...
This tulip table design was inspired by Three of Hearts Design: a furniture company devoted to making furniture with novel designs. Two representatives of the company requested a mechanism that could fold to look like a tulip and unfold to become a functioning table. The purpose of this project was to design a single degree of freedom mechanism to unfold three inner petals and three outer petals to create a flat, table-like surface.
Design and CAD
Our device meets the project requirements by using a four-bar slider crank mechanism. The mechanism unfolds the outer and inner petals with a downward force on one of the outer petals. To allow for an flat surface when unfolded, the inner petals fold and fit inside of the sides formed by the outer petals. These petals fold on two hinges located on the center of each inner petal. These hinges rotate about a steel shaft that acts one of the links of each four-bar mechanism. Four U-joints were added later that connect to the inner petals and cause them to fold on their hinges while the entire table folds upward. These U-joints emulate spherical joints and allow the inner petals to be approximately the same size as the outer petals when unfolded. These joints act create a 3D four-bar linkage, similar to the 2D linkages analyzed in this class.
The four-bar slider crank mechanism is comprised of machined steel shafts for links and pinned joints, 3D printed PLA connection pieces, and laser cut acrylic for the longest links and the petal. In addition to those parts, the entire table has 54 bushings, 6 aluminum rods, and 39 snap rings.
The slider piece was 3D printed with PLA and moves along a single shaft that acts as the stem of the tulip table. By 3D printing the sliding mechanism, we were able to cut down on material costs and create the complicated geometry shown in the CAD model. Additionally, 3D printing allowed us to easily create the complicated slider geometry and rapidly iterate between designs. The slider supports the mechanisms and connects the four-bar linkages to the stem of the table.
The device is powered by a downward force exerted on one outer petal or the 3D printed slider. At the request of 3 of Hearts Design, we have not included any necessary motors or electronics in our design. When the downward force is applied, the table and it’s inner petals unfold to create a flat, table-like surface.
The final prototype consists of six completed linkage mechanism with three pairs of interlocking petals. The specific shape of the petals are not final because the goal of this project is to achieve the correct table opening motion, not to design the geometry of the table petals.
Analysis
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.
Manufacturing and Prototyping
To transition from the SolidWorks model to a prototype, we needed to find a good balance between material cost, ease of fabrication, and design precision. To avoid the slipping of circular bearings inside of our components, the team decided to use bushings, each with two flat outer edges to constrain rotation of the bushings when in use. To prevent any translational movement of the bushings on their corresponding shafts, we machined grooves into the steel shafts for snap rings on the outside edges of all of the bushings.
The machining process of the pinned joints involved cutting 3, 6mm, 12 inch shafts into 18, 1 ½ inch sections using a table saw in the UT Machine Shop. The edges of each piece were grinded down to remove any sharp edges or burrs left over from cutting. Next, 1/32 inch thick snap ring grooves were lathed down 0.08 inches, each 3/16 inches from each outside edge of the shaft.
Laser cut acrylic was used on the longest links of the four-bar slider linkages. Laser cutting was chosen for its ease of manufacturing and quick prototyping capabilities. Additionally, acrylic is available at very little cost in the UT MakerSpace. The team decided that machining all of the slider crank links would be time consuming and not cost effective. After cutting the links, the team discovered that laser cutting did not yield precise holes to press fit the bushings. To hold the bushings in place, the team cut snap ring grooves into all shafts that held the bushings to the acrylic links.
To connect the petals to the four-bar slider crank mechanisms, we 3D printed pin joint foundations to adhere to each petal using PLA. Additional parts printed using PLA include the hinges used to fold the inner petals, the slider, joints between linkages, and the U-joints which cause the inner petals to fold.
Conclusion and Areas of Improvement
Ultimately, the team was able to achieve our objective of creating a flower-like table that can unfold to create a flat surface. The build quality of this prototype could be improved by having tighter tolerance on machined shafts to ensure symmetry and smooth motion. Loose tolerances of the machined shafts caused discrepancies in the necessary spacers needed to constrain the movement of the shafts and bushings.
Additionally, the device could be made more stable by adding supports to the linkages, or making them out of a sturdier material such as aluminum. Machining aluminum parts, rather than extruded PLA or laser cut materials would achieve a sturdier design and tighter tolerances. To provide additional stability, two points of contact on the bottom collar should be used rather than one. Additionally, using off the shelf U-joints would eliminate much of the assembly time required to create this prototype.
Finally, to continue the design of this table, the petal and hinge geometry needs to be refined to create a continuous, flat surface. The team achieved our goal of creating a functioning linkage mechanism that unfolds a tulip to create a flat surface. However, geometric tweeks will need to be made to create a functional and aesthetically pleasing table.