Statics, Dynamics and Mechanical Engineering 1、Introduction Mechanics: Science which describes and predicts the conditions of rest or motion of bodies under the action of forces. The field of Classical Mechanics can be divided into three categories : . 1) Mechanics of Rigid Bodies 2) Mechanics of Deformable Bodies 3) Mechanics of Fluids Rigid-body mechanics ( General mechanics ) Statics deals with bodies that are in equilibrium with applied forces. [ Such bodies are either at rest or moving at a constant velocity. ] . Dynamics deals with the relation between forces and the motion of bodies. [ Bodies are accelerating. ] Notes ? Rigid-body mechanics is based on the Newton’s laws of motion. . ? These laws were postulated for a particle, which has a mass, but no size or shape. . ? Newton’s laws may be extended to rigid bodies by considering the rigid body to be made up of a large numbers of particles whose relative positions from each other do not change. Newton’s Laws of Motion 1st law. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. 2nd law. If the resultant force acting on a particle is not zero, the particle will experience an acceleration proportional to the magnitude of the force and in the direction of this resultant force. 3rd law. The mutual forces of action and reaction between two particles are equal in magnitude, opposite in direction, and collinear. 2.1 Vectors ? Scalar : Any quantity possessing magnitude (size) only, such as mass, volume, temperature. ? Vector : Any quantity possessing both magnitude and direction, such as force, velocity, momentum. The calculation of a vector must be in a reference frame. A scalar is independent of reference frames. Given two vectors, the vectors will only be equal if both the magnitude and direction of both vectors are equal. In Cartesian coordinate system, an arbitrary vector can be written in terms of unit vectors as Addition of Two Vectors Subtraction of Two Vectors Inner Product of Two Vectors Vector Product of Two Vectors 2.2 Forces Force is a vector quantity, a force is completely described by:1.Magnitude2.Direction3.Point of Application External force : Forces caused by other bodies acting on the rigid body being studied. ( Ex.-- weight, pushing, pulling. ) Internal force : Those forces that keep the rigid body together. Force in 3D A force F in three-dimensional space can be resolved into components using the unit vectors : The vectors i, j, k are unit vectors along the x, y and z axes respectively. . 2.3 Moments The moment of force F about point O is defined as the vector product : where r is the position vector drawn from point O to the point of application of the force F. . The right-hand rule is used to indicate a positive moment. ( torque ) 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 regard 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 ) General plane motion(3 DOF)( the sum of a translation and a rotation ) Motion about a fixed point (3 DOF) General motion (6 DOF) Equations of motion for rigid bodies : Where m is the mass of the rigid body, a is the acceleration of the body’s center of mass, I is called the mass moment of inertia (in kg·m2), and α is the angular acceleration of the center of mass (in rad/s2). 3.3 Solving a Dynamics Problem Free body diagrams Equations of motion The acceleration and angular acceleration must be indicated on the diagram. 4、Summary Rigid-body mechanics, which includes statics and dynamics, is a branch of science that deals with forces and motion of bodies that do not deform under the applied loads. In a free-body diagram, the body under considera- tion is isolated from its surrounding, and loads acting on the body are shown. The direction and magnitudes of the loads must be properly indicated or the analysis will fail.