# Chapter 2: Concurrent force systems. Department of Mechanical Engineering

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1 Chapter : Concurrent force sstems

2 Objectives To understand the basic characteristics of forces To understand the classification of force sstems To understand some force principles To know how to obtain the resultant of forces in D and 3D sstems To know how to obtain the components of forces in D and 3D sstems

3 Characteristics of forces orce: Vector with magnitude and direction Magnitude a positive numerical value representing the size or amount of the force Directions the slope and the sense of a line segment used to represent the force Described b angles or dimensions A negative sign usuall represents opposite direction Point of application A point where the force is applied A line of action a straight line etending through the point of application in the direction of the force The force is a phsical quantit that needs to be represented using a mathematical quantit

4 Eample direction j i 1000 N α magnitude Line of action Point of application

5 Vector to represent orce A vector is the mathematical representation that best describes a force A vector is characterized b its magnitude and direction/sense Math operations and manipulations of vectors can be used in the force analsis

6 ree, sliding, and fied vectors Vectors have magnitudes, slopes, and senses, and lines of applications A free vector The application line does not pass a certain point in space A sliding vector The application line passes a certain point in space A fied vector The application line passes a certain point in space The application point of the vector is fied

7 Vector/force notation The smbol representing the force bold face or underlined letters The magnitude of the force lightface (in the tet book, italic) A A or A A

8 Classification of forces Based on the characteristic of the interacting bodies: Contacting vs. Non-contacting forces Surface force (contacting force) Eamples:» Pushing/pulling force» rictions Bod force (non-contacting force) Eamples:» Gravitational force» Electromagnetic force

9 Classification of forces Based on the area (or volume) over which the force is acting Distributed vs. Concentrated forces Distributed force The application area is relativel large compare to the whole loaded bod Uniform vs. Non-uniform Concentrated force The application area is relativel small compare to the whole loaded bod

10 What is a force sstem? A number of forces (in D or 3D sstem) that is treated as a group: A concurrent force sstem All of the action lines intersect at a common point A coplanar force sstem All of the forces lie in the same plane A parallel force sstem All of the action lines are parallel A collinear force sstem All of the forces share a common line of action

11 The eternal and internal effects A force eerted on the bod has two effects: Eternal effects» Change of motion» Resisting forces (reactions) Internal effects» The tendenc of the bod to deform develop strain, stresses If the force sstem does not produce change of motion» The forces are said to be in balance» The bod is said to be in (mechanical) equilibrium

12 Eternal and internal effects Eample 1: The bod changes in motion a Not fied, no (horizontal) support Eample : The bod deforms and produces (support) reactions The forces must be in balance ied support Support Reactions

13 Principle for force sstems Two or more force sstems are equivalent when their applications to a bod produce the same eternal effect Transmissibilit Reduction A process to create a simpler equivalent sstem to reduce the number of forces b obtaining the resultant of the forces Resolution The opposite of reduction to find the components of a force vector breaking up the resultant forces

14 Principle of Transmissibilit Man times, the rigid bod assumption is taken onl the eternal effects are the interest The eternal effect of a force on a rigid bod is the same for all points of application of the force along its line of action

15 Resultant of orces Review on vector addition Vector addition R A B B A B Triangle method (head-to-tail method) Note: the tail of the first vector and the head of the last vector become the tail and head of the resultant principle of the force polgon/triangle Parallelogram method Note: the resultant is the diagonal of the parallelogram formed b the vectors being summed A R B R A

16 Resultant of orces Review on geometric laws Law of Sines Laws of Cosines α β γ cos cos cos ac c b a ac c a b ab b a c A B C c a b β γ α

17 Resultant of two concurrent forces The magnitude of the resultant (R) is given b R R cosγ cosφ Pa attention to the angle and the sign of the last term!!! The direction (relative to the direction of 1 ) can be given b the law of sines sin β sinφ R

18 Resultant of three concurrent forces and more Basicall it is a repetition of finding resultant of two forces The sequence of the addition process is arbitrar The force polgons ma be different The final resultant has to be the same

19 Resultant of more than two forces The polgon method becomes tedious when dealing with three and more forces It s getting worse when we deal with 3D cases It is preferable to use rectangular-component method

20 Eample Problem -1 Determine: The resultant force (R) The angle between the R and the -ais Answer: The magnitude of R is given b R R lb The angle α between the R and the 900-lb force is given b 0 sinα sin( o α The angle therefore is (900)(600) cos ) 0

21 Eample Problem - Determine The resultant R The angle between the R and the -ais

22 Another eample If the resultant of the force sstem is zero, determine The force B The angle between the B and the -ais

23 orce components

24 Resolution of a force into components The components of a resultant force are not unique!! R A B ( G I) C D E H The direction of the components must be fied (given)

25 How to obtain the components of a force (arbitrar component directions)? Parallel to u Parallel to v Steps: Draw lines parallel to u and v crossing the tip of the R Together with the original u and v lines, these two lines produce the parallelogram The sides of the parallelogram represent the components of R Use law of sines to determine the magnitudes of the components u v o sin 45 sin 5 o 900 sin110 o u v 900sin 45 0 sin sin 5 o sin110 o 0 677N 405N

26 Eample Problem -5 Determine the components of 100 kn along the bars AB and AC Hints: Construct the force triangle/parallelogram Determine the angles α, β, γ Utilize the law of sines

27 Another eample Determine the magnitude of the components of R in the directions along u and v, when R 1500 N

28 Rectangular components of a force What and Wh rectangular components? Rectangular components all of the components are perpendicular to each other (mutuall perpendicular) Wh? One of the angle is 90 o > simple Utilization of unit vectors Rectangular components in D and 3D Utilization of the Cartesian c.s. Arbitrar rectangular

29 The Cartesian coordinate sstem The Cartesian coordinate aes are arranged following the right-hand sstem (shown on the right) The setting of the sstem is arbitrar, but the results of the analsis must be independent of the chosen sstem z

30 Unit vectors A dimensionless vector of unit magnitude The ver basic coordinate sstem used to specif coordinates in the space is the Cartesian c.s. The unit vectors along the Cartesian coordinate ais, and z are i, j, k, respectivel The smbol e n will be used to indicate a unit vector in some n- direction (not,, nor z) An vector can be represented as a multiplication of a magnitude and a unit vector A is in the positive direction along n B is in the negative direction along n A A e n Ae n B B e n Be n A e n A A A

31 The rectangular components of a force in D sstem While the components must be perpendicular to each other, the directions do not have to be parallel or perpendicular to the horizontal or vertical directions j i i j j i 1 tan sin cos

32 z z z z z cos cos cos cos cos cos The rectangular components in 3D sstems z n n z z k j i e e k j i z j i z z k i k j e n z k j i e z n cos cos cos

33 Dot Products of two vectors A B B A A B cos AB cos A Special cosines: It s a scalar!!! B Cos 0 o 1 Cos 30 o ½ 3 Cos 45 o ½ Cos 60 o 0.5 Cos 90 o 0

34 Dot products and rectangular components The dot product can be used to obtain the rectangular components of a force (a vector in general) A A n n A e A n e n n Acos n (magnitude) (the vectorial component in the n direction) A n ( A e ) e n n The component along e n A t A A n The component along e t Remember, e n and e t are perpendicular

35 Cartesian rectangular components The dot product is particularl useful when the unit vectors are of the Cartesian sstem (the i, j, k) j i i j k j i z sin ) cos(90 cos 90- Also, in 3D, j j i i j i ) ( ) (

36 More usage of dot products Dot products of two vectors written in Cartesian sstem A B A B A B A B The magnitude of a vector (could be a force vector), here A is the vector magnitude z z A A A cos0 A A A A A The angle between two vectors (sa between vectors A and B) cos 1 A B AB Az B AB z A z A z

37 The rectangular components of arbitrar direction zn z n n n z n n n z n n t t n n z z cos cos cos ) ( e k e j e i e k j i e e e k j i k j i e zn n n n cos cos cos z j i z z k i k j e n zn n n n t Can ou show the following?

38 Summarizing. The components of a force resultant are not unique Graphical methods (triangular or parallelogram methods) combined with law of sinus and law of cosines can be used to obtain components in arbitrar direction Rectangular components are components of a force (vector) that perpendicular to each other The dot product can be used to obtain rectangular components of a force vector obtain the magnitude of a force vector (b performing selfdot-product) Obtain the angle between two (force) vectors

39 Eample Problem -6 ind the and scalar components of the force ind the and scalar components of the force Epress the force in Cartesian vector form for the - and - aes

40 Eample Problem -6 ' ' cos cos β β cos6 11N 450sin 6 397N 450cos3 38N 450sin 3 38N o ' o cos(90 ) cos(90 β ) Writing the in Cartesian vector form: β (11i 397j) N (38e ' 38e ) ' N

41 Eample Problem -8 B ind the angles,, and z ( is the angle between OB and ais and so on..) The,, and scalar components of the force. The rectangular component n of the force along line OA The rectangular component of the force perpendicular to line OA (sa t )

42 Eample Problem -8 B To find the angles: ind the length of the diagonal OB, sa d d m Use cosines to get the angles z cos cos cos o o o The scalar components in the,, and z directions: z cos cos cos ( 1.86i j 1.86k)kN z 1.86kN kN 1.86kN

43 Eample Problem -8 To find the rectangular component n of the force along line OA: Needs the unit vector along OA Method 1 : ollow the method described in the book Method : utilize the vector position of A (basicall vector OA) OA r A 3 i 1j 3k Remember, that an vector can be represented as a multiplication of its magnitude and a unit vector along its line of application ra 3i 1j 3k eoa r A i 1j 3k 0.688i 0.30j 0.688k 4.36

44 Eample Problem 8- OA e OA The scalar component of along OA OA OA (1.86i j 1.86k) (0.688i 0.30j 0.688k) kN The vector component of along OA OA ( e ) e 1.6(0.688i 0.30 j 0.688k) OA 14.86i 4.97j 14.86k The vector component of perpendicular to OA t OA ( i 1.18j k) The scalar component of perpendicular to OA OA (1.86i j 1.86k) (14.86i 4.97j 14.86k) t t ( i 1.18j k) ( ) 1.18 ( ) 1. 50kN Check: OA t kN

45 Resultants b rectangular components The Cartesian rectangular components of forces can be utilized to obtain the resultant of the forces 1 1 Adding the vector components, we obtain the vector component of the resultant R 1 Adding the vector components, we obtain the vector component of the resultant 1 R 1 The resultant can be obtained b performing the vector addition of these two vector components R R R R i R j

46 Resultants b rectangular components The scalar components of the resultant R 1 ( 1 ) i Ri R 1 ( 1 ) j R j The magnitude of the resultant R R R The angles formed b the resultant and the Cartesian aes 1 R cos R cos All of the above results can be easil etended for 3D sstem 1 R R

47 Please do eample problems -9, -10, and -11

48 HW Problem -0 Determine the non-rectangular components of R

49 HW Problem -37 Determine the components of 1 and in - and - sstems

50 HW Problem -44 Epress the cable tension in Cartesian form Determine the magnitude of the rectangular component of the cable force Determine the angle α between cables AD and BD Tpo in the problem!!! B(4.9,-7.6,0) C(-7.6,-4.6,0) Don t worr if ou don t get the solution in the back of the book

51 HW Problem -46 Determine the scalar components Epress the force in Cartesian vector form Determine the angle α between the force and line AB

52 HW problems -55 Given: lb, 300 lb, 3 00 lb Determine the resultant Epress the resultant in the Cartesian format ind the angles formed b the resultant and the coordinate aes

53 HW Problem -49 Given T 1 and T are 650 lb, Determine P so that the resultant of T 1, T and P is zero

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