The Aging Spring
The Aging Spring
The dynamical system governed by the following time-varying ordinary differential equation is a variation of the classic forced mass-spring-dashpot system with mass , dashpot constant , constant stiffness , and forcing term . Aging springs are characterized by a stiffness decaying with time. Here an exponential decay from an initial value at time zero to an asymptotic value is considered.
m+c+k(t)x=acos(2πft)
..
x
x
m
c
k
acos(2πft)
k(t)=+(-)exp(-ϵt)
k
1
k
0
k
1
k
0
k
1
Mathematica can integrate this equation exactly, but the solution in general is a huge expression involving BesselJ and Gamma functions, together with their integrals; it is very difficult to imagine their behavior as time advances.