Suppressing undesired vibration of 3 r robot arms using impact dampers

International Journal of Recent advances in Mechanical Engineering (IJMECH) Vol.3, No.4, November 2014
DOI : 10.14810/ijmech.2014.3412 135
SUPPRESSING UNDESIRED VIBRATION OF 3R
ROBOT ARMS USING IMPACT DAMPERS
Mohammad Amin Rashidifar1
And Darvish Ahmadi2
1
Department of Mechanical Engineering, Islamic Azad University, SHADEGAN Branch,
FATHOLMOBIN BLV, SHADEGAN, Khuzestan 6431863758, Iran
Department of Mechanical Engineering, Islamic Azad University, SHADEGAN Branch,
FATHOLMOBIN BLV, SHADEGAN, Khuzestan 6431863758, Iran
Abstract
The behavior of a vibration system suppressed with an impact damper is investigated, where the impact
damper is simplified as a combination of spring and viscous damping. The analytical answer for the
contact duration is developed using iteration perturbation technique. Iteration perturbation is a new
analytical technique which can solve strongly nonlinear equations. In the presented work an impact
damper is used to reduce the undesired vibration of an industrial robot arm. For this reason a 3R robot is
equipped with a simple model of a single unit impact damper which is constructed using spring, mass and
viscous damper. The output forces of robot with and without impact damper are compared and effects of
using the impact damper on correct application of the robot are shown.
Keywords
Impact damper, Iteration perturbation, 3R robot
1. INTRODUCTION
An impact damper is a lightweight auxiliary device which consists of a container with a loose
mass. In impact dampers, the secondary mass (loose mass) absorbs kinetic energy of the main
vibratory system and converts it into heat through inelastic collisions between the secondary mass
and the enclosure. This damper is effective for discrete and continuous systems in resonant
operation and produces maximum displacement reduction during two impacts per cycle of
motion. Applications of the impact dampers to reduce excessive vibration of industrial machinery
have been extensively investigated by researchers [1]. The reason for this interest lies in the
following features of an impact damper:
• Small and simple in construction.
• Easy to mount on the main vibratory systems.
• No need to adjust parameters of an impact damper to the vibratory characteristics of the main
vibratory systems.
• Effective application to suppress excess vibration.
In the past few years, behavior of impact dampers has been investigated experimentally,
analytically and numerically [6-11].

136
The effects of using impact dampers on better control of mechanical systems are investigated for
many years [2]. For example, effectiveness of these systems on suppressing undesired vibrations
of a cantilever beam was considered by many researchers [3]. But application of these systems on
more complicated systems for example robot arms was not explored. Son et al. [12] proposed
active momentum exchange impact dampers to suppress the first large peak value of the
acceleration response due to a shock load.
In this paper the governing equation of motion of a vibratory system which is equipped with an
impact damper is solved using a combination of nonlinear normal modes method and iteration
perturbation [4, 5]. Iteration perturbation is a new perturbation technique coupling with the
iteration method. It can solve strongly nonlinear equations, accurately. Contact duration can be
achieved by employing the discussed method and is used for finding motion of primary mass
contacting the barriers. Therefore, the contact time is taken into consideration. This feature is
important for a resilient rather than a rigid impact damper when the noise issue is concerned.
Finally, the attenuating the undesired vibration of the tip of a robot arm is investigated. For this
reason, a 3R robot is modeled in SIMMECHANICS toolbox of MATLAB. The robot is equipped
with a simple model of a single unit impact damper which is constructed using spring, mass and
viscous damper. The output forces of robot with and without impact damper are compared.
2. MATHEMATICAL MODELING
A simple model of a vibratory system consist of an oscillator with linear stiffness K, mass M,
viscous damping C and an impact damper with mass m and clearance d is considered. A
schematic model of this system is shown in figure 1.
Figure1. Schematic of an impact damper
Motion of the vibratory system can be divided to two separate motions:
• Motion of impact damper between barriers.
• Motion of impact damper when it collides with barriers.
The differential equations of motion between impacts are shown as:
‫ݔܯ‬ሷ + ܿ‫ݔ‬ሶ + ݇‫ݔ‬ = 0
݉‫ݕ‬ሷ = 0
(1)

137
So, the motion of the primary mass between impacts i and i+1 can be written as follows:
‫ݔ‬ሺ‫ݐ‬′ሻ = ݁ିకఠ௧
[൤
௫ሶ೔ೌ
ඥଵି఍మఠ
+
క
ඥଵି఍మ
‫ݔ‬௜൨ sin൫ඥ1 − ߞଶ߱‫ݐ‬′
൯ + ‫ݔ‬௜ cos൫ඥ1 − ߞଶ߱‫ݐ‬′
൯
(2)
Where
߱ଶ
= ට
௞
௠
, ߦ =
௖
ଶ√௞௠
ܽ݊݀ ‫ݐ‬′
= ‫ݐ‬ − ‫ݐ‬௜
(3)
And motion of the impact mass can be formulated as follows:
‫ݕ‬ = ‫ݕ‬ሶ௜௔‫ݐ‬′
+ ‫ݕ‬௜௔ (4)
The subscript i,iୟ, iୠ represent the value of variables at contact, just after and just before the ith
contact, respectively. Impact occurs only when
|‫ݔ‬ − ‫|ݕ‬ =
ௗ
ଶ
(5)
Velocities after ith
impact are determined by using the concept of coefficient of restitution and
conservation of momentum. These give:
‫ݔ‬ሶ௜௔ = ቀ
ଵିఓ௘
ଵାఓ
ቁ ‫ݔ‬ሶ௜௕ + ߤሺ
ଵା௘
ଵାఓ
ሻ‫ݖ‬ሶ௜௕ (6)
‫ݕ‬ሶ௜௔ = ቀ
ଵା௘
ଵାఓ
ቁ ‫ݔ‬ሶ௜௕ + ሺ
ఓି௘
ଵାఓ
ሻ‫ݖ‬ሶ௜௕ (7)
Where µ is the mass ratio, m/M, and e is the coefficient of restitution. When contact occurs, the
vibratory system can be shown using a two degree of freedom system. This system is shown in
figure 2, schematically.
Figure2. Schematic while the impact mass colliding the primary mass.
Therefore, the governing equations of motion of the vibratory system should be written by two
coupled equations which are shown in the following equations.

138
‫ܯ‬
ௗమ௫
ௗ௧మ + ݇‫ݔ‬ + ܽሺ‫ݔ‬ − ‫ݕ‬ሻଷ
+ ܾሺ‫ݔ‬ − ‫ݕ‬ሻ = 0 (8)
݉
ௗమ௫
ௗ௧మ + ݇‫ݔ‬ + ܽሺ‫ݔ‬ − ‫ݕ‬ሻଷ
+ ܾሺ‫ݔ‬ − ‫ݕ‬ሻ = 0 (9)
In the above equations, the damping of system is considered negligible. It is reasonable because
the contact duration with respect to the oscillation period of the primary mass is negligible.
Nonlinear normal modes method (NNM) is used for solving the above displacement coupled
equations. Consider ‫ݔ‬ = ‫ݐ߱ݏ݋ܿܣ‬ and ‫ݕ‬ = ‫ݐ߱ݏ݋ܿܤ‬ and substituting them into equations (8) and
(9).Vanishing the resonant term results in:
஻
஺
=
௞ିெఠమ
௠ఠమ
(10)
Therefore equation (8) changes to:
‫ݔܯ‬ሷ + ݇‫ݔ‬ + ሺ1 − ‫ܥ‬ሻଷ
‫ݔ‬ଷ
+ ܾሺ1 − ‫ܥ‬ሻ‫ݔ‬ = 0 (11)
The above equation is a Doffing form of nonlinear equations, which can solve using iteration
perturbation. This equation can be re-written as follows:
‫ݔ‬ሷ௡ାଵ + ቀ
௄ା௕ି௕஼
ெ
ቁ ‫ݔ‬௡ାଵ + ߳‫ݔ‬௡ାଵሺ
௔ሺଵି஼ሻయ
ெ
ሻ‫ݔ‬௡
ଶ
= 0 (12)
While ߳ is an artificial parameter, which is originally is equal to one. Consider:
‫ݔ‬ = ‫ݔ‬଴ + ߳‫ݔ‬ଵ + ⋯ (13)
Assume ‫ݔ‬଴ = X sinωt as the first iteration. Therefore the extended form of angular frequency
can be written in the following form:
௄ା௕ି௕஼
ெ
+ ߳
௔௑మሺଵି஼ሻయ
ଶெ
= ߱ଶ
+ ߳‫ܥ‬ଵ + ⋯ (14)
Consider sentences which have same order of߳. So the result for x1 can be written as follows:
‫ݔ‬ሷଵ + ߱ଶ
‫ݔ‬ଵ + ‫ܥ‬ଵ‫ݔ‬଴ −
௔௑మሺଵି஼ሻయ
ଶெ
‫ݔ‬଴ܿ‫ݐ߱2ݏ݋‬ = 0 (15)
Substituting the first iteration into the above equation and vanishing the resonant term.
߱ଶ
=
ଵ
ெ
൬‫ܭ‬ + ܾሺ1 − ‫ܥ‬ሻ + ܽ ቀ
ଷ
ସ
ܺଶ
ሺ1 − ‫ܥ‬ሻଷ
ቁ൰ (16)
Substituting the above equation into equation (10) results in:
− ቀ
ଷ
ସ
ܺଶ
ܽߤቁ ‫ܦ‬ସ
+ ሺ
ଷ
ସ
ܺଶ
ܽሺ ߤ − 1ሻ‫ܦ‬ଷ
− ሺ ܾߤሻ‫ܦ‬ଶ
+ ሺ ሺܾ − ‫ܭ‬ሻߤ − ܾሻ‫ܦ‬ + ‫ߤܭ‬ = 0 (17)
Where
1 − ‫ܥ‬ = ‫ߤ ݀݊ܽ ܦ‬ =
௠
ெ
(18)

139
So, the coefficient C can be achieved by employing equation (18) and the period of the vibration
can be obtained using equation (16) .Contact duration is equal to half of the period of vibration of
impact mass when it collides with the barriers. Therefore the contact duration is equal to:
‫ݐ‬௔ =
ଶగ
ට
భ
ಾ
൬௄ା௕ሺଵି஼ሻା௔ሺ
య
ర
௑మሺሺଵି஼ሻయሻ൰
(19)
And motion of primary mass between contacts can be formulated as follows:
‫ݔ‬ሺ‫ݐ‬′ሻ = ݁ିకఠ௧′
[
ೣሶ ೔ೌ
ഘ
ାకሺ௫೔್ା௏௧ೌሻ
ඥଵିకమ
sin൫ඥ1 − ߦଶ߱‫ݐ‬′
൯ + ሺ‫ݔ‬௜௕ + ܸ‫ݐ‬௔ሻ cos൫ඥ1 − ߦଶ߱‫ݐ‬′
൯ (20)
Where ܸ is the velocity of the primary mass when it collides with the impact mass. it is equal to:
ܸ =
௠
௠ାெ
‫ݕ‬ሶ௜௕ +
௠
௠ାெ
‫ݔ‬ሶ௜௕ (21)
A computer program in MATLAB was written to simulate equations (2) and (20) in real time.
Consider the below parameters for the vibratory system with an impact damper.
Table1. Parameters for the vibratory system with an impact damper
Masses
M=2 ݇݃
m=0.1 ݇݃
Damping C=0.01
ே.௦
௠
Stiffness K=3000
ே
௠
Coefficient of
restitution
e=0.4
Figure 3 shows the displacement response of the main mass corresponding to the clearance
d=0.01m when it is subjected to the initial displacement x0=y0=-0.007m. In this figure great effect
of using impact damper for suppressing undesired vibration is clear.
Figure3. Displacement response of the main mass with impact damper and without impact damper
Figure 3 shows significant effect of using the impact damper for suppressing undesired vibration
of the main mass in free vibration but what is the effect of this system on forced vibration.

140
3. APPLICATION OF AN IMPACT DAMPER ON A 3R ROBOT:
To begin this section, we pose the question: what is the effect of using an impact damper on
vibration of a robot arm. For finding the answer of this question, a 3R robot is modeled using
SimMechanics toolbox of MATLAB. The robot is equipped with a simple model of a single unit
impact damper which is constructed using spring, mass and viscous damper. Two degrees of
freedom of the robot are allocated to position control and another one is used for force
controlling. Both of force and position are controlled using PID controllers and these separate
controllers are synchronized using a hybrid control subsystem. Figure 3 shows the presented
robot, schematically.
Figure4. Schematic of the 3R robot
(a) (b)
Figure4. Force response of the robot tip corresponding to clearance d=0.04m and (a) m=0.4 Kg, (b)
m=0.1kg; _, with impact damper and ---, without impact damper
Figure 5 shows effectiveness of impact damper in case of forced vibration. In this figure, it is
obvious that using the simple presented impact damper can attenuate the undesired vibration of
robot tip and causes it works more accurate.

141
4. CONCLUSION
This paper presents new analytical results for applying the impact damper to reduce structure
vibration. A simple model of impact damper is constructed using spring, mass and viscous
damper. The important feature to be carried out in the analysis of this model is that the
deformation of an impact damper during the collision with the main mass can be formulated;
therefore, the contact time is taken into consideration. This feature is important for a resilient
rather than a rigid impact damper. The results clearly show that the impact damper can reduce the
undesired vibration of primary mass in free vibration. Finally, the effect of impact damper for
suppressing vibration of a robot tip is investigated. Using impact damper causes the robot works
better than before. So, it is concluded that single unit impact dampers can suppress undesired
vibration of forced vibratory systems.
REFERENCES
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Journal of Mechanical Sciences, 2003. 45(4): p. 589-604.
[3] Cheng, J. and H. Xu, Inner mass impact damper for attenuating structure vibration. International
Journal of Solids and Structures, 2006. 43(17): p.5355-5369.
[4] He, J.H., Iteration perturbation method for strongly nonlinear oscillations. Vibration Control, 2001.
7(5): p. 631.
[5] He, J.H., Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of
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[8] Duncan, M.R., C.R. Wassgren, and C.M. Krousgrill, The damping performance of a single particle
impact damper. Journal of Sound and Vibration, 2005. 286(1–2): p. 123-144.
[9] Blazejczyk-Okolewska, B., Analysis of an impact damper of vibrations. Chaos, Solitons & Fractals,
2001. 12(11): p. 1983-1988.
[10] Oledzki, A.A., Siwicki, I., Wisniewski, J., Impact dampers in application for tube, rod and rope
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