Distributed and Localized Active Vibration Isolation in Membrane Structures

Abstract

Once a membrane starts vibrating, suppressing the vibration is very difficult. Thus, the present study primarily aims at isolating a membrane from major disturbance sources, that is, from support structures. The present study introduces weblike suspension cables around a membrane and develops in theory a vibration-isolation strategy applied only along the cables. First, collocated small actuators/sensors are attached at the interfaces of the cables and the membrane to realize a distributed cable-tension control. Second, linear theory-based localized controllers are designed for suspension-cable substructural models. The feedback laws for these two kinds of controllers are derived employing a partitioned equation of motion. The resultant control system is lightweight, simple, low order, robust, and redundant. A series of transient analyses using a geometrically nonlinear finite-element method corroborates the effectiveness of the proposed vibration-isolation strategy. Nomenclatureactuator influence matrix for substructure controller B = actuator influence matrix for interface force canceling controller C = constraint Boolean matrix D = partitioned damping matrix D g = assembled damping matrix E = Young's modulus f = function f = partitioned nodal external force vector f D = nodal d'Alembert's force vector f g = global nodal external force vector f u = control force vector G h = feedback gain matrix G h1 = gain matrix for displacement G h2 = gain matrix for velocity h = thickness I = identity matrix J = scalar performance/cost index K = partitioned stiffness matrix K g = assembled stiffness matrix K i = stiffness matrix of ith substructure L = assembly Boolean matrix M = partitioned mass matrix M g = assembled mass matrix P = Riccati matrix for optimal control Q = state weighting matrix q = partitioned nodal displacement vector q g = global nodal displacement vector R = control input weighting matrix t = time u = control input u h = control input of substructure controller u = control input of interface force canceling controller v 0 = amplitude of prescribed velocity x = state vector x, y, z = Cartesian coordinates b = scalar weighting of strain energy b = scalar weighting of kinetic energy T i = tension change in ith interface force canceling controller T i = generated tension in ith interface force canceling controller " i = strain change in collocated sensors/actuators at ith boundary node f = residual transmission force between substructures t = time step size " = design strain b = localized Lagrange multiplier vector bi = ith component of b = Poisson's ratio = total energy functional = density Subscripts h = homogenous localized-substructure controller ic = inner-catenary cable m = membrane opc = outer-perimeter cable tie = tie cable = interface force canceling controller (cable-tension controller)

Fractional order control of the two-dimensional wave equation

“…One common practice is first to approximate the continuous medium by spatially discrete models, which can be achieved either by modal truncation, by the Finite Element (FE) method or even by experimentally obtained frequency response, and then to control it by state space techniques. This is the approach, for example, in Balas (1978) and Hu (2008) for the control of 1D flexible structures and in Kukathasan and Pellegrino (2002) and Sakamoto, Park, and Miyazaki (2006) for the control of membranes. However, the use of approximated models carries some disadvantages.…”

Dynamic model and active vibration control of a membrane antenna structure

“…Sakamoto et al. (2005, 2006) proposed a vibration suppression strategy of membrane structures which meets rigorous requirements for membrane space structures; the optimal feedback laws for two kinds of active controllers were derived using portioned structural modeling. Sultan and coworkers (Yipaer and Sultan, 2013; Ferhat and Sultan, 2014, 2015a, 2015b) investigated the modern control design of a square membrane structure with clamped edges and four piezoelectric bimorph actuators.…”

Decentralized adaptive fuzzy vibration control of smart gossamer space structure

“…Gorinevsky and Hyde (2002) and Gorinevsky et al (2001) used embedded actuators to control the shape of a membrane reflector. Sakamoto et al (2005) and Sakamoto and Park (2006) attached small actuators at the interfaces of the cables of a membrane structure to control the vibrations of the membrane. Renno et al (2008) augmented two macro-fiber composite (MFC) actuators and sliding mode control method to control the vibration of a membrane strip.…”