Moment Of Inertia - Memphis
Moment of InertiaWhatisacommi eunnecessary.Moment of InertiaWhen we calculated the centroid of ashape, we took the moment generated bythe shape and divided it by the total areaof the shape. This gave us a distance, which was thedistance to the centroid of the shape 2Moment of Inertia by IntegraionMonday, November 19, 20121
Moment of InertiaThe moment of inertia is actually thesecond moment of an area or mass aboutan axis Notice that it is not a distance, it is amoment of a moment That may sound strange l 3It shouldMoment of Inertia by IntegraionMonday, November 19, 2012Moment of InertiaThere is really nothing that can easily beused to describe the moment of inertia For an area, it will have units of length4which is very difficult to map to a physicalquantity 4Moment of Inertia by IntegraionMonday, November 19, 20122
Moment of Inertia The symbol for the moment of inertia is Iwith a subscript describing about whichaxis the moment is being calculated The moment of inertia about the x-axiswould be Ix, about the y-axis, Iy 5There is also a moment of inertia about theorigin, known as the polar moment ofinertia designated as JOMoment of Inertia by IntegraionMonday, November 19, 2012Moment of InertiaThe moment of inertia is a physicalproperty and determines the behavior of amaterial under certain loading anddynamic conditions Remember, we are taking the moment ofthe moment (the second moment) of anarea about an axis Keep this in mind and you won’t have anytrouble here 6Moment of Inertia by IntegraionMonday, November 19, 20123
Moment of Inertia 7The first moment of a shape about an axiswas calculated by taking the moment armto the local centroid of the shape andmultiplying that by the area of the shapeMoment of Inertia by IntegraionMonday, November 19, 2012Moment of InertiaThe second moment will be generated in asimilar manner We will take a moment arm from the axisto the centroid of the shape, square thatmoment arm, and multiply that product bythe area 8Moment of Inertia by IntegraionMonday, November 19, 20124
Moment of InertiaFor a moment of inertia about (around) ay-axis, the moment arm will be measuredperpendicular to the y-axis, so it will be anx-distance So for Iy we would have Iy x A29Moment of Inertia by IntegraionMonday, November 19, 2012An Example 10Consider the following figureMoment of Inertia by IntegraionMonday, November 19, 20125
An Example 11We will start with the Iy, or the moment ofinertia about the y-axisMoment of Inertia by IntegraionMonday, November 19, 2012An Example 12To take a moment about the y-axis, we willneed to have a moment arm that has an xdistanceMoment of Inertia by IntegraionMonday, November 19, 20126
An Example Again, we will begin by generating adifferential area, dA4mydxy2 4xytop-ybottomy 1 2x44mx13Moment of Inertia by IntegraionMonday, November 19, 2012Point to Note Youmust be careful that theside of the rectangledescribing the differentialarea that does not have thedifferential component isparallel to the axis aboutwhich you are taking themoment of inertia14Moment of Inertia by IntegraionMonday, November 19, 20127
Point to Note Ifyou do not set up theproblem this way, thecalculations are a bitdifferent as you have seenfrom the example we did inclass.15Moment of Inertia by IntegraionMonday, November 19, 2012An Example In this case, the height is parallel to the yaxis4mydxy2 4xytop-ybottomy 1 2x44mx16Moment of Inertia by IntegraionMonday, November 19, 20128
An Example If this isn’t so, the method breaks down4mdxyy 4x2ytop-ybottomy 1 2x44mx17Moment of Inertia by IntegraionMonday, November 19, 2012An Example Once we have the differential area, welocate the moment arm from the axis4mydxy 4x2ytop-ybottomxy 1 2x44mx18Moment of Inertia by IntegraionMonday, November 19, 20129
An Example Now the second moment of this differentialarea will be the moment arm squaredtimes the differential area4m2x dAdxyy2 4xytop-ybottomy x1 2x44mx19Moment of Inertia by IntegraionMonday, November 19, 2012An Example In this example the differential area dA isthe height of the rectangle times the widthof the rectangle4mdA ( yTOP yBOTTOM ) dxy x2 dA 2 x dx4 dxy2 4xytop-ybottomxy 1 2x44mx20Moment of Inertia by IntegraionMonday, November 19, 201210
An Example The moment of inertia of the differentialarea is the square of the moment armtimes the differential area4mI y A x dA2I yAdxyy2 4x x2 x 2 x dx4 2ytop-ybottomy x1 2x44mx21Moment of Inertia by IntegraionMonday, November 19, 2012An Example The moment of inertia for the completeshape, Iy, is the sum of all the moments ofinertia of the differential areas4mI y x 2 dAyAI y x 2 ( yTOP yBOTTOM ) dxAIy 4m0m22 xx2 2 x 4 2 dx Moment of Inertia by Integraiondxy2 4xytop-ybottomxy 1 2x44mxMonday, November 19, 201211
An Example Notice that we are calculating Iy but thedistances are in the x-direction, be carefulto remember thisI y x 2 dA4mAI y x 2 ( yTOP yBOTTOM ) dxdxyy2 4xAIy 4m0m xx2 2 x 4 223ytop-ybottom dx y x1 2x44mxMoment of Inertia by IntegraionMonday, November 19, 2012An Example Evaluating the integral, we have 52 x 4 I y 2 x dx0m4 4m7221 x5I y 2x 75 4y0mI y 73.14 51.20 0 0I y 21.94m 4244m4mMoment of Inertia by Integraiondxy2 4xytop-ybottomxy 1 2x44mxMonday, November 19, 201212
An Example Using the same method, we can calculatethe Ix25Moment of Inertia by IntegraionMonday, November 19, 2012An Example Start by drawing the differential area4myy2 4xxright-xlefty dy1 2x44mx26Moment of Inertia by IntegraionMonday, November 19, 201213
An Example Draw the moment arm from the x-axis4myy2 4xxright-xlefty 1 2x44mydyx27Moment of Inertia by IntegraionMonday, November 19, 2012An Example The second moment for this differentialarea is4my2xright-xlefty dy1 2x44myy dAy2 4xx28Moment of Inertia by IntegraionMonday, November 19, 201214
An Example The second moment for this differentialarea is4m2y2( xRIGHT xLEFT ) dyy2 2 y 2 y dx4 29yy2 4xxright-xlefty dy1 2x44myy dAxMoment of Inertia by IntegraionMonday, November 19, 2012An Example The Ix for the composite area is the sum ofthe Ix’s for the individual differential areas 52 y 4 I x 2 y dy0m4 4my4m30xright-xlefty dy0mI x 73.14 51.20 0 0I x 21.94my2 4x1 2x44my2 7 1 y5Ix 2 y 2 75 44mx4Moment of Inertia by IntegraionMonday, November 19, 201215
An Example The polar moment of inertia, JO, is the sumof the moments of inertia about the x and yaxis4myJO I x I yJ O 21.94m4 21.94m4J O 43.88m431y2 4xy 1 2x44mxMoment of Inertia by IntegraionMonday, November 19, 2012An Aside 32Just for your information, you are notrequired to know this method, you can usea double integral to find the moment ofinertiaMoment of Inertia by IntegraionMonday, November 19, 201216
An Aside The difference is how you describe thedifferential area, in this case the differentialarea would be4myy2 4xdA ( dx )( dy )dyy dx1 2x44mx33Moment of Inertia by IntegraionMonday, November 19, 2012An Aside The second moment of this differentialarea about the y-axis would be4myy2 4xdyx2 dA x2 ( dx )( dy )dxy 1 2x44mx34Moment of Inertia by IntegraionMonday, November 19, 201217
An Aside As we sum the differential areas throughthe composite, we are integrating in twodirections, x and y4mydy x dA x ( dx )( dy )2y2 4x2Ay dx1 2x44mAx35Moment of Inertia by IntegraionMonday, November 19, 2012An Aside 4m0mSince we have an x2, we can choose tothe y-direction as the inner integral andmove y from bottom to topyTOP2 x yBOTTOM ( dx ) ( dy )4m 0m 4m0m364xx244my x ( dx ) ( dy )2dydxy 1 2x44m4xx y ( dx ) x22y2 4x4xMoment of Inertia by IntegraionMonday, November 19, 201218
An Aside Making the inner integration, we have4my 4m0m xx2 4 x 4 237y2 4xdy dx y dx1 2x44mxMoment of Inertia by IntegraionMonday, November 19, 2012An Aside Which is the same form as we had beforefor Iy4myIy 4m0m xx 4x 4 22 dx y2 4xdydxy 1 2x44mx38Moment of Inertia by IntegraionMonday, November 19, 201219
HomeworkProblem 10-1 Problem 10-2 Problem 10-7 39Moment of Inertia by IntegraionMonday, November 19, 201220
31 Moment of Inertia by Integraion Monday, November 19, 2012 An Example ! The polar moment of inertia, J O, is the sum of the moments of inertia about the x and y axis y x yx2 4 1 2 4 yx 4m 4m 44 4 21.94 21.94 43.88 Ox y O O JII Jm m Jm 32 Moment of Inertia by Integraion Monday, N
I : moment of inertia about any parallel axis I CM: moment of inertia about an axis through its center of mass M : total mass h : distance from a parallel axis to the center of mass. BTW: The moment of inertia of any object about an axis through its center of mass is the minimum moment of
3 Moment of Inertia - Composite Area Monday, November 26, 2012 Radius of Gyration ! The radius of gyration, k, is the square root of the ratio of the moment of inertia to the area I x x y y O xy O k A I k A J II k AA 4 Moment of Inertia - Composite Area Monday, November 26, 2012 Para
MOMENT OF INERTIA The moment of inertia, or the second moment of area, is defined as I y x2 dA I x y2 dA The polar moment of inertia J of an area about a point is equal to the sum of the moments of inertia of the area about any two point. I z J I y I x (x 2 y2) dA
Moment of Inertia is always positive. Units of Moment of Inertia are length raised to the fourth power, such as in4 or m4. Polar Moment of Inertia The second moment of Area A with respect to the pole O or the z-ax
dependent on the wheel inertia. So it is crucial that the inertia of the tire-wheel combination is simulated. The typical problem is that the dynamometer inertia is greater than the tire-wheel inertia. So the simulation must apply a force to the dyno inertia to compensate for the inertia difference
Moment of inertia of an object is an indication of the level of force that has to be applied in order to set the object, or keep the object, in motion about a defined axis of rotation. Moment of inertia, which is a derivative of Newton’s second law, is sometimes referred to as the second moment
Memphis Center Air Traffic Manager 6 Memphis Center Support Manager for Training 7 Memphis Center Organizational Chart 8 . 1985 as an air traffic control specialist at Memphis ARTCC. She has held several positions at Memphis including support specialist and front line manager. She has
List of Plates Plate 1 Tea break! 4 Plate 2 Outline of robbed out wall visible in Trench 2c. Taken from the N. 8 Plate 3 W facing fireplace [2055], during excavation. Taken from the SW. 9 Plate 4 General view of fire place and rake out area following excavation, Trench 2c. Taken from the SW. 9 Plate 5 Stake [2091], set into natural sand (2072). Taken from the N 10
