STATICS. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Centroids and Centers of Gravity.

Transcription

1 5 Distributed CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Texs Tech Universit Forces: Centroids nd Centers of Grvit Contents Introduction Center of Grvit of D Bod Centroids nd First Moments of Ares nd Lines Centroids of Common Shpes of Ares Centroids of Common Shpes of Lines Composite Pltes nd Ares Smple Problem 5.1 Determintion of Centroids b Integrtion Smple Problem 5.4 Theorems of Pppus-Guldinus Smple Problem 5.7 Distributed Lods on Bems Smple Problem 5.9 Center of Grvit of D Bod: Centroid of Volume Centroids of Common D Shpes Composite D Bodies Smple Problem

2 Introduction The erth exerts grvittionl force on ech of the prticles forming bod. These forces cn be replce b single equivlent force equl to the weight of the bod nd pplied t the center of grvit for the bod. The centroid of n re is nlogous to the center of grvit of bod. The concept of the first moment of n re is used to locte the centroid. Determintion of the re of surfce of revolution nd the volume of bod of revolution re ccomplished with the Theorems of Pppus-Guldinus. 5 - Center of Grvit of D Bod Center of grvit of plte Center of grvit of wire M M xw xw x dw W W dw 5-4

3 Centroids nd First Moments of Ares nd Lines Centroid of n re Centroid of line x xw x dw At x t xa x A Q first moment wit h respect to Q x first moment wit h respect to x x xw x dw L x xl x dl L dl dl 5-5 First Moments of Ares nd Lines An re is smmetric with respect to n xis BB if for ever point P there exists point P such tht PP is perpendiculr to BB nd is divided into two equl prts b BB. The first moment of n re with respect to line of smmetr is zero. If n re possesses line of smmetr, its centroid lies on tht xis If n re possesses two lines of smmetr, its centroid lies t their intersection. An re is smmetric with respect to center O if for ever element t (x,) there exists n re of equl re t (-x,-). The centroid of the re coincides with the center of smmetr. 5-6

4 Centroids of Common Shpes of Ares 5-7 Centroids of Common Shpes of Lines 5-8 4

5 Composite Pltes nd Ares Composite pltes X W xw Y W W Composite re X A x A Y A A 5-9 Smple Problem 5.1 For the plne re shown, determine the first moments with respect to the x nd xes nd the loction of the centroid. SOLUTION: Divide the re into tringle, rectngle, nd semicircle with circulr cutout. Clculte the first moments of ech re with respect to the xes. Find the totl re nd first moments of the tringle, rectngle, nd semicircle. Subtrct the re nd first moment of the circulr cutout. Compute the coordintes of the re centroid b dividing the first moments b the totl re

6 Smple Problem 5.1 Find the totl re nd first moments of the tringle, rectngle, nd semicircle. Subtrct the re nd first moment of the circulr cutout. Qx mm Q mm 5-11 Smple Problem 5.1 Compute the coordintes of the re centroid b dividing the first moments b the totl re mm x A X A mm X 54.8 mm Y mm A A mm Y 6.6 mm 5-1 6

7 Determintion of Centroids b Integrtion xa A x x dxd dxd x el el Double integrtion to find the first moment m be voided b defining s thin rectngle or strip. xa A xel x dx el dx xa A xel x el x dx x dx xa xel r 1 cos r A el r 1 sin r d d 5-1 Smple Problem 5.4 SOLUTION: Determine the constnt k. Evlute the totl re. Determine b direct integrtion the loction of the centroid of prbolic spndrel. Using either verticl or horizontl strips, perform single integrtion to find the first moments. Evlute the centroid coordintes

8 Smple Problem 5.4 SOLUTION: Determine the constnt k. k x b b k k b x or x b Evlute the totl re. A b 1 1 b b x dx x dx Smple Problem 5.4 Using verticl strips, perform single integrtion to find the first moments. Q Q x x el b b b xdx x x x 5 b 4 0 b x dx 1 b el dx x dx

9 Smple Problem 5.4 Or, using horizontl strips, perform single integrtion to find the first moments. Q xel x b 1 b d 0 b 4 b x xd d 0 1 Qx el xd d 1 b b b d 1 0 b Smple Problem 5.4 Evlute the centroid coordintes. xa Q b b x 4 x 4 A Q x b b b

10 Theorems of Pppus-Guldinus Surfce of revolution is generted b rotting plne curve bout fixed xis. Are of surfce of revolution is equl to the length of the generting curve times the distnce trveled b the centroid through the rottion. A L 5-19 Theorems of Pppus-Guldinus Bod of revolution is generted b rotting plne re bout fixed xis. Volume of bod of revolution is equl to the generting re times the distnce trveled b the centroid through the rottion. V A

11 Smple Problem 5.7 SOLUTION: Appl the theorem of Pppus-Guldinus to evlute the volumes or revolution for the rectngulr rim section nd the inner cutout section. The outside dimeter of pulle is 0.8 m, nd the cross section of its rim is s shown. Knowing tht the pulle is mde of steel nd tht the densit of steel is kg m determine the mss nd weight of the rim. Multipl b densit nd ccelertion to get the mss nd ccelertion. 5-1 Smple Problem 5.7 SOLUTION: Appl the theorem of Pppus-Guldinus to evlute the volumes or revolution for the rectngulr rim section nd the inner cutout section. Multipl b densit nd ccelertion to get the mss nd ccelertion kg m mm 10 m mm 60.0 kg 9.81m s m V m 60.0 kg W mg W 589 N 5-11

12 Distributed Lods on Bems W L 0 wdx A A distributed lod is represented b plotting the lod per unit length, w (N/m). The totl lod is equl to the re under the lod curve. OP W xdw L OPA x xa 0 A distributed lod cn be replce b concentrted lod with mgnitude equl to the re under the lod curve nd line of ction pssing through the re centroid. 5 - Smple Problem 5.9 A bem supports distributed lod s shown. Determine the equivlent concentrted lod nd the rections t the supports. SOLUTION: The mgnitude of the concentrted lod is equl to the totl lod or the re under the curve. The line of ction of the concentrted lod psses through the centroid of the re under the curve. Determine the support rections b summing moments bout the bem ends

13 Smple Problem 5.9 SOLUTION: The mgnitude of the concentrted lod is equl to the totl lod or the re under the curve. F 18.0 kn The line of ction of the concentrted lod psses through the centroid of the re under the curve. 6 kn m X 18 kn X.5 m 5-5 Smple Problem 5.9 Determine the support rections b summing moments bout the bem ends. 6 m 18 kn.5 m 0 M A 0 : B B 10.5 kn 6 m 18 kn6 m.5 m 0 M B 0 : A A 7.5 kn 5-6 1

14 Center of Grvit of D Bod: Centroid of Volume Center of grvit G W j W j rg r W G W W j r W j j rw j dw rgw rdw Results re independent of bod orienttion, xw xv 5-7 xdw xdv W V dw For homogeneous bodies, W V nd dw dv dv zw zv zdv zdw Centroids of Common D Shpes

15 Composite D Bodies Moment of the totl weight concentrted t the center of grvit G is equl to the sum of the moments of the weights of the component prts. X W xw Y W W Z W zw For homogeneous bodies, X V xv Y V V Z V zv 5-9 Smple Problem 5.1 SOLUTION: Form the mchine element from rectngulr prllelepiped nd qurter clinder nd then subtrcting two 1-in. dimeter clinders. Locte the center of grvit of the steel mchine element. The dimeter of ech hole is 1 in

16 Smple Problem Smple Problem 5.1 X Y Z xv V zv V V V 4.08 in 5.86 in X in in 5.86 in Y in in 5.86 in Z in. 5-16