Unformatted text preview: Dynamics Review Coordinate Systems Rectangular: r = xi + yj + zk r = x 2 + y 2 + z 2 Path (nt): v = vet ( s=rq Cylindrical: r=reR v = vxi + vyj + vzk v = v x + v y + v z 2 2 2 a = axi + ayj + azk a = a x + a y + a z 2 2 2 & a = atet + anen where a = v and a = n v & v = rq r r r &e v = r R + r &e qq
& a = rq& ) t v 2 r & = rq 2 r v v & r q q& &q a = a r e r + a e , where a = & - r & 2 and a = r & + 2r & q q r q Particles Rectilinear or Tangential Motion: a = d 2 s dv dv = = v 2 dt ds dt 2 2 If acceleration is constant: v=v0+at x=x0+v0t+at /2 v =v0 2 +2a(xx0) If x,y,z motion is uncoupled (e.g. projectiles), analyze each motion independently FMA Method: Technique: Draw FBD and MAD! Set up Equations of Motion. Use coord system which works best for application. FMA with Rect Coords: SFx=max SFy=may FMA Path Coords: SFn=man SFt=mat SFz=maz (SFz=maz) SFz=maz. FMA Cylindrical/Polar Coords:
Work & Energy
s 2 SFR=maR SFq=maq U = F ds = F cos ds = Area under Fs curve. Scalar! q t
s 1 Work of Constant Force: Work due to Gravity: Work due to Spring: U=FDd (Dd = displacement in direction of force) U= wDy U= 1/2k(d2 2 d1 2 ) Principle of Work and Kinetic Energy: Use to solve for Dv when Ds is known. U12 = T2T1 Scalar! U'12=DT+DVg+DVe 2 Vg = mgh Ve = (1/2)kd 2 T=(1/2)mv Technique: MVD helps to find total kinetic energy. FBD helps to find all forces which cause work. Principle of Conservation of Momentum and Impulse: Technique: Draw FBD and Momentum Diagrams Linear Impulse and Momentum: Use to find Dv when Dt is known v v F = DG DG=mDv dt Angular Impulse and Momentum: v v v v M dt = DH = r m Vector! (Break into x,y,z components) v Impact Typical Problem: Determine velocities after impact. mAvA1 + mBvB1 = mAvA2 + mBvB2 v B - v A v sep Coefficient of Restitution for Direct Impact: e = = v v A - v B app 2 1 2 1 Coefficient of Restitution for Oblique Impact, Use components along line of impact: v sep v Bn - v An e = = v v An - v Bn app 2 2 1 1 Analysis of Impact Problems: ^ a. Draw FBD during impact. Use P for Impulsive Forces b. Draw Momentum Diagrams before and after hit. c. Solve Impulse and Momentum equations. Rigid Bodies Plane Motion Kinematics General: a = d 2q d w d w = =w 2 dt ds dt Fixed Axis Rotation: Vector Form: v v v v = w r v v v v v v a = w ( r ) + a r w Scalar Form: v = r = v / r w
a n = r 2 = w a = r a t v 2 = v w r Relative Motion of Rigid Bodies v v v v B / A = w r / A B Normal and Tangential Accelerations: v v v v B / A = w r / A B v v v v v a B / A = w v B / A = w ( r / A ) w B v v a B / A = a r / A B n t Method of Relative Velocities v v v v v v v B = v A + v B / A = v A + w r / A B No Slip: v0=Rw a0 = Ra (only at center!!) Method of Instant Centers: Instant Center is point at which all other points are in pure rotation about it. Use to determine w and absolute velocities of any point on object. Can not be used for accelerations. Method of Relative Accelerations:
v v v v v v a B = a A + a B / A = a A + a B / A n + a B / A t v v v v v a B / A n = w v B / A = w ( r / A ) w B v v a B / A t = a r / A B Rigid Bodies Kinetics Mass Moment of Inertia I = r 2 dm m I = mk 2 I a = I + md 2 =>Parallel Axis Theorem k=Radius of Gyration FMA Method Rigid Bodies Plane Motion y Draw FBD and MAD. y FBD: IGa x MAD: mx my x SFx = m x a SF = m y a y S G = I M a SM A = I + m d a a Slip / NoSlip: For NoSlip, Fmax = msN must be greater than required force per F=ma. If don't know, assume noslip, then check Fmax = mkN. If wheel is NoSlip, ao=ar. Use Relative Acceleration to find aG, if necessary. Work/ Energy Rigid Bodies Draw a FBD to determine all forces that cause work. A force only causes work if it causes movement.
s 2 v v U = F d s s 1 U = FDd (d is in direction of constant force) (Work is positive if force is in direction of motion.) U = mgDy =>Work due to Gravity U = 1/2 k(x2 2 x2 2 ) =>Work due to spring Remember that x is always taken from unstretched length.
U = Md q
q1 q 2 Work of a Couple Plane Motion (Work is positive if moment is in direction of rotation.) Power = dU/dt = Mw h=output power / input power x 100% Kinetic Energy: 1 1 T = S ( m 2 + I 2 ) v w 2 2 Principle of Work and KE: U 1 - 2 = T 2 - T Add work and KE for all components. Internal forces usually cancel. 1 Conservation of Energy: 2 U'12=DT+DVg+DVe Vg = mgh Ve = k x (U'12 = Work done by other forces besides gravity and springs) Show gravity datum. Remember that x = stretched spring length unstretched spring length Impulse and Momentum Rigid Bodies v r Linear Momentum: G = m v Angular Momentum: Angular Momentum is the Moment of Momentum. H G = I w H p = I + m d w v y Momentum/MVD Diagram example: mvGy x Ig w mvGx Principle of Impulse and Momentum: S F dt = m x 2 - m x 1 v v x
t 1 t 2 S F dt = m y 2 - m y 1 v v y t 1 t 2 S M dt = DH t 1 t 2 Draw FBD and Momentum Diagram. Impact: ^ Draw FBD during impact, and designate impact forces with hat, as in P . Nonimpact forces are negligible compared to impact forces. Draw Momentum Diagram immediately before and after impact. Solve Impulse/Momentum equations or Conservation of Momentum. Can isolate single body or system of bodies for FBD and Momentum diagrams. ...
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