Chapter 1, Part 1: Kinematics of Particles PDF

CHAP TER 1: KINEMATICS OF PARTICLES MEC420: PART 1 : IN TRODUCTION AND REVIEW OF DYNAMICS VECTOR ANALYSIS IN DYNAMICS WHAT IS DYNAMICS? A branch of mechanics which deals with the motion of bodies under the action...

CHAP TER 1: KINEMATICS OF PARTICLES MEC420: PART 1 : IN TRODUCTION AND REVIEW OF DYNAMICS VECTOR ANALYSIS IN DYNAMICS WHAT IS DYNAMICS? A branch of mechanics which deals with the motion of bodies under the action of forces. Divided into two parts; ❑ –Kinematics: Study of motion without referring to the forces which cause the motion. ❑ –Kinetics: Study of motion by relating the action of forces that cause the motion or Vice Versa. WHY IT IS IMPORTANT? ▪ Basic to the analysis and design of moving structures or fixed structures subject to shock loads ▪ To design robotic devices and automatic control systems. ▪ To control the movement of rockets, missiles, spacecraft, and other transport. ▪ To design machinery elements such as turbines, pumps, engines, machine tools, hoists, lifts, etc. BASIC TERMINOLOGY ▪ Rigid body: a body whose dimensions are significant and its shape is unchanged (i.e. the relative movement between points is negligible) ▪ Particle: a body of negligible dimension ▪ Statics: A study of rigid bodies at rest (static equilibrium) ▪ Dynamics: A study of rigid bodies in motion (i.e. in dynamic equilibrium) ▪ Length: applied to the linear dimension of a straight or curved line 4 BASIC TERMINOLOGY ▪ Area: the two-dimensional size of the shape or surface ▪ Volume: the three-dimensional size of the space occupied by the substance ▪ Force: the vector action of one body on another whether by contact or no contact (at a distance) such as the force of gravity and magnetic force. ▪ Mass: the amount of matter in a body or quantitative measure of inertia (or resistance) to change in motion of a body ▪ Weight: the force with which a body is attracted toward the center of the Earth 5  TYPES OF COORDINATE SYSTEMS The motion of particle P can be described by specifying its coordinates. These coordinates can be measured from fixed reference axes or Global reference axes (absolute-motion analysis) or from moving reference axes or Local reference axes (relative- motion analysis) Analysis of the motion in 3-D space: performed using 1. Rectangular coordinates, x-y-z 2. Cylindrical coordinates, r--z 3. Spherical coordinates, R-- 4. Normal and Tangential coordinates (path variables), n-t 6 REVIEW ON VECTOR DYNAMICS ▪ Definition Vector & Scalar ▪ Vector Algebra Vector Addition Product of scalar and vector Cross Product, Dot Product Unit Vectors ▪ Rectangular Components of Vector ▪ Vector Calculus Differentiation & Integration 7 DEFINITIONS Scalar ▪ A scalar is a quantity that has only magnitude and no direction. ▪ Examples: mass, distance, speed, length, energy Vector ▪ A vector is a quantity that has both magnitude and direction. ▪ Examples: velocity, acceleration, force, and displacement 8  VECTOR ADDITION ▪ The addition of vectors is according to the parallelogram law. ▪ The sum of two vectors P and Q is obtained by using P and Q as two sides of the parallelogram. ▪ Vector addition is associative and commutative. P + Q = Q + P ……Commutative P + (Q + R) = (P + Q) + R….associative ▪ The negative of a given vector is a vector having the same magnitude but opposite direction. ▪ Therefore; –P + (-Q) = -Q + (-P) = -(P+Q) It is clear that P + (-P) = P - P = 0 9 PRODUCT OF SCALAR AND VECTOR ▪ Product of scalar k and a vector P have the same direction as P (if k is positive) or opposite direction (if k is negative) and a magnitude equal to the product of magnitude P and the absolute value of k. ▪ It follows the common rules such as k (P + Q) = k P + k Q KP=Pk K ( l P) = ( k l ) P (k+l)P=kP+lP 10 CROSS PRODUCT OF 2 VECTORS ▪ Cross product also known as vector product since it produces a vector ▪ It is denoted as C = A x B or A ٨ B ▪ It satisfies the following conditions; The line of action of C is perpendicular to the plane containing A and B. Magnitude of C given by the formula C = A B sin θ ▪ Direction of C based on” right-hand rule” where the right-hand finger is curled in the rotation to bring vector A in line with vector B. Positive direction of vector C is indicated by the thumb. 11 DOT PRODUCT OF 2 VECTORS ▪ Dot product is also known as a scalar product since it produces a scalar ▪ It is denoted as C = A B ▪ It shows that; The magnitude of C is given by the formula C = AB cos θ C is a projection of vector A on the vector B times the magnitude of B or vice-versa. 12 UNIT VECTORS 13 UNIT VECTORS ▪ Unit vector is a vector having 1 unit magnitude. –It has the same direction as the main vector –It is a ratio between the main vector and its magnitude; Therefore A = 5 a 14 PRODUCTS OF UNIT VECTOR i i=j j=k k=1 i j=j k=k i=0 iʌi=jʌj=kʌk=0 i ʌ j = k, j ʌ k = i, k ʌ i = j j ʌ i = -k, k ʌ j = -i, i ʌ k =-j 15 RECTANGULAR COMPONENTS OF VECTOR In a 3D system, vectors are divided into X, Y and Z axes with their unit vector as i, j, and k respectively. As in the figure shown, vector A is denoted by components Ax, Ay and Az and written as; A = Ax i + Ay j +Az k 16 RECTANGULAR COMPONENTS OF VECTOR 17 RECTANGULAR COMPONENTS OF VECTOR PRODUCT 18 OMEGA THEOREM VECTOR REPRESENTATION RESOLUTION VECTOR RESOLUTION VECTOR ADDITION OF SEVERAL VECTORS ADDITION OF SEVERAL VECTORS RESOLUTION OF A PLANAR VECTOR (2D) RESOLUTION OF A SPATIAL VECTOR (3D) 28 REVIEW ON CALCULUS 29 REVIEW ON CALCULUS 30 31 REVIEW ON CALCULUS 32 REVIEW ON CALCULUS 33 REVIEW ON CALCULUS 34 35