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Rolling Ball Sculpture as a Mechanical Design Challenge

Authors:

Abstract

Rolling ball sculpture, even the simple one, can be viewed as mechanical design challenge. If sculpture is made of poplar plywood, then bending and twisting of track causes stresses which can destroy rails of track. Another aspect is kinematic and dynamics of rolling ball. Sections of track where the rails is closer together will cause the ball to roll faster, but the ball is more likely to fall off the track. Centripetal force acting on the ball on spiral path increases own intensity with square of velocity and might cause ball to fall off. All these aspects will be analyzed in the paper.
7th International Conference
„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021
June 24-26. 2021 Sarajevo, Bosnia and Herzegovina
ROLLING BALL SCULPTURE AS A MECHANICAL DESIGN
CHALLENGE
Alma Žiga1, Derzija Begic-Hajdarevic2
1(University of Zenica, Faculty of Mechanical Engineering,
aziga@mf.unze.ba)
2(University of Sarajevo, Faculty of Mechanical Engineering,
begic@mef.unsa.ba)
-----------------------------------------------------------------------------------------------------------
ABSTRACT:
Rolling ball sculpture, even the simple one, can be viewed as mechanical design challenge.
If sculpture is made of poplar plywood, then bending and twisting of track causes stresses
which can destroy rails of track. Another aspect is kinematic and dynamics of rolling ball.
Sections of track where the rails is closer together will cause the ball to roll faster, but the
ball is more likely to fall off the track. Centripetal force acting on the ball on spiral path
increases own intensity with square of velocity and might cause ball to fall off. All these
aspects will be analyzed in the paper.
Keywords: rolling ball sculpture, stress analysis in plywood semicircle console,
kinematic and dynamics of ball rolling on a track.
-----------------------------------------------------------------------------------------------------------------------
1. INTRODUCTION
A rolling ball sculpture (sometimes referred to as a marble run, ball run, gravitram,
kugelbahn, or rolling ball machine) is a form of kinetic art. Even though a rolling ball
sculpture can range from simple to extremely complex, it is always grounded to the simple
movement; a ball rolling on a track. When ball is positioned at the top of the track, and is
let to go, the gravity becomes the motive force. Utilization of creative track designs to
harness the energy of the rolling ball leads to amazing things that can be achieved. The
rolling ball can be used to captivate, demonstrate, and educate on many levels. From small
single track sculptures, to room-filling complex installations, each sculpture works in
harmony with both the environment and with the experiences of the viewers [1]. Adding
the laser cut parts, gears, handle, slots for lifting the balls, the simple movement has been
elevated to a work of mechanical art.
Idea for sculpture, described in this paper, was conceived during watching YouTube
channel Build Amazing Big Marble Run Machine [2]. Design was obtained in CAD
software SolidWorks (Fig. 1.). All parts of sculpture were made from poplar plywood,
4 mm thick. After design and analysis all parts were cut from plywood sheet by laser
cutter (Fig. 2.). Dimensions of Front and Back plate are 280x290 mm. Plates are 16 mm
set apart. Between them is big toothed wheel with slots for balls to drop in when rolled
down the track. Big wheel meshes with three small gears, one of which has handle to set
big wheel in motion and to lift balls to the beginning of the track. The track forms left-
hand, helix spiral with two revolutions. Spiral track has central dimensions: radius 120
mm and height 160 mm. Cross section of rail is rectangle 8x4mm. Rails is spaced for 8
mm (Fig. 7c). Balls are made of steel and have diameters of 11.4 mm. For assembly, track
was made of four, half-revolution spirals. As the two rails of the track should be bent and
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka
7th International Conference
„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021
June 24-27. 2021 Sarajevo, Bosnia and Herzegovina
twisted to obtain 40 mm deflection, stress analysis should be done for inner rail, with
smaller diameter. Another design concern is distance between rails as it means ball
stability and ratio of its translational to rotational velocity. And the last analysis is dynamic
of ball rolling on track.
Fig. 1. SolidWorks design of rolling ball sculpture: 1-Front plate, 2-Back Plate,
3-Toothed wheel with slots, 4-Small gear, 5-Handle, 6-Track and 7-Ball
Fig. 2. Rolling ball sculpture
1
2
6
3
5
4
7
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika
7th International Conference
„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021
June 24-26. 2021 Sarajevo, Bosnia and Herzegovina
2. DEFLECTION OF INNER RAIL AS A HALF-CIRCLE CONSOLE
The deflection of the half-circle console end (at the point of application of the force P and
in the direction of the force, Fig. 3) will be determined, as it is done in papers: [3], Horibe
T. & Mori K. (2015), [4], Dahlberg T. (2004) and [5], Žiga A. et al. (2018). The force P
is normal to the xy plane.
Fig. 3. Half-circle console
Fig. 4. Moments equilibrium
Fig. 4 shows cross-section of the beam situated at angle
ϕ
. The bending moment Mb and
torsional moment Mt are acting at this cross-section. The shear force has been omitted in
the figure, since its influence on beam deflection can be neglected. The equilibrium of
moments is used, equations are obtained and solving for Mb and Mt gives:
[ ]
[ ]
cos ( )sin
( )cos sin
b
t
M Py R x
M P Rx y
ϕϕ
ϕϕ
=− +−
= −+ +
(1)
Elastic strain energy stored in the beam is:
22
00
t
11
dd
22
LL
bt
U Ms Ms
EI GK
= +
∫∫
(2)
Where: L is the length of the half-circle console, E is modulus of elasticity, G is shear
modulus, I is second moment of cross-sectional area, Kt is the cross-sectional factor of
torsional rigidity.
Using Castigliano’s theorem, the deflection of the console end, at the load P, can be
calculated:
00
11
2 d 2d
22
LL
bt
bt
t
MM
UM s Ms
P EI P GK P
δ
∂∂
= = +
∂∂ ∂
∫∫
(3)
With Mb, Mt,
Mb/
P and
Mt/
P from (1), deflection is:
(4)
Expressions for
sin ,cos ,d , ,d /dxsy y
ϕϕ
can be obtained by geometry (Fig. 3.) and
replaced in Equation (4). The integration over ds from 0 to L becomes integration over dx
x
x
y
R
y
ϕ
s
P
M
b
M
t
ϕ
dx
dy
ds
ϕ
τ
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka
7th International Conference
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June 24-27. 2021 Sarajevo, Bosnia and Herzegovina
from R to -R. With substitution of dimensionless integration variable t=x/R, the integrals
have limits 1 and -1.
3 32
2
11
11
(1 )1
1d d
(1 )
t
PR PR t t
tt t
EI GK t
δ
−−
−+ −
= −− + +
∫∫
(5)
33
1,5708 4,71239
t
PR PR
EI GK
δ
= +
(6)
Deflection of half-circle console consists of two parts: due to bending and due to torsion.
So in order to evaluate contribution of these parts to whole deflection, isotropic wooden
material will be considered. Poplar veneer has three modulus of elasticity and three values
of Poisson ratio. These values have been taken from the paper [6], Brezović, Mladen
Vladimir, J. and Stjepan, P. (2003). One mean value is calculated for modulus of elasticity
and Poisson ratio: E = 3633 MPa,
ν
= 0.35. Shear modulus is calculated by the expression:
G=E / [2(1+
ν
)] and has a value: G = 1346 MPa. Dimensions of half-circle, inner track are:
radius-R=112 mm and cross-sectional dimensions (bxh) - 8x4 mm. Axial moment of
inertia is
33 4
/12 8 4 /12 42.67 mm
x
I bh=⋅=⋅=
.
The factor of torsional rigidity is
33 4
2
0,229 8 4 117.25 mm
t
K cb h= ⋅ = ⋅⋅ =
, where c2 is
coefficient for rectangular bar in torsion, taken from the book [7], Beer, F. P. (2014),
Mechanics of Materials.
Solving equation (6) for unknown force, where, at the end of half-circle console, deflection
is
δ
=40 mm, gives force value of P=0.7117 N. There, part of deflection due to bending is
10.1 mm and part of deflection due to torsion is 29.9 mm. So for isotropic wooden half-
circle console, torsion has much more influence on deflection than bending.
Coordinates of the console clamp are:
, cos 112 mm, sin 0.xR yR
ϕπ ϕ ϕ
===−==
Using equation (1), bending and torsional moments in the clamp are:
0, 159.42Nmm
bt
MM= =
.
Bending and torsional stresses in the clamp are:
22
1
60 5.063 MPa
bt
bend torsion
MM
bh c bh
στ
= = = =
(7)
Where coefficient c1 depends only upon the ratio b and h. For b/h = 8/4 = 2, coefficient is:
c1=0,246.
With stress element
σ
x = 0,
σ
y =
σ
bend =0 and
τ
x =
τ
torsion = 5.063 MPa, principal stresses
are:
2
2
1,2
5.063MPa
22
xy xy xy
σσ σσ
στ
+−

= ± +=±


(8)
Ekvivalent, von Mises stress is:
22
1 12 2 8.77MPa
vonMises
σ σ σσ σ
= − +=
(9)
For numerical analysis of deflection, the inner rail was modeled and meshed (Fig. 5). The
left side was clamped and at the right side, vertical translation of outer, lower vertex was
set to be 40 mm. The study gave maximal, von Mises stress of the value 8.791 MPa, near
the clamp (Fig. 6).
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika
7th International Conference
„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021
June 24-26. 2021 Sarajevo, Bosnia and Herzegovina
Fig. 5. Boundary conditions and
meshing
Fig. 6. Von Mises stress in inner track due to
deflection
3. TRACK RAILS SPACING AND BALL KINEMATICS
The spacing between the two rails of the track is an optimization between the security of
the ball on the track and ball spin relative to its linear velocity.
Consider a ball rolling over a horizontal, frictional surface (Fig. 7a). Let vC be the
translational velocity of the ball's center of mass, and let
ω
be the angular velocity of the
ball about an axis passing through its center of mass. Consider the point B of contact
between the ball and the surface. The velocity vB of this point is made up of two
components: the translational velocity vC, which is common to all elements of the ball,
and the tangential velocity vt =
ω⋅
R due to the ball's rotational motion. Thus, vB = vCvt =
vC
ω⋅
R. Suppose that the ball rolls without slipping. In other words, suppose that there is
no frictional energy dissipation as the ball moves over the surface. This is only possible if
there is zero net motion between the surface and the bottom of the ball, which implies
vB = 0 or vC = vt or vC =
ω⋅
R. The ratio of translational velocity to the tangential velocity of
the bottom of the ball is: vC / vt =1.
However, if the point that the ball is rolling on changes to two points, the ratio changes. A
certain angle θ is subtended by the radius of ball contact point with the vertical (Fig. 7c).
Let
ω
be the angular velocity of the ball. Let vC be the translational velocity of the ball's
center of mass: vC =
ω⋅
b (Fig. 7b), where b is the height of the center of mass from the axis
connecting the points of contact. Consider the point B at the bottom of the ball. The
velocity vB of this point is made up of two components: the translational velocity vC and
the tangential velocity vt =
ω⋅
R due to the ball's rotational motion. Thus, vB = vCvt =
ω⋅
b
ω⋅
R. The ratio of translational velocity to the tangential velocity of the bottom of the
ball is: vC / vt = b / R = cos
θ
.
Sections of track where the rails were closer together would cause the ball to roll faster, at
the cost of stability, as the ball was more likely to fall off the track. Increasing the distance
between the rails would cause the ball to roll slower, but would increase the odds that the
ball stayed on the track.
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka
7th International Conference
„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021
June 24-27. 2021 Sarajevo, Bosnia and Herzegovina
Fig. 7. A ball rolling over a rough surface: a) one point contact b) two point contact c)
distance between rails.
For the given track, distance between rails is d = 8 mm. For radius of steel ball R = 5.7
mm, angle subtended by the radius of ball contact point with the vertical is
θ
= 44.56°.
The ratio of translational velocity to tangential velocity of bottom ball point is vC / vt =
b / R = cos
θ
= 0.71. This angle yields a good balance between security on the track and
translational velocity to tangential velocity ratio.
3. BALL DYNAMICS DURING ROLLING DOWN THE TRACK
Fig. 8. A ball rolling down: a) natural coordinate system b) all forces and moments
acting on the ball
Using d'Alembert's principle, rolling ball can be transformed into an equivalent static
system by adding the so-called "inertial forces" and "inertial torques" or moments [8]. The
ball can then be analyzed as a static system subjected to this "inertial forces and moments"
and the external forces and reactions. Assuming that friction is negligible, forces that act
on rolling ball are (Fig. 8b): inertial forces due to tangential and centripetal acceleration,
0.1
0.0
0.1
x,m
0.1
0.0
0.1
y,m
0.15
0.10
0.05
0.00
z,m
θ
θ
ω
v
C
b
R
ω
= v
C
/ b
v
t / vC = cosθ
vB= vC- vt
ω
v
C
R
ω
= v
C
/ R
v
t / vC = 1
v
t
v
t
C
B
a)
v
B
= v
C
- v
t
=0
b
R
c)
d
v
t
v
t
b)
B
C
v
B
B
N
T
F
a in
mg
F
cp
in
Fo
F
i
M
in
θ
θ
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika
7th International Conference
„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021
June 24-26. 2021 Sarajevo, Bosnia and Herzegovina
inertial moment due to angular acceleration, reaction forces due to contacts with inside
and outside rail, and active force of gravity.
The local, normal and tangential (natural) coordinate system for certain points along the
path are shown in Fig. 8a, using Mathematica software [9]. For given a parameterized path
r(s), definition of unit vectors in tangential, normal and binormal directions are:
'(s)
'(s) '(s) ''(s)
'(s) '(s) '(s) ''(s)
T
TN B
T
u
r rr
uu u
r u rr
×
= = = ×
 
 
  
(10)
Path as a vector is
0.04
(s) 0.12cost,0.12sint,
rt
π

= −


, where t is in interval
{ }
,0, 4t
π
for
two revolutions spiral.
Static equilibrium conditions in normal and binormal coordinate system are:
0 sin sin 0
0 cos cos . 0
in
N i o i cp
Bi o i B
F FFF
F F F mg u
θθ
θθ
Σ= + =
Σ= + + =

(11)
Fcpin is centripetal force,
2in
cp
F mv
ν
=
, where
ν
is a curve of the path:
3
'(s) ''(s)
'(s)
rr
r
ν
×
=

.
Velocity of ball can be calculated using energy conservation. The system is closed, so
energy must be conserved. Initially the ball is at rest, so at this instant it contains only
potential energy. When it travels along the track, it has potential energy and kinetic energy
(translational and rotational).
22
() 1
(0) (s) ( )
22
zz c
mvs
mgr mgr I s
ω
= + +⋅
(12)
Where m is the mass of the object, Ic is the moment of inertia about ball’s center of mass
2
1
2
c
I mR=
, rz(s) is the height of the center of mass at position s and b is the height of the
center of mass from the axis connecting the points of contact (Fig.7b).
Using the definition of angular velocity
()vs
b
ω
=
, we can relate it to v(s). Then, above
equation gives ball’s center velocity:
[]
2
2 (0) (s)
()
zz
mg r r
vs I
mb
=
+


(13)
Fig. 9a shows ball velocity vector as ball rolls down. It can be seen its increase in intensity,
due to conversion of potential energy. Fig. 10b shows values of velocity during two
revolutions of spiral path. The velocity square is a function of the vertical coordinate for
a given path point rz(s).
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka
7th International Conference
„NEW TECHNOLOGIES DEVELOPMENT AND APPLICATION“ NT-2021
June 24-27. 2021 Sarajevo, Bosnia and Herzegovina
Fig. 9. Ball velocity a) vector drawn on path b) intensity
Solving equilibrium conditions in normal and binormal directions (eq.11) gives intensity
of path reactions. These are: reaction on outside rail Fo and reaction on inside rail Fi.
Reaction on outside rail increases due to increase of centripetal force. Opposite, reaction
on inner rail decreases and near the end (3/4) of third half-revolution it becomes zero.
Because of this all third part of path was needed to be fenced as was shown on Fig. 2.
Fig. 10. Track reactions on outside and inside rail a) vectors drawn on path b)
intensity
4. CONCLUSION
Rolling ball sculpture gives possibility to apply knowledge that is being acquired during
Mechanical courses on technical faculty. Another aspect is artistic component of design.
Every student can express its vision of sculpture. Sculpture can be conceived as an idea,
modeled in SolidWorks, analyzed and then ‘digitally’ produced by laser cutter. Another
interesting aspect toward finished product is an assemblage. No matter how design is detail
and every aspect is analyzed, there are always some unpredicted circumstances that should
0.1
0.0
0.1
x,m
0.1
0.0
0.1
y,m
0.15
0.10
0.05
0.00
z,m
2
3
4
s
0.2
0.4
0.6
0.8
1.0
1.2
v, m
s
0.1
0.0
0.1
x,m
0.1
0.0
0.1
y,m
0.15
0.10
0.05
0.00
z,m
2
3
4
s
0.02
0.02
0.04
0.06
0.08
0.10
F, N
Fo
Fi
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžika
7th International Conference
„NEW TECHNOLOGIES, DEVELOPMENT AND APPLICATION“ NT-2021
June 24-26. 2021 Sarajevo, Bosnia and Herzegovina
be overcome. In this structure, third part of spiral track needed to be fenced, because ball
was jumping out of this part. Analysis was shown that in this part, inner reaction was
become zero due to centripetal force, so ball was in contact only with outer rail of track.
4. REFERENCES
[1] Boes, Eddie. “Rolling Ball Sculptures by Kinetic Artist Eddie Boes.” Eddie's Mind,
www.eddiesmind.com/.
[2] Mini Gear, (2018, July 14). Build Amazing Big Marble Run Machine - DIY.
YouTube. https://www.youtube.com/watch?v=LF0aDmlM5XU&feature=youtu.be
[3] Horibe, T., Mori, K. (2015). In-plane and Out-of-plane Deflection of J-shaped
Beam. Journal of Mechanical Engineering and Automation, 5(1), 14-19.
[4] Dahlberg, T. (2004). Procedure to calculate deflections of curved beams.
International journal of engineering education, 20(3), 503-513.
[5] Žiga, A., Cogo, Z., Kačmarčik, J. (2018). Out-of-plane deflection of J-shaped beam.
Proceedings of the 21th International Research/Expert Conference, TMT 2018.
pp.269-272.
[6] Brezović, M., Jambreković, V., Pervan, S. (2003). Bending properties of carbon
fiber reinforced plywood. Wood research (Bratislava), 48(4), 13-24.
[7] Beer, F. P., Johnston, R., Dewolf, J. & Mazurek, D. (2014). Mechanics of Materials,
7th Edition, McGraw-Hill.
[8] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John
Wiley & Sons.
[9] Wolfram, S. (1999). The MATHEMATICA® book, version 4. Cambridge university
press.
CORRESPONDANCE:
Alma Žiga, Ass. D.Sc. Eng.
University of Zenica
Faculty of Mechanical Engineering
St. Fakultetska 1
72000 Zenica, Bosnia and Herzegovina
E-mail: aziga@mf.unze.ba
Derzija Begic-Hajdarevic, Full professor
University of Sarajevo
Faculty of Mechanical Engineering
Vilsonovo setaliste 9
71000 Sarajevo, Bosnia and Herzegovina
e-mail: begic@mef.unsa.ba
Editors: Isak Karabegović, Ahmed Kovačević, Sead Pašić,Sadko Mandžuka
... When this intricate design is combined with children imagination, the magic watch is made. The paper is a part of the project Mechanical toys intended for Stem education, described in the references [1,2,3]. In this paper, the magic of opening the mechanical flower placed in the watchcase will be explained. ...
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In-plane and Out-of-plane Deflection of J-shaped Beam
  • T Horibe
  • K Mori
Horibe, T., Mori, K. (2015). In-plane and Out-of-plane Deflection of J-shaped Beam. Journal of Mechanical Engineering and Automation, 5(1), 14-19.
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  • Eddie Boes
Boes, Eddie. "Rolling Ball Sculptures by Kinetic Artist Eddie Boes." Eddie's Mind, www.eddiesmind.com/.
Build Amazing Big Marble Run Machine - DIY
  • Mini Gear
Mechanics of Materials, 7th edn
  • F P Beer
  • R Johnston
  • J Dewolf
  • D Mazurek