Dynamics of Non-Smooth Systems
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1 Dynamics of Non-Smooth Systems P Chandramouli Indian Institute of Technology Madras October 5, 2013 Mouli, IIT Madras Non-Smooth System Dynamics 1/39
2 Introduction Filippov Systems Behaviour around hyper-surface Numerical Implementation Case Studies Mouli, IIT Madras Non-Smooth System Dynamics 2/39
3 What is a non-smooth system Physical systems often operate in different modes Transition from one mode to another idealized as instantaneous Time scales of transition usually small Examples Dry friction Switches Switch from slip to stick or vice-versa Systems with impact/backlash Switch from zero to finite stiffness Diodes in electrical circuits Switch from short-circuit when forward biased to open-circuit when reverse biased Mouli, IIT Madras Non-Smooth System Dynamics 3/39
4 Continuous and Smooth Functions A function f (x) is continuous at x X if for all ɛ > 0 there exists δ > 0 such that for y B(x; δ) implies f (x) f (y) ɛ Roughly speaking this means for a single-valued function one can draw the function on a piece of paper without taking our hands off the paper A function f (x) is smooth if it is continuously differentible upto any order in x Mouli, IIT Madras Non-Smooth System Dynamics 4/39
5 Non-smooth Type Continuous vector field but non-smooth mẍ + cẋ + kx = f 0 cos(ωt) f (x) { f (x) = 0, if x < x c k c (x x c ), else Support Derivatives of f (x) discontinuous and hence f (x) non-smooth Mouli, IIT Madras Non-Smooth System Dynamics 5/39
6 Discontinuous Continuous vector field but non-smooth mẍ + kx = f 0 cos(ωt) f (ẋ) µ, ẋ > 0 f (ẋ) [ µ, µ], ẋ = 0 µ, ẋ < 0 Friction f (ẋ) discontinuous Called Filippov type or differential inclusion Mouli, IIT Madras Non-Smooth System Dynamics 6/39
7 Aircraft Engine Vibration Imbalance forces due to fan blade loss Vibration of engine caused Blades start to rub against stator Contact with friction occurs Will the engine survive for rest of flight? Simulation needs to provide the answer Mouli, IIT Madras Non-Smooth System Dynamics 7/39
8 Example Let us look at an example ẋ = f (x) = 1 2 sgn(x) For initial conditions x(0) 0 one can obtain a solution { 3t + C 1, x < 0 x(t) = t + C 2, x > 0 C 1 and C 2 are determined from IC Both these solutions reach x = 0 in finite time If it arrives at x = 0 it cannot leave x = 0 Why? PhasePlane Mouli, IIT Madras Non-Smooth System Dynamics 8/39
9 Example System Dynamics When x < 0 the velocity ẋ > 0 However if x > 0 the velocity ẋ < 0 Hence when the system trajectory reaches x = 0 it implies that ẋ = 0 There is no discontinuity of the variables as a function of time Therefore the system continues to remain at x = 0 This does not agree with our usual notion that sgn(0) = 0 Then ẋ(0) = 1 0 Mouli, IIT Madras Non-Smooth System Dynamics 9/39
10 Set Valued Functions We can deal with the situation if we replace the function concept with a set value function { 1}, x < 0 Sgn(x) = [ 1, 1], x = 0 {1}, x > 0 In the above set [a, b] implies the interval {x R a x b} The set {a, b} implies the elements are a and b The differential equation is then replaced by the differential inclusion ẋ F(x) or ẋ 1 2 Sgn(x) Mouli, IIT Madras Non-Smooth System Dynamics 10/39
11 Solution With the use of set-valued functions x(t) = 0 is a unique global solution to the differential inclusion ẋ 1 2 Sgn(x) With initial condition x(0) = 0 The example used is a one-dimensional one Let us now try to generalize to a multi-dimensional case Mouli, IIT Madras Non-Smooth System Dynamics 11/39
12 Multi-dimensional Case We assume that the discontinuous right hand side is limited to one hyper-surface The space R n is split into two sub-spaces V and V + by a hyper-surface S R n = V S V + The hyper-surface S is defined by a scalar indicator function h(x(t)) When x(t) is on S then h(x) = 0 The normal to the hypersurface is given by n(x) = h(x) Mouli, IIT Madras Non-Smooth System Dynamics 12/39
13 Different Spaces The different spaces can now be written in terms of the indicator function V = {x R n h(x(t)) < 0} S = {x R n h(x(t)) = 0} V + = {x R n h(x(t)) > 0} The vector field f(x, t) is assumed to be smooth and locally continuous provided x / S { f (x, t), x V ẋ(t) = f(x, t) = f + (x, t), x V + The above system of differential equations does not define the dynamics if x S Mouli, IIT Madras Non-Smooth System Dynamics 13/39
14 Multi-dimensional Inclusion We do a set-value extension like the single dimension case f (x, t), x V ẋ(t) F(x, t) = co(f, f + ), x S f + (x, t), x V + The co represents a convex set which means co(f, f +) = {qf + (1 q)f +, q [0, 1]} The solution of Filippov systems are not unique Two different types of behaviour around S will be demonstrated Mouli, IIT Madras Non-Smooth System Dynamics 14/39
15 Example 2 Let us look at the following system x 1 = sgn(x 2 c) x 2 = sgn(x 2 c) The phase plane for this case is shown here In this case the following holds (n T f ) (n T f + ) > 0 for x S Transversal Mouli, IIT Madras Non-Smooth System Dynamics 15/39
16 Example 2 Let us look at the following system x 1 = sgn(x 2 c) x 2 = 2 4 sgn(x 2 c) The phase plane for this case is shown here Aslide In this case the solution is attracted to the hyper-surface and move along the surface It is called a sliding mode This implies n T f = 0 One can express f = αf + + (1 α)f Calculate α = nt f n T (f f + ) Mouli, IIT Madras Non-Smooth System Dynamics 16/39
17 Final remarks Solution to the differential inclusion with x 0 S is locally unique in forward time if Projections of the vector fields are on the same side of S (n T f ) (n T f +) > 0 Projections point to S (n T f ) > 0 and (n T f +) < 0 Mouli, IIT Madras Non-Smooth System Dynamics 17/39
18 Numerical Implementation: Switch Model One could use smoothening of the sgn(x) function Arctan approximation 2 π arctan γx with γ quite large Often leads to stiff differential equations Use of switch model helps solve these systems without encountering stiff equations A tolerance band near the indicator function hyper-surface is made Band Mouli, IIT Madras Non-Smooth System Dynamics 18/39
19 Procedure Check if h(x) < tol If it is so then we are in the band Now compute the projections of the vector fields n T f and n T f + If n T f > 0 and n T f + < 0 we have an attractive sliding mode The opposite of this is the repulsive sliding mode which I have not discussed To ensure convergence from the edge of the band to the middle for the sliding mode we let ḣ = 1 τ h Mouli, IIT Madras Non-Smooth System Dynamics 19/39
20 Calculations Now since ḣ = h x x t one can simplify as ḣ = n T (αf + + (1 α)f ) From this one can solve for α as α = nt f + h τ n T (f f +) Choose the parameter tol sufficiently small If the projections are of the same sign then simply continue integrating Mouli, IIT Madras Non-Smooth System Dynamics 20/39
21 Other Methods Switch model is not suitable if many friction interfaces or contact/friction interfaces are present Event driven integration methods are used Contact/friction is solved using Linear Complementarity Programming (LCP) approach Solution gives the next mode of the trajectory of the system Mouli, IIT Madras Non-Smooth System Dynamics 21/39
22 3 DOF Brake Model We now look at an example problem to apply the Filippov Method Mouli, IIT Madras Non-Smooth System Dynamics 22/39
23 Result velocity displacement Mouli, IIT Madras Non-Smooth System Dynamics 23/39
24 SD Oscillator Realization of the Oscillator Non-dimensional equation is x + 2ζx + x(1 1 x 2 +α 2 ) = f 0 cos Ωτ RestForce Mouli, IIT Madras Non-Smooth System Dynamics 24/39
25 Typical Result Results obtained for α = 0 using switch model and event based time integration Mouli, IIT Madras Non-Smooth System Dynamics 25/39
26 Friction Interface of Turbine Blade Variable normal load model Four states: Stick, two slip states and separation possible mü + c u + ku + f u = F u (t); m v + f v = F v (t) Mouli, IIT Madras Non-Smooth System Dynamics 26/39
27 Nonlinear Forces k u (u u 0 ) + f 0, stick µ(n 0 + k v v), positive slip f u = µ(n 0 + k v v), negative slip 0, separation { n0 + k v v, contact f v = 0, separation Mouli, IIT Madras Non-Smooth System Dynamics 27/39
28 Four States Mouli, IIT Madras Non-Smooth System Dynamics 28/39
29 Response with different Pre-loads Mouli, IIT Madras Non-Smooth System Dynamics 29/39
30 Concluding Remarks I have not talked about stability of periodic solutions here This is done from the fundamental solution matrix Φ(x, t) The eigenvalues of this matrix are the Floquet multipliers More recent work available using shooting methods for Filippov systems We are now looking at using a time-variational method for such systems Faster computations as we seek periodic solutions in time domain Mouli, IIT Madras Non-Smooth System Dynamics 30/39
31 Major References Leine, R. I. and Nijmeijer, H., 2004, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer-Verlag: Berlin Moreau, J. J. and Panagiotopoulos, P. D., 1988, Non-smooth Mechanics and Applications, CISM Courses and Lectures, vol. 302, Springer: Wien. Glocker, Ch., 2001, Set-Valued Force Laws, Dynamics of Non-smooth Systems, Lecture Notes in Applied Mechanics, vol. 1, Springer-Verlag:Berlin. Mouli, IIT Madras Non-Smooth System Dynamics 31/39
32 Switch Types Return Mouli, IIT Madras Non-Smooth System Dynamics 32/39
33 Support with Clearance Back Mouli, IIT Madras Non-Smooth System Dynamics 33/39
34 Support with Clearance Return Mouli, IIT Madras Non-Smooth System Dynamics 34/39
35 Phase Plane Trajectory Back Mouli, IIT Madras Non-Smooth System Dynamics 35/39
36 Transversal Intersection Return Mouli, IIT Madras Non-Smooth System Dynamics 36/39
37 Attractive Sliding Mode Back Mouli, IIT Madras Non-Smooth System Dynamics 37/39
38 Switch Model Band Return Mouli, IIT Madras Non-Smooth System Dynamics 38/39
39 Restoring Force Back Mouli, IIT Madras Non-Smooth System Dynamics 39/39
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