equations of motion: m s z s = F s u, m u m u z u = u + k t (z r z u ). z u road disturbances k t z r SICE Annual Conference, Fukui, Japan, 4 August

5 Invariant Equation The following equation holds independently of u. m s z s + m u z u = F s + k t (z r z u ) This represents behaviour that the suspension designer cannot influence. Consequence: any one of the following disturbance transmission paths determines the other two. sprung mass (z s ) road (z r ) suspension deflection (z s z u ) tyre deflection (z s z u ) J.K. Hedrick and T. Butsuen, Invariant properties of automotive suspensions, Proc. Instn. Mech. Engrs., 204 (1990), pp SICE Annual Conference, Fukui, Japan, 4 August

6 Invariant Points ω 1 = kt m u tyre-hop invariant frequency road (z r ) sprung mass (z s ) 10 0 For any suspension: ẑ s ẑ r (jω 1 )= m u m s Magnitude Frequency (rad/sec) ω 1 50 Phase (degrees) Frequency (rad/sec) ω 1 SICE Annual Conference, Fukui, Japan, 4 August

7 Further Work 1. What is the complete freedom on a given transfer function? [1] 2. What is the minimum number of sensors required to achieve a given behaviour? [1] 3. Are there conservation laws? [1] 4. Can disturbance paths for ride and handling be adjusted independently? [2] [1] M.C. Smith, Achievable dynamic response for automotive active suspension, Vehicle System Dynamics, 24 (1995), pp [2] M.C. Smith and G.W. Walker, Performance limitations and constraints for active and passive suspensions: a mechanical multiport approach, Vehicle System Dynamics, 33 (2000), pp SICE Annual Conference, Fukui, Japan, 4 August

8 Conservation Laws grip transfer function (road tyre deflection) 0dB magnitude Area formula: Area of amplification is equal to the area of attenuation. True for any suspension system (active or passive). rad/sec SICE Annual Conference, Fukui, Japan, 4 August

9 Active Suspension Design (1) Modal Decomposition (2) Decoupling of Ride and Handling c s m s A m u k t k s z s z s z u m s c s s+k s u + K + [1] R.A. Williams, A. Best and I.L. Crawford, Refined Low Frequency Active Suspension, Int. Conf. on Vehicle Ride and Handling, Nov. 1993, Birmingham, Proc. ImechE, , C466/028, pp , [2] K. Hayakawa, K. Matsumoto, M. Yamashita, Y. Suzuki, K. Fujimori, H. Kimura, Robust H Feedback Control of Decoupled Automobile Active Suspension Systems, IEEE Transactions on Automat. Contr., 44 (1999), pp [3] M.C. Smith and F-C. Wang, Controller Parameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control, IEEE Trans. on Contr. Syst. Tech., 10 (2002), pp SICE Annual Conference, Fukui, Japan, 4 August

10 Passive Suspensions (Abstract Approach) Try to understand which vehicle dynamic behaviours are possible and which are not without worrying initially how the behaviour is realised. This is a black-box approach. Classical electrical circuit theory should be applicable. Driving-Point Impedance Z(s) = ˆv(s) î(s) v i Electrical Network i SICE Annual Conference, Fukui, Japan, 4 August

11 Classical Electrical Network Synthesis Definition. Anetworkispassive if for all admissible v, i which are square integrable on (,T], T v(t)i(t) dt 0. Theorem 1. Anetworkispassive if and only if Z(s) ispositive-real, i.e. Z(s) is analytic and Re(Z(s)) 0inRe(s)> 0. Im forbidden Z(jω) Re SICE Annual Conference, Fukui, Japan, 4 August

12 Fundamental Theorem of Electrical Network Synthesis Theorem 2. Brune (1931), Bott-Duffin (1949). Any rational function which is positive-real can be realised as the driving-point impedance of an electrical network consisting of resistors, capacitors and inductors. Positive Real Function Circuit Realisation Classic reference: E.A. Guillemin, Synthesis of Passive Networks, Wiley, SICE Annual Conference, Fukui, Japan, 4 August

13 Electrical-Mechanical Analogies 1. Force-Voltage Analogy. voltage force current velocity Oldest analogy historically, cf. electromotive force. 2. Force-Current Analogy. current force voltage velocity electrical ground mechanical ground Independently proposed by: Darrieus (1929), Hähnle (1932), Firestone (1933). Respects circuit topology, e.g. terminals, through- and across-variables. SICE Annual Conference, Fukui, Japan, 4 August

14 Standard Element Correspondences (Force-Current Analogy) v = Ri resistor damper cv = F v = L di dt inductor spring kv = df dt C dv dt = i capacitor mass m dv dt = F i i v 2 Electrical v 1 F F v 2 Mechanical v 1 What are the terminals of the mass element? SICE Annual Conference, Fukui, Japan, 4 August

15 The Exceptional Nature of the Mass Element Newton s Second Law gives the following network interpretation of the mass element: One terminal is the centre of mass, Other terminal is a fixed point in the inertial frame. Hence, the mass element is analogous to a grounded capacitor. Standard network symbol for the mass element: F v 2 v 1 =0 SICE Annual Conference, Fukui, Japan, 4 August

16 Table of usual correspondences Mechanical Electrical F F i i v 2 v 1 v 2 v 1 spring inductor F v 2 v 1 = 0 mass i v 2 v 1 i capacitor F F i i v 2 v 1 v 2 v 1 damper resistor SICE Annual Conference, Fukui, Japan, 4 August

17 Consequences for network synthesis Two major problems with the use of the mass element for synthesis of black-box mechanical impedences: An electrical circuit with ungrounded capacitors will not have a direct mechanical analogue, Possibility of unreasonably large masses being required. Question Is it possible to construct a physical device such that the relative acceleration between its endpoints is proportional to the applied force? SICE Annual Conference, Fukui, Japan, 4 August

18 One method of realisation rack pinions terminal 2 gear flywheel terminal 1 r 1 = radius of rack pinion γ = radius of gyration of flywheel r 2 = radius of gear wheel m = mass of the flywheel r 3 = radius of flywheel pinion α 1 = γ/r 3 and α 2 = r 2 /r 1 Equation of motion: F =(mα 2 1α 2 2)( v 2 v 1 ) (Assumes mass of gears, housing etc is negligible.) SICE Annual Conference, Fukui, Japan, 4 August

19 The Ideal Inerter We define the Ideal Inerter to be a mechanical one-port device such that the equal and opposite force applied at the nodes is proportional to the relative acceleration between the nodes, i.e. F = b( v 2 v 1 ). We call the constant b the inertance and its units are kilograms. The stored energy in the inerter is equal to 1 2 b(v 2 v 1 ) 2. The ideal inerter can be approximated in the same sense that real springs, dampers, inductors, etc approximate their mathematical ideals. We can assume its mass is small. M.C. Smith, Synthesis of Mechanical Networks: The Inerter, IEEE Trans. on Automat. Contr., 47 (2002), pp SICE Annual Conference, Fukui, Japan, 4 August

20 A new correspondence for network synthesis F v 2 Mechanical F v 1 df dt = k(v 2 v 1 ) Y (s) = k s spring i v 2 Electrical v 1 di dt = 1 L (v 2 v 1 ) i Y (s) = 1 Ls inductor F v 2 F F = b d(v 2 v 1 ) dt Y (s) = bs v 1 v 1 inerter i v 2 i = C d(v 2 v 1 ) dt i Y (s) = Cs capacitor F F Y (s) = c i v 2 v 1 i Y (s) = 1 R v 2 v 1 F = c(v 2 v 1 ) damper i = 1 R (v 2 v 1 ) resistor Y (s) =admittance = 1 impedance SICE Annual Conference, Fukui, Japan, 4 August

21 A New Approach to Vibration Absorption m mω 2 0 y M x k 1 k 1 /ω 2 0 inerter M x k 3 c 1 k 2 c z z Conventional vibration absorber Solution using inerters SICE Annual Conference, Fukui, Japan, 4 August

22 The Inerter applied to Passive Vehicle Suspensions F s m s z s Y (s) m u z u The design of a passive suspension system can be viewed as the search for a suitable positive-real admittance Y (s) to optimise desired performance measures. k t z r SICE Annual Conference, Fukui, Japan, 4 August

23 Traditional Suspension Struts Theorem 3. The driving-point admittance Y (s) of a finite network of springs and dampers only has all its poles and zeros simple and alternating on the negative real axis with a pole being rightmost. Im Re (Y (jω)) Corollary. Forω 0: 20 db Bode-slope( Y (jω) ) 0dB, 90 arg(y (jω)) 0. 0dB arg(y (jω)) +90 ω ω 90 SICE Annual Conference, Fukui, Japan, 4 August

24 Realisations in Foster form Theorem 4. Any admittance comprising an arbitrary interconnection of springs, dampers (and levers) can be realised in the following form: k k 1 k 2... k n c 1 c 2 c n SICE Annual Conference, Fukui, Japan, 4 August

25 Three Candidate Admittances I. One damper: Y 1 (s) =k (T 2s +1) s(t 1 s +1) where T 2 >T 1 > 0andk>0. II. Twodampers: Y 2 (s) =k (T 4s +1)(T 6 s +1) s(t 3 s +1)(T 5 s +1) where T 6 >T 5 >T 4 >T 3 > 0andk>0. III. Same degree as two damper case but general positive real: Y 3 (s) =k a 0s 2 + a 1 s +1 s(d 0 s 2 + d 1 s +1) where d 0, d 1 0andk>0. (Need: β 1 = a 0 d 1 a 1 d 0 0, β 2 := a 0 d 0 0, β 3 := a 1 d 1 0 for positive-realness.) SICE Annual Conference, Fukui, Japan, 4 August

26 Design Comparison m s = 250 kg, m u =35kg,k t = 150 knm 1 Problem: maximise the least damping ratio ζ min among all the system poles subject to a static spring stiffness of k h = 120 knm 1. Results: Y 1 and Y 2 : ζ min = Y 3 : ζ min = zr zs Phase Magnitude ω (rad/sec) t (seconds) ω (rad/sec) Step response: Y 1, Y 2 (solid) and Y 3 (dot-dash). Bode plot of Y 3 (s) showing phase lead. SICE Annual Conference, Fukui, Japan, 4 August

27 Brune Realisation Procedure for Y 3 (s) A continued fraction expansion is obtained: Y 3 (s) = k a 0s 2 + a 1 s +1 s(d 0 s 2 + d 1 s +1) = k s + 1 s 1 + k b 1 c c 4 b 2 s k c 3 k b c 4 where k b = kβ 2, c 3 = kβ 3, c 4 = kβ 4, b 2 = kβ 4 d 0 β 1 β 2 β 4 := β2 2 β 1 β 3. Numerical values in previous design: and b 2 k = 120 knm 1, k b =, b 2 = kg, c 3 =9.8 knsm 1, c 4 =45.2 knsm 1 SICE Annual Conference, Fukui, Japan, 4 August

28 Darlington Synthesis Realisation in Darlington form: a lossless two-port terminated in a single resistor. Y 3 (s) V I 1 I 2 V 1 2 R Ω Lossless network k 2 k 3 λ 1 λ 2 k 1 b c Corresponding mechanical network realisation of Y 3 (s) has one damper and one inerter but employs a lever. SICE Annual Conference, Fukui, Japan, 4 August

29 Simple Suspension Struts k b c k c k c k c b k b (c)layouts1 (d) layout S2 (e)layouts3 (f) layout S4 parallel series M.C. Smith and F-C. Wang, Performance Benefits in Passive Vehicle Suspensions Employing Inerters, 42nd IEEE Conference on Decision and Control, December, 2003, Hawaii, to appear. SICE Annual Conference, Fukui, Japan, 4 August

30 Series Arrangements with Centring Springs k b k b k 1 c k 1 c k k 1 c k k 1 c k k 1 b k k 2 b k 1 b k 2 b (g) layout S5 (h) layout S6 (i) layout S7 (j) layout S8 SICE Annual Conference, Fukui, Japan, 4 August

31 Performance Measures Assume: Road Profile Spectrum = κ n 2 (m 3 /cycle) where κ = m 3 cycle 1 = road roughness parameter (typical British principal road) and V =25ms 1. Define: J 1 = E [ z 2 s(t) ] ride comfort = r.m.s. body vertical acceleration J 3 = E [ (k t (z u z r )) 2] grip = r.m.s. dynamic tyre load SICE Annual Conference, Fukui, Japan, 4 August

32 Optimisation of J 1 (ride comfort) J % J x 10 4 static stiffness x 10 4 static stiffness (a) Optimal J 1 (b) Percentage improvement in J 1 Key: layout S1 (bold), layout S3 (dashed), layout S4 (dot-dashed), layout S5 (dotted) and layout S6 (solid) SICE Annual Conference, Fukui, Japan, 4 August

33 Optimisation of J 3 (dynamic tyre loads) J % J x 10 4 static stiffness x 10 4 static stiffness (a) Optimal J 3 (b) Percentage improvement in J 3 Key: layout S1 (bold), layout S2 (bold), layout S3 (dashed), layout S4 (dot-dashed), layout S5 (dotted) and layout S7 (solid) SICE Annual Conference, Fukui, Japan, 4 August

34 Rack and pinion inerter made at Cambridge University Engineering Department mass 3.5 kg inertance 725 kg stroke 80 mm SICE Annual Conference, Fukui, Japan, 4 August

35 Damper-inerter series arrangement with centring springs SICE Annual Conference, Fukui, Japan, 4 August

36 Laboratory Testing Schenck hydraulic ram Cambridge University Mechanics Laboratory SICE Annual Conference, Fukui, Japan, 4 August

37 Experimental Results Bode plot of admittance magnitude phase (degrees) Hz Hz experimental data ( ) theoretical without inerter damping ( ) theoretical with inerter damping ( ) SICE Annual Conference, Fukui, Japan, 4 August

38 Alternative Inerter Embodiments I screw nut flywheel screw nut gears flywheel SICE Annual Conference, Fukui, Japan, 4 August

39 Alternative Inerter Embodiments II hydraulic T1 t t1 T lever arm rotary See Cambridge University Technical Services Ltd patent PCT/GB02/03056 for further details. SICE Annual Conference, Fukui, Japan, 4 August

40 Conclusion A new mechanical element called the inerter was introduced which is the true network dual of the spring. The inerter allows a complete synthesis theory for passive mechanical networks. Performance advantages for problems in mechanical vibrations and suspension systems have been described. It is expected that future work will continue to explore both the theoretical and practical advantages as well as its applicability in collaboration with interested commercial partners. SICE Annual Conference, Fukui, Japan, 4 August