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Statics (MET 2214) Prof. Simin Nasseri Equilibrium of a Rigid Body

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Statics (MET 2214) Prof. Simin Nasseri Summary Reviewing the Free-Body Diagram Reactions at Supports & ConnectionsReactions at Supports & Connections Equilibrium of a Particle (2D and 3D spaces)

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Statics (MET 2214) Prof. Simin Nasseri Review -Three Dimensional Vectors The vector components are written as where, i - the unit vector along the x direction j - the unit vector along the y direction k - the unit vector along the z direction

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Statics (MET 2214) Prof. Simin Nasseri Review -Three Dimensional Vectors Vector Components: In two dimensions, a force can be described using a magnitude |F| and three angles, θ x, θ y, and θ z. The components of the vector are F x, F y, and F z.

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Statics (MET 2214) Prof. Simin Nasseri Review -Three Dimensional Vectors Vector Components: The three angles, θ x, θ y, and θ z are defined as:

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Statics (MET 2214) Prof. Simin Nasseri Review - Three Dimensional Vectors Vector Components: These vector cosines are

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Statics (MET 2214) Prof. Simin Nasseri Review - Three Dimensional Vectors Vector Components: Substitute into the vector The magnitude is |F| and unit vector is

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Statics (MET 2214) Prof. Simin Nasseri Review -Three Dimensional Vectors Vector Components: The unit vector can be written as: The magnitude is |F| and unit vector is

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Statics (MET 2214) Prof. Simin Nasseri Review - Resultant Forces The components of vectors are used to find the resultants acting on object. Using the unit vectors, the components of forces are

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Statics (MET 2214) Prof. Simin Nasseri Review -Equilibrium of a Particle in Space The components of the forces in equilibrium

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Statics (MET 2214) Prof. Simin Nasseri Free Body Diagrams The first step in solving a problem is drawing a free-body diagram (FBD). Drawing the FBD is the most crucial and important step in solving any problem. It defines weight of the body, the known external forces, and unknown external forces. It defines the constraints and the directions of the forces. If the FBD is drawn correctly the solving of the problem is trivial.

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Statics (MET 2214) Prof. Simin Nasseri Free Body Diagrams Construction of a free body diagram. Step 1: Step 2: Step 3: Decide which body or combination of bodies are to be shown on the free-body diagram. Prepare drawing or sketch of the outline of the isolated or free body. Carefully trace around the boundary of the free-body and identify all the forces exerted by contacting or attracting bodies that were removed during isolation process.

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Statics (MET 2214) Prof. Simin Nasseri Free Body Diagrams Construction of a free body diagram(cont.) Step 4:Choose the set of coordinate axes to be used in solving the problem and indicate their directions on the free-body diagram. Place any dimensions required for solution of the problem on the diagram.

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Statics (MET 2214) Prof. Simin Nasseri Example Problem A 12-ft length of steel pipe weighing 600-lb is lifted by a crane cable CD. Determine the tension in cables AC and BC. 30 o

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Statics (MET 2214) Prof. Simin Nasseri Example Problem F CD =600 lb 600 lb Is this free body diagram going to help? 30 o This FBD is not useful for finding the tension in cables AC or CB. However, you can use it to find the tension CD which will be equal to the weight of the bar. In fact, this can be considered as the FBD of the whole ABC triangle.

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Statics (MET 2214) Prof. Simin Nasseri Example Problem 600 lbs T CA T CB 30 o C

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Statics (MET 2214) Prof. Simin Nasseri FBD - Examples What is the free-body diagram of the weight?

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Statics (MET 2214) Prof. Simin Nasseri FBD - Examples No! The diagram should have the given angles /dimensions. T DA T DB T DC W Is this free body diagram going to help?

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Statics (MET 2214) Prof. Simin Nasseri Example Problem In a ship-unloading operation, a lb automobile is supported by a cable. A rope is tied to the cable at A and pulled in order to center the automobile over its intended position. The angle between the cable and the vertical line is 2 o, while the angle between the rope and the horizontal line is 30 o. What is the tension in the rope? Draw the free-body diagram.

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Statics (MET 2214) Prof. Simin Nasseri Example Problem

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Statics (MET 2214) Prof. Simin Nasseri Example Problem

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Statics (MET 2214) Prof. Simin Nasseri Example Problem Use the equilibrium to solve the problem 30 o 2o2o 3500lb T AC T AB

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Statics (MET 2214) Prof. Simin Nasseri Example Problem Use x component to get a relationship for T AB Solve for tensions

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Forces associated with joints and connections are unlike the forces we’ve been working with so far in this course. The rules for forces and moments acting at joints and contacts, will specify where the forces act; and they will specify that the forces and moments can only act along certain directions. The magnitude of the force is always unknown.

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections x Y z FxFx FyFy FzFz MxMx MyMy MzMz In 3-D space, there are 6 degrees of freedom: 3 translations and 3 rotations FxFx FyFy MzMz x Y 2-D space: 3 degrees of freedom: 2 translations and one rotation

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections 3D Clamped, or welded joints The following few slides are from Division of Engineering Brown University This joints constrains: 3 forces & 3 Moments, all DOFs How many degrees of freedom are restrained by this joint?

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections 2D versions of the clamped joint This joint constrains: 2 forces & 1 Moment (all 3 DOFs) Welded joint

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Pin joints How many degrees of freedom are constrained by this joint?

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Statics (MET 2214) Prof. Simin Nasseri Pin joints Reactions at Supports & Connections This pin constrains: 3 forces & 2 Moments

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Statics (MET 2214) Prof. Simin Nasseri 2D pinned joints Reactions at Supports & Connections 2 forces are constrained.

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Statics (MET 2214) Prof. Simin Nasseri Roller and journal bearings, Type 1 (Bearings are used to support rotating shafts). The bearing shown is like a pin joint: it allows rotation about one axis, but prevent rotation about the other two, and prevents all relative displacement of the shaft. Reactions at Supports & Connections How many reaction forces and moments do you consider for this joint?

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Statics (MET 2214) Prof. Simin Nasseri Roller and journal bearings, type 1 Reactions at Supports & Connections We have 3 reaction forces & 2 reaction moments

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Statics (MET 2214) Prof. Simin Nasseri Roller and journal bearings, type 2 Some types of bearing allow the shaft both to rotate, and to slide through the bearing. Reactions at Supports & Connections How many reaction forces and moments do you consider for this joint?

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Roller and journal bearings, type 2 There are 2 reaction forces & 2 reaction moments

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Statics (MET 2214) Prof. Simin Nasseri Swivel joint Like a pinned joint, but allows rotation about two axes. Reactions at Supports & Connections How many reaction forces and moments do you consider for this joint?

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Swivel joint 3 forces & 1 Moment There must be 3 components of reaction force, and 1 component of reaction moment.

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Reaction forces: Prevents any relative motion. Reaction moments. Allows free rotation about all 3 axes. Ball and socket joint

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Statics (MET 2214) Prof. Simin Nasseri Ball and socket joint Reactions at Supports & Connections 3 forces There must be three components of reaction force. No reaction moments can be present.

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Statics (MET 2214) Prof. Simin Nasseri Slider with pin joint Reactions at Supports & Connections Allows relative motion in one direction, and allows relative rotation about one axis.

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Slider with pin joint 2 forces & 2 moments

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Statics (MET 2214) Prof. Simin Nasseri Slider with pin joint Reactions at Supports & Connections Reaction forces: Motion is prevented in two directions, but allowed in the third. There must be two components of reaction force, acting along directions of constrained motion. Reaction moments: Relative rotation is prevented about two axes, but allowed about a third. There must be two components of reaction moment. 1 force

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Slider with swivel joint

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Statics (MET 2214) Prof. Simin Nasseri Reactions at Supports & Connections Slider with swivel joint Reaction forces: Relative motion is prevented in two directions, but allowed in the third. There must be two components of reaction force acting to prevent motion. Reaction moments: Rotation is permitted around two axes, but prevented around the third. There must be one component of reaction moment. 2 forces & 1 moment

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Statics (MET 2214) Prof. Simin Nasseri Reactions (Summary) Reaction Equivalent to a Force with Known Line of Action – Number of Unknowns = 1

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Statics (MET 2214) Prof. Simin Nasseri Reactions (Summary Reactions (Summary) Reactions Equivalent to a Force with Unknown Direction – Number of Unknowns = 2

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Statics (MET 2214) Prof. Simin Nasseri Reactions (Summary) Reactions Equivalent to a Force with Unknown Direction and a couple – Number of Unknowns = 3

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Statics (MET 2214) Prof. Simin Nasseri Reactions (Summary) Reaction Equivalent to a Force with Known Line of Action – Number of Unknowns = 1

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Statics (MET 2214) Prof. Simin Nasseri Reaction Equivalent to a Force with Known Line of Action – Number of Unknowns = 1 Reactions (Summary)

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