Vibration of two-dimensional rod structures (Swinging Graph)
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To analyse and solve problems of structure-borne sound, simple computation models are frequently used, as detailed modelling, e.g. with finite elements, is complex. A verstile idealisation of reality leads to a system of straight rods, which are arranged in a plane and can be exited to vibrate (a quasi »multi-one-dimensional« structure). If already the real structure consists of rods and beams, the application of the model is obvious. But also in case of an appropriate arrangement of plates, idealisation to such a »rod structure« is reasonable, if it suffices to consider wave propagation only in the plane of the rod structure.
The model of the vibrating rod structure was continuosly applied for practical problems at the Fraunhofer IBP [1, 2]. However, a frequent application in acoustics and vibration technology - not at least for teaching purposes - was first of all constricted by fact that really flexible and user-friendly software was not available. This lack was eliminated by the development of the programme "Swinging Graph", equipped with a Windows user interface. The denomination of the programme alludes to the mathematical graph theory, where a graph consists of points (also calles "nodes" or "vertices") and lines ("edges") between points. In contrast to the graphical representation of mathematical graphs, however, lines (rods) in SwingingGraph must be straight.
A rod is characterised by material properties (density, Young's modulus and loss factor) as well as thickness and cross section. Longitudinal as well as bending waves including their near fields are admitted. The rod deflection due to bending waves exclusively accurs in the plane of the rod structure. If both possible propagation directions of the waves in rods of finite lenght (between two nodes) are considered, the motion of the rod is uniquely described by six amplitudes. For semi-infinite rods, running from one node to infinity, it is supposed that there is no sound energy arriving from infinity, meaning that only three amplitudes are required. All rods ought to be thin with the result of usual frequency-independent longitudinal wave velocities and frequency-dependent bending wave velocities.
Each node has three degrees of freedom: the two displacement components in the rod structure plane and the rotation around the axis through the node vertical to this plane (kinematic variables; in SwingingGraph: the related velocities). Correspondingly an external force (two components) and an external momentum can act on each node (dynamic variables). For each degree of freedom either the kinematic of the dynamic variable with magnitude and phase can be specified, e.g.a velocity in x-direction, an external force in y-direction and an external momentum, whereby sinusoidal time dependencies with a common frequency are a prerequisite. If the kinematic variable is set to zero, the node is held fixed in the corresponding degree of freedom (special case of kinematic excitation). And vice versa, if the dynamic variable is set to zero, the node can move freely in that degree of freedom (special case of dynamic excitation). Further properties can be assigned to each node, independent of rod characteristics: mass, moment of inertia, and whether the rods converging in a node are connected rigidly or flexibly. In the latter case, no external momentum can act on the node.
Conditions at the nodes, i. e. at the end points of the rods, determine the amplitudes and thus the vibrations of the whole rod structure. Mathematically speaking, this means to solve a linear equation system. Continuity of displacement and rotation (with the exception of flexible connection) as well as the balance of force and momentum is required at each node.
Thera are no fundamental difficulties in confering the model to a three-dimensional rod structure. In this case, additional bending waves and torsion waves must be taken into consideration. But the Programming of clear graphical representations requires remarkably more time and higher effort than in two dimensions.
The computation of the response of a rod structure to the specified excitations as function of frequency ("Response Function") gives a first glance at the vibration behaviour and follows the determination of resonance frequencies. The velocity component at a node or the mean velocity of a rod may be selected as response parameter.
Vibration forms and intensities (vibrational energy flows) can be computed at user-defined frequencies. The parameters of the graphs can be selected within a wide range, be it the spatial or temporal resolution in case of animation, be it the separate or combined representation of longitudinal and bending wave shares in case of intensities. The latter allows studying the conversion between longitudinal and bending waves, sometimes decisive for sound propagation.
Fig. 1 shows the first modes of a tuning fork, which is composed of eight straight homogeneous rods and possesses an additional mass of 3 g at the left end (total lenght: 105 mm). Exitation is effected by a transversal force at the right lower end.
The eigenfrequencies were determined by means of response functions presented in Fig. 2.
Fig. 3 describes how vibrational energy is propagated starting from the transfer point and dissipated.
The programme SwingingGraph supplements the series of computation tools [3, 4], especially for the field of structure-borne sound transmission. There is a multitude of possible applications:stairs, flanking transmission, design of stud profiles of plasterboard walls, vibrational excitation of structures, to mention just a few.
 Horner, J. L.; White, R. G.: Prediction of vibrational power transmission through bends and joints in beam-like structures. J. Sound Vib. 147 (1991)1, p. 87-103.
 Rosenhouse, G.; Ertel, H.; Mechel, F. P.: Theoretical and experimental investigation of structureborne sound transmission through a "T" joint in a finite system. J. Acoust. Soc. Am. 70 (1981) 2, p. 492-499.
 Maysenhölder, W.: LAYERS - ein Werkzeug zur Untersuchung der Schalldämmung von Platten aus homogenen anisotropen Schichten. IBP-Mitteilung 26 (1999), Nr. 347 (German)
 Maysenhölder, W.: HYPERAKUS - ein Werkzeug zur Untersuchung der Schalldämmung von periodisch strukturierten Wänden. IBP-Mitteilung 25 (1998), Nr. 330 (German)