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Chapter 5 –3 Center and Centroids

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Chapter 5 –3 Center and Centroids ENGR 0135 Chapter 5 –3 Center and centroids Department of Mechanical Engineering Centers and Centroids Center of gravity Center of mass Centroid of volume Centroid of area Centroid of line Department of Mechanical Engineering Center of Gravity A point where all of the weight could be concentrated without changing the external effects of the body To determine the location of the center, we may consider the weight system as a 3D parallel force system Department of Mechanical Engineering Center of Gravity – discrete bodies The total weight isWW i The location of the center can be found using the total moments 1 M Wx W x xW x yz G i i G W i i 1 M Wy W y yW y zx G i i G W i i 1 M Wz W z z W z xy G i i G W i i Department of Mechanical Engineering Center of Gravity – continuous bodies The total weight Wis dW The location of the center can be found using the total moments 1 M Wx xdW x xdW yz G G W 1 M Wy ydW y ydW zx G G W 1 M Wz zdW z zdW xy G G W Department of Mechanical Engineering Center of Mass A point where all of the mass could be concentrated It is the same as the center of gravity when the body is assumed to have uniform gravitational force Mass of particles 1 n 1 n 1 n n xC xi m i C y yi m i C z z mi i m i m m i m i m i i Continuous mass 1 1 1 x x dm yy dmz z dm m dm G m G m G m Department of Mechanical Engineering Example:Example: CenterCenter ofof discretediscrete massmass List the masses and the coordinates of their centroids in a table Compute the first moment of each mass (relative to the planes of the point of interest) Compute the total mass and total first moment Compute the center Department of Mechanical Engineering CenterCenter ofof massmass –– listlist ofof massmass andand thethe coordinatescoordinates Labels Mass xi (m) yi (m) zi (m) (kg) A 1 0.3 .24 0.0 B 2 0.15 0.4 0.0 C 1 0.3 0.4 0.27 D 2 0.3 0.0 0.27 E 1 0.0 0.2 0.27 Department of Mechanical Engineering CenterCenter ofof discretediscrete massmass –– calculationcalculation ofof thethe centercenter 1st moment of mass Mass # Mass xi (m) yi(m) zi(m) mix i miy i miz i (kg) A 1 0.3 0.24 0.0 0.3 0.24 0.0 B 2 0.15 0.4 0.0 0.3 0.8 0.0 C 1 0.3 0.4 0.27 0.3 0.4 0.27 D 2 0.3 0.0 0.27 0.6 0.0 0.54 E 1 0.0 0.2 0.27 0.0 0.2 0.27 total 7 1.5 1.64 1.08 The center 1.5/7 1.64/7 1.08/7 xc yc zc This method applies to discrete weights, lines, areas etc Department of Mechanical Engineering of VolumesCentroids Volumes made of sub vols 1 n 1 n 1 n n xC xi V i C y yi V i C zz Vi i V i V V i V i V i i xi , yi , zi ub volumes= centroids of the s Vi = volumes of the segments Continuous volumes 1 1 1 x x dV y y dVz z dV V dV C V C V C V Department of Mechanical Engineering oCentroids f Areas Areas made of segments 1 n 1 n 1 n n xC xi A i C y yi A i C zz Ai i A i A A i A i A i i xi , yi , zi engtseea smarof the = centroids Ai = Areas ofents the segm Continuous areas 1 1 1 x x dA y y dAz z dA A dA C A C A C A Department of Mechanical Engineering of Lines (xCentroids c, yc, zc) Lines made of segments 1 n 1 n 1 n n xC xi L i C y yi L i C zz Li i L i L L i L i L i i xi, yi, zi = centroids of the line segments Li = length of the segments Continuous lines 1 1 1 x x dL y y dLz z dL L dL C L C L C L Department of Mechanical Engineering Tables of special volumetric bodies, areas, and lines These tables are helpful when the centroid of a composite body (composed of volumes, areas, or lines) is in question In the following table, the centroids of the body are relative to the given origin O Department of Mechanical Engineering Department of Mechanical Engineering Department of Mechanical Engineering Department of Mechanical Engineering Department of Mechanical Engineering Continuous bodies – crucial tasks Choosing the coordinate system Determining the differential element for the integration Determining the lower and upper limits of the integral Carefully perform the integration (may require integration table) Department of Mechanical Engineering Example: dA = differential element = b dy This is not the only choice of the differential element !! Department of Mechanical Engineering Example: Many possibilities of differential elements and coordinate system Department of Mechanical Engineering Please read example problems 5-17 and 5-18 5-17 Centroid of line segments 5-18 Centroid of a cone Department of Mechanical Engineering Problem 5-80: Centroid? Department of Mechanical Engineering CentroidCentroid ofof anan areaarea –– areaarea integrationintegration Key components: – The differential element and its definition – The limits of the integration – The moment arms Department of Mechanical Engineering Centroid of an area – vertical differential element The area of the differential element x x h y y b b b1 dA hdx 2 1 a a dx 2 dA h 1 Department of Mechanical Engineering Centroid of an area – vertical differential element The limits of the integration – Lower limit x = 0 – Upper limit x = a x=a x=0 Department of Mechanical Engineering Centroid of an area – vertical differential element Performing the integral to obtain the area a a a x 1 2 3 / 2 A dA 1 b dx b x x 0 0 a a 3 0 1 23 / 2 2 ab b a a ab1 a 3 3 3 dA Department of Mechanical Engineering Centroid of an area – Getting the 1st moment of area about y axis - My My needs a moment arm parallel to x-axis The arm is from the y axis to the centroid of the element, here for the element it is x M s dA xdA y x dA q a x x xhdx 1 bx dx 0 0 a Department of Mechanical Engineering Centroid of an area – Getting the 1st moment of area about y axis (My) and the x coordinate of the centroid Performing the integration for the 1st moment of area a a x M xdA xhdx 1 bx dx y 0 0 a a 1 2 1 25 / 2 1 2 1 25 / 2 2 1 b x x b a a ba 2 a 5 0 2 a 5 10 Calculating the x coordinate of the centroid xdA a/2 b 10 xC 0 .a 3 dA ab/ 3 Department of Mechanical Engineering Centroid of an area – Getting the 1st moment of area about x axis - Mx Mx needs a moment arm parallel to y The arm is from the x axis to the centroid of the element 1 M y y dA x 2 1 2 2 The centroid of the rectangular element dA is [ x, (y1 +y2 )/2] 1 (y1+y2)/2 y1 x Department of Mechanical Engineering Centroid of an area – Getting the 1st moment of area about x axis (Mx) and the y coordinate of the centroid Performing the integration for moment area 1 M y y dA x 2 1 2 1 x 1 x x b b dA b bb b dx 2 a 2 a a a b2 a x b2 1 x2 ab2 1 dxx 2 0 a 2 2a 0 4 Calculating the y coordinate of the centroid 2 Mx ab/ 4 yC 0 . 75b dA ab/ 3 Department of Mechanical Engineering Problem 5-79: Centroid? Department of Mechanical Engineering Problem 5-79: Solution dAv xc, yc=x, y/2 Department of Mechanical Engineering Problem 5-79: Solution dAv xc, yc=x, y/2 Department of Mechanical Engineering Problem 5-79: Solution dAv xc, yc=x, y/2 Department of Mechanical Engineering CentroidsCentroids ofof compositecomposite bodiesbodies Possible elemental bodies: – Basic areas – Basic volumes – Line segments Similar method to centroid of discrete mass Pay attention to the centroid of the elemental bodies Department of Mechanical Engineering CentroidCentroid ofof aa compositecomposite areaarea The composite = A square - a full circle - a quarter circle Department of Mechanical Engineering CentroidsCentroids ofof thethe elementalelemental areasareas Area 1 120mm 120mm 160/ 4r/3 Area 2 Area 3 4r/3 60mm See Table 5-1 60mm Department of Mechanical Engineering CalculationCalculation ofof thethe centroidcentroid relativerelative toto OO Label Area xi (mm) yi (mm) Aix i (1000 Aiy i (1000 mm3) mm3) 1 57600 120 120 6912 6912 2 -11309 100 80 -1130.9 -904.72 3 -11309 240- 240- -2138.2 -2138.2 160/ 160/ Total 34982 3642.8 3869.1 The centroid 104.1 110.6 Department of Mechanical Engineering CentroidCentroid ofof compositecomposite volumevolume andand lineline Similar method to composite area can be applied (use volume and length instead of area) Use Table 5-1 and 5-2 to determine the centroid of the elemental bodies Department of Mechanical Engineering DecompositionDecomposition ofof thethe lineline bodybody Straight line segments Semicircular arc Department of Mechanical Engineering How about this? Department of Mechanical Engineering DistributedDistributed loadsloads onon structuralstructural membersmembers Tasks: – Find the resultant – Find the location of the resultant Distributed loads: – Continuous distribution xR R involves some area integral – Composite of simple distribution – A combination of the two Department of Mechanical Engineering DistributedDistributed loadload The magnitude L R dR () w x dx 0 The location L xw() x dx xdR d 0 R R Department of Mechanical Engineering CompositeComposite ofof simplesimple distributeddistributed loadload R1 R R2 3 1 2R .
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