Bubbles on Waves
Jill Crisman
1. Overview
Our kinetic sculpture, Bubbles on Waves, is an abstract representation of the sun setting on the ocean, into
the waves. The interesting thing about this work is that it is a classic rolling ball sculpture but has a softer,
less mechanical look than most rolling ball works. The motion of the sculpture is integrated into the theme
of the sculpture. The sun to the left in Bubbles on Waves rotates clockwise; picking up clear acrylic balls in
its rays, and moves the balls to the top of the sculpture. The balls, or bubbles, then roll on the waves,
falling from wave to wave until they reach the bottom of the sculpture where they are picked up again by
the sun. For an interesting effect, we would like to distribute the balls evenly through the sculpture. The
sculpture will be made of birch plywood and wood stains will be applied to give the sun and waves a light
golden tone and a darker wooden tone for the backboard.
2. Detailed Analysis
One of the challenges of this sculpture is to design the waves such that the balls keep rolling over the wave
(i.e. are not caught in a valley) and are caught by the next wave. To create the shapes of the waves in the
sculpture, we create a simulation of the balls rolling on waves shapes. We adjust the shapes to get shapes
that are functional and aesthetically pleasing. We present the results of the simulation in Section 2.1.
To appropriately power the sculpture, we will design a simple mechanical motor system that obtains torque
from a falling weight and controls its angular speed using a damper. To choose the component of this
power system, we need to compute the torque necessary to drive the sculpture and the angular velocity at
which we want to the sculpture to operate to distribute the balls evenly throughout the sculpture. The
computation of the angular velocity is given in Section 2.2 and the required torque is computed in Section
2.3.
Finally, in Section 2.4, we describe the motor assembly and compute the components that need to match
the torque and angular velocity we desire for the sculpture.
2.1. Designing the Wave Shapes
To design the shapes of the waves, we want to use shapes that are easily drawn in Solidworks (to create a
template for the waves for fabrication) and in Working Model 2D (to test the motion in simulation.) We
discovered that the curved polygon shape in Working Model 2D creates the same shape as the spline shape
in Solidworks, and therefore, we chose this representation to create the shape of the waves.
Where the top and bottom waves overlap the sun, we want to wave to follow the path of the ball as it is
rotated on the sun. This will help the transfer of the ball between the sun and the waves. Therefore we
pick spline points that match the motion of the ball on the sun.
To create the splines for the other parts of the wave, we need to specify the vertices of the polygon. We
started by laying out a regular grid of points every 5 inches along the horizontal direction where the slopes
between the points had a regular pattern (e.g. down 1 ½”, up ½”, etc.) We then tried rolling a ball on these
shapes in Working Model and adjusted the vertices until the ball didn’t roll from the ends of the waves nor
did it get caught in the valley of a wave. The following figures show the vertex values that were selected
from this experimentation.
Top Wave (AB)
Hi X Hi Y dy
Lo X Lo Y
-5
7.5
0
9
1.5
-2
7.5
5
7.5 -1.5
3
6
10
7.5
0
8
6
15
6 -1.5
13
4.5
20
6.5
0.5
18
5
25
5 -1.5
23
3.5
30
5
0
Second Wave (CD)
Hi X Hi Y dy
Lo X Lo Y
40
5
35
2.5 -2.5
36
1
30
3
0.5
32
1.5
25
1.5 -1.5
27
0
20
1.5
0
Bottom Wave (GH)
Hi X Hi Y dy
Lo X Lo Y
40 -2.5
35 -5.5
-3
36
-7
30 -5.5
0
32
-7
25 -7.5
-2
27
-9
20
-7
0.5
22 -8.5
15 -8.5 -1.5
17
-10
10
-8
0.5
12 -9.5
5 -9.5 -1.5
7
-11
0
-10 -0.5
2 -11.5
7.5
7.5 17.5
Third Wave (EF)
Hi X Hi Y dy
Lo X Lo Y
11
0
16
-2
-2
15 -3.5
21
-2
0
19 -3.5
26 -3.5 -1.5
24
-5
31 -3.5
0
The picture on the front of this proposal shows Bubbles on Waves with these wave shapes.
2.2. Desired Output Angular Velocity
To get the desired angular velocity of our sculpture, we will compute the time that it takes for a ball to roll
from the top to the bottom of the sculpture. We can then equate this time with the time it takes for ½ of the
revolution of the sun (the time that it takes to lift a ball.) By keeping these quantities approximately equal,
we can keep balls rolling evenly throughout the sculpture. We compute the time for a ball to roll from the
top of the sculpture to the bottom by approximating the waves of the sculpture by ramps or inclined planes.
From physics, we know that if a ball of radius r and mass m rolls from the top of an inclined plane of
height, h, and length l:
r
l
h

that
1 2 1 2
mv  I
2
2
mgh 
Since the inertia, I, of the ball (or solid sphere) is:
I
2
mr 2
5
and the angular velocity  of the ball is:

v2
r2
then substituting:
mgh 
7
mv 2
10
Now the velocity of the ball at the end of the ramp can be solved from the above equation to be:
v
10
10
gh 
gl sin 
7
7
The center of mass of a rolling body moves with a uniform acceleration, a of
a
5
g sin 
7
but the total distance traveled (i.e. the length of the ramp) is:
l
so
1 2
at
2
t
2l
a
We can approximate the path of the ball in our sculpture by a series of four ramps and apply the above
equations. A representation of the linear model is shown below:
A
B
C
D
R
E
F
G
H
The circle represents the radius of the sun, R, without the waves that hold the bubbles. The other letters (AH) represent endpoints of the line segments of the linear approximations of the waves. We will assume that
the origin is at the center of the sun, and then the following table shows the actual values of the endpoints
of the lines and the calculations of the length and the angle of the ramp.
 (rad) a (m/s)
Wave
X1(in) Y1 (in) X2 (in) Y2 (in) l (in)
l (m)
t (sec)
AB
0
9
30
5
30.27 0.7687435
0.13
0.93
1.29
CD
40
5
20
1.5
20.30 0.5157201
0.17
1.21
0.92
EF
11
0
31
-3.5
20.30 0.5157201
0.17
1.21
0.92
GH
40
-2.5
0
-10
40.70 1.0337051
0.19
1.29
1.27
TOTAL
4.40
Therefore the total time for a single ball to roll down four ramps from rest is approximately 4.4 seconds.
From the Working Model simulation that we created (from Section 2.2) using the actual wave shapes of the
sculpture, we get a time estimate of 5 seconds. This is close enough to our approximation, that we will use
this value in designing the speed of the sculpture. Then we need a ½ of a rotation of the sun to take about 5
seconds, giving an angular velocity of approximately 6 rpm (as computed in the following table.
Time for ball to reach bottom
Angular Velocity
Angular Velocity
t
wo=0.5/t
wo=30/t (rpm)
5 sec
0.1 rev/sec
6 rpm
2.3. Torque Required by the Sculpture
In this section, we estimate the torque that is required to lift balls to the top of the sculpture.
Lifting Balls
To compute the amount of torque necessary to lift a ball, we can model the ball as a point mass located 9.5
inches from the axis of rotation (the radius from the center of the sun to the center of the ball.) Therefore
the torque required to lift a single ball is its weight multiplied by the radius to the axis of rotation. The
following table summarizes these calculations.
Density
Ball Radius
Volume
Weight of ball
Radius of ball motion
Torque
Torque of 4 balls
Torque with losses
d
r
V=4/3 PI r^3
W=V*d
R
t=WR
t=4WR
t=1.10(4WR)
0.043
0.5
0.523599
0.022515
9.5
0.21389
0.85556
1.026672
lb/in^3
in
in^3
lb
in
in-lb
in-lb
in-lb
From MatWeb, we find that the density, d, of extruded acrylic is 0.043 lb/in3 (1.19 gr/cc). We have
selected 1” clear extruded acrylic balls to be the bubbles in our sculpture and these have a radius of ½”.
We also know that the volume V of a sphere of radius r is:
4
V  r 3
3
Multiplying the volume by the density, we can compute the weight of the balls to be 0.0225 lbs.
We also know that the distance between the center of mass of the ball and the axis of rotation R is 9.5
inches. Therefore the torque is simply the weight multiplied by R and is equal to 0.214 in-lbs. If the sun is
lifting (or pushing) four balls about the same axis of rotation (a conservative estimation), then the total
torque is four times the torque for a single ball.
In addition, we will model frictional losses as an additional 10% torque required. Therefore, we will
assume that our sculpture needs to have 1.0 in-lb of torque to operate.
Inertia of the Sculpture
To compute the inertia and mass of the sun, we will approximate the sun as a short cylinder of height ¾”
and a radius of 10” which is conservatively, the outside radius of the sun. Therefore, the inertia, I, is:
I
1
MR 2
2
The equations to compute the actual mass and inertia of our sculpture are given in the following table.
Description
Density of Birch
Outside Radius of Sun
Thickness of Sun
Volume of Birch Sun
Weight of Sun
Mass of Sun
Inertia of Cylinder
Variable
d
R
t
V=PI R^2 * t
W=Vd
g
M=W/g
I=0.5MR^2
Value
0.0199
10
0.75
236.361945
4.703602706
386.088
0.012182722
0.609136091
Units
lb/in^3
in
in
in^3
lb
in/sec^2
lb sec^2/in
lb in sec^2
We use the density of birch wood found on MatWeb Birch Density. The radius of the sun we use for the
approximation is the outside radius of 10”. The sun is ¾” thick giving a volume of 236 in 3. The weight of
the sun is the volume multiplied by the density; therefore the weight of our birch plywood sun is
approximately 4.7 lbs. Since we are using English units, to compute the mass of the sun, we need to divide
the weight by the acceleration of gravity on earth. This gives us a mass of 0.0122 lb-sec2/ins. Therefore,
the inertia of the cylindrical approximation of the sun is 0.61 lb-in-sec2.
In our sculpture, we will not worry about the torque necessary to overcome the inertia. Since a viewer will
be winding to sculpture to start it, he or she could also give the sculpture a small push to overcome the
initial inertia to get it moving.
2.5. Summary
For our sculpture to move as desired, we need to have our mechanical motor assembly produce about 1 lbin of torque and rotate at 6 rpm.
3. Generating the Desired Motion
Finally, we look at a generic design of a mechanical motor system to produce the desired torque damped to
the needed angular velocity. At this time, we will use a simple weight on a string and damper motor
assembly to drive the sculpture.
In Section 3.1, we describe a simple torque and damper motor system and, in general, how compute the
parameters of the motor system to give the desired output torque and angular velocity to the sculpture. A
table of available rotary dampers from McMaster Carr is given in Section 3.2. Then we compute the
parameters that we need for Bubbles on Waves in Section 3.3.
3.1. Torque and Damper Motor
Let’s consider a very simple motor drive system that contains a source of torque (such as a falling weight or
a torsion spring), a rotary damper (purchased from a catalog), and the output shaft. Let’s also assume that
the torque source (S) is connected to the damper (D) through a gear train and that the damper is connected
via a gear train to the output shaft (O). A diagram of this system could be as follows:
The figure shows the bottom view of a motor assembly. A string (attached to the weight) would be wound
around the source torque spool (shown to the left) which is attached to a gear. This gear attaches to a gear
located on the shaft of the damper, which in turn connects to the shaft that drives the sculpture.
The torque source supplies the torque to be provided to the system. The damper constrains the angular
velocity of the system. The angular velocity of the damper has a linear relationship between the torques
applied to the damper, i.e.

Torque Rating
K = Torque Rating / Max Speed

Maximum Speed
This implies that the larger the torque applied to the damper, the faster it will run until it hits its torque
rating and its maximum speed. Running the damper at higher torques than the rating may cause damage to
the damper so that it will no longer function correctly. Since the damper constrains relationship between its
torque and speed, the damper defines the relationship between the output torque and output rotational
velocity. We will use this in selecting a damper for our system.
Now, we select the components (damper, gears, torque source) of the motor assembly system to give us the
desired torque and angular velocity on the output shaft for our sculpture. First, we need to compute the
torque and angular velocity needed to power the sculpture. Let’s call these known variables, o and o.
STEP 1: Pick gear ratio between damper and output
First we will pick the gear ratio between the damper and the output G do based on the desired output velocity
that we would like. We know that here is a maximum allowable angular velocity we can drive the damper
without breaking down the fluid in the damper, i.e.
 d   d max
So we will pick a gear ratio to make sure that we are operating at 75% or less of the maximum allowable
angular velocity by:
God 
0.75 *  d max
o
STEP 2: Pick the damper constant and damper
To pick a damper, we want to match the ratio of the desired output torque and velocity with the damper the
torque-angular velocity ratio, K. We know that the damper constrains its torque and angular velocity to
have the following relationship:
d
K
d
The relationship between the torque of the damper and the output torque is given by the gear ratio:
o
 God
d
Similarly, the relationship between the angular velocity of the damper and the output angular velocity is
given by the inverse gear ratio:
o
1

 d God
Therefore:
o
2 
2
 God d  God K
o
d
So we should choose the damper that has the K closest to:
K
o
2
God  o
STEP 3: Pick the gear ratio between the output and the source torque.
Finally, we choose the gear ratio between the output and the torque source to give the correct output torque
to the system, i.e.,
Gos 
o
s
3.2. Potential Dampers
For the McMaster-Carr rotary dampers,
 d max  10rpm
The following table summarizes the dampers available from McMaster Carr :
Torque Rating
in-oz
(specs)
0.28
0.56
0.83
1.39
2.78
4.17
6.94
13.89
17.36
20.83
Torque Rating
in-lb
(in-oz/16)
0.0175
0.035
0.051875
0.086875
0.17375
0.260625
0.43375
0.868125
1.085
1.301875
K
in-lb/rpm
(torque rating/wdmax)
0.00175
0.0035
0.0051875
0.0086875
0.017375
0.0260625
0.043375
0.0868125
0.1085
0.1301875
3.3. Calculations for Bubbles on Waves
The following table shows the procedure derived in Section 3.1 applied to the parameters of our sculpture.
Max angular velocity of damper
Desired angular velocity
Desired output torque
Gear ratio between damper and output
Choosen gear ratio
Damper constant
Choosen damper constant
Choosen damper torque rating
Gear ratio between source and output
wdmax
wo
to
God=0.75*wdmax/wo
God
K=to/(God^2*wo)
K
td max
Gos=to/ts
10
6
1
1.25
1
0.17
0.13
17.36
1
rpm
rpm
in-lb
in-lb/rpm
in-lb/rpm
in-oz
From these calculations, we can power our sculpture with a very simple drive system. We will create a
source torque from a weight on a string rotating a shaft. If we design this to deliver 1 in-lb of torque (what
we computed our sculpture needs to move) then we can eliminate the gear train and have the weight and
string mounted directly on the output shaft. In addition, since the speed at which we want to run our
sculpture is perfectly within the range of the operating speed of the dampers, we do not need a gear train
between the damper and the output shaft either. Therefore, for Bubbles on Waves, torque source, damper
can all share a single shaft (i.e. the output shaft.)
4. Fabrication
We will fabricate the prototype sculpture completely out of wood (with the exception of the damper,
bubbles and weight.) We will begin with the faceplate and a stand (for ease of testing without mounting on
the wall.) Then we will construct the sun and put it on a test shaft and test its rotation and lifting of the
balls. Next we will fabricate the waves, from top to bottom, testing the ball motion on each wave as it is
completed. Finally, we will design the motor assembly, adjusting the weight of the motor to create the
needed torque for the sculpture. The details of this fabrication are given in the next section.
4.1. Faceplate
The faceplate is 53x26” rectangle cut from a sheet of ½” plywood with a ¼” hole drilled for the sun’s shaft
12” from the top and left of the sculpture. We will build a stand to hold the sculpture upright during
construction.
4.2. Sun Assembly
We will cut two sun shapes from 1/4” birch plywood using a band saw. The following figure shows the
overall sun shape and the detailed shap of the each ray.
We will cut two of these sun shapes with rays, one for the front and back faces of the sun. We will also cut
a center disk for the sun that is a circle of radius 8” from a ¼” sheet of plywood. These three pieces will be
aligned and glued together. A 3/8” thick spacer will be added to the back of the sun to allow set screws
hole to be drilled for mounting on the shaft and to space the distance between the faceplate and the sun. A
¼” diameter hole will be drilled 1” into the center of the assembly to be later fit to the shaft.
4.3. Waves
The waves will cut from 2 sheets of 3/4” birch plywood.
To fabricate the waves, a template of these shapes will be printed and traced to cut the wave shapes from
two ¾” sheets of birch plywood. We will cut out the middle waves entirely, but cut only the top curve of
the waves for the top and bottom waves, cutting a straight horizontal edge as a guide on these pieces.
The top and bottom waves transfer the ball bubbles to and from the sun. Because of this, the waves and the
sun need to overlap. To accomplish this, we will hollow out the center of the top and bottom waves to
allow the sun to rotate freely in the center of these pieces. The following figure shows the hollowed wave,
the wave thickness, and the thickness of the sun from a top view.
Then we will hollow out these portions of the top and bottom waves using a band saw. Finally we will cut
the bottom shapes of the waves. A router will be used to put in a circular grove for the ball as shown in the
following cross-section then glue the front and back pieces of the wave together to get a cross section as
shown below.
After fabricating the waves, we can place each wave individually on the faceplate, testing to ensure that the
balls transfer correctly to and from the sun and roll freely through the sculpture.
4.4. Motor Assembly
A bottom view of the weight and damper driven motor assembly that we will use for our sculpture is shown
below.
The rectangle at the bottom of the figure is the back of the motor assembly. The two sidepieces are
mounted to the back of the motor assembly and will be screwed into the back of the faceplate of the
sculpture to hold the entire motor assembly in place. At the bottom center of the figure is the damper. A
coupling is mounted to the damper so that a setscrew or a pin can be attached to the shaft of the damper.
The other side of the coupling has a setscrew or pin to attach to the shaft of the sun. In addition, spool caps
will be fabricated to form a spool directly on the shaft for the winding of the string holding the weight of
the sculpture.
5. Budget
An estimated budget for prototype of the sculpture is:
Part
Purpose
Vendor
4' x 8' x 3/4" birch
plywood
4' x 8' x 1/2" birch
plywood
4' x 8' x 1/4" birch
plywood
Pine boards
Clear acrylic 1" dia
balls (10)
Damper
Waves
Home Depot
$45.00
Total
Cost
1 $45.00
Backboard
box
Sun
Home Depot
$35.00
1
$35.00
Home Depot
$25.00
1
$25.00
Misc
Bubbles
Home Depot
McMaster Carr
$5.00
$13.02
1
1
$5.00
$13.02
McMaster Carr
$28.13
1
$28.13
Home Depot
Home Depot
$2.00
$25.00
1
1
$2.00
$25.00
Motor
Assembly
1/4" dia dowel
Shaft
Misc: Screws, glue, Hom Depot
stain
TOTAL
Part Num Unit Cost
1383K58
#
$178.15
Since all of the smaller components are made from plywood, they will be made from the pieces of wood
remaining from cutting the larger pieces of the sculpture.