The design of a planar Kinetic Reciprocal Frame

Transcription

1 The design of a planar Kinetic Reciprocal Frame Dario PARIGI 1*, Mario SASSONE 2 1* Ph.D. student in Architecture and Building Design, Politecnico di Torino Viale Mattioli 39, I Torino, Italy dario.parigi@polito.it 2 Assistant professor, Dep.of Structural and Geotechnical Eng., Politecnico di Torino Abstract In this paper is presented the project of a high-tech planar Kinetic Reciprocal Frames. Reciprocal frame structures are based on the geometric configurations studied in the past by Leonardo da Vinci. These structures, particularly interesting for their potential in architectural applications, features kinematic properties which differs from the more common pin-joint assemblies, because bars joins not only in the end but also in intermediate points. While recent applications in architecture seem to emphasize the simplicity and the naiveness of this kind of construction, trough timber construction, for instance, we are firmly convinced that RF can be of interest when approached in more technological way. Our research is then specifically oriented to Kinetic Reciprocal Frames (KRF), in which a high tech construction can improve consistently the performances, allowing the design of innovative structures. Kinetic structures as KRF can be a powerful design solution for example to meet the complex requirements of time-depending performances in green buildings like the light control in roofs and facades. The project presented is the result of two distinct but intertwined phases: the analytical phase, dedicated to the definition of the problem from the mathematical point of view; and the prototyping phase, dedicated to the validation and extension of the analysis with the use of physical models. Keywords: reciprocal frames, kinetic structures 1.Introduction Reciprocal frames (RF) and multiple reciprocal frames (MRF) are structures composed by mutually supported elements, arranged to form, respectively, one or more closed circuits of forces [1]. They differ from better known truss assemblies because bars, in RF and MRF, join to each other not only at the ends but even at intermediate points. There are many examples of structures conceived following the reciprocity principle, starting from the technique adopted in Japan by the monk Chogen ( ), for the construction of temples. In Europe, during Middle Age, a Plane Reciprocal Frame (PRF) have been proposed for building the floor of big rooms using short beams, as described by the medieval architect Villard de Honnecourt ( ). The wider and most [4]: he explored various patterns of beams grillages (Figure 1, Figure 2), and studied three dimensional arch structures for domes and bridges, in which short elements are

2 supported by each others. Starting from Honnencourt's work, Sebastiano Serlio in 1537 first introduced a multiple plane reciprocal frame (MPRF) as a general solution for the construction of large floor by means of short elements simply supported to each other. A complete historical overview on ancient and contemporary realisations can be found in Rizzuto et al.[7]. Figure 1 sketch and scheme Figure 2 sketch and scheme 1.1 Towards Kinetic Reciprocal Frame One main feature of Reciprocal frames is that bars joins not only in the ends but also in intermediate points. This feature suggest that intermediate points position can vary through the length of the supporting bars, thus a sliding hinge can be placed instead of a rotational hinge. For example in the fan of Figure 3 rotational hinges are substituted with sliding hinges: the structure turns then from kinematically determinate to kinematically indeterminate (Figure 4): the relative motion between bars is allowed because bars can slide into one another. If the kinematic understanding of single fan can be easily predicted, the behaviour of network needs a more complete algebraic formulation [6]. One first indication of the behaviour is given by the degrees of freedom (DOF) of the structure: in the following paragraph is presented the scheme to build the kinematic matrix, and the relation of its rank with the DOF. Figure 3 Figure 4

3 2. The kinematic determinacy of Reciprocal Frames networks The degrees of freedom of a structure are the number of independent parameters required to completely specify the configuration of the mechanism [9]. It is equal to the degrees of freedom of all the moving bodies diminished by the degrees of constraints imposed by all the joints. In two dimension the relation is: m= 3*b-c (1) where b is the total number of bodies, c is the number of degrees of constraints, and m is the number of the resulting degrees of freedom. However, as it is well known [6], equation (1) is not sufficient to determine the DOF of a structure: it is necessary to introduce the rank r of the kinematic matrix and the expression (1) is substituted with expression (2) : m= 3*b-r (2) In the next paragraph an algorithm for the construction of the kinematic matrix of reciprocal frames is introduced. 2.1 Kinematic matrix of RF The analysis of the equilibrium matrix and its transpose the kinematic matrix gives us the full details of any state of self stress and modes of inextensional deformation that the assembly may possess, as shown by the work by Pellegrino and Calladine carried on pin-joint frameworks [6] [8]. However, the more convenient way to model reciprocal frames is as a rigid body assembly, given the need of modelling the forces carried by the bars in intermediate point of the bars and not only in the joints as occur in pin-joint assemblies. As a result, the sets of variables of the kinematic matrix varies according to this scheme: A[coefficient of the kinematic matrix]*[set1]=[set2] Pin-joint assemblies Rigid bodies assemblies set 1= node displacement vector set 1= bar dispolacement vector set 2= elongation of the bars set2= node displacement vector 2.2 The kinematic matrix construction scheme For any rigid body in a two dimensional space assembly it is possible to write the kinematic equations with the use of three generalized coordinates for every rigid body (three is minimum number of coordinates to describe the position in the space of rigid bodies), and represent the displacements of a reference point. If the topology of the net is regular, the kinematic equations can be written by an algorithm that calculates the node displacement from the Cartesian coordinates of the assembly [5].

4 The system of equation take the following form in the matrix notation: ; (1) where A is the coefficient matrix, s is the number of external and internal constraints, b is the number of rigid bodies of the assembly, D is the generalized coordinates vector, n is the vector of node displacement. The generalized coordinates are ordered in the D vector (2) (3) 3 Case study: Kinetic Reciprocal Frame The Kinetic Reciprocal Frame (KRF) here presented is a periodic regular structure in which all fans change their position with a single degree of freedom. The structure moves from the position of

5 Figure 5 to the position of Figure 6 while fans slides on one another. It is based on a reciprocal frame unit of Figure 6, composed by 18 bars, then simplified accordingly to the scheme of Figure 8. Figure 5

6 Figure 6 Figure 7 Figure 8 The project is the result of two intertwined phases: the analytical phase, dedicated to the definition of the problem from the geometrical point of view; the testing phase, dedicated to the validation with the use of prototypes. The mechanism is actuated by the rotation of one of the fan in the net (Figure 9) and its behaviour remains the same independently from the number of fans repeated in the pattern. As a reaction to the driving rotation, every fan rotate around its centre, where the three bars converges. Figure 9 The one degree of freedom configuration carries the advantage of an energy efficient actuation: the pattern can be actuated by a single motor, reducing the costs: it can also be placed in anyone of the fan of the assembly, allowing to adapt to different design requirements. 3.1 The parametric definition of the structure The structure of

7 Figure 10 is drawn by means of Cartesian coordinates. The position of every bar is defined by a point (x,y) and and angle. i j Figure 10 ij k Figure 11). To every bar of the k repetitive rules or randomly or the two combined (Figure 12). The coordinates of the contact nodes are obtained by calculating the intersection of the bars given the starting point A and B and the angle of the bars and (Figure 13). The drawing of the structure by means of Cartesian coordinates allows to set up the algorithm for the automatic construction of the kinematic matrix [5]. From the kinematic matrix rank is it possible to test the degrees of freedom of the structure while comparing the results with the built prototypes. j j+1 i+1 k+5 k+4 k k+3 i k+1 k+2 Figure 10 - Figure 11 -

8 A (x,y) B (x,y) Figure 12 Parametric angles (Matlab) Figure 13 cartesian coordinates of the nodes 3.2 The kinematic matrix It is possible to build the kinematic matix of the assembly by taking advantage of the parametric construction of the pattern. Through the use of the Cartesian coordinates of every point it is possible to write iteratively the constraints equations of the pattern. The displacement of the nodes of every fan is expressed through the generalized coordinates [5], so as to have the minimum number of unknowns. The kinematic matrix will allow in the next stages allow to determine the DOF of the network and to make further consideration on the kinematic behaviour [6]. 3.2 The geometrical description of the network The shape of the fans that forms the network were simplified in the previous paragraphs to a three straight arms fan. However, the kinematic property of the pattern depends on the curved shape of the arms (Figure 16). We consider the rotation of Bar 1 and Bar 2 of the same angle, while the Bar 1 slides on Bar 2 more coincident with Bar 2 ( Figure 14). Bar 2 needs to have a shape that allows to span the distance marked after the rotation. The shape is obtained by rotating the point A around the hinge of Bar 2 by the same angle that describe its position relative to hinge of Bar 1 (Figure 15). By repeating the process for every bar in the fan we obtain the configuration of Figure 17.

9 A A 2 A Figure 14 A Figure 15 Figure The role of prototyping Figure 17 A first straw prototype showed that the fans assembled in the unit of Figure 19 were not able to rotate to the position of Figure 20 unless by applying a force that caused a deformation in the flexible elements. This kinematic behaviour detects in fact the need of the fans to have curved arms instead of straight ones. This prototype made also clear that the external constraints could be reduced: the rotational hinges, placed at the center point of every fan when the three arms joins, can be cut by half in an alternate pattern as by Figure Prototype details description

10 The prototype (Figure 21, Figure 22) was built through the use of a laser cutter machine. The pieces are shaped with a small tolerance that allows to diminish the friction between the elements during the motion, but in the same time to avoid a loose mechanism, in which the movements between each elements is transmitted with a delay. The prototype motion is actuated by a servo motor controlled by an Arduino board. The servo motor angle is a function of the light sensor: this allows to demonstrate intuitively the motion by simply shading with the hand the sensor area, and moreover suggest an application of the structure as a mechanism for the light control for roofs and facades. External constraints Figure 18 Reduntant external constraints

11 Figure 19 Figure 20 Figure 21 Figure 22 4 Possible applications and future developments. The structure has the property of changing the relative areas of the sectors during the deployment phase. As a consequence, by covering with opaque and transparent membranes the different sectors it is possible to obtain a mechanism for the light control in facades and roofs (Figure 23, Figure 24 Figure 23 Figure 24

12 5 Conclusion The paper describes in detail the process used to design a kinetic reciprocal frames network, from the geometrical description to the prototype testing. The one-degree of freedom configuration allows applications that features an energy efficient actuation (inextensional mechanisms are activated by a single actuator). The presented pattern can be extended indefinitely in both directions maintaining the same kinematic behaviour, allowing to span even non-regular areas. The next steps are directed to design non-regular patterns removing any type of symmetry by a further generalization of the problem, and then to apply the results in three dimensional ones. References [1] Di Carlo B, The wooden roofs of Leonardo and New Structural Research. Nexus Journal, 2008; 10: [2] Garcia de Jalon, J, Bayo E, Kinematic and Dynamic Simulation of Multybody Systems, the Reat-Time Challenge, Springer-Verlag, New York, 1994 [3] Kirkegaard PH, RK Nielsen S, MBS Analysis of Kinetic Structures using ADAMS, proceedings of the IASS- International Symposium, Valencia, [4] Marinoni A, Il Codice Atlantico della Biblioteca Ambrosiana, Giunti, Firenze, 2000 [5] Parigi D, Sassone M, Napoli P Kinematic and static analysis of plane reciprocal frames, proceedings of the IASS- International Symposium, Valencia, [6] Pellegrino S, Calladine C R. Matrix analysis of Statically and Kinematically Indeterminate Frameworks. International Journal of Solids and Structures, 1986; 22: [7] Rizzuto J, Saidani M, Chilton J, Multi-reciprocal element (MRE) space structure systems, Space Structures 5, 2002; Telford ed. [8] Strang G., Linear Algebra and its applications, Thomson Brooks/Cole, Belmont, CA, 2006 [9] Tsai, L.-W. (2001). Mechanism design: enumeration of kinematic structures according to function. CRC Press.