Page 425 - Dynamics of Mechanical Systems

0593_C11_fm Page 406 Monday, May 6, 2002 2:59 PM

406 Dynamics of Mechanical Systems

θ β
1 1

θ
2 β
2

θ β
3 3
P(M) P(M)
FIGURE P11.5.5 FIGURE P11.5.6
A triple-rod pendulum with a concentrated end A triple-rod pendulum with concentrated end mass
mass. and relative orientation angles.
P11.5.6: Repeat Problem P11.5.5 for the relative orientation angles β , β , and β shown in
2
3
1
Figure P11.5.6.
P11.5.7: See Problem P11.5.5. Let there be linear torsion springs at the pin joints of the
triple-rod pendulum. Let these springs have moduli k , k , and k , and let the resulting
3
1
2
moments generated by the springs be parallel to the pin axes and proportional to the
relative angles β , β , and β between the rods as shown in Figure P11.5.6. Find the
1
2
3
contribution of the spring moments to the generalized active forces for the angles θ , θ ,
1
2
and θ .
3
P11.5.8: Repeat Problem P11.5.7 for the relative orientation angles β , β , and β .
3
2
1
P11.5.9: An elongated box kite K, depicted in Figure P11.5.9A, is suddenly subjected to
turbulent wind gusts creating forces on K as represented in Figure P11.5.9B. Let the wind
forces be modeled by forces F , F ,…, F , whose directions are shown in Figure P11.5.9B
8
2
1
and whose magnitudes are:
F = 5 lb, F = 6 lb, F = 6 lb, F = 7 lb,
1 2 3 4
F = 8 lb, F = 10 lb, F = 13 lb, F = 8 lb
5 6 7 8
Let the orientation of K be defined by dextral orientation angles α, β, and γ, and let the
angular velocity ωω ωω of K be expressed as (see Problem P11.4.3):
( ˙ ) ( ˙ )
˙
˙
ωω= ˙ αcc + βs n 1 + βc γ − ˙ αc s n 2 +( αs β + ) γ n 3
β γ
β γ
γ
where n , n , and n are unit vectors parallel to the edges of K as shown in Figure P11.5.9B.
1
2
3
Finally, let the velocity v of the tether attachment point O of K be:
v = 3 n − 2 n + 5 n ft sec
1 2 3
Find the generalized forces F , F , and F due to the wind forces F , F ,…, F .
β
α
γ
2
8
1
P11.5.10: Repeat Problem P11.5.9 by first replacing F , F ,…, F by a single force F passing
1
2
8
through O together with a couple with torque T; then, use Eq. (11.5.7) to determine F ,
α
F , and F .
γ
β