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### Engineering mechanics by A.Vinoth Jebaraj

• 2. Force Action of one body on the other (push or pull) Point of Application Direction Magnitude
• 3. What is the need of knowing MECHANICS? Mechanics Deals with forces
• 4. Mechanics Mechanics of Rigid Bodies Mechanics of Deformable Bodies Mechanics of Fluids Statics Dynamics kinematics kinetics
• 5. Studying External effect of forces on a body such as velocity, acceleration, displacement etc. Studying Internal effect of forces on a body such as stresses (internal resistance), change in shape etc. Rigid body mechanics Deformable body mechanics
• 6. Statics  Deals with forces and its effects when the body is at rest Dynamics  Deals with forces and its effects when the body is in moving condition Truss Bridge IC Engine
• 7. Rigid body mechanics Actual structures and machines are never rigid under the action of external loads or forces. But the deformations induced are usually very small which does not affect the condition of equilibrium. Negligible deformation (no deformation) under the action of forces. Assuming 100% strength in the materials. Large number of particles occupying fixed positions with each other.
• 8. Particle Mechanics  Treating the rigid body as a particle which is negligible in size when compared to the study involved. (very small amount of matter which is assumed as a point in a space). Example: studying the orbital motion of earth
• 9. Types of forces Concurrent coplanar forces Collinear forces Non Concurrent coplanar (Parallel) Concurrent non-coplanar
• 10. Components of a Force Plane ForcePlane Force Space ForceSpace Force
• 11. Couple Two equal and opposite forces are acting at some distance forming a couple
• 12. How Rotational Effect will change with distance?
• 13. Free body diagram Isolated body from the structure of machinery which shows all the forces and reaction forces acting on it.
• 15. Parallelogram law: Two forces acting on a particle can be replaced by the single component of a force (RESULTANT) by drawing diagonal of the parallelogram which has the sides equal to the given forces. Parallelogram law cannot be proved mathematically . It is an experimental finding.
• 16. The two vectors can also be added by head to tail by using triangle law. Triangle law states that if three concurrent coplanar forces are acting at a point be represented in magnitude and direction by the sides of a triangle, then they are in static equilibrium.
• 17. Lami’s Theorem states that if three concurrent coplanar forces are acting at a point, then each force is directly proportional to the sine of the angle between the other two forces.
• 18. Lami’s theorem considering only the equilibrium of three forces acting on a point not the stress acting through a ropes or strings The principle of transmissibility is applicable only for rigid bodies not for deformable bodies
• 19. F1 F2 F5 F4 F3 A B E D C Polygon Law of Forces “If many number of forces acting at a point can be represented as a sides of a polygon, then they are in equilibrium”
• 21.
• 22.
• 23.
• 25.
• 26.
• 27.
• 28.
• 29. Friction Friction is a force [Tangential force] that resists the movement of sliding action of one surface over the other.
• 30. Few examples where friction force involved
• 31. Theory of Dry Friction Uneven distribution of friction force and normal reaction in the surface. Microscopic irregularities produces reactive forces at each point of contact. The distance ‘x’ is to avoid “tipping effect” caused by the force ‘P’ so that moment equilibrium has been arrived about point ‘O’.
• 32.
• 33.
• 34. Limiting static frictional force: when this value is reached then the body will be in unstable equilibrium since any further increase in P will cause the body to move. At this instance, frictional force is directly proportional to normal reaction on the frictional surface. Where μs  coefficient of static friction When a body is at rest, the angle that the resultant force makes with normal reaction is known as angle of static friction.
• 35. Where μk  coefficient of kinetic friction When a body is in motion, the angle that the resultant force makes with normal reaction is known as angle of kinetic friction.
• 36.
• 37. Laws of Dry Friction
• 38. Laws of Dry Friction
• 39. Laws of Dry Friction
• 41. Trusses  Stationary, fully constrained structures in which members are acted upon by two equal and opposite forces directed along the member. Frames  Stationary, fully constrained structures in which atleast one member acted upon by three or more forces which are not directed along the member. Machines  Containing moving parts, always contain at least one multiforce member.
• 42. Applications of Trusses Electric Tower BridgeRoof support Cranes
• 43.  A framework composed of members joined at their ends to form a rigid structure is called a truss.
• 44.
• 45. Rigid Structure Rigid  Non-collapsible and deformation of the members due to induced internal strains is negligible. Axially Loaded Members
• 46. Types of Trusses Plane Trusses Bridge Trusses Roof Trusses Space Trusses
• 47. Internal and External Redundancy External Redundancy  More additional supports Internal Redundancy If m + 3 = 2j, then the truss is statically determinate structure If m + 3 > 2j, then the truss is redundant structure (statically indeterminate structure) [more members than independent equations] If m + 3 < 2j, then the truss is unstable structure (will collapse under external load) [deficiency of internal members]  For statically determinate trusses, ‘2j’ equations for a truss with ‘j ‘ joints is equal to m+3 (‘m’ two force members and having the maximum of three unknown support reactions)
• 50. Zero Force Members  These members are not useless.  They do not carry any loads under the loading conditions shown, but the same members would probably carry loads if the loading conditions were changed.  These members are needed to support the weight of the truss and to maintain the truss in the desired shape.
• 52. When a particle moves along a curve other than a straight line, then the particle is in curvilinear motion. Curvilinear Motion. Velocity of a particle is a vector tangent to the path of the particle
• 53. Acceleration is not tangent to the path of the particle The curve described by the tip of v is called the hodograph of the motion
• 54. Tangential and Normal Components Tangential component of the acceleration is equal to the rate of change of the speed of the particle. Normal component is equal to the square of the speed divided by the radius of curvature of the path at P.
• 55.
• 56.
• 57. Radial and Transverse components The position of the particle P is defined by polar coordinates r and θ. It is then convenient to resolve the velocity and acceleration of the particle into components parallel and perpendicular to the line OP.  Unit vector er defines the radial direction, i.e., the direction in which P would move if r were increased and θ were kept constant.  The unit vector eθ defines the transverse direction, i.e., the direction in which P would move if θ were increased and r were kept constant.
• 58. Where -er denotes a unit vector of sense opposite to that of er Using the chain rule of differentiation, Using dots to indicate differentiation with respect to t
• 59. To obtain the velocity v of the particle P, express the position vector r of P as the product of the scalar r and the unit vector er and differentiate with respect to t: Differentiating again with respect to t to obtain the acceleration,
• 60. The scalar components of the velocity and the acceleration in the radial and transverse directions are, therefore, In the case of a particle moving along a circle of center O, have r = constant and
• 61. Kinetics of Particles Work Energy Method  Work of a force & Kinetic energy of particle. In this method, there is no determination of acceleration. This method relates force, mass, velocity and displacement. Work of a Constant Force in Rectilinear Motion
• 62. Work of the Force of Gravity
• 63. Work of the Force Exerted by a Spring
• 64. Kinetic Energy of a particle Consider a particle of mass m acted upon by a force F and moving along a path which is either rectilinear or curved. When a particle moves from A1 to A2 under the action of a force F, the work of the force F is equal to the change in kinetic energy of the particle. This is known as the principle of work and energy.
• 65. Dynamic Equilibrium Equation ΣF - ma = 0 The vector -ma, of magnitude ‘ma’ and of direction opposite to that of the acceleration, is called an inertia vector. The particle may thus be considered to be in equilibrium under the given forces and the inertia vector or inertia force. When tangential and normal components are used, it is more convenient to represent the inertia vector by its two components -mat and –man.
• 66. Principle of Impulse and Momentum Consider a particle of mass m acted upon by a force F. Newton’s second law can be expressed in the form where ‘mv’ is the linear momentum of the particle. The integral in Equation is a vector known as the linear impulse, or simply the impulse, of the force F during the interval of time considered. Vectorial addition of initial momentum mv1 and the impulse of the force F gives the final momentum mv2. Definition: A force acting on a particle during a very short time interval that is large enough to produce a definite change in momentum is called an impulsive force and the resulting motion is called an impulsive motion.
• 67. When two particles which are moving freely collide with one another, then the total momentum of the particles is conserved.
• 68. KINEMATICS OF RIGID BODIES  Investigate the relations existing between the time, the positions, the velocities, and the accelerations of the various particles forming a rigid body. Various types of rigid-body motionVarious types of rigid-body motion Translation A motion is said to be a translation if any straight line inside the body keeps the same direction during the motion. Rectilinear translation (Paths are straight lines) Curvilinear translation (Paths are curved lines)
• 69. Rotation about a Fixed Axis  Particles forming the rigid body move in parallel planes along circles centered on the same fixed axis called the axis of rotation. The particles located on the axis have zero velocity and zero acceleration Rotation and the curvilinear translation are not the same.
• 70. General Plane Motion  Motions in which all the particles of the body move in parallel planes. Any plane motion which is neither a rotation nor a translation is referred to as a general plane motion. Examples of general plane motion :
• 71. Motion about a Fixed Point  The three-dimensional motion of a rigid body attached at a fixed point O, e.g., the motion of a top on a rough floor is known as motion about a fixed point. General Motion  Any motion of a rigid body which does not fall in any of the categories above is referred to as a general motion. Example:
• 72. Translation (either rectilinear or curvilinear translation) Since A and B, belong to the same rigid body, the derivative of rB/Ais zero When a rigid body is in translation, all the points of the body have the same velocity and the same acceleration at any given instant. In the case of curvilinear translation, the velocity and acceleration change in direction as well as in magnitude at every instant.
• 73. Rotation about a fixed axis Consider a rigid body which rotates about a fixed axis AA’ ‘P’ be a point of the body and ‘r’ its position vector with respect to a fixed frame of reference. The angle θ depends on the position of P within the body, but the rate of change Ѳ is itself independent of P. The velocity v of P is a vector perpendicular to the plane containing AA’ and r.
• 74. The vector It is angular velocity of the body and is equal in magnitude to the rate of change of Ѳ with respect to time. The acceleration ‘a’ of the particle ‘P’ α is the angular acceleration of a body rotating about a fixed axis is a vector directed along the axis of rotation, and is equal in magnitude to the rate of change of ‘ω’ with respect to time
• 75. Two particular cases of rotation Uniform Rotation  This case is characterized by the fact that the angular acceleration is zero. The angular velocity is thus constant. Uniformly AcceleratedRotation  n this case, the angular acceleration is constant
• 76. General plane motion The sum of a translation and a rotation
• 77. Absolute and relative velocity in plane motion Any plane motion of a slab can be replaced by a translation defined by the motion of an arbitrary reference point A and a simultaneous rotation about A. The absolute velocity vB of a particle B of the slab is
• 78. The velocity vA corresponds to the translation of the slab with A, while the relative velocity vB/A is associated with the rotation of the slab about A and is measured with respect to axes centered at A and of fixed orientation
• 79. Consider the rod AB. Assuming that the velocity vA of end A is known, we propose to find the velocity vB of end B and the angular velocity ω of the rod, in terms of the velocity vA, the length l, and the angle θ.
• 80. The angular velocity ω of the rod in its rotation about B is the same as in its rotation about A. The angular velocity ω of a rigid body in plane motion is independent of the reference point.
• 81. Absolute and relative acceleration in plane motion
• 82. For any body undergoing planar motion, there always exists a point in the plane of motion at which the velocity is instantaneously zero. This point is called the instantaneous center of rotation, or C. It may or may not lie on the body! Instantaneous Centre As far as the velocities are concerned, the slab seems to rotate about the instantaneous center C. If vA and vB were parallel and having same magnitude the instantaneous center C would be at an infinite distance and ω would be zero; All points of the slab would have the same velocity. If vA = 0, point A is itself is the instantaneous center of rotation, and if ω = 0, all the particles have the same velocity vA.
• 83. Concept of instantaneous center of rotation At the instant considered, the velocities of all the particles of the rod are thus the same as if the rod rotated about C.
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