Non-holonomic constraints inducing flutter insability in structures under conservative loadings

Abstract

Non-conservative loads of the follower type are usually believed to be the source of dynamic instabilities such as flutter and divergence. It is shown that these instabilities (including Hopf bifurcation, flutter, divergence, and destabilizing effects connected to dissipation phenomena) can be obtained in structural systems loaded by conservative forces, as a consequence of the application of non-holonomic constraints. These constraints may be realized through a `perfect skate' (or a non-sliding wheel), or, more in general, through the slipless contact between two circular rigid cylinders, one of which is free of rotating about its axis. The motion of the structure produced by these dynamic instabilities may reach a limit cycle, a feature that can be exploited for soft robotics applications, especially for the realization of limbless locomotion.

Figures (14)

• Figure 1: A non-holonomic constraint forbidding relative velocity $\dot{{\bf r}}_C$ of the instantaneous contact point $C$ along direction ${\bf e}(t)$, Eq.(\ref{['numerouno']}). This constraint is obtained through slipless contact between two rigid (massless and circular) cylinders, one of which is free of rotating about its axis while the other is not. Note that the reaction $p(t)$ transmitted between the two cylinders is aligned parallel to ${\bf e}$, so that it provides null work during every motion and the conservativeness of a system is not altered.
• Figure 2: Left: the two rigid circular cylinders (of which only one is free of rotating about its axis) in slipless contact, Fig.\ref{['figotta']}, and connected to a double pendulum, realize two types of non-holonomic constraints, called in the following ' skate' and ' violin bow' constraints, the former (the latter) transmitting a reaction tangential to the structure (a reaction on a fixed line) similar to the force postulated by Ziegler (Reut). Right: the original version of the Ziegler's and Reut's structures, loaded through non-conservative forces.
• Figure 3: Dynamics of a visco-elastic double pendulum subject to a ' skate' non-holonomic constraint (realized with a non-sliding wheel) at one end and to a dead load at the other. When the applied dead force lies within the flutter region, a complex motion is generated (parts a-f), leading the structure to reach a limit cycle. Such behaviour is visible in the trajectory of the structure's end shown in the lower part, exhibiting sharp corners induced by the non-holonomic constraint. Note that instability permits motion to a structure that would be at rest in the trivial configuration as the friction parallel to the wheel axis is assumed to be infinite (non-slip condition).
• Figure 4: An elastic double-pendulum (in its quasi-static trivial equilibrium, upper part, and in a deformed configuration during motion, lower part) subject to a ' skate' non-holonomic constraint at its right end and a dead load $F$ applied on its left end. The mechanical system is conservative because the sum of elastic and kinetic energies as well as the potential of the load $F$ remains constant in time. The non-holonomic constraint at the final end transmits an unknown time-dependent reaction force $p(t)$, which is maintained tangential to the second rod, as in the Ziegler case.
• Figure 5: Two undeformed structures differing only in the non-holonomic constraint applied at the right end. Each structure is discretized as a chain made up of $N$ rigid bars of mass $m_i$ and connected to each other through rotational visco-elastic springs (with viscosity parameter $c_i$ and stiffness $k_i$, $i=1, ..., N$). The first rigid bar on the left ($i=1$) is connected to a rigid block sliding along the $x-$axis. The two non-holonomic constraints are inclined at an angle $\beta_0$ with respect to the last bar. These constraints are a ' skate' (upper part) or a ' violin bow' (lower part) and constrain the velocity to have a null component in the direction orthogonal to the skate (realized for instance with a non-sliding wheel) or parallel to a non-sliding freely rotating cylinder (which is in turn in contact with the ' violin bow'). In the case $\beta_0=0$, the two structures reduce to the non-holonomic counterpart of the non-conservative Ziegler and Reut columns, respectively. Note that the last rigid element of the column reported in the lower part is ' T--shaped', so that the freely-rotating cylinder can continue to transmit a force to the structure during motion.