Damped memory
The initial-condition contribution decays exponentially.
u_memory=e⁻ᵃᵗu₀
Damping, Forcing, and Equilibrium
How systems forget the past and reach balance
Damping removes memory of the initial condition, while sustained forcing builds a long-term response.
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The essential idea: for positive damping, the balance between −au and b produces the equilibrium u*=b/a.
One Concept · One Example
Damping, Forcing, and Equilibrium
Presented by Charlotte Moser · Coming soon
What to notice
- Larger a shortens the memory time.
- Constant forcing alone produces linear growth.
- Together, damping and forcing can balance.
The mathematical structure
Forced equilibrium
The forcing contribution grows toward the steady state.
u_force=(b/a)(1−e⁻ᵃᵗ)
Watch balance emerge
Vary damping, forcing, and the initial condition. The total trajectory is decomposed into its two physical contributions.
total u(t)initial memoryforced response
Damping controls how quickly the past is forgotten; forcing controls the equilibrium that remains.