BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 39
The friction forces are FfA
= μN
A, FfB = μN
B
and FfC = μNC
because the
friction is considered at limit state.
The system of the six equatio ns of equilibrium is then symbolically
solved.
3. The Solution of the System Equilibrium Equations
In order to obtain the symbolic solution of the system of equilibrium
equations ( 1), (2) and (3), the following two matrices are built in Mathcad
(Maxfield, 2009)
⎛ 1 −1 −μ⎞
⎜
⎟
M = −μ −μ
1
⎜− 11 11+ h − μΦ −q
⎟
⎝
⎠
(11)
⎛ 0 ⎞
⎜
⎟
v: = F + Fe
. (12)
⎜( F + Fe
)( d + Φ)
⎟
⎝
⎠
The symbolic solution is obtained by the command
Insolved( M,v) =
2 2 3 2 2
⎡ 2 μ ( Fh+ Feh + FμΦ + FeμΦ + 2Fμd+ 2 Feμd) + ( F+ Fe)( μ h−μ Φ − h+ 2μ 11 + μΦ
+ 2 μq) -( μ − 1)( Fh + Feh+ FμΦ
+ FeμΦ
+ 2Fμd + 2 Feμd
) ⎤
⎢
⎥
2 3 2
2 μμh ( − μ Φ − h + 2μ 11 + μΦ
+2 μq)
Fh + Feh+ FμΦ
+ FeμΦ
+ 2Fμd + 2Feμd
⎢
2 3 2
⎥
⎢
μ h −μ Φ − h+ 2μ 11 + μΦ
+2μq
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎢
⎥
⎥
⎢
2 3 2 2
( F + Fe )( μ h − μ Φ−η+ 2μ 11 + μΦ + 2 μq) -( μ − 1)( Fh + Feh+ FμΦ + FeμΦ
+ 2Fμd + 2Fe
μd)
⎥
⎢
−
⎥
2 3 2
⎢
2 μμh ( − μ Φ − h + 2μ 11 + μΦ
+2 μq)
⎥
= ⎢
⎥
⎢
⎢
⎥
⎥
The three elements of the previous matrix are the unknowns N A , N B and N C .
Hence, the normal force is
NC
Fh + F h + FμΦ
+ F μΦ
+ 2Fμd + 2F μd
NC =
μ h −μ Φ −h + 2μ 11+ μΦ
+ 2μq
e e e
.
2 3
2
(13)
The cam follower is blocked if the force NC
is infinite, that is when the
denominator of the fraction is zero. In this case the distance h is
2 11
h =−
2 3
μ − μ + μ +2μ
Φ Φ q
2
μ −1
(14)
In fact, the distance h must be greater then the value given by Eq. (14) in
order for the cam to be able to rotate.