The Orbital Wave Project

 

An alternative to the Piston Wave Project is explored using rotating disks that show unique patterns. 

Pendulum Waves & Piston Waves

Pendulum Wave is a popular physics demonstration that uses a series of pendulums of different length, initially oscillating in synchrony that evolves interesting wave patterns before it returns to perfect unison.

A variation on the core principle of Pendulum Waves was discussed in The Piston Wave Project of this Hobby Physics blog, shown below. Please see that post for more details.

Both of these demonstrations share a common principle: The frequencies $f_i$ of oscillation are a mathematical progression, where integer i is the designated number for each pendulum, ranging from 0 to $N_p - 1$, and $N_p$ is the number of pendulums. The progression is: $f_i = f_0 \frac{K + i}{K}$, where $f_0$ is the lowest frequency (longest pendulum), and K is an integer. 

Another commonality is the inherent sensitivity of these demos to errors in the dimensions (lengths of the pendulums or radii of the pulleys). This can lead to accumulated errors in position after a number of oscillations that distort the undulation patterns.

As described in The Piston Wave Project, anything that oscillates, whose frequency of oscillation can be controlled to 0.1% or better is a candidate to illustrate the undulation effect. 

The Orbital Wave Machine

Another Method of Displaying the Undulation Wave

In the Piston Wave design, rotating each large pulley by an angle $\theta_i$ moves a pin at a radius r to a position $(x_i, y_i) = (r\cos{\theta_i} , r\sin{\theta_i})$, as shown above. The pistons only show the vertical coordinate $y_i$. Another option would be to show each pulley pin as it rotates, similar to a planetary or “Orrery” mechanical model. A simulation video of the pin position of all 16 disks as the pulleys are rotated is shown below.

Simulation Video (Excel macro)

• Np = 16, K = 33
• $\frac{R_0}{R’} = 1.9329$

                Piston Wave                                                                        Orbital Wave

Note that at certain values of N’ (turns of the small pulley) the disks group together at evenly spaced angles. The faster disks catch up to the slower ones, aligning at specific moments.

Simulation Special Cases

These plots show $y, \theta$, and $(x,y)$ after $N’$ rotations of the small pulleys. If we define $N’_{full}$ as the number of turns of the small pulleys when all of the disks realign to the original position, we see from these simulations that groupings occur when $\frac{N’_{full}}{N’} = m$, where m is the number of angular groups.

Condition for Angle Groups

Previous discussion noted that $\theta_i = 2\pi N’ R’\frac{i + K}{R_0 K}$ and $N’_{full} = \frac{R_0 K}{R’}$. Defining $\rho = \frac{N'_{full}}{N'}$, this gives $\theta_i = 2\pi \frac{i + K}{\rho}$. For m number of angle groups, where $m = 1 \rightarrow N_p$, each of the disks are at $\theta_i$ equal to some integer multiple j of $\frac{2\pi}{m}$, where $j = 1 \rightarrow m-1$. Grouping then occurs whenever $2\pi \frac{i + K}{\rho} = 2\pi \frac{j}{m}$. Since $2\pi (i + K)$ is equivalent to $2\pi$ for each disk, this gives angular grouping at $\rho = \frac{m}{j}$. The unique cases are at values of j that do not evenly divide into m. 

The table below shows the case of $N_p = 16, m = 1 \rightarrow 16$ groupings, and for each m there are values of j that produce these m groups. Note that if m is a prime number there are m-1 possibilities for j.

The plot below shows the case for m = 10, which occurs when $\rho = \frac{N’_{full}}{N’} = \frac{m}{j}$, where j = 1, 3, 7, or 9.

Orbital Wave Design

• As with the Piston Wave design, a crank turns a set of fixed radius (R’) pulleys on a common shaft. 
• Each of these pulleys are connected by a rubber cord to independently rotating pulleys of radius $R_i$ that are attached to a tube shaft. 
• The shafts are concentric and are coupled at the top portion to a disk on a rod. 

Design Details

• Shafts are 0.2mm thick brass or stainless-steel concentric tubes, OD (mm) = 10, 9.5, 9, 8.5, 8, 7.5, 7, 6.5, 6, 5.5, 5, 4.5, 4, 3.5, 3 + center brass rod OD = 2.5.
• Large pulleys have brass inserts OD 0.75, glued to each shaft.
• The orbiting disks and shaft clamps are Acetal, joined by 1/16” brass rods, press fit.
• The top most disk is clamped to the innermost 2.5mm rod shaft, coupled at the bottom (largest) pulley with the slowest rotation.
• The crank is coupled via beveled gears to a stack of R’ small pulleys. The R' pulleys were made by cutting parallel grooves in a single aluminum rod. The stack is tapped for ¼-20 at each end for threaded axels. 
• A PTFE washer punched from a 0.1mm sheet is placed between the large pulleys for lubrication.

Components

Shaft + clamps (earlier brass rod clamps were replaced by black Acetal to minimize shaft compression):

R pulleys

As with the Piston Wave, the $R_i$ pulley diameters were measured and tuned using a rotation counter method (video below).

Final Assembly

On the left: The Orbital Wave Machine is partly assembled. There is a bracket below the large pulley, fitted with a bearing for the 2.5mm diameter brass rod shaft. All shafts > 2.5mm are brass tubes with diameter increments of 0.5mm, each with a wall thickness of 0.2mm. The top plate contains a bearing fitting the 10mm tube diameter. A brass ring clamp is shown at the top to hold a horizontal rod and disk. It was discovered that when all such clamps were in place, the shaft tubes became over compressed, preventing free rotation.

On the right: To reduce shaft compression, the brass ring clamps were replaced by 1/8” thick press-fit rings made from black acetal copolymer, and half of the shafts were replaced with 304 stainless steel. Brass rods (1/16” diameter) join the clamps to white acetal disks (see diagram below). The pulley cords are also in place to couple the two pulley stacks, each cord welded from poly cord cut to length to achieve sufficient tension.

Performance

Front view forward and reverse:

Top view forward and reverse:

$\Rightarrow$ While basic operation is evident, the rough grouping and lack of exact alignment differs from the simulation.

Performance Analysis

• Despite numerous variations including adjustment of pulley tension, lubrication, and shifting from brass to SS tube shafts, the rotating disks do not fully regroup when the crank direction is reversed and fail to realign precisely. 

• The smallest pulley at the top with 10mm diameter shaft through the plate bearing was strongly coupled to the pulley shaft below it, so its ring clamp was removed (bottom of the disk stack). 

• Since the process to make and measure the pulley diameters was the same for the orbital and piston versions, it appears that cord slippage is a likely cause for the orbital configuration. 

• One explanation is that the lateral pull from the cords warps the concentric shafts, increasing the rotation friction which the pulleys cannot overcome. 

• At higher cord tension, there is greater torque from the cord, but more shaft friction due to bending. Less tension reduces shaft friction but also leads to cord slippage. Thicker tubes (0.5mm wall thickness for example) would probably work better but were unavailable in equal diameter increments to make 16 shafts. 

Alternative Designs

Planetary Designs

Another option that was considered was a planetary design, shown below. However, longer extension arms exaggerate errors in the pully rotation angle, increasing the alignment error. As with the current construction, concentric shafts are also subject to warping from the pulley cord tension, leading to cord slippage.

Pulley Extensions Design

To avoid the use of concentric tube shafts that lead to pulley cord slippage, this design uses a single rod shaft for the R pulleys and radial extensions to indicate the angle of each pulley. The top plate would be clear polycarbonate so the viewer can look down on the radial extensions to see the orbital patterns. This is a preferred alternative, and hopefully the subject of a future post.

The Wave Machines

Acknowledgements

Thanks to:

• Keith for numerous machinist tips.

• James for suggesting an orbital option.

• Barb for many constructive suggestions on the presentation.

• John for blogging guidance.