By Christian Studer
This booklet is principally fitted to humans either in examine and who're attracted to the numerical simulation of mechanical structures with unilateral contacts and friction. Such platforms are available in a variety of fields of technical program; think of for instance contacts among wheels and floor in motor vehicle dynamics, strolling machines, restrict stops and friction in arbitrary form of mechanisms or contacts among millions of grains in granular mechanics. The e-book focuses generally on inflexible multibody platforms dynamics with unilateral contacts and friction, notwithstanding, the proposed non-smooth modeling procedure is far extra common and in addition well matched for non-rigid applications.
Four major issues are addressed:
- Non-smooth modeling of unilateral contacts and friction through set valued strength legislation,
- Mathematical review of those legislation utilizing basic cone inclusions with linked projective equations,
- Numerical time-stepping integration tools for non-smooth platforms,
- Implementation in a software program code.
The publication closes the distance among hugely really good mathematical literature and engineering software, supplies an exceptional evaluation on good demonstrated equipment in non-smooth mechanics and opens new views for the longer term improvement during this promising box of lively research.
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Extra info for Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics
16) 22 3 Non-Smooth Mechanics Either the relative acceleration γ˙i is zero and the friction element remains sticking with |λi | < ai , or the friction element begins to slide, where the relative acceleration γ˙i points in the opposite direction of the sliding force λi . e. Stribeck friction. 16) is combined with an external force, which depends on the relative velocities γi . This external force has no set-valued character and its value is zero for γi = 0, see Sect. 2 or . e. a unilateral contact with friction.
This criterion can be evaluated for a global choice of ri = r ∀i, see also . Let μ G and μ TJOR be vectors which contain the eigenvalues of G and TJOR , respectively. Note that μ G 0 because G is at least positive semidefinite. 54) which allows for writing the Lipschitz constant as a function of μ G , L = ρ (TJOR ) = max(|1 − r max(μ G )|, |1 − r min(μ G )|). 55) The Lipschitz constant is minimal for 1 − r max(μ G ) = 1 − r min(μ G ) ⇒ r= 2 . 56) Note that if the matrix G is only positive semidefinite, then at least one eigenvalue is equal to zero.
30) j=1 which we call the JORprox scheme. Note that this JORprox scheme corresponds to the Jacobi-relaxation (JOR) scheme  λhν +1 = λhν − ωh m ( ∑ Ghk λkν + ch ), Ghh k=1 h = 1 . . 2 Augmented Lagrangian Approach m ∑ k=1 k=h 37 Ghk <1 Ghh ∀h = 1 . . m. 32) The scalar ωh is the relaxation parameter which must be chosen as 0 < ωh < 2. 31) in order to solve the linear system Gλ + c = 0 and adds a projection to ensure that the new update of the force λ νi +1 is within the set Ci of admissible forces.