2.4 Couples A couple is formed by 2 forces F and -F that have equal magnitudes, parallel lines of action and opposite direction The moment of a couple is a vector M perpendicular to the plane of the couple and equal in magnitude to the product Fd. Notes aA couple will not cause translation only rotation a The moment of a couple is independent of the point about which it is computed a Two couples having the same moment M are equivalent. They have the same effect on a given rigid body The direction of a couple is given by the right-hand rule. Therefore, a positive couple generates rotation in a counterclockwise sense 2.5 Equilibrium of a Rigid Body Conditions for rigid-body equilibrium where Forces are" external forces"( body force, applied force, support reaction Moment may be taken about any center of rotation"o 2.6 Free Body Diagrams( FBD Three steps in drawing a free body diagram 1. Isolate the body, remove all supports and connectors 2. Identify all external forces acting on the body. 3. Make a sketch of the body, showing all forces acting on it 2.7 Solving a Statics Problem STEPS 1. Draw a free body diagram 2. Choose a reference frame. Orient the axes 3. Choose a convenient point to calculate moments around 4. Apply the equilibrium equations and solve for the unknowns 2.8 Frictional Forces In problems involving the contact of two bodies, if the contact is not smooth, a reaction will occur along the line of contact This reaction is a force of resistance called the friction. Frictional forces inhibit or prevent slipping Provided that there is no slipping at the contact surface and that the body is not accelerating, experimental studies have shown that the frictional force is related to the normal contact force by the equation: F Where F is the static frictional force and N is the normal contact force. The constant us is called the coefficient of static friction If the body is accelerating, then the frictional force has a value less than the static value. This frictional force, F, is called the kinetic frictional force and is related to the normal force as F=uk M where uk is the coefficient of kinetic friction, Values of uk are as much as 25% smaller than values for As 3. Dynamics Dynamics= Kinematics Kinetics 1). Kinematics, branch of dynamics concerned with describing the state of motion of bodies without regar d to the causes of the motion. displacement, velocity, acceleration, and time 2). Kinetics, branch of dynamics concerned with causes of motion and the action of forces. work, power, energy, impulse, .. Direct dynamics: Calculation of kinematics from forces applied to bodies Inverse dynamics: Calculation of forces and moments from kinematics of bodies and their inertial properties Applications: Analysis of cams, gears, shafts, linkages, connecting rods, etc 3.1 nematics Types of rigid-body motion Translation (3 degrees of freedom) Rotation about a fixed axis(1 DOF)(angular velocity w, angular acceleration a)2.4 Couples A couple is formed by 2 forces F and -F that have equal magnitudes, parallel lines of action and opposite direction. The moment of a couple is a vector M perpendicular to the plane of the couple and equal in magnitude to the product Fd. Notes @ A couple will not cause translation only rotation. @ The moment of a couple is independent of the point about which it is computed. @ Two couples having the same moment M are equivalent. They have the same effect on a given rigid body. The direction of a couple is given by the right-hand rule. Therefore, a positive couple generates rotation in a counterclockwise sense. 2.5 Equilibrium of a Rigid Body Conditions for rigid-body equilibrium : where: • Forces are “external forces” ( body force, applied force, support reaction ) • Moment may be taken about any center of rotation “o” 2.6 Free Body Diagrams ( FBD ) Three steps in drawing a free body diagram: 1. Isolate the body, remove all supports and connectors. 2. Identify all external forces acting on the body. 3. Make a sketch of the body, showing all forces acting on it. 2.7 Solving a Statics Problem STEPS: 1. Draw a free body diagram. . 2. Choose a reference frame. Orient the axes. 3. Choose a convenient point to calculate moments around. . 4. Apply the equilibrium equations and solve for the unknowns. . 2.8 Frictional Forces In problems involving the contact of two bodies, if the contact is not smooth, a reaction will occur along the line of contact. This reaction is a force of resistance called the friction. Frictional forces inhibit or prevent slipping. Provided that there is no slipping at the contact surface and that the body is not accelerating, experimental studies have shown that the frictional force is related to the normal contact force by the equation : F = µs N Where F is the static frictional force and N is the normal contact force. The constant µs is called the coefficient of static friction. If the body is accelerating, then the frictional force has a value less than the static value. This frictional force, F, is called the kinetic frictional force and is related to the normal force as F = µk N where μk is the coefficient of kinetic friction. Values of μk are as much as 25% smaller than values for μs . 3. Dynamics Dynamics = Kinematics + Kinetics 1). Kinematics, branch of dynamics concerned with describing the state of motion of bodies without regar d to the causes of the motion. [ displacement, velocity, acceleration, and time ] . 2). Kinetics, branch of dynamics concerned with causes of motion and the action of forces. . [ work, power, energy, impulse, …] Direct dynamics:Calculation of kinematics from forces applied to bodies. Inverse dynamics:Calculation of forces and moments from kinematics of bodies and their inertial properties. Applications : Analysis of cams, gears, shafts, linkages, connecting rods, etc. 3.1 Kinematics Types of rigid-body motion : Translation (3 degrees of freedom) Rotation about a fixed axis (1 DOF) (angular velocity ω, angular acceleration α )