FE-regularization of non-smooth vibrations due to friction and impacts

Abstract

Friction and impacts during oscillations lead to discontinuities of the velocity and of the internal forces in the time-domain and to changes in the number of degrees of freedom, Ibrahim (1994). The analytical procedure for the integration of such non-smooth motions is to compute the history dependent separation times and to patch together a sequence of solutions for successive smooth problems, Popp (1998). However, this very accurate procedure has limits even for a relatively low number of generalized coordinates because of the required computational effort. Regularization techniques as usually used with FE allow to avoid the exact computation of all discontinuities by smoothing. But there is a big uncertainty in the choice of the regularization parameters needed for a sufficiently correct description of the oscillations under investigation. Stationary solutions of two forced massspring oscillators are used to calibrate the regularization parameters by comparing analytical results with regularized ones. This allows to compute the self-excitation of a continuous system and to prove the phenomena with known experimental data.A non-smooth discrete system changes its mechanical properties during the course of time. Examples are the possibility of sliding or sticking at contact points or the change from a motion in contact to a free motion or vice versa. Generally there exists a set of possible states which can differ in their degrees of freedom. At the beginning of a motion it is neither known how many possible states will become active in the future, nor the sequence of the active states is predetermined. However, we have to note, that each state represents itself a smooth problem. If X denotes the vector of generalized coordinates, the total solution in the time domain is given in general by a sequence of smooth solutionsLet us assume, the solution X i corresponding to a distinct element of the set of possible states is known by integration up to a certain time t > t i . Then a condition must be given, to determine the end of this state at t iþ1 . This condition is found by checking switching conditions for each t ! t i . Knowing t iþ1 a switching decision allows to choose the new state X iþ1 out of the set of all possible states. The complete information about the mechanical quantities at the end of the state X i is used as initial conditions for X iþ1 . By sequential application the total solution is patched together at the points of discontinuities t i .The spatial discretization of a continuum by the FEmethod leads to equations of motions with a finite number of degrees of freedom. Therefore all dynamic contact problems lead to non-smooth mechanical systems. In principle, analytical solutions can be computed as described above. In practice such an exact solution can only be obtained if the number of all possible states is relatively low, which leads essentially to the restriction to a small number of coordinates and contact points, Vielsack and Kammerer (1999). FE discretizations, however, lead to large number...

Parameter estimation for finite element analyses of stationary oscillations of a vibro-impacting system

“…This method involves introducing a contact stiffness and subsequently choosing a penalty parameter. This procedure can also be applied to oscillating systems where impacts occur during motion [6]. The observed energy loss due to impact can be accounted for by introducing a contact damper which implies the choice of a second penalty parameter.…”

Clapping in delaminated sandwich-beams due to forced oscillations

“…The classical penalty method can be extended by introducing penalty damping in addition to the penalty stiffness. Thus, this procedure requires the choice of two penalty parameters, which have no physically specifiable quantity [7]. Moreover, these parameters are mutually dependent with the time step of integration [10].…”

On the numerical simulation of non-smooth, resonant vibrations of delaminated structures

“…The classical penalty method, however, can be extended by introducing penalty damping in addition to the penalty stiffness. Thus, this procedure requires the choice of two penalty parameters, which essentially have no physically specifiable quantity [6]. Moreover, these parameters are mutually dependent on the time step of integration [3].…”

“…possible contact points (n N elements) M C : actually closed contact points (n C elements) M M : sudden impact with immediate separation (n M elements) M P 0 : beginning of permanent contact (n P 0 elements) M PP : persistent permanent contact (n P P elements) (6) The condition for the occurrence of a contact event g N,k C ≤ 0 can be kept unmodified in relation to eq. (4).…”