A Rocking/Rolling Half-Disk Vibratory System

Paper ID #18222

A Rocking/Rolling Half-Disk Vibratory System

Dr. Jeffrey C. Hayen, Oregon Institute of Technology

Jeffrey Hayen joined the faculty in the MMET Department at the Oregon Institute of Technology (OIT)in 2011. Before arriving at OIT, Jeffrey served as a Professor of Engineering, Mathematics, and Physicsat Southwestern Oregon Community College for 16 years. Prior to that experience, he worked in theaerospace industry as a thermodynamicist and propellant analyst for high-performance upper-stage rock-ets at the Space Systems Division of the General Dynamics Corporation. He also has conducted researchconcerning structural dynamics and control for the Kajima Corporation of Japan, and he currently providestechnical analyses and performs computational simulations for the United Launch Alliance in Denver. Jef-frey earned his B.S. and M.S. degrees in Mechanical Engineering from San Diego State University, andhis Ph.D. degree in Applied Mechanics and Physics from the California Institute of Technology.

c©American Society for Engineering Education, 2017


Introduction

A topic routinely covered in an elementary physics or (to a greater extent) mechanical vibrations

course is the case of an object exhibiting small-amplitude oscillations with no significant energy

dissipation (at least for several cycles of oscillation). This situation closely approximates simple

harmonic motion, and it enables a milder introduction to the study of vibratory systems. Because

appreciable damping mechanisms are not present, only the concepts/characteristics of amplitude,

period, and frequency are needed for the discussion of such motion.

An experiment has been devised for a mechanical engineering course (taught multiple times by

the author) which is utilized to demonstrate a vibratory system. The object examined consists of

a stable aluminum half disk that rocks/rolls back and forth if given an initial displacement from

its equilibrium position and released from an initial state of rest (see reference diagrams at the

end of this article). It is the inaugural experiment undertaken for this course, and it is typically

performed during the second week of an academic term since the necessary background material

is reviewed during the laboratory session of the first week. This activity lends itself well to this

placement within the laboratory schedule because it involves the unforced (or ‘free’) vibration of

a single-degree-of-freedom (SDOF) system that is almost undamped, so it essentially illustrates

simple harmonic motion. Also, it does not depend upon advanced concepts or principles which

are covered at later stages in the course, yet it is easy to perform and does not require expensive

equipment or elaborate instruments.

Before they perform the experiment, the students enrolled in this course are initially led through

the derivation of an equation governing the motion of the system, which is a non-linear ordinary

differential equation (ODE). However, because this derivation is based upon Rayleigh’s energy

method [1, 2], and because of the energy-conservation property of simple harmonic motion, the

essential parameter characterizing the vibration can be determined from this equation without the

need to formally solve the ODE obtained (although courses on rigid-body dynamics and ODEs

are prerequisites for this course). This important procedure is briefly outlined in the next section

of this article.

This derivation also provides an excellent opportunity for students to review and reinforce their

understanding of, as well as proficiency in calculating, such kinetic properties as the location of

the mass center G and the mass-moment of inertia GI for a non-trivial geometric solid object.

The students must utilize the results of these calculations directly with the equations involved in

order to obtain the formula for the theoretical undamped natural frequency nω of oscillation for

the rocking/rolling half disk:

n

8

(9 16)

g

Rω

π=

− (1)

From this formula, the theoretical period can be evaluated as well, and the experiment essentially

consists of comparing the actual period with the theoretical period over a time interval of about 6

to 8 oscillations for the half disk. The effect of the initial angular displacement for the half disk

is also investigated so that students become aware of the ultimately unavoidable consequences of

damping effects and the non-linearity in the governing equation.

The experiment itself can be conveniently conducted in a classroom with minimal apparatus: the

aluminum half disk, a clean dry table surface, a chronometer, and a printable/erectable reference

protractor that facilitates placement of the half disk at any one of several pre-determined initial

angular displacements (to be specified) prior to its release. It is anticipated that this experiment

will be briefly demonstrated during the formal presentation of this article.

Admittedly, just analyzing the rocking/rolling half disk is not a novel exercise [3–5]. However,

acquiring such an object and then testing its dynamic behavior in order to compare theoretically

predicted results with experimentally obtained results appears to be an original laboratory activity

involving the application of multiple topics covered in previous engineering mechanics courses.

Rayleigh’s Energy Method for Simple Harmonic Motion

Consider the case of a SDOF vibratory system whose (angular) displacement variable is denoted

by θ , and suppose that this system is given the initial conditions oθ θ= and oθ θ=� � at time 0t = .

Under the assumption that this system exhibits simple harmonic motion, θ is determined [6] by

1 n 2 ncos( ) sin ( )C t C tθ ω ω= + (2)

where, for the initial conditions stated above, it can be shown [7] that

1 o 2 o n,C Cθ θ ω= = � (3)

For the subsequent development below, it is very useful to recast Eq. (2) as

ncos( )A tθ ω ϕ= − (4)

A and ϕ respectively denote the amplitude and phase angle associated with the oscillation, and

it can be shown [8] via the cosine subtraction and tangent half-angle formulas that

1 212 2

1 2 1

1

2 tan if,

if

CC A

A C C A C

C A

ϕ

π

−

≠ − = + = +

= −

(5)

It should be mentioned that the expression for ϕ given in Eqs. (5) is quite convenient for general

initial conditions, and it does not require a ‘quadrant correction’ to be performed (which, when it

occurs, is a consequence of the range imposed by the universal definition for the inverse-tangent

function), as is often required when the traditional formula 2 1tan C Cϕ = is utilized. Next, from

Eq. (4), observe that

2 2 2 2 2 2 2n n n n

2 2n

sin ( ) cos( )[ ] [ ]{ }A t t

A

D

θ ω θ ω ω ϕ ω ϕ

ω

+ = − + −

=

�

�

(6)

where D is a constant. The motivation for considering this particular expression is based upon

the underlying property that energy is conserved for simple harmonic motion. Because Eq. (6) is

valid at every instant in time, it is realized that this constant may be evaluated at any convenient

time, such as 0t = when θ and θ� are supposedly known, in which case 2 2 2o n oD θ ω θ= +� . As will

be seen, if an equation of motion (typically obtained by accounting for total mechanical energy)

can be formulated for a vibratory system and subsequently manipulated into the form of Eq. (6),

then it is possible to identify the undamped natural frequency nω without formally solving this

equation. Taken together, these properties constitute Rayleigh’s energy method [1, 2].

The results for simple harmonic motion presented above are completely general, but it should be

acknowledged that oθ� will be precisely zero for the laboratory activity considered in this article,

especially since it would be very difficult with the apparatus utilized to intentionally produce or

accurately measure a specific value for oθ� . Thus, oA θ= , 2 2n oD ω θ= , and 0ϕ = for all cases

encountered in this experiment. Also, it will be subsequently assumed that o 0θ > , so oA θ= .

Laboratory Activity Guide for the Experiment

The laboratory activity guide developed for this experiment appears at the end of this article, and

it is primarily intended for students. However, it contains some useful resource information for

instructors who would potentially adopt this activity for a course with laboratory experiments, or

for a classroom demonstration conducted during a lecture session. This guide should be perused

at this stage of the article before proceeding with the discussions continued below.

Sample Data Collected from a Laboratory Session

The tables provided below contain sample data collected during a laboratory session along with

values obtained or predicted for the period P and frequency nω over a duration of n oscillatory

cycles. This data corresponds to an aluminum half disk for which 1.875 inR = and 1.375 inb = .

Table 1. Sample Data Collected for Half-Disk Oscillation

oθ n 1( )t∆ 2( )t∆ 3( )t∆ t∆

10° 6 2.99 s 3.01 s 3.00 s 3.00 s

20° 6 3.15 s 3.11 s 3.13 s 3.13 s

30° 6 3.29 s 3.26 s 3.23 s 3.26 s

Table 2. Natural Period and Frequency for Half-Disk Oscillation

oθ experP n exper( )ω n theor( )ω theorP Rel. Error

10° 0.500 s 12.57 rad/s 11.58 rad/s 0.542 s 7.75%

20° 0.522 s 12.04 rad/s 11.58 rad/s 0.542 s 3.69%

30° 0.543 s 11.57 rad/s 11.58 rad/s 0.542 s 0.18%

Note: Relative Error is evaluated as theor exper theor 100%[| | ]P P P− × .

Notice the trend in the deviations between the experimental and theoretical values of P and nω

as the amplitude oθ is altered. The differences should be the result of two principal influences:

• Departure from the small-amplitude assumption

• Overall effect of energy-dissipation mechanisms

Initially, it was a surprise to the author that smaller values of oθ yield larger deviations between

experiment and theory since the small-amplitude approximation is more closely obeyed for such

values. However, when performing the experiment, the half-disk motion is observed to dampen

out more quickly in cases of smaller values of oθ . It is currently believed that the larger angular

momentum (with peak value G n oI ω θ ) of the half disk in cases of larger values of oθ suppresses

the impeding effect of energy dissipation to a greater extent, bringing theory and experiment into

better alignment. In the case of o 30θ = � , Table 2 clearly shows that excellent agreement exists.

It is also important to mention that students in the laboratory are encouraged to conduct a number

of preliminary trials of performing the experiment until they gain confidence and proficiency in

the procedure before collecting the actual data upon which the final results are based. This effort

underlies the data displayed in Table 1.

Pedagogical Effectiveness and Learning Enhancement

As alluded to above, this laboratory activity is always well received with no apparent difficulties

for the students enrolled in the mechanical vibrations course taught by the author. Admittedly, it

involves a quite-simple experiment procedure and calculations. Nevertheless, laboratory session

experience has revealed that the students greatly appreciate a relatively-mild initial activity that

emphasizes the most basic concepts of vibratory systems: amplitude, period, and frequency. The

class members in the course generally consist of senior-level students who ideally work in groups

of three individuals to ensure their proper attention on and participation in the activity.

Many benefits emerge from the worthwhile and valuable efforts required of the students in order

to obtain the theoretical results which are then compared with the experimental results, and they

include opportunities to perform the tasks indicated below:

• Locate the mass center of a non-trivial geometric solid object.

• Evaluate the mass-moment of inertia for this same object.

• Apply the parallel-axis theorem to obtain this moment of inertia.

• Utilize an energy-based method to analyze rigid-body motion.

• Explore the dimensional consistency of a derived relation.

• Recognize the limitations imposed by simplifying assumptions.

The exercise in performing these tasks is just as important as conducting the experiment because

it has been observed by the author (when teaching dynamics and vibrations courses) that many of

these topics are not fully understood in the context of ‘paper-and-pencil’ problem solving efforts.

However, when incorporated into a laboratory activity, a stronger connection between theory and

practice is established within the minds of the students, which is no doubt due to the ‘hands-on’

nature of the activity together with a venue to review and reinforce concepts and principles which

already have been introduced, and to apply them to a physical system.

Another positive aspect of this experiment is that it offers a very accessible exposure to simple

harmonic motion in the laboratory. Although examples involving highly-idealized systems that

exhibit simple harmonic motion abound in textbooks, it is difficult to find an actual system that

behaves (approximately, to greater/lesser degrees) in this manner. The ‘key’ characteristic of the

rocking/rolling half disk is that it can roll without slipping, so frictional resistance due to sliding

motion is not induced, which significantly reduces the damping effects. Other energy-dissipation

mechanisms at play are briefly described in the laboratory activity guide provided below.

It should be mentioned that, on occasion in the laboratory, the half disk is observed to slip when

released from larger initial angular displacements. Subject to certain simplifying assumptions, it

is possible to perform an analysis to estimate the critical release angle at which slip is impending;

this analysis is provided at the end of this article for the benefit of interested readers. However, it

is normally not required to be addressed by students when the experiment is performed. In fact,

it has been found that the slipping phenomenon generally can be avoided when an ordinary plain

sheet of white paper is affixed to the support surface over which the half disk rocks/rolls.

As remarked in the laboratory activity guide, the result given in Eq. (1) for nω does not depend

upon ρ , b , or M . For future implementations of this experiment, it is envisioned that a variety

of half disks could be investigated which all share the same radius R but may be fabricated from

different materials (and even possess different uniform thicknesses). But, via Eq. (1), each disk

would still possess the same natural frequency nω , provided that the small-amplitude, negligible

energy-dissipation, and rolling-without-slipping conditions are nearly obeyed.

Final Remarks

The author would like to eventually expand the Theoretical Derivations/Relations section of the

laboratory activity guide to include a complementary derivation for obtaining the formula for nω .

This development would be strictly based upon the Newtonian method of analysis, and it would

yield the standard linearized differential equation governing simple harmonic motion:

2n 0θ ω θ+ =�� (7)

As with the energy method, nω can be identified from this equation without actually solving the

equation. With just over two decades of experience as an engineering, mathematics, and physics

educator, the author believes that there is an enduring benefit in exposing students to more than

one approach for achieving the desired objective (in this context, obtaining the formula for nω ).

The full development for this alternate analytical approach is included in the Newtonian method

supplement that also appears (with the laboratory activity guide) at the end of this article. These

appendices will assist instructors who adopt this activity for an experiment or demonstration.

References

[1] S. S. Rao, Mechanical Vibrations, 4th Edition, Pearson/Prentice-Hall, Upper Saddle River, NJ, 2004, pp.

134–139.

[2] W. J. Palm, Mechanical Vibration, Wiley, New York, 2007, pp. 131–135.

[3] F. P. Beer, E. R. Johnston, Vector Mechanics for Engineers: Statics and Dynamics, 6th Edition, McGraw-

Hill, New York, 1997, pg. 1239.

[4] B. H. Tongue, Principles of Vibration, 2nd Edition, Oxford University Press, New York, 2002, pg. 54.

[5] R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics, 14th Edition, Pearson/Prentice-Hall, Hobo-

ken, NJ, 2016, pg. 661.

[6] S. S. Rao, Mechanical Vibrations, 4th Edition, Pearson/Prentice-Hall, Upper Saddle River, NJ, 2004, pg.

128.

[7] S. S. Rao, Mechanical Vibrations, 4th Edition, Pearson/Prentice-Hall, Upper Saddle River, NJ, 2004, pg.

129.

[8] J. C. Hayen, “Trigonometric Treasures in Mechanical Engineering”, presented at the Oregon Mathematical

Association of Two-Year Colleges (ORMATYC) Conference, April 2013, Lincoln City, Oregon.

[9] R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics, 12th Edition, Pearson/Prentice-Hall, Upper

Saddle River, NJ, 2010, pg. 447.

APPENDIX A: LABORATORY ACTIVITY GUIDE

Mechanical Vibrations (MECH 480) Laboratory Activity 1 – An Experimental Investigation

Free Vibration of a ‘Nearly-Undamped’ SDOF System


Description

Consider a solid half-disk object of radius R , thickness b , and mass density ρ , as is depicted in

the diagrams displayed below; see Figures 1–3. When placed upon a horizontal support surface

and displaced from its equilibrium state, the half disk will manifest oscillatory excursions due to

the effect of gravity, which acts downward along the vertical direction.

An experimental investigation of the dynamic behavior of this object offers opportunities to:

• Analyze a vibratory system which may be modeled as a SDOF system.

• Observe a system that approximately exhibits simple harmonic motion.

• Perform free-vibration tests to obtain system response characteristics.

• Compare theoretically predicted and experimentally obtained results.

• Write a technical memorandum/report to document efforts and results.

Apparatus

• An aluminum half-disk object (to be provided by the instructor)

• A planar horizontal support surface that is rigid, clean, and dry

• A protractor (or other reference scheme, such as indicated in Figure 4) to precisely set the

angular displacement of the half disk when it is released from an initial state of rest

• A stopwatch or chronometer to accurately measure elapsed time

• A calibrated ruler (for distance) and a weight scale (for mass)

Preliminary Remarks

• For the theoretical model developed in conjunction with this investigation, assume that:

• All energy-dissipation mechanisms present may be appropriately neglected.

• The half disk has a well-defined radius R and a uniform thickness b .


• The half disk is comprised of a homogeneous material, which then implies that the disk

has a uniform density ρ , and that its mass center coincides with its geometric centroid.

• The half disk rocks/rolls over the support surface without slipping due to static friction.

• The half disk experiences small-amplitude angular oscillations, for which

6 6

rad radπ πθ− ≤ ≤ (A-1)

• Under this condition, it then can be shown that

sin

cos 1

θ θ

θ

≈

≈; 21

21 cosθ θ− ≈ (A-2)

Experiment Objectives/Tasks

• Perform free-vibration tests to collect data in order to experimentally assess:

• The natural period P and natural frequency nω of the system

• The effect of oθ on producing a non-linear system response

Experiment Procedure/Calculations

1. Measure the radius R , thickness b , and mass M of the half disk.

2. Calculate the total volume ∀ and mass density ρ of the half disk.

3. Perform separate tests based upon the initial displacements: o 10 , 20 , 30θ = � � �

• Conduct three (repeated) tests for each oθ considered.

• For each of these tests, strictly ensure that o 0θ =� .

4. For each test conducted, collect data for n and t∆ .

• n denotes the number of complete cycles of oscillation observed.

• t∆ denotes the elapsed time measured for these cycles to occur.

5. For each oθ considered, average the t∆ from the three tests to determine t∆ .

• This process reduces the effect of random errors on each t∆ measured.

• Selecting a larger n also reduces errors in the chronometer operation.


6. System Parameter Calculations – Experimental Case:

• P t n= ∆

• n 2 Pω π=

7. System Parameter Calculations – Theoretical Case:

• n 8 (9 16)[ ]g Rω π= − with 232.17 ft/sg =

• n2P π ω=

Theoretical Derivations/Relations (Assignment for the Student)

Perform each of the tasks described below in order to obtain the results indicated:

a) Evaluate d , which identifies the location of the mass center G for the half disk, which is

displayed in Figure 2. Recall from Engineering Mechanics (Statics) that d is obtained from

el0

1 r R

r

d y dmM

=

=

= ∫ (A-3)

in which M denotes the total mass of the half disk, along with

21

2

( )

( )

dm d b r dr

M b R

ρ ρ π

ρ ρ π

= ∀ =

= ∀ =; el

2 ry

π= (A-4)

where ely identifies the location of the mass center elG for a representative half-cylindrical shell

element displayed in Figure 1. Also, r is a polar coordinate utilized for this calculation. Then,

by completing the integration and simplifying the result, show that

4 (3 )d R π= (A-5)

b) Evaluate CI , which is the mass-moment of inertia of the half disk about an axis through the

disk center (of curvature) C and normal to the plane of the disk. Recall that CI is obtained from

2C

0

r R

r

I r dm=

=

= ∫ (A-6)

c) Utilize the parallel-axis theorem in the form 2C GI I M d= + to show that

2 21G 2

( )I M R d= − (A-7)


d) Utilize the energy (Rayleigh’s) method to obtain an equation of motion that governs the

angular displacement θ of the half disk as a function of the elapsed time t associated with the

vibratory motion. Begin with the conservation of mechanical energy equation:

2 2 1 1K U K U+ = + (A-8)

where the subscripts 1 and 2 respectively denote the release (initial) state and a representative

(general) state of the half disk during its vibratory motion. Since C and G are not fixed points

in space, it is necessary to evaluate the energy terms for the half disk as indicated below:

2 21 1G G2 2

G

(1 cos )

( )[ ]

K I M v

U M g y R d

M g d

θ

θ

= +

= − −

= −

�

(A-9)

Before applying these relations to evaluate the energy equation, observe from Figure 3 that

G C

G C

sin

cos

x x d

y y d

θ

θ

= −

= − (A-10)

Based upon these relations, show that

G

G

cos

sin

( )

( )

x R d

y d

θ θ

θ θ

= −

=

��

��

(A-11)

Based upon these relations, show that

2 2 2G G G

2 2 22 cos( )

v x y

R R d dθ θ

= +

= − +

� �

� (A-12)

Then, combine these (and previous) results into the energy equation to show that

2 23o2

2 cos 2 (cos cos )( )R R d g dθ θ θ θ− = −� (A-13)

where the initial condition o 0θ =� has been invoked. Next, utilize the condition and assumptions

indicated in Eqs. (A-1) and (A-2) to show that Eq. (A-13) approximately becomes

2 2 2 23o2

2( )R R d g d g dθ θ θ− + =� (A-14)

e) The equation indicated above approximately (but accurately) governs the vibratory motion

of the half disk for small-amplitude oscillations. This ordinary differential equation should not

be solved but instead should be recast into the standard form


2 2 2n Dθ ω θ+ =� (A-15)

where D is a constant (which should be identified in the derivation). Then, after substitution for

d with the expression obtained above, show that nω is therefore given by

n

8

(9 16)

g

Rω

π=

− (A-16)

Remarkably, nω depends upon only g and R (but not upon ρ , b , or M ). As requested above,

this formula is to be utilized for the theoretical evaluation of nω and P .

It should be emphasized that employing the energy (Rayleigh’s) method via Eq. (A-15) for the

identification of nω is strictly valid only for a vibratory system that behaves (or can be modeled)

as a simple harmonic oscillator.

The actual energy-dissipation mechanisms in this situation, which are always inherently present,

are primarily due to the effects of contact deformation, contact adhesion, and aerodynamic drag

(in the order of most-to-least significant), which will be further discussed in class. However, as

previously mentioned, these effects are to be neglected in this investigation.

Required Elements for Report

• Document the experimental investigation in a technical memorandum/report.

• Follow the report format described in the instruction guide separately provided.

• Organize data collected from each test conducted for each oθ within a suitable table.

• This table serves to portray the data collected in lieu of ‘raw data’ sheets.

• Compare the experimental and theoretical values of nω and P obtained for each oθ .

• Comment on the trend observed in the deviation between these values as oθ is altered.

• Be sure to include all requested derivations and results within the report.

• Provide a complete succession of steps for each of the derivations requested.

• Express each calculated result in terms of its appropriate dimensional units.

• Remember to word process the entire report (manual annotations are not allowed).

• Be sure to include a digital image of the experiment apparatus within the report.


Theoretical Derivations/Relations (Resources for the Instructor)

a) See below:

el0

210

2

32

2 20

0

1

1 2( )

4 4

3

4

3

( )

r R

r

R

RR

d y dmM

rb r dr

b R

rr dr

R R

R

ρ ππρ π

π π

π

=

=

=

=

= =

=

∫

∫

∫

b) See below:

2C

0

2

0

43

00

42 21 1

2 2

212

( )

4

4( )[ ]

r R

r

R

RR

I r dm

r b r dr

rb r dr b

b Rb R R

M R

ρ π

π ρ π ρ

π ρρ π

=

=

=

=

= =

= =

=

∫

∫

∫

c) See below:

2 2C G G C

2 2 2 21 1G 2 2

( )

I I M d I I M d

I M R M d M R d

= + ⇒ = −

⇒ = − = −

d) See below:

G C

G C

sin sin

cos cos

x x d R d

y y d R d

θ θ θ

θ θ

= − = −

= − = −

G

G

(cos ) cos

0 (sin ) sin

( )

( )

x R d R d

y d d

θ θ θ θ θ

θ θ θ θ

⇒ = − = −

= + =

� � ��

� ��


2 2 2G G G

2 2 2 2 2 2 2

2 2 2

2 cos (cos ) (sin )

2 cos( )

[ ] [ ]

v x y

R R d d d

R R d d

θ θ θ θ θ

θ θ

⇒ = +

= − + +

= − +

� �

� �

�

G (cos )

(1 cos )

( ) ( )y R d R d R d

d

θ

θ

⇒ − − = − − −

= −

After cancellation of the common factor M from both sides, utilization of the initial condition

o 0θ =� , and further simplification, the energy equation (still in an exact form) becomes

2 23o2

2 cos 2 (cos cos )( )R R d g dθ θ θ θ− = −�

But cos 1θ ≈ while 22 2

2 2 21 cos 2 sin 2( ) ( )θ θ θθ− = ≈ = , so that 2 2

o o2(cos cos )θ θ θ θ− ≈ − . Thus,

the energy equation can be approximately represented as

2 2 2 23o2

2( ) ( )R R d g dθ θ θ− = −�

e) Equivalently:

2 2 2n Dθ ω θ+ =� ;

2n 23

2

2 2n o

2

g d

R R d

D

ω

ω θ

=−

=

But the denominator of 2nω may be simplified (based upon the result for d ) as

223

22 (9 16)

6

RR R d π

π− = −

Finally (again based upon the result for d ), an expression for 2nω is obtained:

2n

8

(9 16)

g

Rω

π=

−

which then yields

n

8

(9 16)

g

Rω

π=

−

It is a worthwhile exercise to have students verify the dimensional homogeneity of this formula.

Also, when evaluating nω via this formula and the value provided for g , they will be confronted


with the need to perform a units conversion. An error periodically committed by students is that

they may forget to address the consistency and conformity of the units involved, which produces

a quite-different theoretical value for nω in comparison to its experimental value.

Even though the mass M and density ρ are not required for any of the foregoing derivations or

evaluations, they are requested so that the astute students will at some stage wonder “What am I

supposed to do with these parameters?” In the process, it is hoped that they will realize that nω

actually is independent of these quantities — and the significance of this fact. This characteristic

of the vibratory system can be vividly demonstrated if identical half disks are fabricated from a

variety of nearly homogeneous but different materials (e.g., metal, plastic, and wood) and caused

to oscillate, with each of these half disks manifesting the same natural frequency/period.

APPENDIX B: NEWTONIAN METHOD SUPPLEMENT

Mechanical Vibrations (MECH 480) Further Results for a Rocking/Rolling Half Disk

Obtained via the Newtonian Method of Analysis

Equations of Motion for Half Disk

(Consult the laboratory activity guide and Figure 3 prior to examining the developments below.)

Equation for Horizontal Component of Translational Motion

f G

G C C

G

2G

:

sin ;

cos

sin cos

( )

( ) ( )

x xF ma F M x

x x d x R

x R d

x d R d

θ θ

θ θ

θ θ θ θ

+→ = − =

= − =

⇒ = −

⇒ = + −

∑ ��

��

� ��

As a result, the friction force magnitude fF is determined by

2f sin cos( ) ( )[ ]F M d R dθ θ θ θ= − + −� �� (B-1)

Equation for Vertical Component of Translational Motion

n G

G C C

G

2G

:

cos ;

sin

cos sin

( )

( ) ( )

y yF ma F M g M y

y y d y R

y d

y d d

θ

θ θ

θ θ θ θ

+ ↑ = − =

= − =

⇒ =

⇒ = +

∑ ��

��

� ��

As a result, the normal force magnitude nF is determined by

2n cos sin( ) ( )[ ]F M d d gθ θ θ θ= + +� �� (B-2)

Equation for Rotational Motion about Half-Disk Mass Center

2 21G G f G n C G 2

G C G

: ( )

cos ; sin

( )M I F y F x x M R d

y R d x x d

α θ

θ θ

+ = − − = −

= − − =

∑ ��

or, equivalently,

2 21f n 2

cos sin( ) ( ) ( )F R d F d M R dθ θ θ− − = − �� (B-3)


After substitution for fF and nF from Eqs. (B-1) and (B-2), along with further manipulation and

simplification, Eq. (B-3) eventually becomes

2 232

2 cos sin 0( ) ( )( )R R d R g dθ θ θ θ− + + =�� (B-4)

Equation (B-4) exactly governs θ as an implicit function of t . However, this non-linear ODE is

prohibitively difficult to solve, so it is customary to introduce the small-amplitude approximation

in an effort to facilitate further progress in assessing the dynamic behavior of the rocking/rolling

half disk. But Eq. (B-4) is still exact and will be utilized later below for another purpose.

In accordance with the approximations indicated in Eqs. (A-2) from the laboratory activity guide,

Eq. (B-4) approximately becomes

2 232

2 0( ) ( )R R d R g dθ θ θ− + + =�� (B-5)

Equation (B-5) may be slightly further approximated in order to obtain a governing relation from

which the natural frequency nω of oscillation for the half disk can be readily identified. Recall

Eq. (A-15) from the laboratory activity guide:

2 2 2n Dθ ω θ+ =� (B-6)

which, since 2 2n oD ω θ= for the initial conditions considered, may be recast into the form

2 2 2 2n o( )θ ω θ θ= −� (B-7)

Based upon this relation and the small-amplitude approximation characterized by Eqs. (A-1) and

(A-2) of the laboratory activity guide, it is routinely assumed that the term involving the product 2θ θ� may be justifiably neglected in Eq. (B-5), which then yields

232

2 0( )R R d g dθ θ− + =�� (B-8)

or, equivalently,

2n 0θ ω θ+ =�� (B-9)

where 4 (3 )d R π= and the definition for nω is the same as indicated in Eq. (1) from the main

body of this article. It should be recognized that Eq. (B-9), obtained via the Newtonian method,

is completely equivalent to Eq. (B-6), obtained via an energy-based method — as can be verified

when the time-derivative operation is applied to both sides of Eq. (B-6) and the common factor

of 2θ� is divided out from both sides, yielding Eq. (B-9). Thus, it is possible with either of these

methods to identify nω for a simple harmonic oscillator from the principal equation that governs

its displacement without actually solving the ODE so obtained.


Minimum Coefficient of Friction

In order to ensure that the rolling-without-slipping condition (upon which this entire analysis has

been based) is enforced and maintained during the half-disk motion, it is necessary that

f nF Fµ≤ (B-10)

where µ denotes the static coefficient of friction between the horizontal support surface and the

outer (rolling) surface of the half disk. As revealed below, this restriction establishes a minimum

coefficient of friction minµ to produce the expected kinematics.

After substitution for fF and nF from Eqs. (B-1) and (B-2), the relation in Eq. (B-10) then can

be rearranged to isolate µ , requiring that

2

2

cos sin

cos sin

( ) ( )

( ) ( )

[ ]R d d

d d g

θ θ θ θµ

θ θ θ θ

− − +≥

+ +

��

� �� (B-11)

for o oθ θ θ− ≤ ≤ . Clearly, the expression appearing on the right side of this relation constitutes

minµ , and it may be equivalently expressed as

2

min 2

cos sin

cos sin

(1 ) ( )

( ) ( )

[ ]g

R

d d

d d

θ θ θ θµ

θ θ θ θ∗ ∗

∗ ∗

− − +=

+ +

��

� �� (B-12)

with 4 (3 )d π∗ ≡ . In order to evaluate minµ , expressions for 2θ� and θ�� in terms of the variable

θ are needed. Observe that Eq. (A-13) from the laboratory activity guide can be recast as

2 o

32

2 (cos cos )

2 cos

d g

d R

θ θθ

θ∗

∗

−=

−� (B-13)

Furthermore, Eq. (B-4) from above can be recast as

2

32

1 sin

2 cos

( )Rg

d g

d R

θ θθ

θ

∗

∗

+= −

−

�

�� (B-14)

where 2θ� in this latter result is to be evaluated via Eq. (B-13). Moreover, it is very convenient

to introduce the following auxiliary functions to facilitate the evaluation of minµ :

o

32

2o

( )

( )

( ) 2 cos

( ) ( ) ( ) ( )

( ) ( ) sin

Rg

rRg r

r d

q r r r

p r dθ

θ

θ θ

θ θ θ θ θ

θ θ θ θ

∗

∗

≡ −

≡ = −

≡ − =

�

��

(B-15)


When the functions indicated in Eqs. (B-15) are combined with Eq. (B-12), a simpler expression

for minµ is obtained as a function of θ alone:

min

cos ( ) sin ( )

cos ( ) sin ( ) ( )

(1 ) ( )

( ) ( )

d p d q

d q d p r

θ θ θ θµ

θ θ θ θ θ∗ ∗

∗ ∗

− −=

− + (B-16)

It should be recognized that, for a particular oθ , this result provides minµ at each displacement θ

of the half disk during the course of its motion. By computational exploration, it can be deduced

that the largest value of minµ always occurs when oθ θ= (as would be physically expected). In

this case, it is found that

o

4 4o o3 3

min 23 8 4o o2 3 3

1 cos sin

cos sin

( )

( )π π

θ θπ π

θ θµ

θ θ=

−=

− − (B-17)

Thus, requiring that o

min θ θµ µ

=≥ ensures that the rolling-without-slipping condition is enforced

and maintained for o oθ θ θ− ≤ ≤ . It is crucial for the reader to understand that this result:

a) is independent of R , b , g , and ρ (but applies only to a homogenous half disk);

b) has not been based upon the small-amplitude approximation; and, importantly,

c) is valid to the extent that all energy-dissipation mechanisms may be neglected.

Equation (B-17) reveals that o

min 0.065, 0.128, 0.186θ θ

µ=

= for o 10 , 20 , 30θ = � � � , respectively.

Prior to an experimental investigation of the rocking/rolling half-disk motion, a simple test could

be performed to determine the actual µ involved, which then may be combined with Eq. (B-17)

to determine an upper bound for oθ . However, as previously remarked in the main body of this

article, slipping does not occur for cases with o 30θ ≤ � if an ordinary plain sheet of white paper is

affixed to the support surface over which the half disk rocks/rolls.

As a final remark, it is important to emphasize that this result for minµ cannot be obtained via an

energy-based method because the forces associated with fF and nF serve only as constraints for

the half-disk motion but do not perform any actual work, so these forces do not appear within the

equation formulated by such a method. In contrast, the Newtonian method directly incorporates

these forces into the equations of motion assembled from the free-body diagram for the half disk

(see Figure 3), so expressions for fF and nF are available from those equations to subsequently

substitute into Eq. (B-10).

A remarkably similar problem that addresses the issue of potential slipping during the vibratory

motion of the half disk can be found in [9]. However, the analysis that appears in an associated

solutions manual offers less generality than the development and results given above.

A Rocking/Rolling Half-Disk Vibratory System

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