Statik Rijid Cisimler - Eşdeğer Kuvvet Sistemleri - 2 VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Seventh Edition Seventh Edition Ferdinand P. Beer Ferdinand P. Beer Ferdinand P. Beer E. Russell Johnston, Jr. E. Russell Johnston, Jr. E. Russell Johnston, Jr. Lecture Notes: Lecture Notes: Lecture Notes: N. MEYDANLIK N. MEYDANLIK N. MEYDANLIK T T Trakya rakya rakya University University University CHAPTER © 2003 The McGraw-Hill Companies, Inc. All rights reserved. 3 Rigid Bodies: Rigid Bodies: (II) (II) Equivalent Equivalent Systems of Forces Systems of Forces © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 2 İ İ Ç Ç İ İ NDEK NDEK İ İ LER : LER : • Scalar Product of Two Vectors • Mixed Triple Product of Three Vectors ve Moment of a Force About a Given Axis • Moment of a Couple • Resolution of a Force Into a Force at O and a Couple • System of Forces: Reduction to a Force and Couple • Further Reduction of a System of Forces© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 3 Scalar Product of Two Vectors Scalar Product of Two Vectors • The scalar product or dot product between two vectors P and Q is defined as ( ) result scalar cos? PQ Q P = • r r • Scalar products: - are commutative, - are distributive, - are not associative, P Q Q P r r r r • = • ( ) 2 1 2 1 Q P Q P Q Q P r r r r r r r • + • = + • ( ) undefined = • • S Q P r r r Skaler çarpım (veya iç çarpım) denir Example: Determine the dot product of the two vectors shown below. 25º x y A B |A| = 50 |B| = 30 YNTz1360 © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 4 Scalar Product of Two Vectors Scalar Product of Two Vectors • Scalar products with Cartesian unit components, 0 0 0 1 1 1 = • = • = • = • = • = • i k k j j i k k j j i i r r r v r r r r r r r r ( ) ( ) k Q j Q i Q k P j P i P Q P z y x z y x r r r r r r r r + + • + + = • 2 2 2 2 P P P P P P Q P Q P Q P Q P z y x z z y y x x = + + = • + + = • r r r r If A 2i - 4j 5k and B 3i 6k, determine A B = + = + • Example: YNTz36© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 5 Scalar Product of Two Vectors: Applications Scalar Product of Two Vectors: Applications 1. Angle between two vectors: Example: Determine the angle between the two vectors in the last example (repeated below). A 2i - 4j 5k and B 3i 6k = + = + PQ Q P Q P Q P Q P Q P Q P PQ Q P z z y y x x z z y y x x + + = + + = = • ? ? cos cos r r YNTz36.86° © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 6 Example: Problem 3.37 Example: Problem 3.37 Example: Consider the volleyball net shown below. Determine the angle formed by guy wires AB and AC. (YNT: ~43º)© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 7 Scalar Product of Two Vectors: Applications Scalar Product of Two Vectors: Applications 2. Projection of a vector on a given axis: Q Q P Q P Q P Q Q P P P P Q Q P PQ Q P OL P P P z z y y x x OL OL OL + + = • = = = • = • = = r r r r r r ? ? ? cos cos cos along of projection z z y y x x OL P P P P P ? ? ? ? cos cos cos + + = • = r r • For an axis defined by a unit vector: © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 8© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 9 Scalar Product of Two Vectors: Applications Scalar Product of Two Vectors: Applications x y A (3,4) P = 20i + 10j B (10,12) Example: If F = 16i + -14j + 10k N, determine the magnitude of the projection of F along the axis of the pole. Also find the magnitude of F. z y x z z y y x x OL P P P P P ? ? ? ? cos cos cos + + = • = r r (YNT: ~20,7) Example: P vektörünün line A-B üzerindeki izdüşümünü bulunuz. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 10 Mixed Triple Product of Three Vectors Mixed Triple Product of Three Vectors • Üç vektörün karışık üçlü çarpımı, ( ) result scalar = × • Q P S r r r • Karışık üçlü çarpım mutlak değer bakımından kenarları S, P, and Q olan parelel yüzlünün hacmine eşittir, eğer S, P, and Q bir sağ üçlü oluşturuyorsa pozitif sol üçlü oluşturuyora negatiftir ( ) ( ) ( ) ( ) ( ) ( ) S P Q Q S P P Q S P S Q S Q P Q P S r r r r r r r r r r r r r r r r r r × • - = × • - = × • - = × • = × • = × • ( ) ( ) ( ) ( ) z y x z y x z y x x y z xy z z x x z y y z z y x Q Q Q P P P S S S Q P Q P S Q P Q P S Q P Q P S Q P S = - + - + - = × • r r r • Evaluating the mixed triple product, S, P ve Q aynı düzlemde ise karışık üçlü çarpım sıfır olur.© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 11 EKSENE G EKSENE GÖ ÖRE MOMENT RE MOMENT - -UYGULAMA UYGULAMA Somunu sökmek için gerekli moment M A = F.d=(M z ) max. (Kol kuvveti yatay) Kuvvet yatay değilse Uygulanan M A momentinin sadece bir kısmı sökmek için kullanılır. Bu da M Z = M A .cos? = (F.d).cos? =F.d’ © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 12 Moment of a Force About a Given Axis Moment of a Force About a Given Axis z y x z y x OL F F F z y x M ? ? ? = • Moment M O of a force F applied at the point A about a point O, F r M O r r r × = • Scalar moment M OL about an axis OL is the projection of the moment vector M O onto the axis, ( ) F r M M O OL r r r r r × • = • = ? ? • Moments of F about the coordinate axes, x y z z x y y z x yF xF M xF zF M zF yF M - = - = - = 1. Sometimes need the component of a moment about a particular axis 2. Scalar analysis: M a =Fd a where d a is the ? or shortest distance from the force line of action to the axis of interest 3. Vector analysis: M a = ? ? ? ? a • (r x F) 4. This called the triple scalar product A’ nın koordinatları OL ekseninin doğrultman kosinüsleri© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 13 Moment of a Force About a Given Axis Moment of a Force About a Given Axis • Kuvvetin herhangi bir eksene göre momenti, ( ) B A B / A B / A B BL r r r F r M M r r r r r r r r - = × • = • = ? ? • SOUÇ eksen üzerinde alınan B noktasından bağımsızdır. z y x B / A B / A B / A z y x BL F F F z y x M ? ? ? = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 14 Example Example : : M M o o = ?, = ?, M M y y =? =? 143.13, 53.13, 90© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 15 Sample Problem 3.5 Sample Problem 3.5 a) about A b) about the edge AB and c) about the diagonal AG of the cube. d) Determine the perpendicular distance between AG and FC. A cube is acted on by a force P as shown. Determine the moment of P © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 16 Sample Problem 3.5 Sample Problem 3.5 • Moment of P about A, ( ) ( ) ? ? ? ? ? ? - × - = × = ? ? ? ? ? ? - = - = - = - = × = k j P j i a P r M k j P k P j P P j i a j a i a r P r M A F A A F A F A r r r r r r r r r r r r r r r r r r r r ) 2 / ( ) 2 / ( ) 2 / ( ) 2 / ( ? ? ? ? ? ? + + ? ? ? ? ? ? = k j i aP M A r r r r 2 / ? ? ? ? ? ? + + ? ? ? ? ? ? • = • = k j i aP i M i M A AB r r r r r r 2 / 2 / aP M AB = • Moment of P about AB, (M A nın AB üzerindeki izdüşümü)© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 17 Sample Problem 3.5 Sample Problem 3.5 • Moment of P about the diagonal AG, ( ) ( ) ( ) ( ) ( ) 1 1 1 6 aP k j i 2 aP k j i 3 1 M M k j i 2 aP M k j i 3 1 3 a k a j a i a r r M M A AG A A / G A / G A AG - - = + + • - - = • = + + = - - = - - = = • = r r r r r r r r r r r r r r r r r r r r r r ? ? ? 6 aP M AG - = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 18 Sample Problem 3.5 Sample Problem 3.5 • Perpendicular distance between AG and FC, ( ) ( ) ( ) 0 1 1 0 6 3 1 2 = + - = - - • - = • P k j i k j P P r r r r r r r ? Therefore, P is perpendicular to AG. Pd aP M AG = = 6 6 a d=© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 19 © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 20© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 21 The rod shown is supported by brackets at A and B. Determine the moment of the Force F = {-600 i + 200 j - 300 k} which tends to rotate the rod about the AB axis. Example 4.9 Example 4.9 ( ) AB AB AB 2 2 AB AB D ˆ u r F ˆ ˆ r 0.4i 0.2 j ˆ u r (0.4) (0.2) ˆ ˆ ˆ u 0.894i 0.447 j ˆ r r ( 0.2 j )m ˆ ˆ ˆ F { 600i 200 j 300k} N AB M = · × + = = + = + = = = - + - r r r r r © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 22 0.894 0.447 0 0 0.2 0 600 200 300 0.894 [( 0.2)( 300) (0)(200)] 0.447[(0)( 300) (0)( 600)] 0[(0)(200) (0.2)( 600)] 53.67 N m AB AB AB M M M = - - = - - + - - - + - - =- · AB AB AB 53.67 N m ˆ u ˆ ˆ M ( 53.67)[0.894i 0.447 j] ˆ ˆ M 48.0i 24.0 j N m AB M oppposite sense of =- · = - + ? ? = - - · ? ? r r© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 23 Moment of a Couple Moment of a Couple (Bir kuvvet (Bir kuvvet ç çiftinin momenti) iftinin momenti) • • Moment of the couple, Moment of the couple, ( ) ( ) Fd rF M F r M F r r F r F r M B A B A = = × = × - = - × + × = ? sin r r r r r r r r r r r • Kuvvet çiftinin moment vektörü koordinat eksenlerinin orjinin seçiminden bağımsızdır, yani serbest vektör’dür (moment alınan noktadan bağımsızdır). Dçinde bulunduğu düzlemin herhangi bir noktasına uygulanabilir. • Şekilde görüldüğü gibi şiddetleri eşit etki çizgileri parelel ama yönleri zıt olan iki kuvvetten (F ve –F) oluşan bir bileşkeye indirgenemeyen kuvvet sistemine couple (kuvvet çifti) denir. Kuvvetlerin toplamı “0” ama bir noktaya göre momentleri “0” değildir. Cismi döndürmeye çalışır. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 24 E Eş şde değ ğer kuvvet er kuvvet ç çiftleri iftleri Two couples will have equal moments if • 2 2 1 1 d F d F = • the two couples lie in parallel planes, and • the two couples have the same sense or the tendency to cause rotation in the same direction. • • Momentleri ayn Momentleri aynı ı olan kuvvet olan kuvvet ç çiftleri iftleri rijid rijid cisme ayn cisme aynı ı etkiyi yaparlar. etkiyi yaparlar.© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 25 Couples Can Be Represented by Vectors Couples Can Be Represented by Vectors • Bir kuvvet çifti bir rijid cisme etki ettiğinde kuvvet çiftin oluşturan kuvvetlerin nereye etkidiği, şiddetleri ve doğrultularının ne olduğunun önemi yoktur. Sadece kuvvet çiftinin momentinin şiddeti ve doğrultusu önemlidir.Bir kuvvet çifti, şiddeti ve yönü olan bir moment vektörü ile temsil edilebilir. • Couple vectors obey the law of addition of vectors. • Couple vectors are free vectors, i.e., the point of application is not significant. • Couple vectors may be resolved into component vectors. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 26 50© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 27 Example: Finding the Moment of a Couple Example: Finding the Moment of a Couple Example: a) Calculate the moments about points A, B, C, and D generated by the two forces shown below by calculating the moment due to each force separately. b) Then treat the two forces as a couple and calculate the moment. c) Draw the object with the forces replaced by a couple shown as a free vector. 3 mm 4 mm 10 N 10 N A B C D A plate in the shape of a parallelogram is acted upon by two couples. Determine: a) The moment of the couple formed by the two 21-N forces (8.4 m) b) The perpendicular distance between the 12-N forces if the resultant of the two couples is zero. Also find the value of d. (0.7 m , 0.85 m) Example: © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 28 Verilen Bir kuvveti O noktas Verilen Bir kuvveti O noktası ına etkiyen Bir kuvvet ve kuvvet na etkiyen Bir kuvvet ve kuvvet ç çiftine d iftine dö ön nü üş şt tü ürme rme • F kuvvet vektörü O noktasına taşındığında cismin başlangıçtaki dengesi bozulmamalıdır. • Bunun için O noktasında cisme net etkisi olmayan birbirine eşit ve zıt yönde iki kuvvet uygulanır. • Ve bu üç kuvvet de bir kuvvet-kuvvet çifti sistemine indirgenebilir.© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 29 • F kuvvetini A dan farklı O’ noktasına taşımak istersek bu kez de O’ ne göre moment vektörünün (M O’ ) hesaplanması gerekir, F r M O r r r × ' = ' • The moments of F about O and O’ are related, ( ) F s M F s F r F s r F r M O O r r r r r r r r r r r r r × + = × + × = × + = × = ' ' • Bir kuvvet-kuvvet çifti sistemini O dan O’ ne taşırken O noktasındaki kuvvetin O’ noktasına göre momenti de eklenmelidir. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 30 Sample Problem 3.6 Sample Problem 3.6 Şekilde gösterilen kuvvet çiftleri (moment etkisi yaratır) yerine eşdeğer bir tek moment elde ediniz. SOLUTION: • Attach equal and opposite 20 N forces in the +x direction at A, thereby producing 3 couples for which the moment components are easily computed. • Alternatively, compute the sum of the moments of the four forces about an arbitrary single point. The point D is a good choice as only two of the forces will produce non-zero moment contributions..© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 31 Sample Problem 3.6 Sample Problem 3.6 • Attach equal and opposite 20 N forces in the +x direction at A • The three couples may be represented by three couple vectors, ( )( ) ( )( ) ( )( ) mm. 4500 mm 225 20 M . m m 6000 mm 300 20 M mm 13500 mm 450 30 M z y x + = + = + = + = - = - = ( ) ( ) ( )k m. 5 . 4 j m. 6 i m 3.5 1 M r r r r · + · + · - = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 32 Sample Problem 3.6 Sample Problem 3.6 • Alternatively, compute the sum of the moments of the four forces about D. • Only the forces at C and E contribute to the moment about D. ( ) ( ) ( ) ( ) [ ] ( )i 20 k mm 00 3 j mm. 25 2 k 30 j mm. 50 4 M M D r r r r r r r - × - + - × = = ( ) ( ) ( )k m. 5 . 4 j m. 6 i m 3.5 1 M r r r r · + · + · - = • Sonuç önceki ile aynıdır© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 33 Kuvvetler sisteminin bir kuvvet Kuvvetler sisteminin bir kuvvet- -kuvvet kuvvet ç çifti momentine indirgenmesi ifti momentine indirgenmesi • The force and couple vectors may be combined into a resultant force vector and a resultant couple vector, ( ) ? ? × = = F r M F R R O r r r r r • The force-couple system at O may be moved to O’ with the addition of the moment of R about O’ , R s M M R O R O r r r r × + = ' • Two systems of forces are equivalent if they can be reduced to the same force-couple system. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 34 E Eğ ğer bile er bileş şke kuvvet ve moment ke kuvvet ve moment bibirine bibirine dik ise sistem tek bir kuvvete indirgenebilir dik ise sistem tek bir kuvvete indirgenebilir© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 35 Kuvvetler sisteminin tek bir kuvvete indirgenmesi Kuvvetler sisteminin tek bir kuvvete indirgenmesi • Eğer bileşke kuvvet ve bir O noktasındaki moment birbirine dik ise bunlar yeni bir tesir çizgisi üzerinde etki eden tek bir kuvvete indirgenebilir • Bir kuvvetler sistemi için bir bileşke kuvvet ve bir momentin birbirine dik olabilmesi için; kuvvetler 1) ya bir noktada kesişmeli, 2) ya aynı düzlemde olmalı, yada 3) kuvvetler birbirine parelel olmalı. © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 36 Coplanar Force Systems =© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 37 D Dü üzlemdeki kuvvetlerin bir tek kuvvete indirgenmesi zlemdeki kuvvetlerin bir tek kuvvete indirgenmesi • Bileşke kuvvet tesir çizgisi doğrultusunda O noktasına göre ye eşit moment oluşturacak şekilde kaydırılır. R O M r R r • In terms of rectangular coordinates, R O x y M yR xR = - © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 38 Parallel Force Systems© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 39 V Vİ İDA DA R F M oranına vida adımı denir. vida ekseni Bir vektörün başka bir vektörün tesir çizgisi üzerindeki izdüşümü hatırlanırsa ; M Ro nun F R üzerindeki izdüşümü, 2 R R O R R R R O R F M F F M F M F M r r r r • = • = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 40 Sample Problem 3.8 Sample Problem 3.8 For the beam, reduce the system of forces shown to (a) an equivalent force-couple system at A, (b) an equivalent force couple system at B, and (c) a single force or resultant. Note: Since the support reactions are not included, the given system will not maintain the beam in equilibrium. SOLUTION: a) Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about A. b) Find an equivalent force-couple system at B based on the force- couple system at A. c) Determine the point of application for the resultant force such that its moment about A is equal to the resultant couple at A.© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 41 Sample Problem 3.8 Sample Problem 3.8 SOLUTION: a) Compute the resultant force and the resultant couple at A. ( ) ( ) ( ) ( ) j j j j F R r r r r r r N 250 N 100 N 600 N 150 - + - = = ? ( ) j R r r N 600 - = ( ) ( ) ( ) ( ) ( ) ( ) ( ) j i j i j i F r M R A r r r r r r r r r 250 8 . 4 100 8 . 2 600 6 . 1 - × + × + - × = × = ? ( )k M R A r r m N 1880 · - = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 42 Sample Problem 3.8 Sample Problem 3.8 b) Find an equivalent force-couple system at B based on the force-couple system at A. The force is unchanged by the movement of the force-couple system from A to B. ( ) j R r r N 600 - = The couple at B is equal to the moment about B of the force-couple system found at A. ( ) ( ) ( ) ( ) ( )k k j i k R r M M A B R A R B r r r r r r r r r m N 2880 m N 1880 N 600 m 8 . 4 m N 1880 · + · - = - × - + · - = × + = ( )k M R B r r m N 1000 · + =© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 43 Sample Problem 3.10 Sample Problem 3.10 Yukarda üç kablo ile bağlı bir braket görülmektedir. Üç kablo kuvveti yerine A da etki eden bir eşdeğer kuvvet-kuvvet çifti sistemi bulunuz. SOLUTION: • Determine the relative position vectors for the points of application of the cable forces with respect to A. • Resolve the forces into rectangular components. • Compute the equivalent force, ? = F R r r • Compute the equivalent couple, ( ) ? × = F r M R A r r v © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 44 Sample Problem 3.10 Sample Problem 3.10 • Determine the relative position vectors with respect to A. ( ) ( ) ( ) m 100 . 0 100 . 0 m 050 . 0 075 . 0 m 050 . 0 075 . 0 j i r k i r k i r A D A C A B r r r r r r r r r - = - = + = • Resolve the forces into rectangular components. ( ) ( ) N 200 600 300 289 . 0 857 . 0 429 . 0 175 50 150 75 N 700 k j i F k j i k j i r r F B B E B E B r r r r r r r r r r r r r r + - = + - = + - = = = ? ? ( )( ) ( ) N 1039 600 30 cos 60 cos N 1200 j i j i F D r r r r r + = + = ( )( ) ( ) N 707 707 45 cos 45 cos N 1000 j i j i F C r r r r r - = - =© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 45 Sample Problem 3.10 Sample Problem 3.10 • Compute the equivalent force, ( ) ( ) ( )k j i F R r r r r r 707 200 1039 600 600 707 300 - + + - + + + = = ? ( ) N 507 439 1607 k j i R r r r r - + = • Compute the equivalent couple, ( ) k k j i F r j k j i F r k i k j i F r F r M D A D c A C B A B R A r r r r r r r r r r r r r r r r r r r r r v 9 . 163 0 1039 600 0 100 . 0 100 . 0 68 . 17 707 0 707 050 . 0 0 075 . 0 45 30 200 600 300 050 . 0 0 075 . 0 = - = × = - - = × - = - = × × = ? k j i M R A r r r r 9 . 118 68 . 17 30 + + = © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 46 Example Example – – Equivalent Systems Equivalent Systems Example: (Problem 4-113 in Statics, 9th Ed. by Hibbeler) Replace the two forces by an equivalent resultant force and couple moment at point O. Use F = 15 lb. Example: (Problem 4-115 in Statics, 9th Ed. by Hibbeler) The system of parallel forces acts on the top of the Warren truss. Determine the equivalent resultant force of the system and specify its location measured from point A.© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 47 Example Example N.m N.m N.m N.m m © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 48 Quiz Quiz© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 49 P 1 m x © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 50© © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 51 Example 2 Example 2 Replace the force and couple moment system by an equivalent force and couple moment acting at point P. Solution: ° = ? ? ? ? ? ? - - = ? ? ? ? ? ? ? ? = = - + - = ? + = v = - = - ° - = ? ? = ^ + ‹ = - = ° - = ? ? = ?› ? + 73 96 51 170 tan tan ? 178 170 96 51 F F N 170 N 170 140 30 60 N 51.96 N 96 51 30 60 1 - 1 - 2 2 R 2 2 R . ) ( ) . ( sin . cos x y y x y y x x R R R R R y R R x R F F F F F F F F F F -60cos30N -60sin30+ (-140)N -51.96N -170N 73 P P © © © 2003 The McGraw 2003 The McGraw 2003 The McGraw- - -Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics Vector Mechanics for Engineers: Statics 3 - 52 Example 2 Example 2 ( ockwise) (countercl kN.m 2.68 N.m 2676 12 3 140 40 8 30 60 8 12 30 60 = ? = ? + + + ° + - ° = ? = + ? ? ? ? P P P P R R R P R M M M M M ) ( ) ( cos ) ( sin ( ockwise) (countercl kN.m 2.68 N.m 2676 12 3 140 40 8 30 60 8 12 30 60 = ? = ? + + + ° + - ° = ? = + ? ? ? ? P P P P R R R P R M M M M M ) ( ) ( cos ) ( sin P 40Nm P 60cos30N 8m Moment due to horizontal component of 60N P 60sin30N 4m Moment due to vertical component of 60N P 140N 15m Moment due to the force 140N