Small-scale magnetorheological dampers for vibration mitigation ...

Proofs (Cont’d)
Introduction
Hysteretic models
Identification methodology
Future work
Experimental identification
Summary
Proof.
Previous considerations in the Identification Methodology
Ikhouane and Rodellar, 2005b
Assumes knowledge of the limit cycle → (x(τ), ¯F(τ)) by a variable τ ∈ [0, T]
Only the loading part of the input signal is considered τ ∈ [0, T + ]
The equation of the loading part is
¯w(x) = ψ + σ,n (ϕ+ σ,n [−ψσ,n(ρ(Xmax − X min))] + ρ(x − X min ))
Its derivative is
Proof
Theorem 1 shows that
d¯w(x)
= ρ(1 − ¯w(x))
dx
x = X min → τ = 0 → −ψ 1,1 (ρ(X max − X min ))
x = X max → τ = T + → ψ 1,1 (ρ(X max − X min ))
Return
Arturo Rodríguez Tsouroukdissian
Small-scale MR dampers for vibration mitigation