* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download FE_Review_Dynamics - Department of Mechanical Engineering
Modified Newtonian dynamics wikipedia , lookup
Fictitious force wikipedia , lookup
Hooke's law wikipedia , lookup
Tensor operator wikipedia , lookup
Jerk (physics) wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Center of mass wikipedia , lookup
Classical mechanics wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Photon polarization wikipedia , lookup
N-body problem wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Four-vector wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Analytical mechanics wikipedia , lookup
Hunting oscillation wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Routhian mechanics wikipedia , lookup
Equations of motion wikipedia , lookup
Rigid body dynamics wikipedia , lookup
FE Review Dynamics G. Mauer UNLV Mechanical Engineering Point Mass Dynamics X-Y Coordinates v g B (d,h) 0 y A (x0,y0) x horiz. distance = d h v0 g A (x0,y0) y h x horiz. distance d = 20 m B A ball is thrown horizontally from A and passes through B(d=20,h = -20) meters. The travel time t to Point B is Use g = 10 m/s2 (A) t = 4 s (B) t = 1 s (C) t = 0.5 s (D) t = 2 s Use g = 10 m/s2 v0 g A (x0,y0) y h x horiz. distance d = 20 m B y (t ) Use v0 * sin( t 0m/s .5 * g *2t 2 g =) *10 20m 0.5 *10m / s 2 * t 2 t 2s A ball is thrown horizontally from A and passes through B(d=20,h = -20) meters. The travel time t to Point B is (A) t = 4 s (B) t = 1 s (C) t = 0.5 s (D) t = 2 s Use g = 10 m/s2 v0 g A (x0,y0) y h x horiz. distance d = 20 m B Use g = 10 m/s2 A ball is thrown horizontally from A and passes through B(d=20,h = -20) meters at time t = 2s. The start velocity v0 is (A) v0 = 40 m/s (B) v0 = 20 m/s (C) v0 = 10 m/s (D) v0 = 5 m/s v0 g A (x0,y0) y h x horiz. distance d = 20 m B x(t ) v0 * cos( ) * t 20m v0 *1* 2s v0 10m / s Use g = 10 m/s2 A ball is thrown horizontally from A and passes through B(d=20,h = -20) meters at time t = 2s. The start velocity v0 is (A) v0 = 40 m/s (B) v0 = 20 m/s (C) v0 = 10 m/s (D) v0 = 5 m/s 12.7 Normal and Tangential Coordinates ut : unit tangent to the path un : unit normal to the path Normal and Tangential Coordinates Velocity Page 53 v s * ut Normal and Tangential Coordinates Fundamental Problem 12.27 a v 2 * un at * ut The boat is traveling along the circular path with = 40m and a speed of v = 0.5*t2 , where t is in seconds. At t = 4s, the normal acceleration is: (A) constant •(B) 1 m/s2 •(C) 2 m/s2 •(D) not enough information •(E) 4 m/s2 Fundamental Problem 12.27 a v 2 * un at * ut The boat is traveling along the circular path with = 40m and a speed of v = 0.5*t2 , where t is in seconds. At t = 4s, the normal acceleration is: at dv / dt 2 * 0.5 * t At __ t 2s : _ at 2 *1m / s 2 (A) constant •(B) 1 m/s2 •(C) 2 m/s2 •(D) not enough information •(E) 4 m/s2 Polar coordinates Polar coordinates Polar coordinates Polar Coordinates Point P moves on a counterclockwise circular path, with r =1m, dot(t) = 2 rad/s. The radial and tangential accelerations are: •(A) ar = 4m/s2 a = 2 m/s2 •(B) ar = -4m/s2 a = -2 m/s2 •(C) ar = -4m/s2 a = 0 m/s2 •(D) ar = 0 m/s2 a = 0 m/s2 Polar Coordinates Point P moves on a counterclockwise circular path, with r =1m, dot(t) = 2 rad/s. The radial and tangential accelerations are: •(A) ar = 4m/s2 a = 2 m/s2 •(B) ar = -4m/s2 a = -2 m/s2 •(C) ar = -4m/s2 a = 0 m/s2 •(D) ar = 0 m/s2 a = 0 m/s2 disk = 10 rad/s B C r e er Unit vectors Point B moves radially outward from center C, with r-dot =1m/s, dot(t) = 10 rad/s. At r=1m, the radial acceleration is: •(A) ar = 20 m/s2 •(B) ar = -20 m/s2 •(C) ar = 100 m/s2 •(D) ar = -100 m/s2 disk = 10 rad/s B C r e er Unit vectors Point B moves radially outward from center C, with r-dot =1m/s, dot(t) = 10 rad/s. At r=1m, the radial acceleration is: •(A) ar = 20 m/s2 •(B) ar = -20 m/s2 •(C) ar = 100 m/s2 •(D) ar = -100 m/s2 Example cont’d: Problem 2.198 Sailboat tacking against Northern Wind VWind VBoat VWind/ Boat 2. Vector equation (1 scalar eqn. each in i- and j-direction) 500 150 i Given: r(t) = 2+2*sin((t)), _dot= constant The radial velocity is (A) 2+2*cos((t ))*-dot, (B) -2*cos((t))*-dot (C) 2*cos((t))*-dot (D) 2*cos((t)) (E) 2* +2*cos((t ))*-dot Given: r(t) = 2+2*sin((t)), _dot= constant The radial velocity is (A) 2+2*cos((t ))*-dot, (B) -2*cos((t))*-dot (C) 2*cos((t))*-dot (D) 2*cos((t)) (E) 2* +2*cos((t ))*-dot 2.9 Constrained Motion A J L vA = const vA is given as shown. Find vB i B Approach: Use rel. Velocity: vB = vA +vB/A (transl. + rot.) V r = 150 mm The conveyor belt is moving to the left at v = 6 m/s. The angular velocity of the drum (Radius = 150 mm) is 6 m/s (B) 40 rad/s (C) -40 rad/s (D) 4 rad/s (E) none of the above (A) V r = 150 mm The conveyor belt is moving to the left at v = 6 m/s. The angular velocity of the drum (Radius = 150 mm) is 6 m/s (B) 40 rad/s (C) -40 rad/s (D) 4 rad/s (E) none of the above (A) Omit all constants! XB xA A B c yE Xc The rope length between points A and B is: •(A) xA – xB + xc •(B) xB – xA + 4xc •(C) xA – xB + 4xc •(D) xA + xB + 4xc Omit all constants! XB xA A B c yE Xc The rope length between points A and B is: •(A) xA – xB + xc •(B) xB – xA + 4xc •(C) xA – xB + 4xc •(D) xA + xB + 4xc NEWTON'S LAW OF INERTIA A body, not acted on by any force, remains in uniform motion. NEWTON'S LAW OF MOTION Moving an object with twice the mass will require twice the force. Force is proportional to the mass of an object and to the acceleration (the change in velocity). F=ma. Dynamics M1: up as positive: Fnet = T - m1*g = m1 a1 M2: down as positive. F = F = m *g T = m a2 net 2 2 3. Constraint equation: a1 = a2 = a Equations From previous: T - m1*g = m1 a T = m1 g + m1 a Previous for Mass 2: m2*g - T = m2 a Insert above expr. for T m2 g - ( m1 g + m1 a ) = m2 a ( m2 - m1 ) g = ( m1 + m2 ) a ( m1 + m2 ) a = ( m2 - m1 ) g a = ( m 2 - m 1 ) g / ( m1 + m 2 ) Rules 1. Free-Body Analysis, one for each mass 2. Constraint equation(s): Define connections. You should have as many equations as Unknowns. COUNT! 3. Algebra: Solve system of equations for all unknowns J g m M*g*sin*i i 0 = 30 M*g 0 -M*g*cos*j Mass m rests on the 30 deg. Incline as shown. Step 1: Free-Body Analysis. Best approach: use coordinates tangential and normal to the path of motion as shown. J g m M*g*sin*i i 0 = 30 M*g Mass m rests on the 30 deg. Incline as shown. Step 1: Free-Body Analysis. Step 2: Apply Newton’s Law in each Direction: 0 N -M*g*cos*j (Forces _ x) m * g * sin * i m * x (Forces _ y) N - m * g * cos * j 0(static _ only) Friction F = mk*N: Another horizontal reaction is added in negative x-direction. J g m M*g*sin*i i 0 = 30 M*g 0 mk*N N -M*g*cos*j (Forces _ x) (m * g * sin mk * N ) * i m * x (Forces _ y) N - m * g * cos * j 0(static _ only) Problem 3.27 in Book: Find accel of Mass A Start with: (A)Newton’s Law for A. (B)Newton’s Law for A and B (C) Free-Body analysis of A and B (D) Free-Body analysis of A Problem 3.27 in Book: Find accel of Mass A Start with: (A)Newton’s Law for A. (B)Newton’s Law for A and B (C) Free-Body analysis of A and B (D) Free-Body analysis of A Problem 3.27 in Book cont’d Newton applied to mass B gives: (A) SFu = 2T = mB*aB (B) SFu = -2T + mB*g = 0 (C) SFu = mB*g-2T = mB*aB (D) SFu = 2T- mB*g-2T = 0 Problem 3.27 in Book cont’d Newton applied to mass B gives: (A) SFu = 2T = mB*aB (B) SFu = -2T + mB*g = 0 (C) SFu = mB*g-2T = mB*aB (D) SFu = 2T- mB*g-2T = 0 Problem 3.27 in Book cont’d Newton applied to mass A gives: (A) SFx = T +F= mA*ax ; SFy = N - mA*g*cos(30o) = 0 (B) SFx = T-F= mA*ax SFy = N- mA*g*cos(30o) = mA*ay (C) SFx = T = mA*ax ; SFy = N - mA*g*cos(30o) =0 (D) SFx = T-F = mA*ax ; SFy = N-mA*g*cos(30o) =0 Problem 3.27 in Book cont’d Newton applied to mass A gives: (A) SFx = T +F= mA*ax ; SFy = N - mA*g*cos(30o) = 0 (B) SFx = T-F= mA*ax SFy = N- mA*g*cos(30o) = mA*ay (C) SFx = T = mA*ax ; SFy = N - mA*g*cos(30o) =0 (D) SFx = T-F = mA*ax ; SFy = N-mA*g*cos(30o) =0 Energy Methods dW F dr Scalar _ Pr oduct Only Force components in direction of motion do WORK Work of Gravity Work of a Spring The work-energy relation: The relation between the work done on a particle by the forces which are applied on it and how its kinetic energy changes follows from Newton’s second law. A car is traveling at 20 m/s on a level road, when the brakes are suddenly applied and all four wheels lock. mk = 0.5. The total distance traveled to a full stop is (use Energy Method, g = 10 m/s2) (A) 40 m (B) 20 m (C) 80 m (D) 10 m (E) none of the above Collar A is compressing the spring after dropping vertically from A. Using the y-reference as shown, the work done by gravity (Wg) and the work done by the compression spring (Wspr) are y (A) Wg <0, Wspr <0 (B) Wg >0, Wspr <0 (C) Wg <0, Wspr >0 (D) Wg >0, Wspr >0 Conservative Forces A conservative force is one for which the work done is independent of the path taken Another way to state it: The work depends only on the initial and final positions, not on the route taken. Conservative Forces T1 + V1 = T2 + V2 Potential Energy Potential energy is energy which results from position or configuration. An object may have the capacity for doing work as a result of its position in a gravitational field. It may have elastic potential energy as a result of a stretched spring or other elastic deformation. Potential Energy elastic potential energy as a result of a stretched spring or other elastic deformation. Potential Energy Potential Energy y A child of mass 30 kg is sliding downhill while the opposing friction force is 50 N along the 5m long incline (3m vertical drop). The work done by friction is (A) -150 Nm (B) 150 Nm (C) 250 Nm (D) -250 Nm (E) 500 Nm h d (Use Energy Conservation) A 1 kg block slides d=4 m down a frictionless plane inclined at =30 degrees to the horizontal. The speed of the block at the bottom of the inclined plane is (A) 1.6 m/s (B) 2.2 m/s (C) 4.4 m/s (D) 6.3 m/s (E) none of the above Angular Momentum H o r mv G mv Linear Momentum Rot. about Fixed Axis Memorize! Page 336: dr v ωr dt an = x ( x r) at = a x r Meriam Problem 5.71 Given are: BC wBC 2 (clockwise), Geometry: equilateral triangle with l 0.12 meters. Angle 60 180 Collar slides rel. to bar AB. Mathcad EXAMPLE Guess Values: (outward motion of collar is positive) wOA 1 vcoll 1 Vector Analysis: OA rA vCOLL BC rAC Mathcad does not evaluate cross products symbolically, so the LEFT and RIGHT sides of the above equation are listed below. Equaling the i- and jterms yields two equations for the unknowns OA and vCOLL Mathcad Example part 2: Solving the vector equations 16.4 Motion Analysis http://gtrebaol.free.fr/d oc/flash/four_bar/doc/ Approach 1. Geometry: Definitions Constants Variables Make a sketch 2. Analysis: Derivatives (velocity and acceleration) 3. Equations of Motion 4. Solve the Set of Equations. Use Computer Tools. Example Bar BC rotates at constant BC. Find the angular Veloc. of arm BC. Step 1: Define the Geometry Example B J vA(t) A AC O A Bar BC rotates at constant BC. Find the ang. Veloc. of arm BC. Step 1: Define the Geometry (t) O i (t) (t) C Geometry: Compute all lengths and angles as f((t)) All angles and distance AC(t) are time-variant B J vA(t) A Velocities: = -dot is given. O A AC (t) O Vector Analysis: OA rA i (t) (t) C vCOLL BC rAC Analysis: Solve the rel. Veloc. Vector equation conceptually B Seen from O: vA = x OA J vA(t) A i O Vector Analysis: OA rA (t) O A AC (t) (t) C vCOLL BC rAC Analysis: Solve the rel. Veloc. Vector equation Seen from O: vA = x OA B J vA(t) A i O Vector Analysis: OA rA (t) O A AC (t) (t) C vCOLL BC rAC Analysis: Solve the rel. Veloc. Vector equation B vA,rel J Seen from C: vCollar + BC x AC(t) A BC x AC(t) i O Vector Analysis: OA rA (t) O A AC (t) (t) C vCOLL BC rAC Analysis: Solve the rel. Veloc. Vector equation numerically B J vA(t) A i (t) O A AC Vector Analysis: OA rA vCOLL BC rAC Mathcad does not evaluate cross products symbol (t) RIGHT sides of the above equation (t)are listed below O terms yields two equations for the unknownsCOA Enter vectors: 0 l cos ( ) 0 l cos ( ) OA 0 rA l sin ( ) BC 0 rAC l sin ( ) 0 wOA 0 wBC Analysis: Solve the rel. Veloc. Vector equation numerically B A i O A (t) (t) Vector Analysis: OA rA vA(t) AC Here: BC is given as -2 rad/s (clockwise). Find OA O J (t) C vCOLL BC rAC LEFT_i l wOA sin ( ) RIGHT_i l wBC sin ( ) vcoll cos ( ) LEFT_j l wOA cos ( ) RIGHT_j l wBC sin ( ) vcoll sin ( ) J B Veloc. Analysis: Solve the rel. v (t) A Vector equation numerically A i O (t) O A AC (t) (t) Solve the two vector (i and j) equations : Given l wOA sin ( ) l wOA cos ( ) l wBC sin ( ) vcoll cos ( ) l wBC sin ( ) vcoll sin ( ) vec Find ( wOA vcoll) 4.732 vec 0.568 C Recap: The analysis is becoming more complex. B vA(t) A i •Mathcad (t) O (t) O A AC •To succeed: Try Clear Organization from the start J (t) C •Vector Equation = 2 simultaneous equations, solve simultaneously! 16.6 Relative Velocity vA = vB + vA/B vRot = x r Relative Velocity vA = vB + vA/B = VB (transl)+ vRot fig_05_011 Seen from A: vB = vA + AB x rB/A Seen from O: vB = o x r Seen from A: vB = vA + AB x rB/A Seen from O: vB = o x r Rigid Body Acceleration Stresses and Flow Patterns in a Steam Turbine FEA Visualization (U of Stuttgart) aB = aA + aB/A,centr+ aB/A,angular Given: Geometry and VA,aA, vB, AB Find: aB and aAB r* AB2 + r* a J Look at the Accel. of B relative to A: iA vA = const r AB Counterclockw . B vB Given: Geometry and VA,aA, vB, AB aB = aA + aB/A,centr+ aB/A,angular r* AB2 + r* a Find: aB and aAB J Look at the Accel. of B relative to A: iA We know: vA = const r B Centrip. r* AB 2 1. Centripetal: magnitude r2 and direction (inward). If in doubt, compute the vector product x(*r) Given: Geometry and VA,aA, vB, AB aB = aA + aB/A,centr+ aB/A,angular r* AB2 + r* a Find: aB and aAB J Look at the Accel. of B relative to A: iA We know: vA = const r Centrip. r* AB 2 B r* a 1. Centripetal: magnitude r2 and direction (inward). If in doubt, compute the vector product x(*r) 2. The DIRECTION of the angular accel (normal to bar AB) Given: Geometry and VA,aA, vB, AB aB = aA + aB/A,centr+ aB/A,angular r* AB2 + r* a Find: aB and aAB J Look at the Accel. of B relative to A: iA We know: vA = const r Centrip. r* AB 2 B Angular r* a aB 1. Centripetal: magnitude r2 and direction (inward). If in doubt, compute the vector product x(*r) 2. The DIRECTION of the angular accel (normal to bar AB) 3. The DIRECTION of the accel of point B (horizontal along the constraint) We can add graphically: Start with Centipetal Given: Geometry and VA,aA, vB, AB aB = aA + aB/A,centr+ aB/A,angular Find: aB and aAB r is the vector from reference point A to point B J i A r vA = const AB Angular r* a Centrip. r* AB 2 B aB Given: Geometry and VA,aA, vB, AB We can add graphically: Start with Centipetal Find: aB and aAB r is the vector from reference point A to point B J i aB = aA + aB/A,centr+ aB/A,angular Now Complete the Triangle: r* a r* AB2 aB A vA = const r AB Centrip. r* AB2 B Result: a is < 0 (clockwise) aB is negative (to the left) E F A a (t) B H AB The instantaneous center of Arm BD is located at Point: (A) F (B) G (C) B (D) D (E) H BD J (t) O i G D vD(t) Plane Motion 3 equations: S Forces_x S Forces_y S Moments about G fig_06_002 Plane Motion 3 equations: S Forces_x S Forces_y S Moments about G Translatio n : Fx m * x ..................... Fy m * y Rotation : ..... M G I G *a fig_06_002 Parallel Axes Theorem Pure rotation about fixed point P I P IG m * d fig_06_005 2 Constrained Motion: The system no longer has all three Degrees of freedom Describe the constraint(s) with an Equation