Engineering Mechanics: Statics

• Engineering mechanics is the application of mechanics to solve problems involving engineering elements.

Statics

• Statics deals with bodies at rest under the action of forces.
• A body is considered to be at rest when it is not moving or is moving with a constant velocity.
• Key concepts in statics include force, moment, and equilibrium.
• Force is a vector quantity describing the interaction between two bodies. It has magnitude, direction, and a point of application.
• Moment is the tendency of a force to cause rotation about a point or axis. It is the product of the force and the perpendicular distance from the point or axis to the line of action of the force.
• Equilibrium is the state where the net force and net moment acting on a body are zero, resulting in no translational or rotational acceleration.
• Free Body Diagrams (FBDs) are essential in statics.
• An FBD isolates the body of interest and shows all external forces acting on it.
• Supports and connections introduce reaction forces on a body, which must be included in the FBD.
• Common types of supports include pin supports which provide reactions in two orthogonal directions, and fixed supports that provide reactions in two orthogonal directions and a moment.
• Equations of Equilibrium: In two dimensions, the equilibrium equations are ΣFx = 0, ΣFy = 0, and ΣM = 0, where ΣFx and ΣFy are the sums of forces in the x and y directions, respectively, and ΣM is the sum of moments about a point.
• Trusses are structures composed of members connected at joints, designed to support loads.
• Truss members are assumed to be pin-connected, meaning they can only transmit axial forces (tension or compression).
• Method of Joints: This involves analyzing the equilibrium of each joint in the truss by drawing FBDs of the joints and applying the equilibrium equations (ΣFx = 0, ΣFy = 0).
• Method of Sections: This involves cutting through the truss to create sections, allowing for the determination of forces in specific members by analyzing the equilibrium of one of the sections.
• Frames and Machines: Frames are rigid structures containing multi-force members, which are members subjected to more than two forces. Machines are structures designed to transmit and modify forces.
• Analysis of frames and machines involves dismembering the structure and analyzing the equilibrium of individual members.
• Friction: Frictional forces oppose the motion or impending motion between two surfaces in contact.
• Static friction (Fs) is the friction force when there is no relative motion, with a maximum value of Fs,max = μsN, where μs is the coefficient of static friction and N is the normal force.
• Kinetic friction (Fk) is the friction force when there is relative motion, given by Fk = μkN, where μk is the coefficient of kinetic friction.
• Centroids and Centers of Gravity: The centroid is the geometric center of an area or volume. The center of gravity is the point where the weight of an object is considered to act.
• For homogeneous bodies in a uniform gravitational field, the centroid and center of gravity coincide.
• The centroid can be determined using integration or by using composite areas/volumes.

Dynamics

• Dynamics deals with the motion of bodies under the action of forces.
• Kinematics is the study of motion without considering the forces causing it.
• Kinetics relates the forces acting on a body to its motion.
• Kinematics of Particles: Position, velocity, and acceleration are fundamental concepts.
• Position (r) is the location of a particle in space, described by a vector.
• Velocity (v) is the rate of change of position with respect to time, v = dr/dt.
• Acceleration (a) is the rate of change of velocity with respect to time, a = dv/dt.
• Rectilinear Motion: Motion along a straight line.
• Constant Acceleration: Equations of motion for constant acceleration are: v = u + at, s = ut + (1/2)at^2, v^2 = u^2 + 2as, where u is the initial velocity, v is the final velocity, a is the constant acceleration, t is the time, and s is the displacement.
• Curvilinear Motion: Motion along a curved path.
• Components of motion can be described in Cartesian coordinates (x, y, z), normal and tangential coordinates (n, t), or polar coordinates (r, θ).
• Normal and Tangential Components: Velocity is tangent to the path, and acceleration can be resolved into tangential (at) and normal (an) components.at = dv/dt, and an = v^2/ρ, where ρ is the radius of curvature.
• Kinetics of Particles: Newton's Second Law of Motion states that the sum of forces acting on a particle is equal to the mass of the particle times its acceleration (ΣF = ma).
• Work and Energy: Work is done by a force when it causes a displacement.
• Work done by a constant force is U = Fdcosθ, where F is the magnitude of the force, d is the magnitude of the displacement, and θ is the angle between the force and displacement vectors.
• Kinetic Energy (T) is the energy possessed by a body due to its motion, T = (1/2)mv^2.
• Potential Energy (V) is the energy possessed by a body due to its position or configuration.
• Gravitational potential energy is Vg = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above a datum.
• Elastic potential energy (for a spring) is Ve = (1/2)kx^2, where k is the spring constant and x is the deformation of the spring.
• Conservation of Energy: If only conservative forces do work, the total mechanical energy (T + V) is conserved.
• Principle of Work and Energy: The work done by all forces acting on a particle equals the change in its kinetic energy: U = ΔT = T2 - T1.
• Impulse and Momentum: Linear momentum (p) is the product of mass and velocity, p = mv.
• Impulse (I) is the integral of force with respect to time, I = ∫F dt.
• The impulse-momentum theorem states that the impulse acting on a particle equals the change in its linear momentum: I = Δp = mv2 - mv1.
• Conservation of Linear Momentum: If the net external force acting on a system of particles is zero, the total linear momentum of the system is conserved.
• Impact: A collision between two bodies.
• Coefficient of Restitution (e): A measure of the elasticity of the collision, e = (v2' - v1')/(v1 - v2), where v1 and v2 are the velocities before impact, and v1' and v2' are the velocities after impact. e = 1 for perfectly elastic collisions, and e = 0 for perfectly plastic collisions.
• Kinematics of Rigid Bodies: Rigid bodies maintain a constant shape.
• Translation: All points in the body have the same velocity and acceleration.
• Rotation about a Fixed Axis: All points move in circular paths about the axis of rotation.
• Angular Velocity (ω) is the rate of change of angular position, ω = dθ/dt.
• Angular Acceleration (α) is the rate of change of angular velocity, α = dω/dt.
• General Plane Motion: A combination of translation and rotation.
• Instantaneous Center of Zero Velocity (IC): A point in a rigid body (or its extension) that has zero velocity at a given instant. It simplifies the analysis of velocities in general plane motion.
• Kinetics of Rigid Bodies: Mass Moment of Inertia (I) is a measure of a body's resistance to angular acceleration.
• For a discrete system, I = Σmi ri^2, and for a continuous body, I = ∫r^2 dm.
• Parallel-Axis Theorem: I = IG + md^2, where IG is the mass moment of inertia about the center of mass, m is the mass, and d is the distance between the parallel axes.
• Equations of Motion: ΣF = maG and ΣMG = IGα, where ΣF is the sum of external forces, maG is the mass times the acceleration of the center of mass, ΣMG is the sum of external moments about the center of mass, IG is the mass moment of inertia about the center of mass, and α is the angular acceleration.
• Work-Energy Principle for Rigid Bodies: The work done by external forces and moments equals the change in kinetic energy: U = ΔT.
• Kinetic Energy of a Rigid Body: T = (1/2)mvG^2 + (1/2)IGω^2, where vG is the velocity of the center of mass and ω is the angular velocity.