PROBLEM 5
Exercises of Machine Dynamics & System Dynamics
Problem 5
1. Determine the velocity and the acceleration of a particle in Cartesian coordinates described by the following coordinate vector:
r(t)
q(t) =  ϕ(t)  ,
z(t)

x(t)
k(t) =  y(t) 
z(t)



Note:
• Begin with the relationship between the Cartesian coordinates k(t) and the cylindrical
coordinates q(t).
Solution
Given







x(t)
r cos ϕ
cos ϕ − sin ϕ 0
r







k(t) =  y(t)  =  r sin ϕ  =  sin ϕ cos ϕ 0   0 
z(t)
z
0
0
1
z
considering for simplicity
r = r(t), ϕ = ϕ(t)
then the velocity is given as




ṙ cos ϕ − rϕ̇ sin ϕ


v(t) =  ṙ sin ϕ + rϕ̇ cos ϕ 
ż

cos ϕ − sin ϕ 0
vr


=  sin ϕ cos ϕ 0   vϕ 

0
0
1
vz
with
vr = ṙ
vϕ = rϕ̇
vz = ż
Note: The velocity vector amplitude does not chang by changing the coordinate system
used to descript the velocity vector therefore
kCartesian v (t)k2 = kCylindrical v (t)k2
The acceleration is given as
r̈ cos ϕ − ṙϕ̇ sin ϕ − ṙϕ̇ sin ϕ − rϕ̇2 cos ϕ − rϕ̈ sin ϕ

a(t) = 
 r̈ sin ϕ + ṙ ϕ̇ cos ϕ + ṙ ϕ̇ cos ϕ − r ϕ̇2 sin ϕ + r ϕ̈ cos ϕ 
z̈


r̈ cos ϕ − 2ṙϕ̇ sin ϕ − rϕ̇2 cos ϕ − rϕ̈ sin ϕ


=  r̈ sin ϕ + 2ṙϕ̇ cos ϕ − rϕ̇2 sin ϕ + rϕ̈ cos ϕ 
z̈





cos ϕ − sin ϕ 0
ar


=  sin ϕ cos ϕ 0   aϕ 

0
0
1
az
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Exercises of Machine Dynamics & System Dynamics
so with
ar = r̈ − rϕ̇2
aϕ = 2ṙϕ̇ + rϕ̈
az = z̈
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PROBLEM 5
PROBLEM 6
Exercises of Machine Dynamics & System Dynamics
Problem 6
Sometimes it is easier to describe the motion on a curved path in cylindrical coordinates than
in Cartesian coordinates. The position vector in cylindrical coordinates is:
r(t) = r(t)er + z(t)ez
Whereas the unit vector er depends on the angle ϕ and this on its part again depends on the
time, which can be written as
er = er (ϕ(t))
Due to this time dependence of the unit vector, the derivation with respect to time for the velocity and the acceleration are not as simple expressions as in Cartesian coordinates. Determine
these expressions?
Z
Z0
P
ez
r(t)
eφ
ey
Y
er
ex
φ
r
X
Solution
Given
r(t) = r(t)er + z(t)ez
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PROBLEM 6
Exercises of Machine Dynamics & System Dynamics
with r(t) = r and z(t) = z. Then we could write
r(t) = rer + zez
Considering the coordinates shown in the next figure
eφ
ey
er
φ
φ
ex
we have the following
er = cos ϕex + sin ϕey
eϕ = − sin ϕex + cos ϕey
derivating these two equations for time we get
ėr = −ϕ̇ sin ϕex + ϕ̇ cos ϕey
ėϕ
= ϕ̇ sin ϕex + cos ϕey
= ϕ̇eϕ
= −ϕ̇ cos ϕex − ϕ̇ sin ϕey
= −ϕ̇ cos ϕex − sin ϕey
= −ϕ̇er
This final result is very important
ėr = ϕ̇eϕ
ėϕ = −ϕ̇er
The velocity is calculated such as
d
(r(t))
dt
= ṙer + rėr + żez
= ṙer + rϕ̇eϕ + żez
= vr er + vϕ eϕ + vz ez
v(t) =
SS2016 - 3.ed
Exercises of Machine Dynamics & System Dynamics
where
vr = ṙ
vϕ = rϕ̇
vz = ż
The acceleration is given as
d
(v(t))
dt
= r̈er + ṙėr + ṙϕ̇eϕ + rϕ̈eϕ + rϕ̇ėϕ + z̈ez
= r̈er + ṙϕ̇eϕ + ṙϕ̇eϕ + rϕ̈eϕ − rϕ̇2 er + z̈ez
a(t) =
= r̈ − rϕ̇2 er + (2ṙϕ̇ + rϕ̈) eϕ + z̈ez
= ar er + aϕ eϕ + az ez
where
ar = r̈ − rϕ̇2
aϕ = 2ṙϕ̇ + rϕ̈
az = z̈
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PROBLEM 6
PROBLEM 7
Exercises of Machine Dynamics & System Dynamics
Problem 7
Using the chain rule of differentiation we get:
a(t) = r̈(t) = J rq q̈(t) + K rq q̇ Q
Determine K rq and q̇ Q .
Solution
ṙ = v = J rq q̇
=
m
X
∂r
i=1 ∂qi
q̇i
a = v̇ = r̈
"
m
X
∂
=
i=1 ∂t
=
m
X
"
i=1
=
m
X
∂
∂t

q̇i
i=1
∂r
q̇i
∂qi
!#
!
∂r
∂r ∂ q̇i
q̇i +
∂qi
∂qi ∂t
m
X
∂ 2 r ∂qj
∂qi ∂qj ∂t
j=1
!
#

∂r 
+
q̈i
∂qi
with m = 3

=


3 
X

∂ 2r
∂ 2r
∂ 2r
∂r 
q̇i

q̇
+
q̇
q̇
+
q̇
q̇
+
q̈
1
i
2
i
3
i

∂q1
∂qi ∂q2
∂qi ∂q3
∂qi 

i=1 | ∂qi{z
} |
{z
} |
{z
}
j=1
2
=
j=2
j=3
2
2
∂ r 2
∂ r
∂r
∂ r
q̇ + q̇1
q̇2 + q̇1
q̇3 +
q̈1
2 1
∂q1
∂q1 ∂q2
∂q1 ∂q3
∂q1
∂ 2r
∂r
∂ 2r
∂ 2r
+q̇2
q̇1 + 2 q̇22 + q̇2
q̇3 +
q̈2
∂q2 ∂q1
∂q2
∂q2 ∂q3
∂q2
∂ 2r
∂ 2r
∂ 2r
∂r
+ q˙3
q̇1 + q̇3
q̇2 + 2 q̇32 +
q̈3
∂q3 ∂q1
∂q3 ∂q2
∂q3
∂q3
|
K
{z
}
q̇
rq Q
| {z }
J
rq
q̈
q̇12


 2q̇1 q̇2 


i
 2q̇1 q̇3 
∂2r


 q̇ 2 
∂q32


2
}

 2q̇2 q̇3 
q̇32

=
h
∂2r
∂q12
∂2r
∂q1 ∂q2
∂2r
∂q1 ∂q3
|
∂2r
∂q22
{z
K
rq
∂2r
∂q2 ∂q3
|

+
h
|
∂r
∂q1
∂r
∂q2
{z
J
rq
∂r
∂q3
i
}
q̈1


 q̈2 
q̈3
| {z }
q̈
SS2016 - 3.ed


{z
q̇
Q
}
PROBLEM 8
Exercises of Machine Dynamics & System Dynamics
Problem 8
An airplane climbs at a constant speed v and at a constant climb angle β. The airplane is
being tracked by a radar station at point A on the ground.
Given
• H, β, v
• Determine the radial velocity Ṙ and the angular velocity θ̇ as functions of the tracking
angle θ.
Hint
For the following figure the law of sines yields
R
sin
SS2016 - 3.ed
π
+β
2
=
vt
sin
π
−θ
2
=
H
sin (θ − β)
Exercises of Machine Dynamics & System Dynamics
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PROBLEM 8
Exercises of Machine Dynamics & System Dynamics
Problem 9
Solve the same problem (Problem 8) using the cylindrical coordinates.
Given
• H, β, v
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PROBLEM 9
Exercises of Machine Dynamics & System Dynamics
PROBLEM 10
Problem 10
A robot arm consists of two parts, which are simply supported in point A. The lower part
(length L1 ) rotates anti-clockwise with the angular speed ω1 . The upper part (Length L2 ) is
powered clockwise by a flanged e ngine a t t he i ntermediate h inge ( angular s peed ω2 ).
Given
• ω1
• |ω2 | = 2ω1
• θ1 (0), θ2 (0)
• L1 , L2
1. Find the position vector I rOP written in the inertia system.
2. Compute the absolute velocity of the tip P in the inertial system using the jacobian, and
assuming the angles θ1 (0) and angle θ2 (0) at the time t = 0.
3. Compute the absolute acceleration of the tip P in the inertial system using the K rq
and q Q .
SS2016 - 3.ed