Dynamics Crib Sheet

VECTORS

r = x i + y j + z k, v = vx i + vy j + vz k

r = |r| =x2 + y2 + z2 =

r r, v = |v| =

v2x + v

2y + v

2z =

v vvx is component in x-direction (scalar), v, r are magnitudes (scalars).

Unit Vectors - 2D

et = i cos + j sin , en = i cos( + 90) + j sin( + 90) = i sin + j cos

i = i cos + j sin , j = i sin + j cos , et et = 0, et en = 1Get component from dot product: vn = et v, vx = i v, etc.Reconstruct a vector: v = vnen + vtet

Relative Vectors

Position vector of a particle or point A on a rigid body: rARelative position vector: rB/A = rB rAUnit vector in direction of rB/A: eB/A = rB/A/|rB/A|Velocity vector in this direction: v = v eB/A

Cross Product

r v = (rx i + ry j + rz k) (vx i + vy j + vz k)i j krx ry rzvx vy vz

= i ry rzvy vz

j rx rzvx vz+ k rx ryvx vy

= i(ryvz rzvy) j(rzvx rxvz) + k(rxvy ryvx)

2-D: r v = k(rxvy ryvx)

i j = j i = k, i k = k i = j, j k = k j = i

GENERAL - Miscellaneous

g = 9.8 m/s2 = 32.2 ft/s2, 32.2 lbm = 1 slug = 1 lb s2/ft, 1 kg = 1 N s2/m,360 = 2pi radians

sin( ) = sin cos cos sin , cos( ) = cos cos sin sin

Angular Velocity and acceleration, 2-D

= zk = k (+ is ccw), = zk = k, z = ddt, z =

dzdt

Curvilinear Motion

v = v et, a = atet + anen, v = , an =v2

= 2, at =

dv

dt

r = r er, v = vrer + ve, vr = r, v = r, ar = r r2, a = r + 2r

Constant Acceleration (in gravity field, ax = 0, ay = g)

vx = vx0 + axt, x = x0 + vx0t+1

2axt

2, vy = vy0 + ayt, y = y0 + vy0t+1

2ayt

2,

z = z0 + z t, z = z0 + z0 t+1

2z t

2

Friction

Assume contact sticks, thus tangential acceleration is known (usually zero), calculateunknown tangential force Ft (usually = Fx), and normal force Fn (usually = Fy). If|Ft| < s|Fn|, the stick assumption is true - done. Otherwise, its a sliding contact, ac-celeration is unknown, Ft = kFn. Algebraic sign is opposite to the relative motion, andthe same as the sign of Fx from the stick calculation.

KINEMATICS - Moving Frame Equations

Subscript: f = frame (moving coordinate system), g = ground (stationary coordinate sys-tem), P = point in question.

The position vector itself can be the same as seen from the ground or moving frame. However,the time derivative of the vector changes as seen from the ground versus as seen from theframe (moving coordinates)

rP = rP/g = rP/f + rO/g, vP = vP/g =d

dtrP =

(d

dtrP

)g

6=(d

dtrP

)f

= vP/f

The angular velocity and acceleration of the moving frame relative to the ground

= f/g, =d

dt

Velocity vector, as seen from the ground versus the frame (moving coordinates)

vP = vP/g = vP/f + (vf/g)P , (vf/g)P = vO/g + rP/OAcceleration vector, as seen from the ground versus the frame (moving coordinates)

aP = aP/g = aP/f + (af/g)P + aCor, aCor = 2 vP/f

(af/g)P = aO/g + rP/O + ( rP/O)

Three Dimensional Rigid Bodies

B = B/g = B/A + A/g = B/A + A

B =

(d

dtB/g

)g

=

(d

dtB/A

)g

+

(d

dtA/g

)g

=

(d

dtB/A

)g

+A(d

dtB/A

)g

= A B/A

Relative Motion of points on a rigid body

Velocity and acceleration of rigid body (RB) as seen from the ground or from moving coor-dinates (same equations either case)

vB = vA + vB/A, vB/A = rB/A, = RB/g or = RB/f

aB = aA + aB/A, aB/A = rB/A + vB/A, = =(d

dt

)g

or =

(d

dt

)f

KINETICS - PARTICLES Direct Method,

F = Fgrav + Fcont + Fext = m a

Fgrav = mg j, Fcont = Fn en + Ft et = Fcx i + Fcy j,

Kinetics, Work-Energy Method

T is kinetic energy, U12 is work, all units [N-m] or [ft-lb], 1 = initial state, 2 = final state

T1 + V1 + Ufric12 + U

mot12 = T2 + V2, V1 = V

grav1 + V

elas1

U12 = 21

F dr = U grav12 + U elas12 + U fric12 + Umot12If constant force:

U12 = F (s2 s1) = F s12 = Fx(x2 x1) + Fy(y2 y1), U fric12 < 0, Umot12 > 0

V grav1 = m g y1, Ugrav12 = m g (y2 y1) = m g y12

V elas1 =1

2k(`1 `0)2, U elas12 =

1

2k((`2 `0)2 (`1 `0)2

),

F elasx = k (x x0), T1 =1

2m v21,

where x0 or `0 is free length of spring, v1 is velocity magnitude of mass center, k is springconstant [N/m] or [lb/ft]

Kinetics, Impulse-Momentum Method

Imp12 = 21

F dt (FA + FB + . . . )(t2 t1) = G2 G1

Kinetics, Impact

vB2n vA2n = CoR (vB1n vA1n), vB2t = vB1t, vA2t = vA1tKINETICS, SYSTEMS OF PARTICLES

mass center, mean velocity

m = mA +mB + . . . , rG =1

m(mArA +mBrB + . . . ), vG =

1

m(mAvA +mBvB + . . . )

21

MO dt (rA/O FA + rB/O FB + . . . )t = HO2 HO1, rA/O = rA 21

MG dt (rA/G FA + rB/G FB + . . . )t = HG2 HG1

G1 = m vG1, HG1 = rA/G (mAvA) + rB/G (mBvB) + . . .G is linear momentum [N-s], H is angular momentum [N-s-m]

KINETICS - Rigid Bodies Direct Method,

F = FA + FB + = m aG

MG = rA/G FA + rB/G FB + + Mext = IGMO = rA/O FA + rB/O FB + + Mext = IO

IG =1

12mL2 thin beam/rod, IG =

1

2mR2 disk, IG = mR

2 hoop

IO = IG +m r2G/O parallel axis theorem, IG = m k

2G radius of gyration

Work-Energy Method, Rigid Bodies

T is kinetic energy, V is potential energy, U12 is work, all units [N-m] or [ft-lb], 1 = initialstate, 2 = final state

T1 + V1 + Ufric12 + U

mot12 = T2 + V2

U12 = 21

F dr F (s2 s1), U12 = 21

Mzd Mz(2 1)

U fric12 < 0, Umot12 > 0, V1 = V

g1 + V

e1 , V

g1 = m g yG1, V

e1 =

1

2k(`1 `0)2,

T1 =1

2m v2G1 +

1

2IG

21, or T1 =

1

2IO

21

`0 is free length of spring, vG1 is velocity magnitude of mass center, is angular velocitymagnitude of rigid body.Right side KE equation is if body rotates about stationary point O.

Impulse-Momentum Method, Rigid Bodies

mass center

m = mA +mB + . . . , rG =1

m(mArGA +mBrGB + . . . )

Imp12 = 21

F dt (FA + FB + . . . )(t2 t1) = G2 G1

AngImp12G = 21

MG dt (Mext + rA/GFA + rB/GFB + . . . )(t2 t1) = HG2HG1

AngImp12O = 21

MO dt (Mext + rA/OFA + rB/OFB + . . . )(t2 t1) = HO2HO1

G1 = m vG1, HG1 = IG 1, HO1 = IO 1,

G is linear momentum [N-s], H is angular momentum [N-s-m]

Three Dimensional Rigid Bodiesconstant angular velocities

MG = HG = HG,

MO = HO = HO,

3-D equations the same as 2-D, except:

HG = HGxi +HGyj +HGzk, HO = HOxi +HOyj +HOzk

HGx = IGx x IGxyy IGxzz, HGy = IGyxx + IGy y IGyzz, HGz = IGzxx IGzyy + IGz z

HOx = IOx x IOxyy IOxzz, etc.

Ixy = Iyx =

xy dm, etc.; Iz =

(x2 + y2) dm, etc.,

As an example, the equations below are for a thin rod extending in the y-direction fromy = LA to y = LB at x = xC

dm = dym

LB LA , Ixy = Iyx =xy dm =

m

LB LA

LBLA

xCy dy

About principal axes (xyz) the equations are much simpler:

HG = HGx i

+H Gy j +H Gz k

= I Gxx i + I Gy

y j + I Gz

z k

HO = HOx i

+H Oy j +H Oz k

= I Oxx i + I Oy

y j + I Oz

z k