BACKGROUND:
Apparently this question’s answer says this some static systems have memory especially those that hysteresis: Confusion about 'memoryless' meaning
So the word static to me comes from static equilibrium of a system. There are courses in mechanical / civil engineering that study the balance of forces and moments when there is no motion or constant velocity motion, but most of all that the inertial forces are zero i.e. the acceleration is zero (second time rate of change of position). They also go further and study stress and strain in the body in follow on courses that assume static equilibrium.
In many real scenarios, the system is not actually static but in quasi-static equilibrium where for example the loading onto the system is ever so slowly applied that the system’s responds like it is static, where the inertial forces are zero.
One feature of static equilibrium in most mechanical systems is that when the dynamics are turned off then the motion returns to static equilibrium. Think of a spring mass damper system where the weight is balanced by a static spring force, that static position is where the dynamics oscillate about. So a related topic, and to me the exact same, is the Zero State Response (zero initial conditions i.e. zero initial displacement and zero initial velocity), where the spring mass damper system is solely driven by an external input, but once that input dies off the system returns to the IC which is just the static equilibrium state.
So from 3 and 4, while having zero inertial forces (really the second time rate of change of displacement), I have come to believe that systems in a state of quasi-static equilibrium are just systems that are not influenced by previous states.
For example using continuous state space representation, in that quasi-static equilibrium scenario, the state variables like
- state-displacement (or strain state) is equal to the output-displacement
- state-velocity (or time rate of change of strain state) is zero
- output-velocity is not zero nor is it equal to state-velocity as that is zero. Therefore output-velocity is driven solely by the time rate of change of the loading
- there is no dynamics (transient nor steady state) from systems state variables
- once the load is removed the displacement stays at that quasi-static equilibrium position like the Zero State Response like the mass spring damper whose dynamics are turned off and rest at static equilibrium position. But here the system has continuous quasi-static equilibrium positions.
- State Variables for the System in Quasi-Static Equilibrium written in state space notation:
[x;dx/dt] := [strain state; time rate of strain state]
[dx/dt; d2x/dt2] := [time rate of change of strain state, second time rate of change of strain state]
Assign for quasi-static equilibrium:
dx/dt = 0 (this state has to be zero otherwise it influences the time rate of change of strain-output)
d2x/dt2 = 0 (has to be zero for basic static equilibrium)
Outputs:
[y; dy/dt] := [strain-output; time rate of strain-output]
which is solely driven by the input and so is a Relativity Zero State Response at each time increment.