A numerical study is presented, which tailors so-called prestress accumulation-release (PAR) strategy to mitigate free vibrations of frame structures. First, the concept of proposed semiactive technique is outlined and possible applications are specified. In the second part of the work a parametric study is discussed, which illustrates the potential of the method for mitigation of free vibrations induced by impact or other initial load scenarios. Special attention is given to the energy balance including all relevant contributions to the total energy of the considered dissipative system. The proposed technique shows a very high potential in mitigation of free vibrations, exceeding 99% of the reference amplitude after 5 cycles of vibration.
Some engineering structures are exposed to transient dynamic loading which, although not dangerous for the structure itself, may generate harmful or undesirable effects. It has been therefore an engineering problem to eliminate vibrations induced by nondestructive impacts, force impulses generated by working machinery, and so forth. Effective mitigation of such vibration might, for example, help improve the resolution of optical equipment or reduce the noise generated by vibrating structure. Out of three classes of possible solutions, that is, passive, active, and semiactive, there has been growing attention to the semiactive methods which allow for adjusting some mechanical parameter characteristic on one hand and utilise the structural deformation to introduce control forces, on the other hand. One advantage of the latter feature, which is common with passive devices, is that the system does not require external power to directly generate the control forces. The external power is needed to regulate an actuator which in turn changes the magnitude of the control force according to the control unit algorithm and is typically in the order of magnitude of tens of Watts. Symans and Constantinou in [
Another group of techniques which gained attention especially in seismic engineering is utilisation of semiactive friction dampers for energy dissipation. Such dampers can be installed either within a structure as part of additional bracing [
Among many available concepts of tailoring semiactive techniques to mitigate vibration, synchronised switch damping (SSD) techniques generate voltage magnification, and a phase shift between the mechanical strain and the resulting voltage of a piezoelectric element. As a result a force always opposite to the velocity is obtained and the level of dissipation corresponds to the part of mechanical energy converted into electric energy. A review of SSD and other semiactive techniques utilising piezoelectric elements is given in [
Technical application of adaptive shock-absorbers to adaptive landing gears and vehicle suspension is discussed, respectively, in [
In the PAR strategy it is assumed that a structure undergoes free vibrations and that there is a certain device or devices installed in the structure capable of imposing kinematic constraints on some degrees of freedom of the system. For instance, a layered beam could be equipped with a device that allows or constrains the relative slip between layers, or a system composed of masses, and springs is equipped with a device which releases or reattaches a chosen spring to a mass. Given such devices are in place, the strain accumulated in the structure could locally be released which results in conversion of a part of the strain energy to the kinetic energy of local, higher frequency vibrations. In the next phase constraints are reimposed which results in “freezing” of a part of the deformation. Local, higher frequency vibrations introduced after reimposing of the constraints can be effectively damped out with material damping. An interesting example of a passive TMD device for damping portions of kinetic energy locally in order to achieve global mitigation effect is described in [
If the time instant of reimposing constraints is chosen properly, that is, at the moment of maximum relative dislocation between top and bottom beam, it will introduce a prestress in the structure. It should be emphasised at this point that a relatively small energy was used to adjust the actuator device, for example, a piezo actuator that controls the friction in a joint (like one introduced by Gaul and Nitsche [
Some numerical as well as experimental results of application of the PAR technique to layered beams can by found in [
Structure assumed in the simulations (semiactive nodes depicted in green).
Physical model analysed in numerical simulations was a one meter long cantilever beam comprising two layers 0.1 m apart and connecting elements spaced every 0.1 m. Two frictional joints at both ends of each connecting element (depicted in green in Figure
All cantilever members were modelled as steel, prismatic, and rectangular bars with cross-section of 20 × 6 mm.
There are two physical sources of energy dissipation in the assumed model: material damping, friction between surfaces of semi-active nodes.
Rayleigh damping model was assumed, taking the form Coulomb friction model in accordance with [
Control strategy outline.
In the outline of the control strategy presented in Figure
The control strategy could be summarised as follows.
Numerical simulations were carried out with Abaqus/Standard Finite Element Software using finite displacement theory because of large rotations during opening of the panel. Semiactive nodes were modelled with connector elements in which the available component of relative motion was the rotation about
The fundamental mode of vibration of the assumed model with the opened panel was 19.6 Hz for structure with frame nodes, 3.96 Hz for structure with truss nodes.
The first longitudinal eigenfrequency was 1213.4 and 1186.8 Hz, respectively.
In the initial simulation all semiactive nodes (indicated by green spots in Figure
Vertical displacement at the tip of main structure (node 11).
The longitudinal displacements of the top and bottom layer’s tip are shown in Figure
Horizontal displacement of tips of top (node 31) and bottom (node 11) layers: (a) full time history; (b) longitudinal vibrations triggered with nodes activation.
Accumulated slip between contact surfaces and the actuator moment: (a) full time history; (b) time window where zero moment corresponds to the truss mode.
For this particular case the nodes are switched at the point of maximum
Axial force in beam elements (a) before and (b) after switching from frame to truss mode and back. Deformation scale U1x500, U2x40 (opened panel not shown).
The sum of mechanical energy of the system, energy dissipated and the work of external forces done on the structure must remain constant throughout the process. For the analysed system undergoing free vibrations there are following nonzero components of the total energy balance: kinetic energy, strain energy, energy dissipated in viscous processes, including material damping, frictional dissipation at contact surfaces of semiactive nodes.
All of the above contributions are depicted in Figure
Energy balance of the whole system.
Based on the carried out initial simulations the following parameters have been identified to have an important impact on the system performance: material damping, number of semiactive nodes, amount of decrease in the friction generating moment in phase 2.
First, the influence of numerical damping on the solution needs to be analysed. According to [
Influence of numerical damping on the solution.
As mentioned before, mass and stiffness proportional damping coefficients of the Rayleigh model are
Influence of decrease of stiffness proportional damping: (a) in terms of vertical displacement of the bottom beam tip (node 11); (b) in terms of viscous dissipation during phase 1.
Preceding results have been obtained with all semiactive nodes connecting top and bottom beam. In this section the number of semiactive nodes is reduced. Namely, starting from the fixed support, only
Vertical displacement amplitude mitigation with varying number of semiactive nodes.
Number of semiactive nodes | Vertical amplitude reduction | Number of node activations |
---|---|---|
10 | 99.4% | 1 |
7 | 98.3% | 2 |
3 | 90.8% | 6 |
1 | 56.7% | 9 |
Varying number of semiactive nodes: (a) vertical displacement at node 11; (b)
As indicated in Figure
Vertical displacement at node 11 for different values of
In Table
Amplitude mitigation obtained with different values of friction generating moment decrease in Phase 2.
% of |
% of amplitude reduction after | |
---|---|---|
5 cycles of vibration | 13 cycles of vibration | |
5.0% | 61.9% | 71.6% |
6.0% | 73.2% | 81.3% |
7.5% | 96.2% | 99.4% |
10.0% | 99.1% | 99.2% |
11.5% | 98.5% | 98.3% |
20.0% | 96.3% | 96.4% |
In this paper a method for semiactive mitigation of free vibrations has been presented. In the so-called PAR strategy, the strain energy accumulated in the system during vibration is transferred to kinetic energy of higher frequency, longitudinal vibrations of the structural members, and eventually efficiently dissipated with material dissipation. In addition PAR strategy introduces an elastic control force into the structure which acts in direction opposing the movement. This prestress acts as a braking force to the vibration. Energy accumulated in prestress is gradually dissipated in the second phase of the control procedure with frictional joints of semiactive nodes. The efficiency of this approach has been demonstrated on a case study of a frame structure. Generally the efficiency is very high, although it depends strongly on the number of semiactive nodes installed in the structure. With few semiactive nodes installed it takes more activations of the nodes to obtain the desired effect, whereas if all nodes are semiactive only a single activation suffice. It has also been shown that the global response is dependent on the algorithm of applying the normal force in the frictional connections. Therefore, apart from the simple on-off strategy for applying the contact force in semiactive nodes, also other strategies are possible, based on the feedback with the slip between contact surfaces.
Generally the PAR strategy requires that the stiffness of the structure is temporarily reduced. Time duration of the minimum stiffness mode is however very short as compared with the time of full stiffness mode.
Carried out simulations indicate some difficulties one would encounter during development of the prototype system. It seems challenging to properly reimpose the full stiffness mode since the frequency of the longitudinal vibrations is relatively high, especially as the monitoring of the relative displacement between structural elements should be avoided for the sake of simplicity. This problem however could be overcome since the frequency of the longitudinal vibration is known a priori and could be used to set the time shift of switching between two modes of the semiactive nodes.
Another problem deals with the reliability of the structure. One possible realisation of semiactive nodes could be based on the concept presented in [
Example of reliable PAR node: (a) full stiffness state, no supply voltage required; (b) reduced stiffness state, supply voltage required.
Another example of a reliable semiactive system used in a seismic protection device is described in [
From general point of view the following problem can be formulated: “how to design optimally adaptive structure (equipped with controllable, semiactive PAR joints) able to reduce maximally vibrations caused by predefined impact” and this paper presents one of such desired solutions.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Financial support of Structural Funds in the Operational Programme-Innovative Economy (IE OP) financed from the European Regional Development Fund, Project “Modern material technologies in aerospace industry,” no. POIG.01.01.02-00-015/08-00, is gratefully acknowledged. Financial support of Structural Funds in the Operational Programme-Innovative Economy (IE OP), Project “Innowacyjne technologie dla poprawy bezpieczestwa maego lotnictwa SWING (Safe-Wing),” no. POIG.01.04.00-14-100/09, POIG.04.01.00-14-100/09, is gratefully acknowledged. Financial support of Polish research project, founded by NCN: “Podstawy adaptacyjnej absorpcji udaru (AIA: Adaptive Impact Aborption) oraz studium wykonalnoœci jej zastosowania do redukcji szkd w kolizjach transportowych,” 2012/05/B/ST8/02971, is gratefully acknowledged.