2. Benchmark solution

2.1. Calculation method

The pulsation of the system is noted \({\omega }_{0}\) and is such as \({\omega }_{0}^{2}=\frac{k}{m}\).

Adherence phase

At the moment \(t=0\), the mass \({m}_{1}\) verifies

\({x}_{1}(t)=0\)

\(\dot{{x}_{1}}(t)=0\)

And mass \({m}_{2}\) verifies

\({x}_{2}(t)=0\)

\(\dot{{x}_{2}}(t)=0\).

The fundamental principle of dynamics applied to mass \({m}_{2}\) allows the following equation to be written

\(\ddot{{x}_{2}}+{\omega }_{0}^{2}{x}_{2}=\frac{F}{m}\),

and on mass \({m}_{1}\) the following equation

\({F}_{T}=-k{x}_{2}\)

The general solution for \({x}_{2}\) is in the form:

\({x}_{2}=\tilde{A}\mathrm{cos}({\omega }_{0}t)+\tilde{B}\mathrm{sin}({\omega }_{0}t)+\frac{F}{(m{\omega }_{0}^{2})}\)

where \(\tilde{A}\) and \(\tilde{B}\) are constants.

Taking into account the initial conditions, the displacement of mass \({m}_{2}\) is therefore written

\({x}_{2}=\frac{F}{k}[1-\mathrm{cos}({\omega }_{0}t)]\).

This expression is valid until

\(∥{F}_{T}∥=\mu {F}_{N}\)

In other words, until \({t}_{1}\) verifying the following expression

\(F[1-\mathrm{cos}({\omega }_{0}{t}_{1})]={\mu }_{S}{F}_{N}\)

So the sliding phase starts at \({t}_{1}\) defined by

\({t}_{1}=\frac{1}{{\omega }_{0}}\mathrm{arccos}(1-{\mu }_{S}\frac{{F}_{N}}{F})\)

Sliding phase

At the moment \(t={t}_{1}\), the mass \({m}_{1}\) verifies

\({x}_{1}({t}_{1})=0\)

\(\dot{x}{}_{1}({t}_{1})=0\)

The fundamental principle of dynamics applied to mass \({m}_{1}\) allows us to write the equation

\(\ddot{x}{}_{1}+{\omega }_{0}^{2}{x}_{1}=\frac{-{\mu }_{D}{F}_{N}}{m}+{\omega }_{0}^{2}{x}_{2}\),

And on mass \({m}_{2}\) the equation

\(\ddot{x}{}_{2}+{\omega }_{0}^{2}{x}_{2}=\frac{F}{m}+{\omega }_{0}^{2}{x}_{1}\).

By making the change of variables

\(\{\begin{array}{}X={x}_{1}+{x}_{2}\\ Y={x}_{1}-{x}_{2}\\ {\Omega }_{0}=\sqrt{2}{\omega }_{0}\end{array}\)

The previous system is written

\(\begin{array}{}\ddot{X}=\frac{F-{\mu }_{D}{F}_{N}}{m}\\ \ddot{Y}+{\Omega }_{0}^{2}Y=-\frac{F+{\mu }_{D}{F}_{N}}{m}\end{array}\)

For \(t\ge {t}_{1}\), the general solution of \(X\) is of the form

\(X=\frac{(F-{\mu }_{D}{F}_{N})}{\mathrm{2m}}{(t-{t}_{1})}^{2}+\tilde{C}(t-{t}_{1})+\tilde{D}\).

and the one in \(Y\) is in the form

\(Y=\tilde{E}\mathrm{cos}({\Omega }_{0}(t-{t}_{1}))+\tilde{F}\mathrm{sin}({\Omega }_{0}(t-{t}_{1}))-\frac{F+{\mu }_{D}{F}_{N}}{\mathrm{2k}}\).

Taking into account the initial conditions, the expressions for the constants \(\tilde{C}\), \(\tilde{D}\),, \(\tilde{E}\) and \(\tilde{F}\) are equivalent to:

\(\begin{array}{}\tilde{C}=\dot{x}{}_{2}({t}_{1})\\ \tilde{D}={x}_{2}({t}_{1})\\ \tilde{E}=\frac{F+{\mu }_{D}{F}_{N}}{\mathrm{2k}}-{x}_{2}({t}_{1})\\ \tilde{F}=-\frac{\dot{x}{}_{2}({t}_{1})}{{\Omega }_{0}}\end{array}\)

The moves \({x}_{1}\) and \({x}_{2}\) are deduced from the previous expressions.

Sliding phase with load \(F=0\) from \({t}_{2}\) (arbitrary but greater than \({t}_{1}\) ) until instant \({t}_{3}\) (moment of return to the grip phase)

The fundamental principle of dynamics presented above is still valid. For mass \({m}_{1}\), the verified equation is the same:

\(\ddot{x}{}_{1}+{\omega }_{0}^{2}{x}_{1}=\frac{-({\mu }_{D}{F}_{N})}{m}+{\omega }_{0}^{2}{x}_{2}\),

And for the masses \({m}_{2}\)

\(\ddot{x}{}_{2}+{\omega }_{0}^{2}{x}_{2}={\omega }_{0}^{2}{x}_{1}\)

With the same change of variable as before, the system is written

\(\begin{array}{}\ddot{X}=-\frac{{\mu }_{D}{F}_{N}}{m}\\ \ddot{Y}+{\Omega }_{0}^{2}Y=-\frac{{\mu }_{D}{F}_{N}}{m}\end{array}\)

For \(t>{t}_{2}\), the general solution of \(X\) is of the form

\(X=-\frac{{\mu }_{D}{F}_{N}}{\mathrm{2m}}{(t-{t}_{2})}^{2}+\tilde{G}(t-{t}_{2})+\tilde{H}\)

The general solution of \(Y\) is of the form

\(Y=\tilde{I}\mathrm{cos}({\Omega }_{0}(t-{t}_{2}))+\tilde{J}\mathrm{sin}({\Omega }_{0}(t-{t}_{2}))-\frac{{\mu }_{D}{F}_{N}}{\mathrm{2k}}\)

Taking into account the initial conditions, the expressions for the constants \(\tilde{G}\), \(\tilde{H}\),, \(\tilde{I}\) and \(\tilde{J}\) are equivalent to:

\(\begin{array}{}\tilde{G}=\dot{x}{}_{1}({t}_{2})+\dot{x}{}_{2}({t}_{2})\\ \tilde{H}={x}_{1}({t}_{2})+{x}_{2}({t}_{2})\\ \tilde{J}=\frac{{\mu }_{D}{F}_{N}}{\mathrm{2k}}+{x}_{1}({t}_{2})-{x}_{2}({t}_{2})\\ \tilde{J}=\frac{\dot{x}{}_{1}({t}_{2})-\dot{x}{}_{2}({t}_{2})}{{\Omega }_{0}}\end{array}\)

The moves \({x}_{1}\) and \({x}_{2}\) are deduced from the previous expressions.

2.2. Reference quantities and results

The quantities tested are the kinematics of the masses \({m}_{1}\) and \({m}_{2}\) at different times in the various phases of the modeling. The following transition times are also tested:

• transition from the adhesion phase to the sliding phase (instant \({t}_{1}\));

• transition from the sliding phase to the adhesion phase (instant \({t}_{3}\)).

2.3. Uncertainties about the solution

Exact analytical solution.

2.4. Bibliographical references

1. 5. BOYERE: Modeling of shocks and friction in transient analysis by modal recombination. Code_Aster R5.06.03 reference documentation. September 2009.