from numpy import *
dtr = pi/180
g = 9.81     # m/s2
gfps = 32.2  # ft/s2
ihat,jhat,khat = array((1,0,0)),array((0,1,0)),array((0,0,1))

l,r = 0.3,0.15  # m
wz = 0.5        # rad/s
ws = l/r*wz
Q = arcsin(r/l)
w = wz*khat - ws*array(( 0, cos(Q), sin(Q) ))
rA = l * array(( 0, cos(2*Q), sin(2*Q) ))
vA = cross(w,rA)
print('vA = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s'.format(*vA))
alpha = cross( wz*khat, -ws*array(( 0, cos(Q), sin(Q) )) )
aA = cross( alpha, rA ) + cross( w, vA )
print('aA = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s2'.format(*aA))

vA = (-0.225 i +0.000 j +0.000 k) m/s
aA = (+0.000 i -0.113 j -0.130 k) m/s2

w1,w2 = 0.15,0.6   # rad/s
w1d = 0.8          # rad/s2
r = 40             # ft
Q = 30*dtr         # rad
rhat = array(( 0, cos(Q), sin(Q) ))
w = w1*khat + w2*ihat
rA = r*rhat
vA = cross(w,rA)
print('vA = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s'.format(*vA))
alpha = w1*w2*jhat + w1d*khat
aA = cross(alpha,rA) + cross(w,vA)
print('aA = ({:+.3f} i {:+.3f} j {:+.4f} k) m/s2'.format(*aA))

vA = (-5.196 i -12.000 j +20.785 k) m/s
aA = (-24.113 i -13.250 j -7.2000 k) m/s2

wA,wB = 5,15     # rad/s
R,r = 0.1,25e-3  # m
d = r*(wB-wA)/(wA+wB)
Q = arctan(d/R)
w = wA*R / ( r*cos(Q) - R*sin(Q) )
ws,wp = w*cos(Q), w*sin(Q)
print('angular velocity of gear C about shaft DE is {:.1f} rad/s'.format(ws))
print('angular velocity of shaft DE about y is {:.1f} rad/s'.format(wp))

angular velocity of gear C about shaft DE is 40.0 rad/s
angular velocity of shaft DE about y is 5.0 rad/s

x,y,z = 1.5,2,1  # m
vA = 8           # m/s
r = sqrt(x**2+y**2)
vB = vA * z * r / y**2
print('speed of B is {:.3f} m/s'.format(vB))
# find w:
wz = vB*x/r/(-y-z**2/y)
wx = -vA/y
wy = z*wz/y
print('w = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s'.format(wx,wy,wz))
sqrt((vB**2+vA**2)/(y**2+z**2)), sqrt(wx**2+wy**2+wz**2)

speed of B is 5.000 m/s
w = (-4.000 i -0.600 j -1.200 k) m/s

(4.219004621945797, 4.219004621945797)

# run code from problem 20-31 first
aA = 5   # m/s2
alphax = 1/y*( aA + (wx*vB*y + wy*vB*x)/r )
aB = r/y*( alphax*z + wz*vB*x/r + wx*vA )
rhat = array(( 0, -x/r, y/r ))
print('aB = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s2'.format(*(aB*rhat)))

aB = (-0.000 i +31.500 j -42.000 k) m/s2

w1,w2 = 0.5,0.25  # rad/s
d,r = 2,15        # ft
rd = 1.5          # ft/s
rdd = 0.5         # ft/s2
Q = 60*dtr        # rad
rhat  = array(( 0, cos(Q), sin(Q) ))
ut    = array(( 0,-sin(Q), cos(Q) ))
vA = w1*d*ihat
aA = w1**2*d*jhat
v_rel = w2*r*ut + rd*rhat 
v_turn = w1*r*cross(khat,rhat)
vB = vA + v_turn + v_rel
print('vB = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s'.format(*vB))
a_rel = (rdd-w2**2*r)*rhat + 2*w2*rd*ut
aB = w1**2*d*jhat + w1*cross(khat, v_turn) + 2*w1*cross(khat,v_rel) + a_rel
print('aB = ({:+.3f} i {:+.3f} j {:+.4f} k) m/s2'.format(*aB))

vB = (-2.750 i -2.498 j +3.174 k) m/s
aB = (+2.498 i -2.243 j -0.0039 k) m/s2

x,z,r = 2,1.5,1  # ft
rd = 2           # ft/s
rdd = -0.5       # ft/s2
w1,w2 = 2,7      # rad/s
w1d,w2d = 6,1    # rad/s2
Q = 30*dtr       # rad
vB = -w1*x*jhat
aB = w1**2*x*ihat - w1d*x*jhat
rhat = array(( 0, cos(Q), sin(Q) ))
vCp = w2*r*cross(ihat,rhat) + rd*rhat
ut = cross(ihat,rhat)
v_rel = w2*r*ut + rd*rhat
v_turn = w1*r*cross(khat,rhat)
vC = vB + v_turn  + v_rel
print('vC = ({:+.3f} i {:+.3f} j {:+.3f} k) m/s'.format(*vC))
a_rel = (w2d*r + 2*w2*rd)*ut + (rdd-w2**2*r)*rhat
aC = aB + w1d*r*cross(khat,rhat) + w1*cross(khat,v_turn) + 2*w1*cross(khat,v_rel) + a_rel
print('aC = ({:+.3f} i {:+.3f} j {:+.4f} k) m/s2'.format(*aC))

vC = (-1.732 i -5.768 j +7.062 k) m/s
aC = (+9.876 i -72.832 j +0.3647 k) m/s2

Ch 20 problems¶