Page 555 - Dynamics of Mechanical Systems

0593_C15_fm Page 536 Tuesday, May 7, 2002 7:05 AM

536 Dynamics of Mechanical Systems

Section 15.4 Dynamic Balancing: Arbitrarily Shaped Rotating Bodies
P15.4.1: Suppose n , n , and n are mutually perpendicular unit vectors fixed in a body B,
1
2
3
and suppose that B is intended to be rotated with a constant speed Ω about an axis X
which passes through the mass center G of B and which is parallel to n . Let n , n , and
1
1
2
n be nearly parallel to principal inertia directions of B for G so that the components I of
ij
3
the inertia dyadic of B for G relative to n , n , and n are:
3
2
1
.
 18 −01. 025 
  2
I = −01. 12 −015. slug ft
ij  
 0 25. −0 15. 6  
Show that with this configuration and inertia dyadic that B is dynamically out of balance.
Next, suppose we intend to balance B by the addition of two 12-oz. weights P and P ˆ
placed opposite one another about the mass center G. Determine the coordinates of P
ˆ
P
and relative to the X-, Y-, and Z-axes with origin at G and parallel to n , n , and n .
3
2
1
P15.4.2: Repeat Problem P15.4.1 if the inertia dyadic components are:
 30 −02. −03. 
  2
I = −02. 20 025 kg m
.
ij  
.
 −  03. 025 10  
ˆ
and if the masses of P and are each 0.5 kg.
P
P15.4.3: Repeat Problems P15.4.1 and P15.4.2 if B is rotating about the Z-axis instead of
the X-axis.
Section 15.5 Balancing Reciprocating Machines
P15.5.1: Suppose the crank AB of a simple slider/crank mechanism (see Figures 15.5.1 and
P15.5.1, below) is modeled as a rod with length of 4 in. and weight of 2 lb. At what distance
ˆ r away from A should a weight of 4 lb be placed to balance AB?

B

A
C
ˆ
r
FIGURE P15.5.1 ˆ
A simple slider crank mechanism. m B
P15.5.2: See Problem P15.5.1. Suppose is to be 1.5 in. What should be the weight of the
ˆ r
balancing mass ˆ m ?
B
P15.5.3: Repeat Problem P15.5.1 if rod AB has length 10 cm and mass 1 kg.
P15.5.4: Consider again the simple slider/crank mechanism as in Figure P15.5.4, this time
with an objective of eliminating or reducing the primary unbalancing force as developed
in Eq. (15.5.23). Specifically, let the length r of the crank arm be 4 in., the length of the
connecting rod be 9 in., the weight of the piston C be 3.5 lb, and the angular speed w of