Passive Network Synthesis Revisited
Malcolm C. Smith Department of Engineering University of Cambridge U.K.
Mathematical Theory of Networks and Systems (MTNS) 17th International Symposium Kyoto International Conference Hall Kyoto, Japan, July 24-28, 2006
Semi-Plenary Lecture
1 Passive Network Synthesis Revisited Malcolm C. Smith
Outline of Talk 1. Motivating example (vehicle suspension). 2. A new mechanical element. 3. Positive-real functions and Brune synthesis. 4. Bott-Duffin method. 5. Darlington synthesis. 6. Minimum reactance synthesis. 7. Synthesis of resistive n-ports. 8. Vehicle suspension. 9. Synthesis with restricted complexity. 10. Motorcycle steering instabilities.
MTNS 2006, Kyoto, 24 July, 2006 2 Passive Network Synthesis Revisited Malcolm C. Smith
Motivating Example – Vehicle Suspension
Performance objectives
1. Control vehicle body in the face of variable loads. 2. Insulate effect of road undulations (ride). 3. Minimise roll, pitch under braking, acceleration and cornering (handling).
MTNS 2006, Kyoto, 24 July, 2006 3 Passive Network Synthesis Revisited Malcolm C. Smith
Quarter-car Vehicle Model (conventional suspension)
PSfrag replacemenload ts disturbances
ms
spring damper
mu
kt tyre road disturbances
MTNS 2006, Kyoto, 24 July, 2006 4 Passive Network Synthesis Revisited Malcolm C. Smith
The Most General Passive Vehicle Suspension
ms
Replace the spring and damper with a Z(s) general positive-real impedance Z(s). PSfrag replacements
mu But is Z(s) physically realisable?
kt zr
MTNS 2006, Kyoto, 24 July, 2006 5 Passive Network Synthesis Revisited Malcolm C. Smith
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), H¨ahnle (1932), Firestone (1933). Respects circuit “topology”, e.g. terminals, through- and across-variables.
MTNS 2006, Kyoto, 24 July, 2006 6 Passive Network Synthesis Revisited Malcolm C. Smith
Standard Element Correspondences (Force-Current Analogy)
v = Ri resistor damper cv = F ↔ v = L di inductor spring kv = dF dt ↔ dt C dv = i capacitor mass m dv = F dt ↔ dt
i i PSfrag replacements v2 v1 Electrical
F F
v2 Mechanical v1
What are the terminals of the mass element?
MTNS 2006, Kyoto, 24 July, 2006 7 Passive Network Synthesis Revisited Malcolm C. Smith
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 F for the mass elemenPSfragt: replacements
v2 v1 = 0
MTNS 2006, Kyoto, 24 July, 2006 8 PSfrag replacements
Passive Network Synthesis Revisited Malcolm C. Smith
Table of usual correspondences
Mechanical Electrical
F F i i v2 v1 v2 v1 spring inductor
F i i • v2 v1 v2 v1 = 0 mass capacitor
F F i i v2 v1 v2 v1 damper resistor
MTNS 2006, Kyoto, 24 July, 2006 9 Passive Network Synthesis Revisited Malcolm C. Smith
Consequences for network synthesis
Two major problems with the use of the mass element for synthesis of “black-box” mechanical impedances: 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 pro- portional to the applied force?
MTNS 2006, Kyoto, 24 July, 2006 10 Passive Network Synthesis Revisited Malcolm C. Smith
One method of realisation rack pinions
PSfrag replacements
terminal 2 gear flywheel terminal 1
Suppose the flywheel of mass m rotates by α radians per meter of relative displacement between the terminals. Then:
F = (mα2) (v˙ v˙ ) 2 − 1 (Assumes mass of gears, housing etc is negligible.)
MTNS 2006, Kyoto, 24 July, 2006 11 Passive Network Synthesis Revisited Malcolm C. Smith
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˙ v˙ ). 2 − 1 We call the constant b the inertance and its units are kilograms.
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. 1648–1662.
MTNS 2006, Kyoto, 24 July, 2006 12 PSfrag replacements
Passive Network Synthesis Revisited Malcolm C. Smith
A new correspondence for network synthesis
Mechanical Electrical
F F Y (s) = k i i Y (s) = 1 s v2 v1 Ls v2 v1 dF di 1 = k(v v ) spring = (v2 v1) inductor dt 2 − 1 dt L − F F Y (s) = bs i i Y (s) = Cs • • v2 v1 v2 v1 d(v2 v1) d(v2 v1) capacitor F = b dt− inerter i = C dt−
F F Y (s) = c i i Y (s) = 1 v2 v1 R v2 v1 1 F = c(v2 v1) damper i = (v2 v1) resistor − R −
1 Y (s) = admittance = impedance
MTNS 2006, Kyoto, 24 July, 2006 13 Passive Network Synthesis Revisited Malcolm C. Smith
Rack and pinion inerter made at Cambridge University Engineering Department
mass 3.5 kg ≈ inertance 725 kg ≈ stroke 80 mm ≈
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Damper-inerter series arrangement with centring springs
PSfrag replacements
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Alternative Realisation of the Inerter
PSfrag replacements
screw nut flywheel
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Ballscrew inerter made at Cambridge University Engineering Department Mass 1 kg, Inertance (adjustable) = 60–180 kg ≈
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Electrical equivalent of quarter car model
Fs PSfrag replacements PSfrag replacements m s 1 kt−
Y (s) + z˙r Y (s) − mu zs mu ms Fs zu kt
zr
Force Current Y (s) = Admittance = = Velocity Voltage
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Positive-real functions
Definition. A function Z(s) is defined to be positive-real if one of the following two equivalent conditions is satisfied:
1. Z(s) is analytic and Z(s) + Z(s)∗ 0 in Re(s) > 0. ≥ 2. Z(s) is analytic in Re(s) > 0, Z(jω) + Z(jω)∗ 0 for all ω at which ≥ Z(jω) is finite, and any poles of Z(s) on the imaginary axis or at infinity are simple and have a positive residue.
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Passivity Defined
Definition. A network is passive if for all admissible v, i which are square integrable on ( , T ], −∞ T v(t)i(t) dt 0. ≥ Z−∞ Proposition. Consider a one-port electrical network for which the impedance Z(s) exists and is real-rational. The network is passive if and only if Z(s) is positive-real.
R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.
MTNS 2006, Kyoto, 24 July, 2006 20 Passive Network Synthesis Revisited Malcolm C. Smith
O. Brune showed that any (ratio- nal) positive-real function could be realised as the impedance or admittance of a network comprising resistors, capacitors, inductors and transformers. (1931)
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Minimum functions A minimum function Z(s) is a positive-real function with no poles or zeros on jR and with the real part of Z(jω) equal to 0 at one or more ∪ {∞} frequencies. PSfrag replacements ReZ(jω)
0 ω1 ω2 ω
MTNS 2006, Kyoto, 24 July, 2006 22 Passive Network Synthesis Revisited Malcolm C. Smith
Foster preamble for a positive-real Z(s)
Removal of poles/zeros on jR . e.g. ∪ {∞} s2 + s + 1 1 = s + s + 1 s + 1 ↑ lossless
↓ 1 s2 + 1 2s − = + 1 s2 + 2s + 1 s2 + 1 Can always reduce a positive-real Z(s) to a minimum function.
MTNS 2006, Kyoto, 24 July, 2006 23 Passive Network Synthesis Revisited Malcolm C. Smith
The Brune cycle
Let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0). Write L = X /ω and Z (s) = Z(s) L s. 1 1 1 1 − 1 Case 1. (L1 < 0)
L1 < 0 PSfrag replacements
Z(s) Z1(s) (negative inductor!)
Z1(s) is positive-real. Let Y1(s) = 1/Z1(s). Therefore, we can write
2K1s Y2(s) = Y1(s) 2 2 − s + ω1
for some K1 > 0.
MTNS 2006, Kyoto, 24 July, 2006 24 Passive Network Synthesis Revisited Malcolm C. Smith
PSfragThe Brreplacemenune cyclets (cont.) Then: L1 < 0
L2 > 0 Z(s) Z2(s)
C2 > 0
2 where L2 = 1/2K1, C2 = 2K1/ω1 and Z2 = 1/Y2. Straightforward calculation shows that
Z2(s) = sL3+ Z3(s)
prop↑ er where L = L /(1 + 2K L ). Since Z (s) is positive-real, L > 0 and Z (s) 3 − 1 1 1 2 3 3 is positive-real.
MTNS 2006, Kyoto, 24 July, 2006 25 PSfragPassive Netreplacemenwork Synthesis tsRevisited Malcolm C. Smith
The Brune cycle (cont.)
Then: L1 < 0 L3 > 0
L2 > 0 Z(s) Z3(s)
C2 > 0
PSfrag replacements To remove negative inductor: L L M 1 3
Lp = L1 + L2 Lp Ls L2 Ls = L2 + L3
M = L2
2 Some algebra shows that: L , L > 0 and M = 1 (unity coupling p s LpLs coefficient).
MTNS 2006, Kyoto, 24 July, 2006 26 Passive Network Synthesis Revisited Malcolm C. Smith
PSfrag replacements The Brune cycle (cont.) Realisation for completed cycle: M
Lp Ls Z(s) Z3(s)
C2
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The Brune cycle (cont.) Case 2. (L > 0). As before Z (s) = Z(s) L s 1 1 − 1 PSfrag replacements L1 > 0 (no need for negative inductor!) Z(s) Z1(s)
Problem: Z1(s) is not positive-real! Let’s press on and hope for the best!!
As before let Y1 = 1/Z1 and write
2K1s Y2(s) = Y1(s) 2 2 . − s + ω1
Despite the fact that Y1 is not positive-real we can show that K1 > 0.
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PSfragThe Brreplacemenune cyclets (cont.) Hence: L1 > 0
L2 > 0 Z(s) Z2(s)
C2 > 0
But still Z2(s) is not positive-real. Again we can check that
Z2(s) = sL3+ Z3(s)
prop↑ er where L = L /(1 + 2K L ). 3 − 1 1 1 This time L3 < 0 and Z3(s) is positive-real.
MTNS 2006, Kyoto, 24 July, 2006 29 PSfragPassive Netreplacemenwork Synthesis tsRevisited Malcolm C. Smith
The Brune cycle (cont.)
So: L1 > 0 L3 < 0
L2 > 0 Z(s) Z3(s)
PSfrag replacements C2 > 0
As before we can transform to: M
Lp Ls Z(s) Z3(s)
C2
2 where L , L > 0 and M = 1 (unity coupling coefficient). p s LpLs
MTNS 2006, Kyoto, 24 July, 2006 30 Passive Network Synthesis Revisited Malcolm C. Smith
R. Bott and R.J. Duffin showed that transformers were un- necessary in the synthesis of positive-real functions. (1949)
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Richards’s transformation Theorem. If Z(s) is positive-real then kZ(s) sZ(k) R(s) = − kZ(k) sZ(s) − is positive-real for any k > 0. Proof. Z(s) Z(k) Z(s) is p.r. Y (s) = − is b.r. and Y (k) = 0 ⇒ Z(s) + Z(k) k + s Y 0(s) = Y (s) is b.r. ⇒ k s − 1 + Y 0(s) Z0(s) = is p.r. ⇒ 1 Y (s) − 0 R(s) = Z0(s) after simplification.
MTNS 2006, Kyoto, 24 July, 2006 32 Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.) Idea: use Richards’s transformation to eliminate transformers from Brune cycle.
As before, let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0).
Write L1 = X1/ω1.
Case 1. (L1 > 0)
Since Z(s) is a minimum function we can always find a k s.t. L1 = Z(k)/k. Therefore: kZ(s) sZ(k) R(s) = − kZ(k) sZ(s) − has a zero at s = jω1.
MTNS 2006, Kyoto, 24 July, 2006 33 Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.) We now write: kZ(k)R(s) + Z(k)s Z(s) = k + sR(s) kZ(k)R(s) Z(k)s = + k + sR(s) k + sR(s) 1 1 = + . 1 + s k R(s) Z(k)R(s) kZ(k) Z(k)s + Z(k)
MTNS 2006, Kyoto, 24 July, 2006 34 Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.)
1 1 Z(s) = + 1 + s k R(s) Z(k)R(s) kZ(k) Z(k)s + Z(k)
kZ(k) Z(k)R(s) PSfrag replacements Z(s)
Z(k) Z(k) k R(s)
MTNS 2006, Kyoto, 24 July, 2006 35 Passive Network Synthesis Revisited Malcolm C. Smith
Bott-Duffin construction (cont.) 1 s 1 We can write: Z(k)R(s) = const s2+ω2 + R (s) etc. × 1 1
R1
Z(s)
PSfrag replacements
R2
MTNS 2006, Kyoto, 24 July, 2006 36 Passive Network Synthesis Revisited Malcolm C. Smith
Example — restricted degree
Proposition. Consider the real-rational function
2 a0s + a1s + 1 Yb(s) = k 2 s(d0s + d1s + 1) where d , d 0 and k > 0. Then Y (s) is positive real if only if the following 0 1 ≥ b three inequalities hold:
β := a d a d 0, 1 0 1 − 1 0 ≥ β := a d 0, 2 0 − 0 ≥ β := a d 0. 3 1 − 1 ≥
MTNS 2006, Kyoto, 24 July, 2006 37 Passive Network Synthesis Revisited Malcolm C. Smith
Brune Realisation Procedure for Yb(s)
Foster preamble always sufficient to complete the realisation if β1, β2 > 0. (No Brune or Bott-Duffin cycle is required).
A continued fraction expansion is obtained:
2 a0s + a1s + 1 kb Yb(s) = k 2 s(d0s + d1s + 1) k 1 = + s s 1 + k k PSfrag1 replacements b c + c4 3 1 1 + c4 b2s c3
kβ2 kβ4 kβ4 b where kb = , c3 = kβ3, c4 = , b2 = and 2 d0 β1 β2 2 β4 := β β1β3. 2 −
MTNS 2006, Kyoto, 24 July, 2006 38 Passive Network SynthesisPSfragRevisitedreplacements Malcolm C. Smith
Darlington Synthesis
I1 I2 Realisation in Darlington form: V1 V2 Z1(s) Lossless a lossless two-port terminated network R Ω in a single resistor.
For a lossless two-port with impedance:
Z11 Z12 Z = Z12 Z22 we find 1 2 R− (Z11Z22 Z12)/Z11 + 1 Z1(s) = Z11 1 − . R− Z22 + 1
MTNS 2006, Kyoto, 24 July, 2006 39 Passive Network Synthesis Revisited Malcolm C. Smith
Writing m1 + n1 n1 m1/n1 + 1 Z1 = = , m2 + n2 m2 n2/m2 + 1 where m1, m2 are polynomials of even powers of s and n1, n2 are polynomials of odd powers of s, suggests the identification:
n1 n2 √n1n2 m1m2 Z11 = , Z22 = R , Z12 = √R − . m2 m2 m2 Augmentation factors are necessary to ensure a rational square root.
Once Z(s) has been found, we then write: s Z(s) = sC + C + 1 s2 + α2 2 · · · where C1 and C2 are non-negative definite constant matrices.
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Darlington Synthesis (cont.) Each term in the sum is realised in the form of a T-circuit and a series connection of all the elementary two-ports is then made:
PSfrag replacements PSfrag replacements X (s) X1(s) X2(s) 1 Network 1 1: n X2(s) X (s) 3 I = 0 1: n X (s) 3 Ideal Network 1 Network 2 Network 2 I = 0 Ideal
MTNS 2006, Kyoto, 24 July, 2006 41 Passive Network Synthesis Revisited Malcolm C. Smith PSfrag replacements PSfrag replacements
k2
k3 Electrical and mechanical realisations of the admittance Yb(s) λ1 1 λ2 k− H 2 1: ρ k1 − b 1 k k3 k3−c H 2 1 k− H 2 λ2 1 λ1 k− H R1 Ω 3 1: ρ −1 b c 1 k1− Hk1 k1− H b F b F R1 Ω
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Minimum reactance synthesis
Let L = = C = = 1. If 1 · · · 1 · · ·
M11 M12 M = sL1 M21 M22 is the hybrid matrix of X, i.e. Nondynamic PSfrag replacements v i Z(s) Network 1 = M 1 , X i2 v2 1 sC1 then
1 Z(s) = M M (sI+M )− M . 11− 12 22 21
D.C. Youla and P. Tissi, “N-Port Synthesis via Reactance Extraction, Part I”, IEEE International Convention Record, 183–205, 1966.
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Minimum reactance synthesis
1 Conversely, if we can find a state-space realisation Z(s) = C(sI A)− B + D − such that the constant matrix
D C M = − B A − has the properties
M + M 0 0, ≥ diag I, Σ M = Mdiag I, Σ { } { } where Σ is a diagonal matrix with diagonal entries +1 or 1. Then M is the − hybrid matrix of the nondynamic network terminated with inductors or capacitors, which realises Z(s). A construction is possible using the positive-real lemma and matrix factorisations.
B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.
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Synthesis of resistive n-ports Let R be a symmetric n n matrix. × A necessary and sufficient condition for R to be realisable as the driving-point impedance of a network comprising resistors and transformers only is that it is non-negative definite. No necessary and sufficient condition is known in the case that transformers are not available. A general necessary condition is known: that the matrix is paramount.1 A matrix is defined to be the paramount if each principal minor of the matrix is not less than the absolute value of any minor built from the same rows. It is also known that paramountcy is sufficient for the case of n 3.2 ≤ 1I. Cederbaum, “Conditions for the Impedance and Admittance Matrices of n-ports without Ideal Transformers”, IEE Monograph No. 276R, 245–251, 1958. 2P. Slepian and L. Weinberg, ”Synthesis applications of paramount and dominant matrices”, Proc. National Electron. Conf., vol. 14, Chicago, Illinois, Oct. 611-630, 1958.
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Simple Suspension Struts
PSfrag replacementsPSfrag replacements PSfrag replacemenPSfragts replacements c k1 c b k k k k k b b b b b k k c k c b b k2
k1 k1 k1
k2 k2 k2 Layout S1 Layout S2 Layout S3 Layout S4
parallel series
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Performance Measures Assume: 2 3 Road Profile Spectrum = κ n − (m /cycle) | | 7 3 1 where κ = 5 10− m cycle− = road roughness parameter. Define: ×
2 J1 = E z¨s (t) ride comfort = r.m.s. body vertical acceleration
MTNS 2006, Kyoto, 24 July, 2006 47 Passive Network Synthesis Revisited Malcolm C. Smith
Optimisation of J1 (ride comfort)
2.6 18
2.4 16
2.2 14
2 12
1.8 1 10 J 1 1.6
J 8
PSfrag replacements 1.4 PSfrag replacements % 6 1.2
4 1
0.8 J1 2
0.6 0 % J 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 static stiffness static stiffness
(a) Optimal J1 (b) Percentage improvement in J1
Key: layout S1 (bold), layout S2 (dashed), layout S3 (dot-dashed), and layout S4 (solid).
M.C. Smith and F-C. Wang, 2004, Performance Benefits in Passive Vehicle Sus- pensions Employing Inerters, Vehicle System Dynamics, 42, 235–257.
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F
Fs Passive Network Synthesis Revisited Malcolm C. Smith K(s) Control synthesis formulation zr zs
zu Fs PSfrag replacements mu
kt ms zs ms
F ks w z ks K(s) G(s) F
mu F z˙s z˙u zu − K(s) kt
zr
Ride comfort: w = zr, z = z˙s
Performance measure: J1 = const. Tzˆ szˆ 2 × k r→ s k
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Bilinear Matrix Inequality (BMI) formulation
1 1 Let K(s) = Ck(sI Ak)− Bk + Dk and Tzˆ szˆ = Ccl(sI Acl)− Bcl. − r→ s − Theorem. There exists a positive real controller K(s) such that
Tzˆ szˆ 2 < ν and Acl is stable, if and only if the following problem is r→ s k k 2 feasible for some Xcl > 0, Xk > 0, Q, ν and Ak, Bk, Ck, Dk of compatible dimensions:
T T A Xcl + XclAcl XclBcl Xcl C cl < 0, cl > 0, tr(Q) < ν2, BT X I C Q cl cl − cl T T A Xk + XkAk XkBk C k − k < 0. BT X C DT D k k − k − k − k
C. Papageorgiou and M.C. Smith, 2006, Positive real synthesis using matrix inequalities for mechanical networks: application to vehicle suspension, IEEE Trans. on Contr. Syst. Tech., 14, 423–435.
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A special problem What class of positive-real functions Z(s) can be realised using one damper, one inerter, any number of springs and no transformers?
PSfrag replacements
c
Z(s) X
b
Leads to the question: when can X be realised as a network of springs?
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Theorem. Let (R R R2) s3 + R s2 + R s + 1 Y (s) = 2 3 − 6 3 2 , (1) s(det R s3 + (R R R2) s2 + (R R R2) s + R ) 1 3 − 5 1 2 − 4 1
R1 R4 R5 where R := R4 R2 R6 is non-negative definite. R R R 5 6 3 A positive-real function Y (s) can be realised as the driving-point admittance of a network comprising one damper, one inerter, any number of springs and no transformers if and only if Y (s) can be written in the form of (1) and there exists an invertible diagonal matrix D = diag 1, x, y such that DRD is { } paramount. An explicit set of inequalities can be found which are necessary and sufficient for the existence of x and y.
M.Z.Q. Chen and M.C. Smith, Mechanical networks comprising one damper and one inerter, in preparation.
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Collaboration with Imperial College
Application to motorcycle stability.
At high speed motorcycles can experience significant steering instabilities. Observe: Paul Orritt at the 1999 Manx Grand Prix
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Weave and Wobble oscillations Steering dampers improve wobble (6–9 Hz) and worsen weave (2–4 Hz). Simulations show that steering inerters have, roughly, the opposite effect to the damper. Root-loci (with speed the varied parameter):
Steering damper root-locus Steering inerter root-locus 60 6 60 6 wobble wobble
50 50
40 40 Axis Axis
30 30 Imaginary Imaginary
20 weave 20 weave
10 10
PSfrag replacements PSfrag replacements 0 0 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 −15 −10 −5 0 5 Real Axis Real Axis
Can the advantages be combined?
S. Evangelou, D.J.N. Limebeer, R.S. Sharp and M.C. Smith, 2006, Steering compensation for high- performance motorcycles, Transactions of ASME, J. of Applied Mechanics, 73, to appear.
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Solution — a steering compensator ... consisting of a network of dampers, inerters and springs. Needs to behave like an inerter at weave frequencies and like a damper at wobble frequencies.
PSfrag replacements PSfrag replacements damper damper inerter real real imaginary inerter imaginary
forbidden forbidden
Prototype designed by N.E. Houghton and manufactured in the Cambridge University Engineering Department.
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Conclusion
A new mechanical element called the “inerter” was introduced which is • the true network dual of the spring. The inerter allows classical electrical network synthesis to be mapped • exactly onto mechanical networks. Applications of the inerter: vehicle suspension, motorcycle steering and • vibration absorption. Economy of realisation is an important problem for mechanical network • synthesis. The problem of minimal realisation of positive-real functions remains • unsolved.
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Acknowledgements
Research students Michael Chen, Frank Scheibe Design Engineer Neil Houghton Technicians Alistair Ross, Barry Puddifoot, John Beavis Collaborations Fu-Cheng Wang (National Taiwan University) Christos Papageorgiou (University of Cyprus) David Limebeer, Simos Evangelou, Robin Sharp (Imperial College)
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