Kinematic Analysis (2D and 3D)

Due to the complicated nature of this table’s geometry, link angles are the most critical calculated component in the design.  There is no need for velocity and acceleration calculations. The slider link angle is constrained by the horizontal slider piece. By using a slider crank design, the toggle points of the mechanism can not be met without fully extending the mechanism to make the two long links parallel. Since this is past the range of motion of the table, the team is not concerned with toggle points.

Positional analysis using complex numbers provided a useful tool for basic configuration of the up/down folding of the petals. This motion could technically be considered a 6-bar linkage, however due to the constraints on the mechanism, it could be decomposed into two 4-bar crank slider mechanisms. Basic vector loop analysis yielded geometry that would allow the inner petals to fold up before the outer petals in order to achieve the desired effect.

A much more difficult component of the design was the RSSR spatial four bar (Revolute/Spherical) used to fold the inner petals lengthwise as they folded in towards the center. The actuation for this sub-mechanism was driven by the differential motion from two of the links on the crank slider for the inner petals. Positional analysis was performed based on the mounting points for the spatial 4-bar components which sufficiently constrained the system as needed to develop the 3D geometry. While spherical joints were desired to make the system actuate as smoothly as possible, budgetary concerns led to the use of 3D-printed universal joints (usually called u-joints). For the sake of simplicity the u-joints were assumed to be spherical, however it should be noted that this type of joint does not exactly replicate spherical joint behaviors. The implemented design could be more precisely described as 5-bar mechanisms, however the spherical assumption proved to be reasonably accurate in the final structure.

In order to perform 3D motion analysis, the spatial mechanism was described using the vectors through its two revolute joints. These vectors describe the planes normal to the axes of rotation, and in the local coordinate system of the spatial mechanism, these two planes are statically defined. The axis through the center of the petal was superimposed over the x-axis of the local coordinate system, and then separate 2-D analyses using complex number methods were carried out in each plane. The resulting values could be mapped directly into (x, y, z)-coordinates in the local reference frame.