# How to solve this Laplace integral for an averaging function in an iterative numerical (finite difference) model?

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:

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.

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.