Active Damping of Flexure Mechanisms | Modelling, Design and Control for Large Deflections
Bram Seinhorst is a PhD Student in the department Precision Engineering. (Co)Promotors are prof.dr.ir. D.M. Brouwer; prof.dr.ir. W.B.J. Hakvoort and dr.ir. M. Nijenhuis from the faculty Engineering Technology.
The high repeatability of flexure mechanisms is crucial for precision mechatronics. Traditionally, flexure mechanisms are designed with low moving mass and high support stiffness. This assures that the performance limiting parasitic resonances are high, enabling high feedback bandwidth, good disturbance suppression and fast cycle times. However, with ever increasing performance targets, other means of improving the dynamic performance of flexures are necessary. Besides aiming for higher frequencies, performance can also be improved by damping the parasitic resonances. There are several relatively well established damping techniques, such as viscoelastic damping and tuned mass damping. However, these can be incompatible with cryogenic environments or unsuitable when the parasitic resonances vary with the deflection of a mechanism.
Alternatively, parasitic resonances can be actively damped by integrating piezoelectric material in the flexures. The regions with piezoelectric material function as additional sensors and actuators, coupling with the parasitic resonances. Using control algorithms, these resonances can be damped or suppressed. Although the active damping principle has been around for a while, the application to flexure mechanisms brings novel challenges, in particular when large deflections are involved. As the flexures deform under the nominal motion of the mechanism, the parasitic resonance frequencies will change in frequency and modeshape. As a result, the coupling of the parasitic resonances with the integrated piezoelectric material also varies with the deflection of the mechanism. This dissertation aims to demonstrate the feasibility of active damping in flexure mechanisms and to improve the tools available for the design, modelling and control of active damping in flexure mechanisms, with a particular focus on large deflections.
Flexure mechanisms can be efficiently modelled with large deflection beam elements. These elements result in fast analysis of flexure mechanisms, while appropriately capturing the dominant deflection-dependent behaviour. This enables fast design iterations and robust optimisation of flexure mechanisms. The first part of this dissertation is dedicated to extending this beam-based finite element modelling approach to active flexures. By performing a finite-element analysis of only the cross-section, the behaviour of the entire active flexure can be condensed to a geometrically non-linear beam element. This approach vastly reduces the required computation time and memory compared to other modelling approaches. The geometrically non-linear capability of the approach allows for fast and efficient analysis of the overall dynamics of flexures in deflected state and in particular the change in coupling with the piezoelectric domain. The reduction to beam behaviour can be performed with various levels of fidelity. First, using the variational asymptotic method, it is shown that the 3D governing equations of an active flexure may be condensed to a non-linear beam model of the Timoshenko type, where the resulting model captures the dominant terms in the electrical enthalpy when the slenderness assumption holds. Second, it is shown that a slightly higher fidelity can be achieved by solving the exact deformation of a beam loaded only at its ends, often referred to as Saint Venant’s problem. A decomposition is presented that separates the exact solution to the 3D fundamental governing equations into a non-linear active beam model and cross-sectional deformations that decay with distance from the ends of the flexure. By neglecting the decaying effects, a model is obtained that is similar to the one obtained with the variational asymptotic method. However, with the separation approach the shear stiffness and shear to voltage coupling terms are captured with greater accuracy. Furthermore, the aforementioned methods do not use any a priori simplifying assumptions, such as a plane stress state or planar
deformation of the cross-section, which can cause significant errors when anisotropic active cross-sections are considered. Third, it is shown that the highest fidelity can be obtained when the decay modes are included by means of an interface element that condenses the effective stiffness of the decay modes to an interface stiffness. All the aforementioned approaches are validated against existing (computationally expensive) modelling software using various simple and complex active flexure geometries.
The second part of this dissertation focusses on mechanical design considerations and the control algorithms for active flexure mechanisms. A simple and effective vibration control structure, called positive position feedback control, is used to suppress the parasitic vibrations. Optimal tuning rules that respect a gain margin constraint are derived for this control structure. Together with the proposed modelling approach, this enables the prediction of the optimally achievable damping for a given design. The tuning rules and modelling approach are used to demonstrate the possibilities of an active damping approach in a large deflection flexure based manipulator. A scheduling approach is used to optimally tune the controller over the entire workspace. It is experimentally demonstrated that the first four parasitic resonances of the manipulator can simultaneously be damped over the entire workspace, where a modal damping of 2-7% is achieved. The improved damping enables significantly higher feedback bandwidth for the control of the nominal deflection of the manipulator. Furthermore, it is also demonstrated that the approach significantly improves the disturbance sensitivity of the manipulator and that faster settling can be achieved, resulting in an improved cycle time for an indexing positioning task.
Overall, the efficient modelling approach, combined with the tuning rules enable fast performance prediction, resulting in fast design iterations and design optimisation. It has been experimentally demonstrated that the active damping approach can be used to suppress the first four position-dependent parasitic resonances of a large deflection flexure mechanism.
