Kinematics, Dynamics and Vibrations Dr. Mustafa Arafa Mechanical Engineering Department American University in Cairo [email protected] Kinematics, dynamics and vibration • Kinematics: study of motion (displacement, velocity, acceleration, time) without reference to the cause of motion (i.e. regardless of forces). • Dynamics: study of forces acting on a body, and resulting motion. • Vibration: Oscillatory motion of bodies & associated forces. Outline A. Kinematics of mechanisms B. Dynamics C. Vibration: natural frequency and resonance D. Balancing Kinematics of mechanisms Four-bar mechanism Single Degree of Freedom Slider-crank mechanism Single Degree of Freedom Position analysis Given: a,b,c,d, the ground position, q2. Find: q3 and q4 b B c b A q3 c a q4 q2 d O2 O4 Graphical solution b • Draw an arc of radius b, c centered at A • Draw an arc of radius c, B1 centered at O4 b • The intersections are the A q3 two possible positions for c the linkage, open and a d q4 crossed q2 O2 O4 B2 Analytical solution Obtain coordinates of point A: Ax acosq 2 Ay asinq 2 Obtain coordinates of point B: 2 2 2 b Bx Ax By Ay 2 2 2 c Bx d By These are 2 equations in 2 unknowns: Bx and By See “position analysis” on page 242 Dynamics Types of motion Rectilinear Curvilinear Overview Kinematics: equations for constant velocity and acceleration d Kinetics: Newton’s second law of motion: F mv dt For constant mass: F ma Kinetic energy: 1 2 T 2 mv Potential energy: U mgh Gravity 1 2 U 2 kx Elastic Friction W F P N Basic equations Projectile Projectile y v a g v x a a v Plane Motion of a Rigid Body Fx ma x F y ma y M G I g For rotation about a fixed axis: MIOO 17 Example At a forward speed of 30 ft/s, the truck brakes were applied, causing the wheels to stop rotating. It was observed that the truck skidded to a stop in 20 ft. Determine the magnitude of the normal reaction and the friction force at each wheel as the truck skidded to a stop. 18 Solution 22 v v00 2 a x x 0 302 2a 20 a 22.5 ft s Free-body diagram: F ma x Gx FAB F m 22.5 F ma y Gy W mg NAB N MI GG 7NFFNBBAA 4 4 5 But FNFNAABB, Unknowns: NNAB,, N N m 22.5 AB 0.699 NAB N mg 19 NAB0.35 mg , N 0.65 mg Vibrations Natural frequency and resonance Overview f Equation of motion: m mx cx kx f x k c Natural frequency:n km 2 f rad/s Hz cm/2 n cccritical 1 overdamping Damping Damping cccritical 1 critical damping ratio coeff. cccritical 1 underdamping Free vibrations with no damping Equation of motion: m mx kx 0 x k Solution: x C12cosnn t C sin t C1 and C2 are constants to be determined from initial conditions x0 and v0 x x00cosn t v n sin n t Model of a vibrating system Spring and mass Spring, mass and damper Forced vibration Balancing of machinery Static unbalance Couple unbalance Dynamic unbalance.