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I am trying to implement a mathematical model for vibrational damping described in this article.

They provide an equation for damping force ($F$) as a function of:

  • a spring constant ($k$)
  • a damping coefficient ($η$)
  • displacement of the point being damped ($x$)
  • velocity of that point ($\dot{x}$)
  • mean displacement and velocity for the point ($x_m$ and $\dot{x}_m$)

I understand the basic principle of the force equation. However, my math is not very advanced, and I don't understand how they calculate the mean displacement and velocity ($x_m$ and $\dot{x}_m$), given by the equation for $z_m$.

The relevant excerpt is as follows:


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The equation for $z_m$ (where $z = x$, $z= \dot{x}$) is what I don't understand. I presume $n$ here is a time constant that allows a relative weighting for older or newer values. Is that correct?

What does the expression $t^{n+1}$ represent? Do they mean time raised to exponent of $(n+1)$? Or something else like time at sample $(n+1)$?

What about the integral? Do they mean a Laplace $s$? What do I do with this in a stepwise function? They explain that this ought to be solved using a stepwise iterative approach which is perfect because I am using stepwise finite difference modeling for the system I want to add this to.

In a finite difference model where $Δt$ is the time step per time sample, $n$ is a given coefficient for mean weighting, and where you have the displacement $x$/$x_m$ and velocity $\dot{x}$/$\dot{x}_m$ at the prior time step, in plain English or simple demonstrative code or line by line math, what would be an iterative solution they are suggesting for $z_m$ (ie. the current $x_m$ or $\dot{x}_m$)?

Thanks for any help. It is very much appreciated.

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The variable of the weighted mean value zm would convey a more expressive power if denoted as zmean. The RHS integral in formula (14) of your reference is a "g(x)-weighted average of f(x) on the interval [0, t]", see http://www.sosmath.com/calculus/integ/integ04/integ04.html. In formula (14), f(s) is z2(s) and g(s) = sn. The integral in (14) is not a Laplace transform, but the extended g(x)-weighted mean value, and s is not a dual variable of the Laplace transform, but a dummy variable of integration.

A variable n in (14) is not an index but the exponent of the power function g(s) = sn, and t is also not indexed, but raised to a power n+1. From the fragment you presented in your question, and if they indeed use Newark's method, it does not follow that the authors seek an iterative solution for zm; rather, they use the weighted means as per formula (14) to find the solution for x, x_dotted.

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