Page 581 - Dynamics of Mechanical Systems

0593_C16_fm Page 562 Tuesday, May 7, 2002 7:06 AM

562 Dynamics of Mechanical Systems

2
2
where, as before, ω is dθ/dt and where, from Eqs. (16.12.2) and (16.12.3), dφ/dθ, d φ/dθ ,
3
3
and d φ/dθ are:
dφ =−  h − h  π  ψ
1
2
dθ  2    θ − θ   sin (16.12.8)
2
1
d φ =−  h − h  π  2 ψ
2
1
2
dθ 2  2    θ − θ 1   cos (16.12.9)
2
and

d φ =  h − h  π  3 ψ
3
1
2
dθ 3  2    θ − θ 1   sin (16.12.10)
2
From Eq. (16.12.2) we see that when θ is θ , then ψ is π, and when θ is θ , ψ is 2π. Hence,
2
1
from Eq. (16.12.3), we have:
=
when θθ : φ = h , d φ = , 0 d 2 φ =  h 2 − h 1   π   2 ,
1 1 θ θ 2   θ − θ 
d d 2  2 1
(16.12.11)
d 3 φ
and 3 = 0
d θ
and

=
when θθ : φ = h , d φ = , 0 d 2 φ = −  h 2 − h 1   π   2 ,
2 2 θ θ 2   θ − θ 
d d 2  2 1
(16.12.12)
d 3 φ
and 3 = 0
d θ
Then, by inspection of Eqs. (16.12.5), (16.12.6), and (16.12.7), we see that dh/dt is zero at
2
both θ and θ , but d h/dt has the values:
2
1
2
±  h 2 − h 1   π   2 ,
 2  θ 2 − θ 1 

3
3
at θ = θ (plus sign) and θ = θ (minus sign), and d h/dt is infinite at both θ = θ and
2
1
1
θ = θ . In the following section, we examine the cycloidal profile which has finite values
2
of d h/dt at the transition points.
3