View the step-by-step solution to:

Question

d1.png

d2.png

/>

d3.png

d4.png

d5.png

Find the transfer function between the active cart position (IPO2) as X1 and the input voltage Va

d1.png

IV. Mathematical Modeling
Please complete the modeling of the dynamics.
Ra
La
W
Ktla
Gearbox
ia (t )
+
v(t)
vo (t ) = Kbom
Jm
Pinion
Bm
Gear
DC Motor
N
Rack
Figure 11: DC motor schematic diagram relating applied voltage to position of the motor pinion
gear, 0g
i. DC Motor Current
By using Kirchhoff's Voltage Law on the circuit diagram in Figure 11,
Raia(t)+ La + VBEMF(t) = VA(t)
(1)

d2.png

Here, the conslants RE and Lu represent the armature resistance and inductance+ and the
variables iflfl]. and 15.13.} represent the motor current draw and the applied voltage
respectively. The voltage fed hack into the circuit+ as a result of the motion of wires in a
magnetic field is called the hack electromotive force, or wag-"F. and can be determined from
the equation+ Vssusfl] = "Kali; {2} where N is the gear ratio of the planetary gearbox. K3 is the motor speed coefficient. and {is
is the rotational velocity of the motor pinion gear. ii. DC Motor Gear Angle
The sum of the moments about the rotor ofthe DC motor gives the second equation,
Lydia + 3...“, = r... + r; {3} Here, I," is the rotor inertia, B," is the damping acting on the rotor+ r... and T; are the torques
on the motor the to the magnetic field and the external load applied via the interactions of
the rack and motor pinion gear respectively. The equations for these torques are given by the
following relations, r... = strum {431'
n; = -r.: 5: {41:} where Kt is the motor torque constant provided by the manufacturer+ I; is the reaction force
applied to the motor pinion gear by the rack, and 1:. is the radius of the motor pinion gear.

d3.png

iii. IP02 Cart Position
Consider the two carts with their respective free body diagrams in Figure 12.
12
X1
B2X2
K(x1 - X2)
Mixi
LFJ Cart
By (1 1 - 12)
IP02 Cart
Figure 12: Linearized external and inertial forces affecting both carts
The third equation can be obtained from the sum of the forces in the x-direction on the IP02
cart as,
Mix, + By(x1 - x2) + Ky(x] - x2) + Bjin = fc
(5)
12

d4.png

Here, M, is combined mass of the IP02 cart, weight and additional 'mass' due to the IP02
cabling, B, is the additional damping observed by the active cart, By is the damping caused
by the linear flexible joint between the two-masses, Ky is the stiffness due to the compression
spring, x, is the position of the IP02 cart in the x-direction, x2 is the position of the LFJ cart
in the x-direction, fe is the external force applied to the IP02 cart at the motor pinion, and (
represents time derivatives of the respective variables.
iv. LFJ Cart Position
The fourth and final equation of motion can be obtained from the sum of the forces on the
LFJ cart as,
Maxz + By(x2 - x1) + Bzx2 + Ky(x2 - x) =0
(6)
where, M2 is combined mass of the LFJ cart and weights, and B2 is the additional damping
observed by the passive cart.
v. Combined Transfer Function
To combine these equations of motion via substitution, Eq. 3 can be rewritten to isolate fe
as,
JmNeg + BmNeg = Kia - fed
(7a)
Jmpeg - Bmy Ogt - Ktla = fc
(7b)
By substituting Eq. 7b into Eq. 5, and realizing that the angle of the motor pinion gear may
be written as 0. = =, Eq. 5 may be rewritten as,
Mix, + By(x1 - 12) + Ky(x1 - x2) + Bix = -Jmrs
-Bg - Bmeg+
NKtia
(8a)
(Mi + / m = ) $1 + (B, + By + Bmz 1 1 + Kyx1 - Byx2 - K,x2 = =Kia
(8b)

d5.png

Now, by assuming zero initial conditions and applying the Laplace transform, we can collapse
Eq. 1, 2, 6, and 8b to form a single transfer function describing the input-output relationship
between applied voltage and cart position. The resulting transfer function will be of the fifth
order, and appear as follows,
G(S) =
X2 (s)
bis+bo
Va (s)
ass5+as4+a3s3+a2s+ajs
(9)
The numerator coefficients, b, and the denominator coefficients, a;, are readily found by this
formulation and dictate the open loop zeros and poles respectively.
In Matlab notation, (9) is given by the following numerator, denominator coefficients:
n = [106604.95, 6880865.21]
d = [1.0, 14456.76, 280415.99, 7100018.02, 55372388.44, 0]
13

Recently Asked Questions

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes