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MECH 335 Project # 2Determination of Centres of Mass and Moments of Inertia

(Instructor: Ron Podhorodeski)

1. OVERVIEWAnalysis of the forces involved in mechanical systems requires an understanding of the

dynamic properties of the system components. For a mechanism with rigid links, the proper-ties which must be known prior to any dynamic analysis include the centre of mass location.and the moment of inertia (2nd mass moment) of the links. In this project these quantities areto be determined by both models and by experimentation.

Section 2 reviews the associated mechanics basics. section 3 specifies the requirements ofthe project and section 4 describes the required form of the lab repon.

2. BACKGROUNDBibliography: [1] Beer & Johnston, Vector Mechanics for Engineers or [2] Hibbeler,Engineering Mechanics,' and [3] Mabie & Reinholtz, Mechanics and Dynamics ofMachinery or [4] Martin, Kinematics and Dynamics of Machines.

Figures 4 and 8 are reproduced from [4]. Appendix 1 is reproduced from [2].

2.1 Equations of MotionThe equations of motion for a rigid body (RB) subject to an externally applied force and

moment system can be expressed as

LF=MaG (1)

L ~ = [I1Ga (2)

where L F is the sum of all external forces acting on the RB and L ~ is the sum of allmoments (due to both couples and forces) as seen from the centre of mass G. In eqns. (1) &(2) the mass of the body is denoted by M, the translational acceleration of G byaG the angu-lar acceleration of the RB by a, and the inertia of the RB by [I]G. For general three dimen-sional motion eqns. (1) & (2) represent two vector equations each corresponding to threescalar equations (i.e., six scalar equations in total).

For planar motion, all translational motion is constrained to parallel planes and rotationoccurs only about the direction perpendicular to the motion plane. Eqn. (1) can then beresolved into two scalar equations (e.g., i and j components) and eqn. (2) becomes a singlescalar expression (e.g., related to rotation about the k direction). This allows eqn. (1) and (2)to be rewritten as:

}: F z = MaG (la*)z

}: Fy = M~J (lb*)

}: MG = Izzc

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2.2 Techniques of Centre of Mass and Inertia Determination

Centre Of Mass Detemlination

Several approaches are possible for determining the location of the centre of mass of anobject. These approaches include: mathematical modeling (taking moments); observation ofsymmetries; and resolution by direct physical measurementS; or a combination of these tech-niques.

- Taking MomentsConsider the rigid body illustrated in Fig. 1.

Taking moments about a - a yields

Ma - a = L r dM

This must equal the moment about a - a if all masswere concentrated at G. That is,

Ma - a = rGM = L r dM

and therefore

(3) Fig. 1 - Moment about a - a

Therefore

ab2/2 + (b+4c/31t)(1tC2J2)rG =ab + 1tC2J2

A table of the centroids of common shapes is includedin Appendix 1.

Fig. 2 - A "Composite Body"

Therefore, G lies on a line parallel to, and a distanceT G, from a - a. Fm1her moments can be taken to iso-late G to a single point

Composition:If a body can be easily "divided" into basic ele-

ments, composition can be used to simply resolvingthe location of G .e.g., The location of the mass centre of the. plate illus-trated in Fig. 2 is to be found. The plate is of uniformdensity and has a mass density for the uniform thick-ness ofp, ([p] =MIL2). Find TG

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- Observation of SymmetryIf a body is homogeneous and is synnnetrical about a plane the centre of mass for the

body must lie on the plane of symmeny. Symmetry about two planes isolates G to a line andabout three unique planes to a single point.

- Physical MeasurementsSuspension:

Consider a body suspended by two cables asillustrated in Fig. 3. The force of gravity acting on themass of the body is equivalent to a single force Mgacting through G . Equilibrium requirements for athree force body can be used to isolate G to a line.Suspending the body from different points can be usedto isolate G to a point. .-Weighing: Fig. 3 - Suspension-Based rG Isolation

A planar link is supponed on two scales as illustrated in Fig. 4. Taking moments aboutthe supports allows resolution of the location of G . Note that symmetry about two planes hasbeen used in this case to isolate G to a line. Furthermore, note that the measurement couldbe performed with a single scale and a rest. ;

I

b-

~2

Mg~,..:,"""'~"

""",."..

Fig. 4 - Two Scale Determination of rG

2.3 Moment Of Inertia DeterminationSeveral approaches are possible for determining the inertia (or an appronmation for the

inertia) of a body. These approaches include: ma.iliematical modeling; assumption of anapproximate model; use of a CAD package; and determination by physical measmements.Note that a CAD package uses a mathematical model approach to determine the inertia.

Mathematical Modeling

Definition:Consider the body illustrated in Fig. 5. The

body's moment of inertia (2nd mass moment) about anaxis a - a is

Ia -a = L ,2dM (4)

d.Fig. 5 - 2nd Mass Moment

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Parallel-Axis Theorem:If 10 is the moment of inertia about an axis pass-

ing through the body's mass centre, the moment ofinerria about a parallel axis a distance d from G is

1 = 10 + d2M (5)

Composition:The parallel-axis theorem can be used to find the

inertia of a body composed of several basic elements.For example, consider the pendulum consisting of auniform disc welded on a slender rod as illustrated inFig. 6. The moment of inertia about the fixed cenn-eof rotation can be found as follows:

10 = lo,.,.t + lo~

10.. = IG...,. 2 1 2 a2 1 2+ Mrod(a/2) = UMroda + Mrod4 = 3Mroda

2 1 2 2+ Mdisc(a + c .) = -MdiscC + Mdisc(a + C)2

Iocisc = IG46sc

Therefore,

The moments of inertia for masses of commongeometric shapes are included in Appendix 1.

Fig. 6 - Composite Pendulum

Approximate ModelsOften if high accuracy is not required in an analysis and/or if a body can not be easily

modeled exactly, an approximate model will be assumed. For example, a slender rod is oftenassumed as a first approximation for a long member of fairly unifonn mass distribution.Often a better assumed model can be formed by. a composite assumption. In the assumedcomposite representation basic elements, e.g., slender rods, point masses, spheres, discs,plates, ..., are used. For example, the connecting rod depicted in Fig. 7 is to be approximatelymodeled. Three possible approximations are illustrated.

P,

Slender Rod Slender Rod &

2 Point Masses

- Slender Rod &

...""

Fig. 7 - A Connecting Rod and Three Approximate Models

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Physical MeasurementsAn experimental approach for detennining the moment of inertia of a body is the knife-

edge technique. The method involves suspending the body by a knife-edge as depicted forthe connecting rod in Fig. 8. Displacing the body a small angle °max, releasing it, andmeasming its period of oscillation allows determination of the moment of inertia of the body.The governing equation for this determination will now be derived.

With reference to Fig. 8, consider the body to be at a displacement o. Recalling thedynamics of a body rotating about a fixed centre, we have the equation of motion

LMo = loa

or

(6)

For a small angle sine = e and eqn. (6) becomes

~ + !!!..8!-e = 0 (7)dt2 10

This is a second order linear differential equation witha general solution of

O=Asin[--J¥~ +BCOS[~-Considering the boundary conditions, (0 = Omax @ t=Oand 0=0 @ 1=0) we find

(8)

~

Fig. 8 - "Knife-Edging"About the centre of mass the inertia is

IG =10 -Mr2=Mr[[~rg-r

The period of oscillation of the body is

~ 0T = 21t - (10)Mgr

Therefore, by measuring the period of oscillation themoment of inertia can be determined by

10 =Mgr[~r (11)

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2.4 Dynamically Equivalent Systems (DES)Often solutions to dynamics problems can be simplified by replacing a link with a

dynamically equivalent system (DES). A DES is a composite of elements that will experiencethe same accelerations as the actual body if acted upon by the same force system. That is, theDES has identical dynamic properties as the original member. A common simple representa-tion for planar links is a two point mass DES. Note that a DES is not an approximation, it isa simplified equivalent system

e.g., A planar body has a mass M and an inertia fa. What are the conditions whichmust be satisfied if two masses M 1 and M 2 are to be dynamically equivalent.

M 1 + M 2 = M (since L F = M 3a) (a)M Ih 1 = M zh2 (to preserve G location) (b)

M Ihf + Mzh£ = fa (since L Ma = fa a) (c)There are an infinite number of choices for M l' M 2' h 17 and h2 satisfying (a) through (c).

2.5 Radius of GyrationThe dynamic properties for bodies are sometimes specified in terms of their radius of

gyration. This specification is often used in component specification in paris catalogues. Theradius of gyration k for a body is the distance from G, where if the mass of the body wasconcenttated the moment of inertia would be the same as the original body. One way of ima-gining this equivalent system is a thin hoop of uniformly distributed mass, lying in the planeperpendicular to the direction of rotation, having a radius k and a center corresponding to G.The inertia for this DES is

IG =MP (13)

For example, consider a thin disc of radius r having uniform mass distribution.

IG=1-M,z = Mk22

Therefore1

k=~rNotice that the radius of gyration is only a function of the geometry (including thegeometry of the mass distribution). An aluminum disc of radius r and a steel disc ofradius r have the same radius of gyration.

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3. PROJECT REQUIREMENTSObjective (my objective)

The idea of this project is to expose you to ideas for determining the dynamic propertiesof bodies. In particular. for links of planar mechanisms. From this project you should gainan insight into modeling and experimental techniques for finding the inertia properties of rigidbodies.

ApparatusAvailable for this project are:

- Four planar links of various lengthes (description is included as Appendix 2)

- An assortment of individual components of the links

- a tape measure- a stop watch- a digital scale(s)- two cables and a frame from which to suspend links- a "knife-edge" for pendulum-based determination of inertia

RequirementsI Experiment~ Experimentally determine the inertia fG for the four links.

~ Plot the inertias as a function of link length.~ Discuss the experimental method in terms of:

(a) assumptions made in the derivation of the governing equation for finding the inertia

(b) for which links the method would be the most accurate and why?

(c) possible improvements (if any) which could be made to improve the accuracyand/or to ensure the consistency of the experimental measurements

(d) any other considerations you feel are important

~ Find the radius of gyration for the longest link.

~ Propose a two mass DES for the longest link.

II Approximate Models~ Propose two approximate models for finding the dynamic properties of the links. The

first should be a very simple model using only a slender rod(s) and point mass(es). Thesecond should be a more detailed model using the composition of any basic elementsthat you feel are necessary.

~ Plot the inertia fa found using your approximate models as a function of link length.

~ Discuss your models in terms of:

(a) assumptions made during the choice of model(b) when the models are going to be the most and least accurate and why

(c) improvements that could be made to improve the accuracy of the better model

(d) any other considerations you feel are important

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ill Comparison~ Compare the results of the models of part n and the experimental results of part I. Dis-

cuss in terms of advantages and disadvantages the use of iDOdels and of experimentationfor finding dynamic properties.

4. . REPORT REQUIREMENTSThe report must be written as a shott technical report and should include:

- an abstract (summary)- a brief introduction- report on experimental procedure and results- description of approximate models used- a discussion and comparison of results (as previously described)- a conclusion- references (If any)

LIBAppendix 1 - Centroids and Mass Moments of Inertia (Common Shapes)

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Center of Gravity and Mass Moment of Inertia of Homogeneous Solids

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