...
to
m,x,
+
k,x1
-
k2(x2
-
XI)
+
c,X,
-
~2( X2
-
XI)
=
F,
and for
x,
mg2
+
k2(x2
-
xI)
+
c2(X2
-
X,)
=
F2
Fig.
2. 2
Energv
for
a
group
of
particles
17
1.14
Conservation ... the system shown in Fig. 2. 2 the scalar functions are
k,
2
k2
Y
=
-XI
+
-(x2
-
XJ2
2 2
Ci
-2
C2
2 2
3
=
-XI
+
-(X2
-
X,f
The virtual work done by the ext...
... below
V
[
-
!!
(4
+
2e-”)
z
+
2
(f
rz2
+
B2
]
L
f2
=
-
C
where
and
(2
-
4p)
+
3 12
f3
=
-
e-’LL’L
-
-
2p
[
-4p+
c
[Le
(6.81)
(6. 82)
where
B,
=
e-6p
+ ...
or
c2
=
Elp
as
before.
Fig.
6. 12
148
Impact and one-dimensional wave propagation
Fig.
6 .21
Fig.
6 .22
Maximum strain at
x
=
L
for
various period numbers,
n
1
2
1...
...
The kinetic energy is given by
m
.2
m
-2
I
.2
I
.2
2
2
2
2
The virtual work done by the impact force at A is
T
=
-XI
+
-x2
+
-e,
+
-e2
6W
=
F(-dr,
+
ado,)
and the ...
-
o,o,z]
+
2m(w,i
-
or$)
-efv
=
m[(q
+
o,)y
-
o,.o,z
-
o,o,x]
+
2m(o$
-
oj)
or
Similarly
2
2
2
2 2
-efz
=
m[(o,
+
O,)Z
-
O,O,X
-
O,O,,y]
+
2m(ox9...
...
f
=
21 ,xi
+
2Z2yj
+
21 ,zk
Now
x=lp=-
ZO
=
-
Ox
gradf
=
-
2L
\
(2T)
\
(2T)
etc.,
so
\
(2T)
Kinetics
of
a
rigid body
59
(4.5)
(4-6)
ds
=
del
X
a
+
de2
X
(a ... rotation is about the
z
axis equation
(4.43)
reduces to
1
2
T
=
-
~rn,(a~xj
-
w,yi )2
1
2
=
-
a2Cmi(x2
+
y2)
12
=-a
I,
2
(4.47)
(4.48)
Since the choice
of
axis...
...
_-
P
m, m2
2
2
IF,^^
=
m,w
r,
-
mI?,
=
m2m
r,
-
m2r2
and taking moments about the centre of
mass
(C
of
M)
gives
0
=
m,(r,&
+
2wi,)
=
m2(r2&
+
2wi2)
or
(5.15) ...
P,cos~ )2
P:
E
=
-
Z,e2
+
+
-
+
mgh
cos8
(4.89)
2 21, sin2
0
21 3
Equation
(4.89)
can be rearranged in the following form
1
2
E’
=
-
z,e2
+
v
(e)
where the c...
...
dx
-
0)
=
-V2
2
(6. 122 )
(6. 123 )
(6. 124 )
(6. 125 )
It is now possible to eliminate
0
between equations (6. 120 ) and (6. 12 1). Equation (6. 120 )
can be written
as
2
2
V.lf
-
KC, ...
+
dzk
1 62
Impact
and one-dimensional wave propagation
we get
2
2
COS
(k)
-
2
=
-
-
mw
S
or
2
2( -2 sin2 (W2))
=
-
=
S
giving
o
=
21
(s/m)
sin(W2)...
... terms of their components
Pzr
=
Plr
+
4
P2,
=
PI,
+
4
P 22
=
PI;
4-
u:
or
(Pz)
=
(PI)
+
(4
(8 .22 )
Kinematics
of
a robot arm
2
13
Fig.
8.19
ce,
o
se,
a,ce,
3[A]4 ...
O)T,
a null vector), we may now combine equations
(8 .23 )
and
(8 .24 )
to
give
(8 .25 )
or, in abbreviated form,
(P2)
=
[Tl(dl)
(8 .26 )
Here
(3)
is the
4
X
1
homogeneous...