Practical Determination of Moment of Inertia

7/29/2019 Practical Determination of Moment of Inertia

1/20

International Journal of Mechanical Engineering Education 33/4

A project in the determination of the momentof inertia

Ron P. Podhorodeski and Paul Sobejko

Robotics and Mechanisms Laboratory, Department of Mechanical Engineering, University of

Victoria, PO Box 3055, Victoria, British Columbia, Canada, V8W 3P6

E-mail: [email protected]; [email protected]

Abstract Analysis of the forces involved in mechanical systems requires an understanding of the

dynamic properties of the systems components. In this work, a project on the determination of both the

location of the centre of mass and inertial properties is described. The project involves physical testing,

the proposal of approximate models, and the comparison of results. The educational goal of the project

is to give students and appreciation of second mass moments and the validity of assumptions that are

often applied in component modelling. This work reviews relevant equations of motion and discussestechniques to determine or estimate the centre of mass and second moment of inertia. An example

project problem and solutions are presented. The value of such project problems within a first course

on the theory of mechanisms is discussed.

Keyworks theory of mechanism projects; dynamics; inertia determination; physical testing;

approximate models

Introduction

Centre of mass and moment of inertia

This paper describes a project that is assigned within a first course on the theory ofmechanisms for mechanical engineering students. The project involves determina-

tion of the dynamic properties of rigid bodies in particular, planar mechanismswith rigid links both with experiments and with analysis of student-proposedapproximate models.

Analysis of the forces involved in mechanical systems requires an understandingof the dynamic properties of the systems components. For a planar mechanism with

rigid links, the properties which must be known prior to any dynamic analysisinclude the mass, the location of the centre of mass, and the moment of inertia

(second mass moment) of the links.The centre of mass of a rigid body is a point on which it will be balanced when

subject to the action of a gravitational field.Inertia can be described as an inherent resistance of an object of mass to accel-

eration. For linear motion, the inertial force opposes the bodys linear acceleration,

and it is equal to the negative of the product of the objects mass and its linear accel-eration. Similarly, for rotational motion, a body under angular acceleration will be

subject to a resisting inertial moment. This inertial moment is equal to the negative

of the product of the objects moment of inertia and its angular acceleration. Momentof inertia is a property of an object and for planar problems its use in equations ofrotational motion is analogous to using the mass property in the equations of linear

motion.


2/20

Understanding moment of inertia is not easy because it is not a property that canbe observed as directly as mass. However, the effect of the moment of inertia can

be deduced through an indirect measurement. In the student project described here,the moments of inertia of bodies are determined experimentally through measure-

ment of the time that a body takes to complete a swinging motion. Having obtainedthe period of oscillation, it is possible to calculate the moment of inertia of the swing-

ing body.

Modelling

Mechanical systems are modelled in order to evaluate and refine their design. Thisrefinement is based on an analysis of the models and a study of the effect of chang-

ing the design parameters. Modelling is only an approximation of reality, and theapproximation becomes better the more accurate the model. In the project, students

are expected to devise two models of increasing accuracy in order to see how theirmodels compare with experimental results.

Outline of the content of the remaining sections

First, the equations of motion for dynamic systems are briefly reviewed and the

reader is introduced to the theories used by the students in their project. Then anexample from project as given to the students is outlined: its purpose, potential pro-

cedures and the report requirements. The following section contains sample projectresults, consisting of two sub-sections, which correspond to the experimental and

the modelling results. The Discussion Section comments on the value of the projectin the curriculum, and the paper closes with conclusions.

Background

Overview

In this section, the equations of motion for dynamic systems are briefly reviewed,

and potential techniques for determination of the dynamic properties of bodies areintroduced. The experimental method for determining the moment of inertia to be

used by the students, namely the knife-edge method, is described in detail. The textsby Beer and Johnston [1], Hibbeler [2], Erdman, Sandor and Kota [3], Mabie and

Reinholtz [4], Martin [5], and Uicker, Pennock and Shigley [6] are the main refer-ences for the section.

Equations of planar motion

In this paper, we are concerned with the planar motion of rigid bodies subject to an

externally applied force and moment system. When dealing with planar motion, alltranslational motion is constrained to parallel motion planes and rotation occurs only

about the direction perpendicular to these planes. In general, there are three equa-

tions of motion to consider:

(1)

(2)f may Gy=f max Gx=

320 R. P. Podhorodeski and P. Sobejko



3/20

(3)

where thexandy directions lie within, and z is perpendicular to, the parallel planesof motion. In the above equations, fx and fy are the sums of all external forces

acting on the rigid body in thexandy directions, respectively, and MG is the sumof all moments (due to both forces and couples) as seen from the centre of mass, G.The other terms in the above equations are the mass, m, of the rigid body, the trans-lational acceleration of point G described by aGx and aGy, the angular acceleration of

the rigid body described by az, and its inertia, represented byIGzz.

Techniques for determining the centre of mass

Several approaches are possible for determining the location of the centre of massof an object. These approaches include: mathematical modelling (taking moments);

observation of symmetries; and resolution by direct physical measurement; or a com-bination of these techniques.

Taking moments

Consider the rigid body illustrated in Fig. 1. The mass moment (M) the body gen-erates about linexxdue to its mass, m, is:

(4)

where dm is a mass element, i.e., a thin slice of the rigid body perpendicular to

linexxand a and b are as shown in Fig. 1. The planar distance from the line xx

to the centre of mass ofdm is denoted by rm, and rG is the distance to the centre of

mass of the entire rigid body. Rearranging equation 4, we obtain:

(5)rr m

mG

ma

b

= d

M r m r mx x m Ga

b

= = d

M IG G zz z z= a

Determination of the moment of inertia 321


Fig. 1 Rigid body and its moment.


4/20

Thus, taking moments about two non-parallel lines gives the location of G for aplanar body.

Observation of symmetry

If a body is homogenous and is symmetrical about a plane, then its centre of mass(G) must lie on the plane of symmetry. Symmetry about two planes isolates G to a

line and about three unique planes to a single point.

Physical measurement

Fig. 2 shows a setup in which the location ofG of a rigid body (a connecting rodin this example) is found. First, observing symmetry has shown G to lie on the line

between the two rotational centres of the connecting rod. The location ofG is thendetermined from the scale readings by observing that the force of gravity that acts

on the mass (m) of the connecting rod goes through G and is equal to mg, where gis acceleration due to gravity. This force is balanced by the recorded support forces.

Taking moments allows the location of G to be determined. Note that the experi-ment could also be performed with a single scale and a rest.

Fig. 3 shows another experiment that can be used to determine a line on which G

must be located. Three forces act on the suspended body: the force of gravity, whichacts downwards through G, and the suspension forces, which act along the cables.

Observing the equilibrium requirement for a three-force body that all forces mustintersect a common point can be used to isolate G to a line. Suspending the link

from different points can be used to isolate G to a point.

Techniques for determining the moment of inertia

Several approaches are possible for determining the inertia of a body, or forestimating it. These approaches include: mathematical modelling or the use of

computer-aided design (CAD) software; assumption of an approximate model; andphysical measurement.



Fig. 2 Two-scale determination of the location ofG.


5/20

Mathematical modelling

Referring again to Fig. 1, the moment of inertia (I) of a body about a line is definedby:

(6)

The parallel-axis theorem states that ifIG is the moment of inertia about an axispassing through a bodys centre of mass, G, then the moment of inertiaIpa about a

parallel axis offset a distance dfrom G is equal to:(7)

where m is the mass of the body. The parallel-axis theorem is used extensively whenmodelling the links by the method of composition.

Approximate models

If a bodys inertial properties cannot be easily modelled precisely, an approximatemodel will be assumed. For example, a slender rod is often assumed as a first approx-

imation for a long member of fairly uniform mass distribution. However, often abetter assumed model can be formed by a composite assumption. In an assumed

composite representation, basic elements are used, such as slender rods, pointmasses, spheres, discs, plates, and so on. The moment of inertia for the entire modelis the sum of the moments of inertia of its composite parts, transferred using the

parallel-axis theorem. In Fig. 4, three approximate models of increasing accuracyare presented for the connecting rod.

Physical measurement: knife-edge method

One experimental approach for determining the moment of inertia of a body is the

knife-edge method. This involves oscillating the body suspended by a knife-edge.Fig. 5 shows a link that is oscillating on a knife-edge.

Fig. 6 shows a schematic of the dynamic systems considered in the student project(see below): a rigid body (a planar mechanism link) suspended on a point, O. Gravity

I I d mpa G= + 2

I r mx x ma

b

= 2d



Fig. 3 Suspension-based isolation ofG.


6/20

acts in the downward direction. There are no forces or moments in or out of the

page. For the planar link show in Fig. 6, summing the moments about the fixedcentre, O, yields the following equation of motion:

(8)M IO O= a



Fig. 4 Examples of approximate models of the connecting rod.

Fig. 5 Link on a knife-edge.


7/20

which is expressed in terms ofq (see Fig. 6) as:

(9)

where ris the distance from the suspension point O to the centre of mass, G,IO isthe moment inertia about O (the unknown), and g is the magnitude of accelerationdue to gravity.

For small angles, sin(q) q, and dividing byIO, equation 9 becomes:

(10)

This is a second-order linear differential equation with a general solution [7] of:

(11)

where A and B are constants. Considering the boundary conditions, q = qmax at

t= 0 and the constant valuesA = 0 andB = qmax are found, yielding:

(12)

The link is therefore swinging in a sinusoidal motion, which repeats after

tmgr

IO

=2p

q q=

max

cosmgr

It

O

d

d

q

t= 0,

q = + A mgr

It B mgr

It

O O

sin cos

d

d

2

20

qq

t

mgr

IO+ =

( ) =mgr I t

Osin qqd

d

2

2



Fig. 6 Swinging link in the knife-edge experiment.


8/20

yielding the period of oscillation, T:

(13)

By measuring the period of oscillation of the link, the links moment of inertia aboutpoint O can be determined by squaring and rearranging equation 13:

(14)

By utilising the parallel-axis theorem, IG of the link can be determined from IOand a known location of the centre of mass. That is, sinceIO =IG + r2m, where risthe distance from G to O, the expression forIG becomes:

(15)

Example of the project

The project that the students taking the course on the theory of mechanisms com-

plete in the laboratory is now presented. The purpose of the project is twofold: toexpose students to ideas for determining the dynamic properties of rigid bodies, and

to give the students an idea of modelling and its limitations.

Project background

The students are given four links such as the ones shown in Fig. 7. These four linkswere removed from a reconfigurable mechanism testbed. The testbed is used to allow

students to evaluate designed mechanisms, including quick-return four-bar mecha-nisms featured in other laboratory projects of the course [8].

All links are identical except for their length. Fig. 8 details a typical link and itscomponents. The students are asked to determine the moment of inertia about the

centre of mass (IGzz

) of the four links by experimental methods and by modelling.The students are to propose two models of different complexity for the links and

then compare the model results to the experimental results.Each link consists of two 1/2stainless steel tubes held in aluminium blocks. The

aluminium blocks are of one of two types: a bearing housing or a fixed shaft block,

as shown in Fig. 8. The shaft block allows the coupling of the link to a motor shaft,thereby allowing the link to transmit power. A link can also be made to function as

a completely passive member of a mechanism simply by constructing it from twobearing blocks. The blocks are designed to be clamped onto the stainless steel tubes

by four steel bolts. The bearing is of a roller-bearing type, and it has an extended

hub to which the free-rotating shaft can be secured with two set-screws. The linkdesign is very simple, allows easy adjustment of link length, and provides a highstiffness in the plane of the link. The link lengths are changed each year, effectivelycreating new experimental and model results.

I I r m mrT

g rG O= =

22

2p

I mgrT

O = 2

2

p

TI

mgr

O= 2p




9/20



Fig. 7 Mechanism links to be analysed.

Fig. 8 Mechanism link and its components.


10/20

The first requirement of the project is to design an experiment in which the linksdimensions and dynamic properties are measured. The results of the experimental

section of the project are to be taken as reality, and to them the results of approxi-mate modelling are compared.

Experiment

The students are given some rudimentary instrumentation in order to perform theexperimental part of the project. They have at their disposal: a tape measure(1mm), a stopwatch (0.01s), two digital scales (0.2g), cables and a frame fromwhich to suspend the link mechanisms, and the knife-edge apparatus. The studentsare to design and carry out a procedure for finding the centre of mass, G. The moment

of inertia of the four links is to be found by the knife-edge method.

Approximate modelsAs the second part of the project, the students are asked to propose two approximate

models that will represent the physical links. It is suggested that the first model bea simple one, such a slender rod. The second model is to be a more detailed (or verydetailed) representation of the physical link. For the second model, the use of CAD

packages has been allowed, resulting in very sophisticated and realistic models beingproposed (see Fig. 8, for example).

Results

For the four links, the moment of inertia is to be calculated with the period of oscil-lation obtained using the knife-edge method. The moment of inertia of each link is

to be calculated using the two proposed approximate models. The model results areto be compared with the results obtained by physical measurement.

Discussion points

The students are asked to discuss several issues related to the experiment and the

modelling. The main part of the discussion is focused on the comparison of the mod-elling and the experimental results. Also, with regard to the experiment, the students

are asked: to describe the assumptions used in the knife-edge experiment and theireffect on the accuracy of the results; to discuss for which links the experiment would

be the most accurate; and to give possible changes that could be made to improvethe accuracy of the experiment. With regard to modelling, the students are asked: tostate the assumptions made during the choice of the models; to discuss when and

why the models are going to be the most and the least accurate; and to make sug-gestions for improving the accuracy of the (better) model.

Example project solutions

The student project outlined above has been carried out at the University of Victoriaduring each of the past 10 years. An example from project carried out by the authorsis presented in this section. The experimental procedure used to generate the reportedresults and other techniques applied by students on the course are described. Five




11/20

approximate models are proposed which are indicative of the models that the students

have been proposing. A discussion of the experiment and the modelling is alsoincluded.

Experimental results

Link dimensions and reference coordinate system

The links are shown in Figs 7 and 8. Fig. 9 shows a simplified representation of a

link, along with the dimensions and coordinate system used for the measurementsand analysis. Note that in each link, both rods are of the same length, dL, and bothtouch the x-axis of the link coordinate system. The y-axis is positioned halfway

through the link width-wise and it runs through the centres of both the bearing andthe shaft blocks. The distances dBB and dSB locate the centres of the bearing and the

shaft blocks, respectively. Point O denotes the location of the suspension point thepoint where the hanging link contacts the knife-edge.

Physical measurements

The links are made out of five distinct parts: two rods, two blocks with bolts and abearing. The dimensions and masses of these components, as they have beendesigned and manufactured, are listed in Table 1. Note that the link components have



y

x

Fig. 9 Link dimensions.


12/20

been designed in imperial values, but equivalent metric values have been reported.The varying dimensions and masses of all four links are listed in Table 2.

Other dimensions of note are: dRods, the distance the steel tubes are spaced apart(0.076m); and dS, the shaft blocks centre hole diameter (0.0127m). After the centre

of mass, G (dimension dG), is found, the dimension dS is needed to find r, the

distance from the suspension point O to point G,

Determination of the moment of inertia about the suspension point (IO) for eachlink is a three-step process: first, the mass of the link is found; then the location ofthe centre of mass (dG) is determined; and finally the moment of inertia about the

suspension point is found by the knife-edge method. The moment of inertia aboutthe centre of mass (IG) is then calculated using the parallel-axis theorem.

The location of the centre of mass was found by first observing symmetries, thusisolating G to lie on they-axis of the links coordinate system (see Fig. 9). Note that

the location ofG along thez-axis is not relevant to the problem at hand. The linkswere then simply balanced on the knife-edge such that the link was horizontal, asshown in Fig. 10. In this way the location ofG was measured directly by marking

where the links rods were touching the knife-edge and then measuring the distancebetween the mark and the bottom of the link. Students often determine G using the

two-scale experiment shown in Fig. 2.The period of oscillation of each link was then determined by recording the time

for the link to complete 50 swings, and calculating the average time for a singleoscillation. This averaging diminishes the error due to the human operation of the

stopwatch. The timing is discussed with students, who will typically initially try totime a low number of oscillations (e.g. 10), with one person observing the swingsand shouting Stop to another person with the stopwatch. In order to diminish the

error due to the linearisation of the governing differential equation (the assumption

r dd

dSBS

G= + 2.



TABLE 1 Masses and dimensions of link components

Component Mass Dimensions Symbol

Bearing block (with bolts) 0.138kg 0.1143m 0.05715m mBBearing 0.980kg 0.03969m mbShaft block (with bolts) 0.207kg 0.1143m 0.05715m mSBRod 0.44kg/m 0.0127 m

TABLE 2 Dimensions of links and their overall masses

Link Length (dL, m) dBB (m) dSB (m) Mass (mL, kg)

1 0.153 0.0375 0.1145 0.582

2 0.203 0.0365 0.1645 0.6243 0.457 0.0380 0.4185 0.846

4 0.610 0.0375 0.5715 0.980


13/20

sin(q) q), the swings were made very small. Basically, the link was set swingingsuch that there was just enough movement for observation to be made. Students are

asked their opinion on what would be an appropriate swing amplitude for their tests.After the period of oscillation was found, the moment of inertia about G was cal-

culated with the inertia expressions of equations 14 and 15. Table 3 lists the resultsof these experiments. The errors in the values for the moment of inertia listed in the

last column of Table 3 have been determined by varying the experimental mea-surements across their tolerance range and monitoring the minimum and maximumvalue for the moment of inertia.

Fig. 11 plots the experimentally obtained moment of inertia,IG, of the four links,along with the error bars for the authors results. Student results (for the year of the

tested link lengths) which were within one standard deviation of the authors resultsare also indicated. For the 24 student groups, 22 (92%), 20 (83%), 19 (79%) and 18

(75%) of the student groups were within one standard deviation of the authorsresultsfor the shortest through the longest links, respectively. Groups with outlying results

tended to have made fundamental calculation errors and not experimental errors.Fig. 11 shows that expected results were obtained. Note that the moment of inertia

for each link varies quadratically with the length of the link (recall that moment of

inertia has dimensions of [mass] [length]2).



Fig. 10 Balancing the link to findG.

TABLE 3 Experimental results

Link dG (m) r(m) Tavg (s) IG (kgm2) Error ( kgm2)

1 0.075 0.046 0.655 0.0016 0.000032

2 0.097 0.074 0.758 0.0032 0.000048

3 0.221 0.204 1.176 0.0240 0.000314

4 0.294 0.284 1.367 0.0501 0.000581


14/20

In relative terms, considering measurement tolerance ranges, the authors results

vary by an average of1.48%. Note that, in absolute terms, the maximum varianceoccurs for the longest link, at 0.0006kgm2. However, the relative variance is thegreatest for the shortest link (1.95%).

Modelling

The following models, illustrated in Fig. 12, are based on ones designed by the stu-dents over the years to approximate the links. The models were analysed with the

aid of the sum-of-moments equations, the superposition principle, and the parallel-axis theorem.

Assumptions and common terms

The following are the assumptions made in all of the proposed models:

The location ofG is assumed to lie on the line intersecting the bearing block and

shaft block centres (i.e., finding G is a one-dimensional problem). Symmetries

of the links construction were used in this assumption. The finding of the moment of inertia is a two-dimensional problem. The thick-

ness of the link, and any varying features of the link along thez-axis, are assumednot to affect the quantityIGzz.



Fig. 11 Experimental results with the fow links (#1 to #4).


15/20

The density of the components is assumed to be uniform.

The steel tubes are approximated as solid rods. For the more accurate models, when modelling the bearing and shaft blocks, their

holes, grooves, and steel bolts are not modelled; instead, each block is modelledas a flat plate of uniform density and equivalent block dimensions and mass.

In the equations presented in the following sections there are some frequently usedsymbols. the overall mass and length of a link are denoted by mL and dL, respec-

tively. The mass of the bearing block together with the bearing is denoted by mBb:

(16)m m mBb B b= +



Fig. 12 Proposed models (ae) for the link.


16/20

The vertical distance from they-axis to the centre of mass of the tubes is denotedby dR:

(17)

The dimensions of the bearing and shaft blocks are w (width) and h (height).When a term is annotated with subscript letter(s) in parentheses the letters

indicate the model(s) in which the term is used, ae, as set out in Fig. 12 and below.

Models

(a)Extremely simple model uniform slender rod. With this model, the entire linkhas been replaced with a uniform slender rod of mass mL and of length dL. The centre

of gravity for this slender rod is halfway up the link, i.e., and the momentof inertia is given by:

(18)

(b) Very simple model combined slender rod with two point masses. The effect ofthe different weights of the bearing and shaft blocks is completely ignored in the

previous model. This second model fixes this serious omission by replacing thebearing and shaft blocks with appropriate point masses. A single slender rod approx-

imates both tubes; their combined mass, mR, is calculated from the total link massby subtracting the block masses:

(19)

The centre of mass of the very simple model can be found by taking moments:

(20)

The composite moment of inertia of the link is:

(21)

where the moment of inertia of the single combined slender rod abut its centre of

mass is

(c) Simple model combined slender rod with two plates. This model is similar tothe previous one except that, instead of point masses, the bearing and shaft blocks

have been replaced with rectangular plates of equivalent masses and outside dimen-sions. The bearing and shaft blocks have moments of inertia about their centres of

mass of and respectively. The centre of

mass, dG, is determined with equation 20. After applying the parallel-axis theorem,the composite moment of inertia is:

I m w hGSB

SB= +( )

122 2 ,I m w hG

Bb

Bb= +( )

122 2

I m dG R LR =1

122 .

I I m d d m d d m d dG G R R G SB SB G Bb G BBb R( ) = + ( ) + ( ) + ( )2 2 2

dm d m d m d

mG

R R SB SB Bb BB

Lb c d e, , ,( )

=+ +( )

m m m mR L Bb SBb c,( ) =

I m dG L La( ) =1

122

d dG La( ) =

1

2 ,

d dR L=1

2




17/20

(22)

(d) Complex model two slender rods with two plates. The complex model is an

evolution of the simple model (c): the tubes are now more faithfully represented as

two slender rods, each of mass mr, given by:

(23)

The distance dG is calculated with equation 20. Note that the distance from G to the

centre of each tube is:

(24)

where dRods is the distance between the centres of the tubes.The composite moment of inertia of the link is:

(25)

where is the moment of inertia of one of the tubes about its centre of

mass.

(e) Very complex model two slender rods with two plates and a point mass. This

model is a refinement of the complex model (d), with the bearing block split intotwo elements: a plate representing the bearing block without the bearing mass, and

a point mass representing the bearing. The distance dG is still calculated with equa-tion 20. The composite moment of inertia of the link is:

(26)

where is the moment of inertia of the bearing block plate about

its centre of mass.

Results

The moments of inertia of the link models described above are summarised in Table4. Fig. 13 compares the experimental with the modelling results. From Fig. 13,

several observations can be made:

The extremely simple model (a) gives results 3040% below the experimental

results. A simple slender rod is clearly not sufficient to model the distributed

masses of the actual links. The very simple model (b) is very inaccurate for the shorter two links. For the

shortest link it is even less accurate than model a. However, for the longer two

lengths, model b returns results within 5% of the experimental results.

Im

w hGB

B= +( )

122 2

I I I I m d m d d

m d d m d d

G G G G r r SB SB G

B G BB b G BB

e r SB B( )= + + + ( ) + ( )

+ ( ) + ( )

2 22 2

2 2

I m dG r Lr =1

122

I I I I m d m d d m d dG G G G r r SB SB G Bb G BBd r SB Bb( ) = + + + ( ) + ( ) + ( )2 22 2 2

dd

d dr

RodsR G

d e,( )

= + ( )2

22

m m m m mr L Bb SB Rd e,( ) = ( ) =1

2

1

2

I I I I m d d m d d m d dG G G G R R G SB SB G Bb G BBc R SB Bb( ) = + + + ( ) + ( ) + ( )2 2 2




18/20

The simple model (c) is almost as accurate as the complex model (d) for the

shorter links. The simple model (c) underestimates the moment of inertia for allfour links, with the worst case (the shortest link) being within 6% of the experi-

mental value. This equals the accuracy of the complex model (d), with its worstcase (also being the shortest link) overestimating the inertia by 6%. Note,however, that the complex model (d) is approximately twice as accurate (with

results within 1% of the experimental values) as the simple model (c) for the twolonger lengths.

The very complex model (e) is the most accurate of the consideredrepresentations.



ExperimentalResults Range

Mod

el'srelativeerrorwithrespectto

theexperimentalresult(%)

Fig. 13 Comparison of the modelling results with respect to the experimental results for

all fow links.

TABLE 4 Modelling results for 1a (moment of inertia) (reported in units of kg m2)

Link Experimental result Model a Model b Model c Model d Model e

1 0.0016 0.0011 0.0009 0.0015 0.0017 0.0016

2 0.0032 0.0021 0.0024 0.0030 0.0033 0.0032

3 0.0240 0.0147 0.0230 0.0236 0.0242 0.0241

4 0.0501 0.0304 0.0482 0.0488 0.0496 0.0494


19/20

Discussion

On the laboratory and the format

The laboratory used for the theory of mechanisms class has a reconfigurable mech-

anism testbed which allows the construction and running of different mechanismtypes, including various forms of four bars and six bars. These mechanisms areassembled using the very links that are analysed in the project. In addition, the lab-

oratory has several personal computers running simulation software, includingGNLINK [9], CAMPRF [10] and Working Model [11]. Also found in the laboratory

is a cut-away five-speed manual transmission used in a gear-train analysislaboratory.

Students are divided into groups of three for the laboratories. Each group has

access to the laboratory for approximately one and a half hours per week. Over the13 weeks of the term, students are currently scheduled to complete the following

four laboratory projects: design and analysis of a quick-return mechanism; approx-imate modelling and physical determination of inertial properties; design and analy-

sis of cam and follower systems; and observation and calculation of gear reductionratios. The timing of the specific laboratory projects coincides very well with the

material coverage in the course.

On the manual for the inertia determination project

The laboratory manual for the project basically consists of the information foundwithin the introductory sections of this paper. While the background remains

the same, the link lengths and components are varied from year to year of thelaboratory.

On the curriculum content of the theory of mechanisms course

Having a laboratory manual that briefly outlines different possible techniques,leaving the students choice of method open, requires a creative procedure design

process. The project described here is offered with similar projects on the synthesisof quick-return mechanisms [8] and on cam design [10]. These projects give the stu-dents a strong appreciation of mechanism analysis and design issues and have

allowed the assigning of a significant percentage of accreditation units (AUs) ofengineering design [12] to the course.

The Canadian Engineering Accreditation Board (CEAB) performs accreditationof all undergraduate engineering programmes in Canada. AUs are assigned to the

curriculum content of the courses within the programme under consideration. Cur-rently AUs are divided between mathematics, basic sciences, engineering sciences,

engineering design, and complementary studies. Quoting CEAB accreditation crite-ria and procedures [12]: Engineering design integrates mathematics, basic sciences,engineering sciences and complementary studies in developing elements, systems

and processes to meet specific needs. It is a creative, iterative, and often open-endedprocess subject to constraints. While not strong on complementary aspects,

the project is strong in terms of being creative, iterative, open-ended, and subject toconstraints.




20/20

Conclusions

Having a project related to the determination of moment of inertia is very benefi-

cial to the students. The experience familiarises them with the terminology related

to the dynamic properties of bodies, and with concepts related to the testing andmodelling of masses and moments of inertia. The presented laboratory is used withina first course in mechanism analysis and occurs during the fourth to seventh weeks

of the course to coincide with the course coverage of dynamics of linkages. Havingthe laboratory has been found to strengthen the students understanding of the

dynamic properties of mechanism components.

Acknowledgements

The undergraduate students of the Department of Mechanical Engineering, Univer-

sity of Victoria, are thanked for providing effective feedback on the project.

References

[1] F. P. Beer and E. R. Johnston, Jr, Vector Mechanics for Engineers: Dynamics, 2nd edn (McGraw-

Hill, Singapore, 1990).

[2] R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics, 6th edn (Macmillan, New York,

1992).

[3] A. G. Erdman, G. N. Sandor and S. Kota,Mechanism Design Analysis and Synthesis, Volume 1, 4th

edn (Prentice Hall Upper Saddle River, NJ, 2001).

[4] H. H. Mabie and C. F. Reinholtz, Mechanics and Dynamics of Machinery, 4th edn (John Wiley,New York, 1987).

[5] G. H. Martin, Kinematics and Dynamics of Machines, 2nd edn (McGraw-Hill, New York, 1982).

[6] J. J. Uicker, Jr, G. R. Pennock and J. E. Shigley, Theory of Machines and Mechanisms, 3rd edn

(Oxford University Press, New York, 2003).

[7] D. G. Zill,A First Course in Differential Equations, 5th edn (PWS-Kent Publishing, Boston, MA,

1993).

[8] R. P. Podhorodeski, S. B. Nokleby and J. D. Wittchen, Quick-return mechanism design and analy-

sis projects,Int. J. Mech. Enging. Educ. 32(2) (2004), 100114.

[9] R. P. Podhorodeski and W. L. Cleghorn, Multiple loop mechanism analysis using a microcom-

puter, in Proceedings of the 9th Applied Mechanism Conference (Kansas City, MO, 1985), pp.

V1V7.[10] W. L. Cleghorn and R. P. Podhorodeski, Disc cam design using a microcomputer, Int. J. Mech.

Enging. Educ. 16(4) (1988), 234250.

[11] Knowledge Revolution, Working Model 2D Users Manual (1996).

[12] Canadian Engineering Accreditation Board, Accreditation Criteria and Procedures, (Canadian

Council of Professional Engineers, Ottawa, ON, 2002).



Practical Determination of Moment of Inertia

Documents

Transcript of Practical Determination of Moment of Inertia