Input shaping control of a flexible cantilever beam excited by a moving hub

Introduction: A cantilever beam is a well-known structural element in engineering, which is only

fixed at one end. This structure can be used to describe a manipulator, whose stiffness is large to

ensure rigidity and stability of the system. A flexible cantilever beam provides a light-weight structure and high cost efficiency but generates vibration under high-speed positioning. In this paper,

we aim to control the vibratory behavior of a flexible cantilever beam attached to a moving hub.

Method: The mathematical model of the flexible beam is described by partial differential equations (PDEs) using Euler-Bernoulli beam theory. Then, The PDE model is approximated by using the

Galerkin method, which is resulted in a set of ordinary differential equations (ODEs). Experiment is

used to determine unknown parameters of the system to complete the model. The ODE model enables the control design of three input shapers: (i) Zero-Vibration (ZV), (ii) Zero-Vibration-Derivative

(ZVD), and (iii) Zero-Vibration-Derivative-Derivative (ZVDD), which are employed to drive the flexible beam to the desired position and to reduce vibrations of the beam. Results and conclusion:

The dynamic model is obtained in term of ordinary differential equations. Unknown parameters of

the system are determined by experimental process. Various controllers are designed to eliminate

the vibration of the beam. The simulation is applied to predict the dynamic response of the beam

to verify the designed controllers numerically. Experiment shows the validity of the mathematical

model through the consistency between the simulation and experimental data and the effectiveness of the controllers for the real system. These controllers show several advantages such as no

need of extra equipment; the positioning controller is intact, which means it may be applied to

many existing systems.

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Input shaping control of a flexible cantilever beam excited by a moving hub
s is neglected in the control
design of input shapers.
The behaviors of the flexible beam are simulated with
three commands: (i)The unshaped command is an S-
curve profile with a displacement of 50 mm in 0.1s;
(ii) ZV, ZVD, and ZVDD shapers are implemented to
compare the performance of each shaper.
As shown in Figure 3, the unshaped command causes
a large vibration of the tip, which takes a long time for
suppression. In the case of shaped command , the ZV
shaper achieves the settling time is 0.5 seconds when
the tolerance band 5% is applied. ZVD shaper shows
better performance compared to the ZV shaper when
the amplitude of the vibration is significantly reduced.
Finally, the ZVDD shaper obtained the best perfor-
mance with zero vibration after 0.4 seconds.
Table 1: System Parameters
Symbol Definition Values
L Beam length 0.45 m
h Beam height of area 28.8 x 103 m
b Beam width of area 0.88 x 103 m
r Beam linear density 0.2 kg.m1
EI Beam flexural rigidity 332.714 x 103
Nm2
m Mass of hub 1.5 kg
The experiments are carried out in this research. The
testbed and data acquisition are shown in Figure 4.
The system consists of one single translational DOF,
whose link is a flexible aluminum beam. One beam’s
end is fixed rigidly to a brushless linear motor while
the other end is free.
In the experiment system, an LM series linear servo
motor from Yokogawa performs movements directly.
This feature brings many advantages, the motor’s ve-
locity and the ratio of thrust to weight are significantly
large (0.8 m/s, and 7m/s2, respectively). Also, the po-
sitioning precision of the fixed end’s link is equal to the
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Science & Technology Development Journal – Engineering and Technology, 3(2):416-424
Figure 3: Simulation results of the tip displacement of the flexible beam.
servo motor’s precision (0.5 mm) due to backlash ab-
sence. These factors contribute to high performance
of the whole system. The motor is powered by TM
series servo amplifier from Yokogawa. The command
is fed by a motion control unit, which is Turbo Com-
pact UMAC CPU from Delta Tau® . With Motorola
DSPs Processor, the Turbo CPU is a powerful mo-
tion controller that provides a packed combination of
motion and I/O control for a complex automatic sys-
tem. The integrated features include trajectory gen-
eration, servo, commutation, compensation, motion
program, and so forth. The damping ratio is obtained
by logarithmic decrement d which is defined as the
natural log of the ratio of any two successive peaks.
Thismethod is conducted through the result of the ex-
periment on the underdamped vibration of the beam
(see Figure 5).
The logarithmic decrement d is presented as
d =
1
n
ln

xi
xi+n

= ln

33:19
31:67

= 0:047 (15)
The damping ratio, then, determined by the relation-
ship with logarithmic decrement as follows:
z =
dp
d 2+4p2
= 0:0075 (16)
The natural frequency is determined by plotting the
frequency spectrum (Figure 6) of the free vibration of
the beam with initial disturbance.
The experimental results are shown in Figure 7, it is
shown that experimental data is consistent with pre-
dicted results. With the unshaped command, the vi-
brations of the flexible beam occur for a long time.
Meanwhile, the ZV, ZVD, and ZVDD shapers elim-
inate vibrations significantly. The simple ZV shaper
reduces a large vibration of 9% compared to unshaped
command while the vibration of ZVD and ZVDD
shapers are both lower than 2% in relation to un-
shaped command.
DISCUSSIONS AND CONCLUSION
In this paper, three input shapers ZV, ZVD, and
ZVDD for the simultaneous position and vibration
control of a flexible cantilever beam excited by a mov-
ing hub are proposed. The design of input shapers is
based on the ODEmodel of the flexible beam system,
which is derived by using the Galerkin method. The
validity of the proposed input shapers is verified by
both simulations and experiments.
ACKNOWLEDGEMENT
This work was supported T-CK-2018-07.
LIST OF ABBREVIATIONS USED
PDE: Partial differential equation
ODE: Ordinary differential equation
ZV: Zero vibration
ZVD: Zero vibration derivative
ZVDD: Zero vibration derivative derivative
PPF: positive position feedback
SRF: strain rate feedback
COMPETING INTERESTS
The authors declare that they have no conflicts of in-
terest.
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Science & Technology Development Journal – Engineering and Technology, 3(2):416-424
Figure 4: Experimental testbed.
Figure 5: Free vibration of a viscous damped cantilever beam.
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Science & Technology Development Journal – Engineering and Technology, 3(2):416-424
Figure 6: The experimental frequency spectrum of the beam.
Figure 7: Experimental results shows the displacements of the flexible beam in uncontrolled case and with con-
trolled cases (with ZV, ZVD, and ZVDD shaper).
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Science & Technology Development Journal – Engineering and Technology, 3(2):416-424
AUTHOR CONTRIBUTION
Nguyen Tri Giang Vu has deisgned the controller for
simulation and experiment purposes and has writ-
ten the manuscript. Phuong Tung Pham has inves-
tigated the dynamic model and has examined the ini-
tial experiment. Quoc Chi Nguyen has provided re-
search ideas, has guided the research and has edited
the manuscript.
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Tạp chí Phát triển Khoa học và Công nghệ – Kĩ thuật và Công nghệ, 3(2):416-424
Open Access Full Text Article Bài Nghiên cứu
1Bộ môn Cơ điện tử, Trường Đại học
Bách khoa - Đại học Quốc gia Thành
phố Hồ Chí Minh
2Khoa kỹ thuật Cơ khí, Đại học Quốc gia
Pusan
Liên hệ
Nguyễn Quốc Chí, Bộ môn Cơ điện tử,
Trường Đại học Bách khoa - Đại học Quốc gia
Thành phố Hồ Chí Minh
Email: nqchi@hcmut.edu.vn
Lịch sử
 Ngày nhận: 01-10-2019
 Ngày chấp nhận: 30-01-2020
 Ngày đăng: 16-8-2020
DOI : 10.32508/stdjet.v3i2.605
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mở được phát hành theo các điều khoản của
the Creative Commons Attribution 4.0
International license.
Điều khiển hệ dầm công xônmềm có bệ gá di động bằng phương
pháp input shaping
Vũ Nguyễn Trí Giang1, Phạm Phương Tùng2, Nguyễn Quốc Chí1,*
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TÓM TẮT
Giới thiệu: Dầm công xôn là một kết cấu phổ biến trong các kỹ thuật, với chỉ một đầu được
gắn cứng. Kết cấu này có thể được dùng để mô tả một cánh tay robot, với độ cứng lớn để đảm
bảo độ cứng vững và ổn định của hệ thống. Một dầm công xôn mềm cung cấp một kết cấu nhẹ
và hiệu quả chi phí, nhưng gây ra dao động dưới chuyển động tốc độ cao. Trong bài báo này,
chúng tôi hướng đến việc điều khiển dao động của thanh dầmmềm được gắn trên một bệ gá có
chuyển động thẳng. Phương pháp: Mô hình toán học của dầm công xôn mềm được biểu diễn
bằng các phương trình vi phân đạo hàm riêng (PDE-partial differential equation). Mô hình được
xấp xỉ bằng phương pháp Galerkin với kết quả là một mô hình biểu diễn bằng một hệ phương
trình vi phân thông thường (ODE-ordinary differential equation). Phương pháp thực nghiệm được
sử dụng để xác định các thông số chưa biết của hệ thống. Mô hình ODE tạo điều kiện cho thiết
kế ba giải thuật điều khiển input shaping: (i) Zero-Vibration, (ii) Zero-Vibration-Derivative, và (iii)
Zero-Vibration-Derivative-Derivative, được dùng để di chuyển dầm công xônmềmđến vị trí mong
muốn và giảm thiểu dao động. Kết quả và kết luận: Mô hình động lực của hệ được thiết lập với
các phương trình vi phân thông thường. Các thông số chưa biết được xác định bằng thực nghiệm.
Các bộ điều khiển khác nhau được thiết kế để khử dao động của thanh dầm. Quá trình mô phỏng
dự đoán đáp ứng động lực học để xácminh độ hiệu quả của các bộ điều khiển bằng phương pháp
số. Thực nghiệm chỉ ra sự hợp lệ củamô hình toán thông qua sự thống nhất của dữ liệumô phỏng
và thực nghiệm và sự hiệu quả của các bộ điều khiển với hệ thống thực. Những bộ điều khiển này
có một ưu điểm như: không cần gắn thêm thiết bị; bộ điều khiển chuyển động không bị tác động
nên các bộ điều khiển này có thể áp dụng với nhiều hệ thống có sẵn.
Từ khoá: Dầm công xôn mềm, phương pháp Galerkin, input shaping, khử dao động
Trích dẫn bài báo này: Giang V N T, Tùng P P, Chí N Q. Điều khiển hệ dầm công xôn mềm có bệ gá di
động bằng phương pháp input shaping. Sci. Tech. Dev. J. - Eng. Tech.; 3(2):416-424.
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