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 Bod 3)Mechanics of Fluids Rigid-body mechanics General mechanics Statics deals with bodies that are in equilibrium with applied forces. I 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. I P Rigid-body mechanics is based on the Newtons laws ofmotion These laws were postulated for a particle, which has a mass, but no size or shap 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 Ist 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 g Scalar: Any quantity possessing magnitude(size)only, such as mass, volume, temperature g 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 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 22 Forces Force is a vector quantity, a force is completely described by: 1. Magnitude2 Direction3 Point of 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 Aforce F in three-dimensional space can be resolved into components using the unit vectors
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 ofmotion. ➢ 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 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 24c。 uples A couple is formed by 2 forces F and-F that have equal magnitudes, parallel lines of action and opposite 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 Acouple 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 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 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=us M Where F is the static frictional force and N is the normal contact force. The constant us is called the If the body is accelerating, then the frictional force has a value less than the static value. This frictional force, F, called the kinetic frictional force and is related to the normal force as F=uk where uk is the coefficient of kinetic friction. Values of uk are as much as 25% smaller than values for u 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
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 μ s . 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 w, angular acceleration a 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, / is called the mass moment of inertia(in kg m2), and a is the angular acceleration of the center of mass(in rad/s 2 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 fa Solid Mechanics and Mechanical Engineering Objectives After learning this chapter you should be able to do the following Differentiate between the different types of basic loading conditions . Understand the basic approach of the Finite Element Method( FEM) 1 Introduction During the analysis of an engineering design, a mechanical engineer is often faced with predicting the deformation of a body In some cases, the inverse problem is solved. That is, the maximum amount of desired deformation is known and the load that will produce the deformation is desired Solid Mechanics: Structural Mechanics, Mechanics of Materials, Elastic Mechanics. Plastic Mechanics
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. Solid Mechanics and Mechanical Engineering Objectives After learning this chapter, you should be able to do the following : ❖ Differentiate between the different types of basic loading conditions. . ❖ Understand the basic approach of the Finite Element Method(FEM). 1. Introduction During the analysis of an engineering design, a mechanical engineer is often faced with predicting the deformation of a body. . In some cases, the inverse problem is solved. That is, the maximum amount of desired deformation is known and the load that will produce the deformation is desired. Solid Mechanics : Structural Mechanics、Mechanics of Materials、Elastic Mechanics、Plastic Mechanics
2. Stress and strain Normal Stress, often symbolized by the greek letter sigma, is defined as the force perpendicular to the cross ectional area divided by the cross sectional area. (axial Axial Strain, a non-dimensional parameter, is defined as the ratio of the deformation in length to the original length Strain is often represented by the Greek symbol epsilon( Application Suppose the force is perpendicular to the longitudinal axis. The stress will be a Shear Stress, defined as force parallel to an area divided by the area. Just as an axial stress results in an axial strain, so does shear stress produce a Shear Strain(n) Application 2----Shearing Force Let's consider a shaft, to which an external torgue is applied(such as in power transmission). The shaft is said to be in torsion. The effect of torsion is to create an angular displacement of one end of the shaft with respect to the other. For a shaft of circular cross section, the relationship between the shear stress and torque is where J is the polar moment of inertia Application 3----Transmission Shaft Notes In general, more than one type of stress may be active in a solid body, due to combined loading conditions. (tension, compression, shear, torsion, etc. )When faced with an engineering problem, an engineer must recognize if more than state of stress exists. Because stresses are vector quantities, care must be taken when adding the terms together Application Transmission system of machine tools Notes The simple loading cases considered in this chapter form the basics of the study of strength of material Method is often used to solve problems involving complicated geometries or loading conditions for structural ar 3. Poisson Effect When a tensile load is applied to a uniform beam the increase in the length of the beam is accompanied by decrease in the lateral dimension of the beam The decrease or the increase in the lateral dimension is due to a lateral strain, which is proportional to the strain along the axial direction The ratio of the lateral strain to the axial strain is related to the poisson ratio named after the mathematician who calculated the ratio by molecular theory The minus sign in Equation is needed in order to keep track of the sign in the strain. For example, because tension corresponds to a decrease in the lateral direction, the lateral strain is negative 4 Hookes law Hooke's Law says that the stretch of a spring is directly proportional to the applied force. Engineers say "Stress is proportional to stra This law is formulated in terms of the stress and strain and may be written as where E is a material constant known as Young's modulus Example 1
2. Stress and Strain . Normal Stress, often symbolized by the Greek letter sigma, is defined as the force perpendicular to the cross sectional area divided by the cross sectional area. (axial stress) . . Axial Strain, a non-dimensional parameter, is defined as the ratio of the deformation in length to the original length. Strain is often represented by the Greek symbol epsilon( ). Application 1——（Tension & Compression） Suppose the force is perpendicular to the longitudinal axis. The stress will be a Shear Stress, defined as force parallel to an area divided by the area..Just as an axial stress results in an axial strain, so does shear stress produce a Shear Strain (γ). Application 2——Shearing Force Let’s consider a shaft, to which an external torque is applied (such as in power transmission). The shaft is said to be in torsion. The effect of torsion is to create an angular displacement of one end of the shaft with respect to the other. For a shaft of circular cross section, the relationship between the shear stress and torque is where J is the polar moment of inertia. Application 3——Transmission Shaft Notes In general, more than one type of stress may be active in a solid body, due to combined loading conditions.(tension, compression, shear, torsion, etc.) When faced with an engineering problem, an engineer must recognize if more than state of stress exists.. Because stresses are vector quantities, care must be taken when adding the terms together. Application 4——Transmission system of machine tools Notes The simple loading cases considered in this chapter form the basics of the study of strength of materials. .. The Finite Element Method is often used to solve problems involving complicated geometries or loading conditions for structural analysis. 3. Poisson Effect When a tensile load is applied to a uniform beam, the increase in the length of the beam is accompanied by a decrease in the lateral dimension of the beam. . The decrease or the increase in the lateral dimension is due to a lateral strain, which is proportional to the strain along the axial direction. The ratio of the lateral strain to the axial strain is related to the Poisson ratio, named after the mathematician who calculated the ratio by molecular theory. The minus sign in Equation is needed in order to keep track of the sign in the strain. For example, because tension corresponds to a decrease in the lateral direction, the lateral strain is negative. 4. Hooke’s Law Hooke's Law says that the stretch of a spring is directly proportional to the applied force. Engineers say "Stress is proportional to strain". This law is formulated in terms of the stress and strain and may be written as : where E is a material constant known as Young’s modulus. Example 1