CC BY
0
References
1. K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974.
2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
4. R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2000.
Abstract
In this paper the dynamic response of Zener oscillator with third order derivative is researched by the Newmark method. First, the approximately analytical solution is obtained. Then, we calculate linear vibration of Zener viscoelastic system.
1. Introduction
The Newmark scheme, originally introduced by Newmark [1959], is a classical time-stepping algorithm popular in structural mechanics codes. It has been modified and improved by many other researchers such as Wilson, Hilber, Hughes and Taylor... However, these methods are only used for the system of second order equations.
Many vibration problems in engineering lead the system of differential equations of third order. In this paper we calculate vibration of Zener system by using established Newmark integration method for calculating vibration of third order system.
2. The Newmark method for the third order systems
The Newmark method is a single-step integration formula. The state vector of the system at a time = ^n + h is deduced from the already-known state vector at time ^ through a Taylor expansion of the displacements, velocities and accelerations.
We get the approximation formulas of displacements, velocities and accelerations of system at time ? ,, to approach solving the system of third order differential equations.
©Bui Thi Thuy, Tran Thi Tram, 2024
Bui Thi Thuy
Doctor at Hanoi University of Mining and Geology,
Vietnam
LINEAR VIBRATION OF ZENER SYSTEM USING NEWMARK METHOD
Keywords:
viscoelastic, third order, dynamic, Newmark, Zener.
(1)
9n+l = 9n+h9n
qn+i =9n+h9n
a A
h2 ..
—ll
2
h q'n+rh qn+1,
f 1^3 3
7 -P h39„+ph39n+v
v6 J
(2) (3)
Let us then assume that the equations of dynamics
Mq +Bq + Cq + Kq = f(t), (4)
are linear, i.e., that matrices M, B, C and K are independent of q, and let us introduce the
numerical scheme (1), (2) and (3) in the equations of motion at time tn+l so as to compute qn+l
\_M + ahB + yh2C + /3h3K]qn+l = fn+l - B[_qn + (1 ■- a)hqn]
- C
h9„
l
-Y \h2q„
K
H2
(5)
By solving a system of linear equations (5) we obtain 9n+i. Then, by using Newmark formulas (1), (2) and (3) we get accelerations, velocities and displacements 9n+\,9n+\ and q +1. We determine the initial conditions of q (tQ ) from the given values of q(t$), </(/0 ) and q(tQ )
q(t0) = M-1[f(t0)-Bq(t0)-Cq(t0)-Kq(t0)]. (6)
Let us assume that the non-linear dynamic equations of third order systems have the following form
M(q)q+k{t, q, q, q) = f(t,q, q, q), (7)
We have (Jn+l from equation (3) and substitute into equations (1) and (2), we realize that 9n+i,9n+i,9n+i are represented by qn+1 and the known values of qn,qn,qn,qn By substituting 9n+u 9n+u 9n+1 'nto we obtain the system of non-linear algebraic equations with unknown qn+i. We have values of qn+l through the iteration method Newton. Then, we determine values of qn+l,qn+l and qn+l with the initial conditions of q (t0 ) derived from the equations of dynamics (7)
q0 = M~l(q0)[f(t0,q0, q0, q0)-k(t0, q0, q0, q0)]. (8)
3. Calculating linear vibrations of Zener system
According Newton's second law of motion, we have motion equation of system
mx(t) + p(t) = F(t) (9)
Where X(t) is displacement of mass m, F (t) is external force, p (t) la internal force in viscoelastic materials.
Property of Zener model: total displacement is sum of ingredient displacements (spring k1 and spring kj). Thus, we have
(10) (11)
X — Xj + X2
And
P — Pi — P2 (lx)+ P2 (n) Pi =KXl> Pl (lx) =k2X2-> P2(n) =CX2
Derivative of expression (10)
JC — JCy ~I-
(12) (13)
Fig. - 1a
Fig. - 1b
p, . Pl(n)
Equation (12) yeilds -. Substituting into (13), we obtain
k ' C
X =
Pi , p2(n) _P P Pl(lx) _P , P k2
= —I----- X,
kx c kx c kx c c From (10) and (14) we deduce relation between internal force and displacement
P h ,
c—h p + —p = k-,x + cx
kx kx We have force p from equation (9)
p = F{t) — mx
Substituting equation (16) into equation (15) we obtain the motion equation of Zener model
cmx
x + k^mx + klcx + klknx = cF + {kx +k^F
Assume at time t we have the motion differential equation of system
cmx
+ (к+кЛтх +kcx +kknx =cF +(k+kAF
\1 2 / n In 12 n n V 1 2 / /?
(14)
(15)
(16)
(17)
(18)
Applying formulas (1), (2) and (3), we obtain Xn
[tfj +aha2 +yh2a3 + fih3a4~jxn = fn~a2[xn_x + (\-a)hxn_x~\
-a
-a,,
h
1
xn_x + hxn_x + —xn_x xn_x
(19)
Where ax = cm,a2 = (kx + k2)m, a3 = kxc,a4 = kxk2,fn = (kx +k2)Fn +cFn. By substituting the above value of xn into equations (1-3), we get the approximation formulas of displacements, velocities and accelerations of system at time tn
x„ = Xn_x +(\-a)h xn_x + ah x„,
i 1 л
x„ = x„ , +hxr, , +
/7 /7—1 /7—1
■r
h2xn_x+yh2xn,
2 •••
(20)
V2 У
-2 /л \
jq ( 1
= x„-i +hx„_x + —xn_x + --p h" xn_x + ph' x„. 2 v 6
V6 У
With the initial conditions of x(0),i;(0),x(0), we determine values of 3c(0):
(0) = — \cF (0) + (kx + k2) F (0) - (kx + k2) m x (0) - kx cx - kx k2x (0)1. cm
4. Numerical example
We have tried out the algorithm on an example. We have chosen the initial conditions
m = 5(kg), c = 2(Ns/m), kx = 5(N/m), ^ = 7(^m), h = 0.0l(s),a = 0.5,
r
y = 0.25, p = 1/12, F = 0.5 sin
n
V
nt + — 6 y
(N),
x (0 )= 0 (m), x (0) = 1( mjs), x (0) = 0 (m/ s2) ^ x
The differential equation of motion has following form
lOx+60x + 10x + 35x = 6sin
( 0 ) = (m/s2).
20
i f
n n
nt + — + ncos nt + —
V 6 y V 6 y
The solution of Eq. (21) obtained by the Newmark method is represented in Fig.2.
(21)
40 60
t(s)
Fig. 2 - Time histories of the displacement in Eq. (21)
5. Conclusions
Using the Newmark integration scheme, a numerical algorithm is developed to calculate dynamic response of third order systems. The motion differential equation of Zener model is established and numerical solution is obtained. In the example, a good agreement is obtained by the Newmark method between second order system and third order system.
The single-step Newmark numerical integration algorithm presented here for Zener third order systems is effective and successful. According to this algorithm, a computer program is developed using MATLAB software. References
1. N.M. Newmark,"A Method of Computation for structural Dynamics", ASCE Journal of Engineering Mechanics Division, Vol. 85, pp 67 - 94, 1959.
2. Wilson, E.L., I. Farhoomand and K.J. Bathe, "Nonlinear Dynamic Analysis of Complex Structures", Earthquake Engineering and Structural Dynamics,1, 241-252, (1973).
3. Hughes, Thomas, "The Finite Element Method - Linear Static and Dynamic Finite Element Analysis", Prentice Hall, Inc., (1987)
4. M. Geradin, D. Rixen, Mechanical Vibrations, Wiley, Chichester 1994.
5. M. West, C. Kane, J.E. Marsden, and M. Ortiz, "Variational integrators, the Newmark scheme, and dissipative systems", International Conference on Differential Equations, Berlin, 1999.
© Thuy B.T., 2024
Tran Thi Tram
Master at University of Mining and Geology,
Hanoi,Vietnam
DEVELOPING AN OBJECT IDENTIFICATION SYSTEM USING CAMERA FOR THE SYSTEM CLASSIFICATION OF PRODUCTS USING ROBOT
Annotation
The report summarizes the results of research conducted in the field of object recognition and positioning using cameras for robotic product classification systems. Machine vision is a potential direction for a variety of applications in the design and manufacture of devices for intelligent measurement and control systems.
Keywords:
robot, Machine vision
1. Introduction
Machine vision is a rapidly growing field with many applications in measurement and control (Peter I. Corke, 1996; Ramesh Jain, Rangachar Kasturi, Brian G. Schunck, 1995). Because vision sensor technology has made great strides in size and resolution, many smart devices with vision have appeared in medical technology, national security and defense, and the arts. entertainment. Many authors are interested in research on industrial image processing, especially the study of algorithms for identifying and locating objects in real time. In this research direction, I have built an object identification and positioning system used for product classification, a system for developing industrial image processing systems connected to robot control systems for the purpose of developing industrial image processing systems:
- Research and develop image processing systems for measurement and control.
- Research and develop object identification and positioning systems used for product classification systems.
- Building a camera image recognition and positioning system for industrial and defense applications.
- Building robot control algorithms based on image feedback.
- Develop dynamic image processing software for measurement and control applications.
Images are received by the camera, processed, identify objects, determine location, convert coordinates, using Labview software tool. The results of image recognition and positioning are passed to the robot control software module to control Robix to classify products.
The process of implementing recognition algorithms has been handled on two software, Imaq Vision Builder and Labview. The recognition part from the Camera will calculate the coordinates (position and direction angle) of the object and transfer those coordinates to the software that controls the robot handle to pick up the object.
