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**Lecture #7 Part a 2D Rigid Body Equilibrium**

(Reference Chapter 5, sections 1, 2, 3 and 4 – 2D)

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**Example of Rigid Body Equilibrium**

Can we solve for the Reaction at each wheel on this truck using Translational Equilibrium? Is this problem Statically Indeterminate?

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**Sample Problem – Truck Given: Mass of the truck is 310.56 Slugs.**

Force pulling the truck is 800 Lbs attached at the bumper 1.5 ft off the ground and acting parallel to the ground. The distance between the wheels is 15 ft Center of Gravity is 5 ft in front of the rear wheel There is an equal force on each tire toward the back of the truck to offset the pulling force

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**Free Body Diagrams show all Unknown External Forces**

Unknown external forces are also known as constraints Constraints on Space Diagrams need to be replaced with the correct reactionary forces on the Free Body Diagram In a real world problem, the type of constraint at each reaction point needs to be determined

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**The truck ramp has a weight of 400 lb. **

APPLICATIONS The truck ramp has a weight of 400 lb. The ramp is pinned to the body of the truck and held in the position by the cable. How can we determine the cable tension and support reactions ? How are the idealized model and the free body diagram used to do this? Which diagram above is the idealized model? Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**APPLICATIONS (continued)**

Two smooth pipes, each having a mass of 300 kg, are supported by the tines of the tractor fork attachment. How can we determine all the reactive forces ? Again, how can we make use of an idealized model and a free body diagram to answer this question? Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**CONDITIONS FOR RIGID-BODY EQUILIBRIUM**

(Section 5.1) In contrast to the forces on a particle, the forces on a rigid-body are not usually concurrent and may cause rotation of the body (due to the moments created by the forces). Forces on a particle For a rigid body to be in equilibrium, the net force as well as the net moment about any arbitrary point O must be equal to zero. F = 0 (no translation) and MO = 0 (no rotation) Forces on a rigid body Write scaler equations on board Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**THE PROCESS OF SOLVING RIGID BODY EQUILIBRIUM PROBLEMS**

For analyzing an actual physical system, first we need to create an idealized model (above right). Then we need to draw a free-body diagram (FBD) showing all the external (active and reactive) forces. Finally, we need to apply the equations of equilibrium to solve for any unknowns. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**FREE-BODY DIAGRAMS (Section 5.2)**

Free-body diagram (FBD) 1. Draw an outlined shape. Imagine the body to be isolated or cut “free” from its constraints and draw its outlined shape. Idealized model 2. Show all the external forces and couple moments. These typically include: a) applied loads, b) support reactions, and, c) the weight of the body. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**FREE-BODY DIAGRAMS (continued)**

Idealized model Free-body diagram 3. Label loads and dimensions on the FBD: All known forces and couple moments should be labeled with their magnitudes and directions. For the unknown forces and couple moments, use letters like Ax, Ay, MA, etc.. Indicate any necessary dimensions. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**SUPPORT REACTIONS IN 2-D**

A few example sets of diagrams s are shown above. Other support reactions are given in your textbook (Table 5-1). As a general rule, if a support prevents translation of a body in a given direction, then a force is developed on the body in the opposite direction. Similarly, if rotation is prevented, a couple moment is exerted on the body in the opposite direction. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**Unknown External Forces – 2D**

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**Unknown External Forces – 2D**

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**Draw: A an idealized model and free- body diagram of the foot pedal.**

EXAMPLE Given: The operator applies a vertical force to the pedal so that the spring is stretched 1.5 in. and the force in the short link at B is 20 lb. Draw: A an idealized model and free- body diagram of the foot pedal. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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1. The beam and the cable (with a frictionless pulley at D) support an 80 kg load at C. In a FBD of only the beam, there are how many unknowns? A) 2 forces and 1 couple moment B) 3 forces and 1 couple moment C) 3 forces D) 4 forces CONCEPT QUIZ Answers : 1. C Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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CONCEPT QUIZ 2. If the directions of the force and the couple moments are both reversed, what will happen to the beam? A) The beam will lift from A. B) The beam will lift at B. C) The beam will be restrained. D) The beam will break. Answer 2. A Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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GROUP PROBLEM SOLVING Draw a FBD of the crane boom, which is supported by a pin at A and cable BC. The load of 1250 lb is suspended at B and the boom weighs 650 lb. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**GROUP PROBLEM SOLVING (continued)**

FBD Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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GROUP PROBLEM SOLVING Draw a FBD of member ABC, which is supported by a smooth collar at A, roller at B, and link CD. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**GROUP PROBLEM SOLVING (continued)**

FBD Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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ATTENTION QUIZ 1. Internal forces are not shown on a free-body diagram because the internal forces are_____. (Choose the most appropriate answer.) A) Equal to zero B) Equal and opposite and they do not affect the calculations C) Negligibly small D) Not important 2. How many unknown support reactions are there in this problem? A) 2 forces and 2 couple moments B) 1 force and 2 couple moments C) 3 forces D) 3 forces and 1 couple moment Answers: 1.B 2.D Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

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**Remaining slides = Section 5.3/5.4 –**

2D Rigid Body Equilibrium!!! (Finally)

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**READING QUIZ (Section 5.3/5.4)**

1. The three scalar equations FX = FY = MO = 0, are ____ equations of equilibrium in two dimensions. A) Incorrect B) The only correct C) The most commonly used D) Not sufficient 2. A rigid body is subjected to forces as shown. This body can be considered as a ______ member. A) Single-force B) Two-force C) Three-force D) Six-force Answers: 1. C 2. B Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**The uniform truck ramp has a weight of 400 lb. **

APPLICATIONS A The uniform truck ramp has a weight of 400 lb. The ramp is pinned at A and held in the position by the cable. How can we determine the forces acting at the pin A and the force in the cable ? Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**APPLICATIONS (continued)**

A 850 lb of engine is supported by three chains, which are attached to the spreader bar of a hoist. You need to check to see if the breaking strength of any of the chains is going to be exceeded. How can you determine the force acting in each of the chains? Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**EQUATIONS OF EQUILIBRIUM (Section 5.3)**

A body is subjected to a system of forces that lie in the x-y plane. When in equilibrium, the net force and net moment acting on the body are zero (as discussed earlier in Section 5.1). This 2-D condition can be represented by the three scalar equations: Fx = Fy = 0 MO = 0 where point O is any arbitrary point. Please note that these equations are the ones most commonly used for solving 2-D equilibrium problems. There are two other sets of equilibrium equations that are rarely used. For your reference, they are described in the textbook. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**TWO-FORCE MEMBERS & THREE FORCE-MEMBERS (Section 5.4)**

The solution to some equilibrium problems can be simplified if we recognize members that are subjected to forces at only two points (e.g., at points A and B). If we apply the equations of equilibrium to such a member, we can quickly determine that the resultant forces at A and B must be equal in magnitude and act in the opposite directions along the line joining points A and B. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**EXAMPLE OF TWO-FORCE MEMBERS**

In the cases above, members AB can be considered as two-force members, provided that their weight is neglected. This fact simplifies the equilibrium analysis of some rigid bodies since the directions of the resultant forces at A and B are thus known (along the line joining points A and B). Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**STEPS FOR SOLVING 2-D EQUILIBRIUM PROBLEMS**

1. If not given, establish a suitable x - y coordinate system. 2. Draw a free body diagram (FBD) of the object under analysis. 3. Apply the three equations of equilibrium (E-of-E) to solve for the unknowns. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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IMPORTANT NOTES 1. If there are more unknowns than the number of independent equations, then we have a statically indeterminate situation. We cannot solve these problems using just statics. 2. The order in which we apply equations may affect the simplicity of the solution. For example, if we have two unknown vertical forces and one unknown horizontal force, then solving FX = 0 first allows us to find the horizontal unknown quickly. 3. If the answer for an unknown comes out as negative number, then the sense (direction) of the unknown force is opposite to that assumed when starting the problem. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**EXAMPLE 2D Equilibrium:**

Given: The 4kN load at B of the beam is supported by pins at A and C . Find: The support reactions at A and C. Plan: 1. Put the x and y axes in the horizontal and vertical directions, respectively. 2. Determine if there are any two-force members. 3. Draw a complete FBD of the boom. 4. Apply the E-of-E to solve for the unknowns. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**EXAMPLE (Continued) FBD of the beam:**

AX AY A 1.5 m C B 4 kN FC 45° Note: Upon recognizing CD as a two-force member, the number of unknowns at C are reduced from two to one. Now, using E-o-f E, we get, + MA = FC sin 45 1.5 – 4 3 = 0 Fc = kN or kN + FX = AX cos 45 = 0; AX = – kN + FY = AY sin 45 – 4 = 0; AY = – kN Note that the negative signs means that the reactions have the opposite direction to that shown on FBD. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**CONCEPT QUIZ Answers: 1. D 2. B**

1. For this beam, how many support reactions are there and is the problem statically determinate? A) (2, Yes) B) (2, No) C) (3, Yes) D) (3, No) F 2. The beam AB is loaded and supported as shown: a) how many support reactions are there on the beam, b) is this problem statically determinate, and c) is the structure stable? A) (4, Yes, No) B) (4, No, Yes) C) (5, Yes, No) D) (5, No, Yes) F Fixed support A B Answers: 1. D 2. B Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**Find: Support reactions at B and C.**

GROUP PROBLEM SOLVING 2D Equilibrium Given: The jib crane is supported by a pin at C and rod AB. The load has a mass of 2000 kg with its center of mass located at G. Assume x = 5 m. Find: Support reactions at B and C. Plan: a) Establish the x – y axes. b) Draw a complete FBD of the jib crane beam. c) Apply the E-of-E to solve for the unknowns. Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**GROUP PROBLEM SOLVING (Continued)**

FBD of the beam 2000(9.81) N FAB 5 m Cx 5 4 3 Cy 4 m 0.2 m First write a moment equation about Point C. Why point C? + MC = (3 / 5) FAB 4 + (4 / 5) FAB 0.2 – 2000(9.81) 5 = 0 FAB = N = 38.3 kN Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**GROUP PROBLEM SOLVING (Continued)**

FBD of the beam 2000(9.81) N FAB 5 m Cx 5 4 3 Cy 4 m 0.2 m FAB = N = 38.3 kN Now solve the FX and FY equations. FX = Cx – (4 / 5) = 0 FY = – Cy + (3 / 5) – 2000(9.81) = 0 Solving these two equations, we get Cx = N or kN and Cy = N or kN Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**ATTENTION QUIZ Answers: 1. C 2. B**

1. Which equation of equilibrium allows you to determine FB right away? A) FX = B) FY = 0 C) MA = 0 D) Any one of the above. AX A B FB AY 100 lb 2. A beam is supported by a pin joint and a roller. How many support reactions are there and is the structure stable for all types of loadings? A) (3, Yes) B) (3, No) C) (4, Yes) D) (4, No) Answers: 1. C 2. B Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.3,5.4

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**Sample Problem – 2D Truss**

Solve this for the Unknown Reactions at Point A and B 2m 1.5m 1.5m

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Sample Problem - Table Given that the table weighs 50 lbs, what is the maximum weight of the person sitting at it before it tips over?

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**F5-1: Find Support Reactions**

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24,000 lb 5:20. The train car has a weight of 24,000 lb and a center of gravity at G. It is suspended from its front and rear track by six tires located at A, B, and C. Determine the normal reactions on these tires if the track is assumed to be a smooth surface and an equal portion of the load is supported at both the front and rear tires. NC NA NB

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5.21: Determine the horizontal and vertical components of reaction at pin A and the tension developed in cable BC used to support the steel frame.

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**5-40: The assembly has a weight of 250 lb and center of gravity at G1**

5-40: The assembly has a weight of 250 lb and center of gravity at G1. If it is intended to support a maximum load of 400 lb placed at point G2, determine the smallest counterweight W that should be placed at B in order to prevent the platform from tipping over.

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**5. 47: The motor has a weight of 850 lb**

5.47: The motor has a weight of 850 lb. Determine the force that each of the chains exerts on the supporting hooks at A, B and C. Neglect the size of the hooks and thickness of the beam.

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**What’s better? Attaching the rope at A or B? FBD!!!!!**

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**Example: 2D – 2 force and 3 force members:**

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**Understanding the Question**

The last part of doing Rigid Body Equilibrium problems is understanding what needs solved. This has to be done by careful examination and engineering judgment (for lack of a better word) Practical Examples of Statics:

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**Washing Machine Isolator**

Accelerometer location Left/Rear Accelerometer location Left/Front Accelerometer location Right/Front

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**Washing Machine Isolator**

Vertical acceleration, left front

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Woofer Isolator

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**Face Mask Shock Isolator:**

Function: absorb impact, dissipate energy, heavily damped soft butyl. Butyl elastomer bonded to g/f nylon

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**Case Study –Industrial Caster**

Isolation mount Competitors Ours

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**Drill Depth up to 48” Bit Isolator Chuck Isolator**

2010 SME Annual Meeting & Exhibit Phoenix, Arizona

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**35 mm (1 3/8 in) Isolator Load Requirements:**

Drill Bit Drill Bit Isolator ( Thrust Load = 8,000 to 10,000 lb Bending Load= 125 lbs)= K*d Drill Rod Torsion Load= 300 lb-ft Chuck Isolator HOT SPOT! 2010 SME Annual Meeting & Exhibit Phoenix, Arizona

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