108 V. Ondra and B. Titurus that focus on different aspects of beam and string modelling, application and testing [7–10]. However, very few studies of coupled beam-tendon systems can be found in literature. To the best knowledge of the authors, a similar system has been considered only in a handful of studies. For instance, a string-beam system has been used as a representation of an optic cable coupler in [11] and its bifurcation and chaotic dynamics mathematically investigated in [12]. In these studies, however, the axial tension of the string had no influence on the beam so that the beam dynamics could be considered separately from the string. The objective of this paper is to study the free vibration of a non-rotating beam-tendon system where the coupling between the beam and the tendon is realised at the tip and a single spanwise location. Due to this coupling, the vibration of the beam and tendon must be considered simultaneously as they influence each other. The paper is organised as follows: in Sect. 10.2 the theoretical model of the system is presented and the numerical procedure used to obtain the modal properties is briefly described. The effect of an attachment point and its location on the modal properties and structural stability are then experimentally and numerically investigated in Sect. 10.3. Finally, in Sect. 10.4, the main findings are summarised, and potential implications to the application of an active tendon concept as a vibration control method in rotorcraft are discussed. 10.2 Theoretical Model The considered system can be seen in Fig. 10.1. The system consists of a straight hollow cantilever beam with a double sectional symmetry that is axially loaded by a tendon. The rectangular cross-section is used in the present numerical and experimental studies. The tendon is attached at the tip of the beam and fixed (clamped) at the same place as the beam. Unlike in the previous studies, the beam and the tendon are also connected in, generally, several spanwise locations using attachment fixtures such as the one schematically shown in Fig. 10.1b. It is assumed that each attachment point ensures the equality of the beam and tendon displacements while the slope of the tendon is different on each side of the attachment. In addition, the attachment fixture is assumed to be friction-free, i.e. the magnitude of tension P is not changed when passing through the fixture. Each attachment fixture has a mass mi and is placed at the distance Li from the clamp. Although there is no limit on the number and locations of the attachments, this paper investigates a beam-tendon system with a single attachment fixture only. The equations of motion describing the beam-tendon system are EIw +Pw +m¨w=F, (10.1a) −Pw t +mt ¨wt =0, (10.1b) where EI is the bending rigidity of the beam, P is the applied force which is transmitted by the tendon so it acts as an axial force at the tip of the beam, mis the mass per meter of the beam, mt is the mass per meter of the tendon, F is the distributed excitation force that is only applied to the beam, w(t,x) andwt(t,x) are vertical displacements of the beam and the tendon, respectively, t is time, x is an independent spatial variable measured along the span of the beam, ˙() =∂/∂t and() =∂/∂x. While these two equations are fully uncoupled, the beam and the tendon interact with each other through the tip mass and attachment fixtures. x=0 Li L1 L2 m1 1 2 i N m2 mi mN beam(E,I,m,w) tendon (P,mt,wt) LN P attachment fixture P P a b distributed force (F) Fig. 10.1 Coupled beam-tendon system: (a) description of the system, and (b) idealisation of the attachment fixture
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