Benchmark solution ===================== Calculation method ------------------ The pulsation of the system is noted :math:`{\omega }_{0}` and is such as :math:`{\omega }_{0}^{2}=\frac{k}{m}`. **Adherence phase** At the moment :math:`t=0`, the mass :math:`{m}_{1}` verifies :math:`{x}_{1}(t)=0` :math:`\dot{{x}_{1}}(t)=0` And mass :math:`{m}_{2}` verifies :math:`{x}_{2}(t)=0` :math:`\dot{{x}_{2}}(t)=0`. The fundamental principle of dynamics applied to mass :math:`{m}_{2}` allows the following equation to be written :math:`\ddot{{x}_{2}}+{\omega }_{0}^{2}{x}_{2}=\frac{F}{m}`, and on mass :math:`{m}_{1}` the following equation :math:`{F}_{T}=-k{x}_{2}` The general solution for :math:`{x}_{2}` is in the form: :math:`{x}_{2}=\tilde{A}\mathrm{cos}({\omega }_{0}t)+\tilde{B}\mathrm{sin}({\omega }_{0}t)+\frac{F}{(m{\omega }_{0}^{2})}` where :math:`\tilde{A}` and :math:`\tilde{B}` are constants. Taking into account the initial conditions, the displacement of mass :math:`{m}_{2}` is therefore written :math:`{x}_{2}=\frac{F}{k}[1-\mathrm{cos}({\omega }_{0}t)]`. This expression is valid until :math:`∥{F}_{T}∥=\mu {F}_{N}` In other words, until :math:`{t}_{1}` verifying the following expression :math:`F[1-\mathrm{cos}({\omega }_{0}{t}_{1})]={\mu }_{S}{F}_{N}` So the sliding phase starts at :math:`{t}_{1}` defined by :math:`{t}_{1}=\frac{1}{{\omega }_{0}}\mathrm{arccos}(1-{\mu }_{S}\frac{{F}_{N}}{F})` **Sliding phase** At the moment :math:`t={t}_{1}`, the mass :math:`{m}_{1}` verifies :math:`{x}_{1}({t}_{1})=0` :math:`\dot{x}{}_{1}({t}_{1})=0` The fundamental principle of dynamics applied to mass :math:`{m}_{1}` allows us to write the equation :math:`\ddot{x}{}_{1}+{\omega }_{0}^{2}{x}_{1}=\frac{-{\mu }_{D}{F}_{N}}{m}+{\omega }_{0}^{2}{x}_{2}`, And on mass :math:`{m}_{2}` the equation :math:`\ddot{x}{}_{2}+{\omega }_{0}^{2}{x}_{2}=\frac{F}{m}+{\omega }_{0}^{2}{x}_{1}`. By making the change of variables :math:`\{\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 :math:`\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 :math:`t\ge {t}_{1}`, the general solution of :math:`X` is of the form :math:`X=\frac{(F-{\mu }_{D}{F}_{N})}{\mathrm{2m}}{(t-{t}_{1})}^{2}+\tilde{C}(t-{t}_{1})+\tilde{D}`. and the one in :math:`Y` is in the form :math:`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 :math:`\tilde{C}`, :math:`\tilde{D}`,, :math:`\tilde{E}` and :math:`\tilde{F}` are equivalent to: :math:`\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 :math:`{x}_{1}` and :math:`{x}_{2}` are deduced from the previous expressions. **Sliding phase with load** :math:`F=0` **from** :math:`{t}_{2}` **(arbitrary but greater than** :math:`{t}_{1}` **) until instant** :math:`{t}_{3}` **(moment of return to the grip phase)** The fundamental principle of dynamics presented above is still valid. For mass :math:`{m}_{1}`, the verified equation is the same: :math:`\ddot{x}{}_{1}+{\omega }_{0}^{2}{x}_{1}=\frac{-({\mu }_{D}{F}_{N})}{m}+{\omega }_{0}^{2}{x}_{2}`, And for the masses :math:`{m}_{2}` :math:`\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 :math:`\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 :math:`t>{t}_{2}`, the general solution of :math:`X` is of the form :math:`X=-\frac{{\mu }_{D}{F}_{N}}{\mathrm{2m}}{(t-{t}_{2})}^{2}+\tilde{G}(t-{t}_{2})+\tilde{H}` The general solution of :math:`Y` is of the form :math:`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 :math:`\tilde{G}`, :math:`\tilde{H}`,, :math:`\tilde{I}` and :math:`\tilde{J}` are equivalent to: :math:`\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 :math:`{x}_{1}` and :math:`{x}_{2}` are deduced from the previous expressions. Reference quantities and results ----------------------------------- The quantities tested are the kinematics of the masses :math:`{m}_{1}` and :math:`{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 :math:`{t}_{1}`); * transition from the sliding phase to the adhesion phase (instant :math:`{t}_{3}`). Uncertainties about the solution ---------------------------- Exact analytical solution. Bibliographical references --------------------------- .. _RefNumPara__37574621: 1. E. BOYERE: Modeling of shocks and friction in transient analysis by modal recombination. *Code_Aster* R5.06.03 reference documentation. September 2009.