2. Benchmark solution#

2.1. Calculation method used for the reference solution#

For a system without damping, the differential equation to be solved is written as:

\[\]

: label: EQ-None

{begin {array} {} mddot {r} +kr=mumid {F} _ {n}mid\ r (t=0) = {r} _ {0}ge 0ddot {r} +kr=mumid {r} _ {r} (t=0) = {r} (t=0) = {r} _ {0}{0}ge 0ge 0\ dot {r} (r)

We show [bib1] that the solution to the differential equation is written as:

math:

r (t) =frac {mathrm {mu} | {F} _ {n} |} {k} +left ({r} _ {0} -frac {mathrm {mu} | {F} {mu} | {F} _ {mu}} | {f} _ {n} |} |} {k} |} {k}right)mathrm {cos} ({mathrm {omega}}}} _ {0} | {0}} t) `with:math: {mathrm {omega}} _ {0} =sqrt {frac {k} {m}}} `

The amplitude of the extrema, which all \({t}_{n+1}=\frac{n\pi }{{\omega }_{0}}\) originate, obeys the following law of recurrence:

\(r({t}_{n+1})={(-1)}^{n}\left[{r}_{0}-\frac{\mu \mid {F}_{n}\mid }{k}\right]\mathrm{cos}{\omega }_{0}t\)

with \(n=\mathrm{1,2},\dots ,N\) such as \(∣\frac{r({t}_{n+1})}{{r}_{0}}∣<\frac{\mu \mid {F}_{n}\mid }{k{r}_{0}}\)

The movement stops when \(∣\frac{r({t}_{n+1})}{{r}_{0}}∣<\frac{\mu \mid {F}_{n}\mid }{k{r}_{0}}\) is in position \(r({t}_{n+1})\).

2.2. Benchmark results#

Values of the movements in the \(\mathrm{\theta }=45°\) direction for the times of change of sign of speed \(r({t}_{1}),r({t}_{2}),r({t}_{3}),\mathrm{...}\) established above.

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

  1. AXISA - Analysis methods in nonlinear structural dynamics: contact nonlinearities - Course IPSI from May 28 to 30, 1991