Received 4 November 2013Received in revised form 18 April 2015Accepted 19 May 2015Available online xxxx

This work studies the features of vibrational motion of an orthogonal mechanism with distur-

Keywords:

n leads to a signicant

e their parameters ac-amics of the vibrating

Mechanism and Machine Theory 92 (2015) 153170

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Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmtmachines and the technological processes performed by these machines.Thework in [1] presents a study of vibration transportation in amaterial part, and in summary, this research presents the vibration

transportation through solid substances and the behavior of granular materials and continuous medium undergoing vibration.processes may be constructed with only the use of vibration, yet in other processes, the application of vibratiointensication of the processes and increases the quality rate.

The structural schemes of vibration machines, as a rule, are not complex; however, one needs to determincurately for a successful application. These parameters can only be determined based on researching the dynIn recent years, the vibrating equipment in the mechanical engineering industry has begun to be constructed on the basis ofleverage mechanisms. These mechanisms possess unique abilities to create oscillations of the executive element. The developmentof vibration mechanisms that are based on mathematical modeling results in acceptable and practical results.

Vibration machines and their technological processes are used almost all industrial elds. In one case, a specic technological1. Introduction Corresponding author. Tel.: +7 7272682781.E-mail addresses: kuat_06@mail.ru ( Bissembayev)

http://dx.doi.org/10.1016/j.mechmachtheory.2015.05.010094-114X/ 2015 Elsevier Ltd. All rights reserved.bances, such as restricted power in the presence of a xed load on the horizontal link. Dynamicand mathematical models were prepared, and the operating conditions elds of existence forthe vibrationmechanism in terms of the driving powerwere dened.With stable rotational oper-ating conditions of the mechanism, it was shown that the value of the angular rate of the drivinglink varies about its average value according to the harmonic law. The frequency of the change inthe value of the angular rate equals the average value of this rate, and the amplitude is inverselyproportional to this value. Therefore, the average value of the angular rate depends on the featuresof the driving link and the sources of power. The rotationalmotions of themechanism are demon-strated to be stable. The librational motions of the mechanism were examined. An amplitude-response curve was built, and the conditions that contribute to the amplitude of driving linksoscillations were dened. The law of variation of the amplitude of the librational motion wasestablished. The frequency of the oscillations were shown to depend on the amplitude and theparameters of the power and mechanism source.Because the mathematical model provided good practical results, the results of this research canbe successfully used during the design of vibration equipment with orthogonal mechanisms.

2015 Elsevier Ltd. All rights reserved.Shaking tableOrthogonal mechanismRotating motionLibrational oscillationStabilityOscillations of the orthogonal mechanism with a non-idealsource of energy in the presence of a load on the operating link

. Bissembayev , Zh. IskakovThe Institute of Mechanics and Machine Science named after the Academician U.A. Dzholdasbekov, the laboratory of vibratory mechanisms and machinery,050010 Almaty, Kazakhstan

a r t i c l e i n f o a b s t r a c t

Article history:, iskakov53@mail.ru (Z. Iskakov).

1

The article in [2] examines the horizontal motion of a part in two directions perpendicular to the excited plane while controllingthe dry friction coefcient between the plane and the part. The dependences of the transition of the part from its initial position to thecenter of the stablemotion pathwere dened. Additionally, the dependences of the directional angle of themotion path from themo-ment when the friction was reduced relative to the excitation signal and the duration of time of the decreased friction were dened.The work in [3] cites research that features the vibration transition of a body on a swinging plane, which is applicable to the eld ofautomated part collection. Dynamic and mathematical models of vibration transition were constructed. The operating conditions ofthe body were dened as being dependent on the excitation frequency and the oscillation of the plane, the rolling angle of theplane and the factor of rigidity.

The authors of articles [4] and [5] examine the tasks of the optimal and dynamic synthesis of a swivel-lever guidance mechanismand counterbalancing, and the solution to these tasks helps determine the directional impact of the lever vibrator on a foundation.Numerical interpretation is used to determine the combined target function that contains all conditions that achieve completedynamic balancing.

Balancing of the principal moment of inertia forces based on the mean-square approximation was used in the work in [6], wherethe counterweights were located on links associated with a rack.

Thework in [7] presents a newmethod of determining the four links,which, in turn, allow for one to satisfy all kinematic demands

such as harmonic, polyharmonic, rectilinear, two-component and spatial oscillations, are used in practice. These modes may be

154 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170implemented with the help of leverage mechanisms, which have a wide range of functional abilities.One of the vibrating pieces of equipment, a shaking table with at leverage mechanisms, can be successfully utilized in the

construction industry to compact concrete mixtures, in the chemical, pharmaceutical and food industries to apply vibrating impacton pulps and suspensions, in the mining industry for screening fractions depending on the volume and weight, and for many otherpurposes. One of the problems with the mathematical modeling during the development of vibrating equipment based on leveragemechanisms is the variability of the vibrational characteristics. An apparatus of generalized functions is used in the works in [8]and [8] to determine the solution of equations describing the machine assembly dynamics.

K. Bissembayev and Zh. Iskakov [10] have studied the oscillation of the automatic press shaking table that uses at leveragemechanisms. A mathematical model of the automatic press shaking table that uses an orthogonal mechanism was also developed.

The book in [11] reviews linear and non-linear systems, and it is hypothesized that these systems are inuenced and excited bysources of energy that have a limited capacity.

Further development and the presentation of the characteristics of oscillating systems that interact with the energy source wereobtained in the work in [12]. This work cites the dynamic analysis of a self-oscillating system interacting with the energy sourcesin the presence of non-linear elastic constraints as well as periodic parameter impacts and delays. Non-linear forced and parameteroscillations of the systems interacting with two energy sources are examined.

Article [13] examines stationarymotions of a non-symmetrical spin (unbalanced rotor), whichhas a xed point and is impacted bythe moment of elasticity and the torque. The impact of the dissipation created by the engine with restricted power on the stability ofthe conservative system is studied.

The purpose of this work is to study the dynamics of the orthogonal mechanism of a shaking table with a non-ideal energy sourcein the presence of a xed load (account of load) on the operating link.

2. Kinematic relations

The calculation model for the orthogonal mechanism is shown in Fig. 1. The start of the OXY coordinate system is placed at thecrank rotation axis. Here, we designate the coordinates of the articulated link C (see Fig. 1) through X and Y, and the ranges of the

Fig. 1. A schematic diagram of the orthogonal mechanism of the shaking table.and ensure dynamic balance. The predicted characteristics of the driving (working) link motion and the conditions of dynamicbalance are ensured by means of prescribing the driving link speed change, the disk counterweights and the sizes of the links.

The effectiveness of all the above-mentioned methods is primarily determined by the choice of the model and the oscillatingconditions for the working body of the corresponding machine because the most diverse and complicated types of oscillations,

relatio

whereTh

FroWe

from w

3. Mec

Th[10]. T

155 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170A0 A312cos3 sin2 cos sin cos fMl2 sin 2 P cos kMD cos sin

MDfMglkMD7hanism motion equation

emotion equation of the orthogonalmechanism in the presence of a xed loadon the horizontal link (link 4) is obtained inworkhe motion is as follows:

1 l22 l21X Xme Y Yme

5

hich

l cosl21

l21l22 l

2

l22

s

1

l21l22 l

2

l22cos2

s24

35

l sinl1

l2s

1

l2cos2

s24

35

6coordinate system.1 XmaxXme XmeXmin l2 YmaxYme YmeYmin l 4

m these formulas, it is shown that the ranges of the horizontal and the vertical oscillations coincide.transfer the start of the OXY coordinate system to the point (Xme, Yme), i.e., transform the OXY coordinate system into the OThe ranges of the horizontal and the vertical oscillations of the articulated link of the orthogonal mechanism relative to theaverage values of the coordinates are determined by the following formulas:Xme 2 l2 1 l22l22

Yme Ymax Ymin

2 l1

3Average values of the coordinates X and Y are

Xmax Xmin

l21 l2

sYmin l l1l22 l22

Ymax l l1l, l1, l2 and l3 are the lengths of links 1, 2, 3 and 4, respectively, and is the angle of the crank rotation axis (see Fig. 1).e extreme values of the coordinates x and y are

Xmax X 0 l l21

l21l22 l

2

l22

s

Xmin X l l21

l21 l2

s2X l cos l2 1l1l22 ll22cos2

Y l sin l11

l2

l21cos2

s 1ntal and the vertical oscillations of the orthogonalmechanism are designated as a1 and a2, respectively. The following kinematicns [9] can be recorded from the equations of the closeness of the vector contours as projections onto the coordinate axes.

2 2

s

horizo

where

A0 M 1m3M

l2 J;A3 2Ml2

ll1

P M 1m1M

m2M

m2M

h igl

whereM is the loadweight,m,m1,m2 andm3 are theweights of links 1, 2, 3 and 4, respectively, J is themoment of inertia of link 1, f isthe coefcient of sliding friction, k is the coefcient of rotational friction, andMD is the moment of the propulsive forces.

Under the condition that

mM1;

m1M

1;m2M

1;ll11;

l1l21; f 0; k 0; 8

the eq

this soin the

system

156 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170Fig. 2. The phase portrait of the system (12).Eq. (11) turns out to be nonlinear as the nonlinear member enters the equation without the small parameter. bMl20

;B aMl220wheres equation of motion (9) to obtain a dimensionless form using the formula d= 0dt

cos B 11the oscillation system in the form of a functionMD ; , where is the coordinate of the source of the energys motion. The drivemoment on the shaft of some engines, for example, an engine with direct current and parallel excitation, is determined accordingto the following formula:

MD ab 10

where a and b are the constant coefcients depending on the parameters of the engine. Let us put (10) into (9) and transform theurcesmode of operation depends on themotion of the oscillation system. The task of dening themoment of propulsive forcesform of an explicit function of time becomes impossible.We are forced to show the inuence of the non-ideal energy source on

where g is the gravitational acceleration.The non-ideal energy source cannot ensure the previously prescribed law of changes of the moment of propulsive forces because20 glwhereuation of motion of the system is transformed as follows:

20 cos MDMl2

9

4. Con

4.1. Au

0;into thfrom t

Th

If motioTheref

If t

157 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531702

the difference

h0 sin0 122this case and under the condition that

1 20 1 sin0;Inh0 sin0 12

0T 2

p ZAA

dsin A sin

p

It follows that the orthogonal mechanism considered is not an isochronal system. Going back to the phase portrait of the system,let us consider the casewhen h0 N 1; this situationmay occur if the initial moment conveys themechanism along a deviation 0 and asufciently large initial speed

0, i.e., if1

he point is below = A 0, then h= sin , and we obtain the following equation for the period of the oscillationsd

from (13), from which we obtain

Z2

d2 h sin

pmechanism. We have

d

2 h sin

qenergy. The points 0; n1 represent special types of saddles and correspond to the upper (non-stable) position of thebalance of the mechanism, i.e., to the maximum of the potential energy of the system. Now, let us determine the period of librationof theh0 sin02 b 0 b 0 and, consequently, | sin 0| b 1 and |h0| b 1, then the mechanism produces periodic oscillations (librational

ns), which are shown in Fig. 2 as having a closed elliptic path functions that surround special points that are a type of center.ore, themechanism uctuates around the stable lower position of balance, which corresponds to theminimum of the potential

You can see that the closed curves are circles only near the singular points, which are a type of center with coordinates 22n; 32 2n (where n is a whole number). At distances further from these points, the deeper curves are drawne formed ellipses. In the phase plane, this corresponds to a graphic reproduction of the oscillation process, which differshe harmonic law.e initial stock of the total energy is determined from the initial shut-off of the mechanism and is equal towhere h is the energy constant. The potential energy has isolated minimum values at points 22n; 32 2n and maximumvalues at points 32 2n; 2 2n. The phase portrait of the system is modeled using the numeric method and is shown inFig. 2.. (12) has the rst integral

2 sin h 13Eqmotions described by the equation

cos 0 12Let us construct a phase portrait of the orthogonal mechanism to demonstrate of the high-quality research of librational and rota-tionalservative systems

tonomous conservative systems

will bewould

(see Fi

varyin

Eq.The

Un

158 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170By integrating Eq. (16) under the initial condition

0 : 0;

we will obtain

0 e 17

The average rotational frequency of the mechanism approaches as , i.e.,

lim

llation. Using expression (15), let us transform Eq. (14) to appear as the following:

dd

1

16Therefore, it follows that the angular speed of the engine shaft equals the average value of the orthogonalmechanisms frequencyof osci B

or a`b0

15We determine the average value of the mechanisms frequency of oscillation in the stationary mode from the following:der this condition, the equation of a stationary mode in the motion is the following

B 0d 0(14) may be studied to determine the variable while in transitional modes.condition of existence of stationary modes is

dtain the following after integration:

dd

B

14we obZ20

cosd 0; 12

Z20

d 1;While the average value is considered constant and keeping in mind thatdd

2

Z20

d1

2

Z20

cosd B2

Z20

dthe angle variable and using the formula d d, let us transform Eq. (11) into the following:(11) may be subject to further simplication if we consider that the value d changes very little over one period wheng the angle from 0 to 2. The derivative of the turn angle may be considered equal to its average value. By taking as5. Dissipative systems

Eq. dreas of librational (closed paths) and rotational (escaping paths) motions of the mechanism in the phase plane, are separatistsg. 2, shown on line 3).functions (see Fig. 2, line 2), which correspond to this case, will not cross the O axis. The curves, which go through saddles and sep-arate agreater than zero. The spinning (rotational) motions of themechanism occurring during conveyance of the initial impulse thatensure transition through the upper position with a speed not equal to zero corresponds to this case. The escaping phase

Th

Letto larg

Set

Let

whererotatioally, th

Let

1

159 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531701 20A x 1Z2

cos zdzz zu2z

1 sinzBxu1x

A1 x u1z

B1 x A2 x 2z

u1u1x

A1 x 1z

B1 x B2 x

26

Let us consider the rst equations in the system (26). For the function u to be limited, it is necessary and sufcient thatu1 coszA1 x ;1 xB1 x The equations used to determine the functions ui, i, Ai and Bi will be the following:us assume that the functions ui and i restricted. Further, to satisfy the conditions in Eq. (22), let us assume

ui x;0 i x;0 0 25z 1 B1 x 2B2 x the functions x and z satisfy the following equations:

x A1 x 2A2 x 240

and following the work in [15,16], let us assume that

x x u1 x; z 2u2 x; z z z 1 x; z 22 x; z

23

where 0 : x x ; z 0 22separation of motions [14] may be used to study the system (21).Let us agree to consider the following Cauchy problem for the system (21).Underx0 dxds is accepted as the designation. The result is a system of equations with a slowly changing variable, an angular speed ofn x and a fast changing variable z. The later variable, z, may be interpreted as the angle of rotation of the engine rotor. Addition-e right-hand portions of the equations are periodic functions of the fast variable z. Therefore, asymptotic methods for thez 1 xus assume that = s, where 1. Then, the system of Eq. (20) result in the following:

x0 cos z2Bx0 21z xansform Eq. (11) to appear as the following:

x cosz; 20will trillations of the orthogonal mechanism subject to rotational motion from the engine rotor

us now investigate themotions of system (11), which are shown as unclosed phase paths (see Fig. 2). These paths corresponde excitation energy values.ting

z; z x 196. Osce integral of Eq. (14) will become the following:

01

0 ln0

18

The

Rep

and, co

Usi

and, co

Fin

160 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531702

0 x; z Zz

sin zdz cos z1:and, consequently, we will obtainB2 x 12

Z20

sin zdz 0From which2z

sinzB2 x ally, let us rewrite the last equations in the system (26) by taking into account the results obtained aboveu2 x; z x0

cos zdz x sinz 32nsequently,

ZzA2 x Bx 12

Z20

dz x 12

Z20

cos zdz Bx 31from whichng the results from Eqs. (28), (29) and (30), let us rewrite the third equation in the system (26)

u2z

BxA2 x u1z

xB1 x 29

nsequently,

1 0 30u1 z sin z 28

eating the same reasoning for the second equations in the systemwhere C x is an arbitrary function of x. However, the condition in Eq. (25) gives us C x 0. Then, we can writeu1 x; z Zz0

cos zdz C x andrefore,

A1 0 27

Continuing this procedurewe can easily determineA3, etc. Restricting our calculations to only terms of the orderO(2), let usmakeEqs. (23) and (24) appear as the following:

x x sin z 2x sin zz z 2 cos z1 x0 2Bxz0 1 x

33

By integrating the third and the fourth equations in (33) and by taking into account the initial conditions in Eq. (22), we will ob-tain

x x0e2Bs

x 2Bs 34

Let

Thvaluesline (con the4, desc

Th

Figthe av

161 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170Fig. 3. Graphs of the dependency of the rotation angle of the driving link of mechanism z from time . 1-2 lines are drawn subsequent to the analytical andcomputational results.+ x taking into account expression (36), and curve 3 is constructed based on the results of the numeric calculations.. 6 shows oscillograms of the horizontal (curve 1) and the vertical (curve 2) oscillations of the orthogonalmechanism relative toerage values of the X and Y coordinates. The dotted line (curve 3) shows the results of the numeric calculations.the transitional process of changing the average shaft rotation speed angle from time. Curve 2 is obtained based on the equation=

e graphic demonstration of the solutions to (36) is presented in Figs. 3, 4 and 5. The calculationsweremade using the followingfor the parameters: = 0.08,=2, x0 = 3. The dependences of z and x on time are shown in Figs. 3 and 4. The continuousurve 1) is constructed based on of the results of the analytical consideration, and a dotted line (curve 2) is constructed basedresults of the numeric calculations of the system of equations given in (20). Similar curves, which are shown in Figs. 3 andribe the similarities between the results of the analytical and numerical solutions.e engine shaft angular speed versus time curves are shown in Fig. 5. Curve 1 is constructed based on Eq. (17) and describesz x0 1e

B

us rewrite Eq. (35) by returning to the variable s and by using 1

x x0e 112

sinz

1sin z

z x0

1e 12 cos z1 36z s 0B

1e

Based on the expression (34), the remaining equations of the system in (33) will appear as the following:

x x0e2Bs 1 2 sinz

sin z

z s x0B

1e2Bs

2 cos z1

z s x0 1e2Bs

35

7. Stab

No

where0; z=

Usi

Eq.

Fig. 4. Graphs of the dependency of the deviation of the angular speed of the driving link of mechanism x from its mean value from time . Lines 1-2 were drawn sub-sequent to the analytical and computational results.

162 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170and by excluding the additive component from the rst derivative from Eq. (39), we will obtain

k sin20 0 41Fig. 5. Gangularp e12 40p p sin p 0 39

By intruding a new variable , which is determined usingqq sin pp q 38

(38) may be presented in the formz

ng Eq. (37) in (20) and after linearization relative to the variables q and p we will obtainx 1 sinx and z are the stationary solutions of Eq. (20) to the second order of smallness, which with initial conditions of =0 : x0 =0, becomes the following:x x qz z p 37stability is determined by Eq. (36). For this purpose, let us prepare a variation of the system of equations given in (20). Let us assumethatility of the rotational motion of the orthogonal mechanism of the engine shaft

w, let us consider the stability of the stationarymodes of the engine shafts rotationalmotion in the orthogonalmechanism. Thisraphs of the transitional process of the rotation of the mechanisms driving link. Curves 1 and 2 describe the dependency of the mean and the instantaneousspeed versus time .

Ac

coefc

Th

Eq

Forsolutionism a

163 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531708. Osc

Th

Fro

If sditiona stable solution, the characteristic exponent may only adopt purely imaginary values. In this case, we have 2 b 0 for such an. Radices of Eq. (44) have negative real parts, i.e., stationary rotational motions of the engine shaft in the orthogonal mecha-re stable.4 2 4

2 1 2 2

1

1 2 2 11 2 2

s440 0 4

Let us present the solution of a biquadratic equation in the following form 0 k 40 4 43

. (43) is transformed to the following

4 2 2 k

2 2 k 2

1 02 2 2 2 2 1 0 k sin20 cos 2 cos

erefore,22 2

1 is a new parameter subject to determination, and is a characteristic exponent. Using Eq. (42) in Eq. (41) and setting theients equal using functions sin 0 and cos 0, which are equal to zero, we will obtain

202k

cos 20 sin

1sin ;wherecording to the Floquet theory, the specic solution of Eq. (41) becomes the following

e sin 0 42where

k 142;0

12

Fig. 6. Oscillograph charts of the horizontal (curve 1) and the vertical (curve 2) oscillations of the orthogonal mechanism.illations of the orthogonal mechanism under librational motion of the engine shaft

e static balance of the system in (20) is determined using the following equation:

cos zc B 45

mwhich

zC arccosB 2nn 0;1;2 46

ubject to the initial conditions =0 : z=0, =0 B N 1, the engine shaft of the mechanism has rotational motion and the con-B b 1 corresponds to librational motion.

Let

LetFor thi

Thi

Ch

and i

164 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531701;1;1;1;2;;1;1 1;11;12; :::::::::::;1;11;21;22; :::::::::::;2;1::::::::::::::::::::::::::::::::::::

0 ::::::::::::::::::::::::::::::::1;1;1;2;;1;

;11 ::::::::::::::::::::::::::

1;2;3;;

;;1 ::::::::::::::::::::::::::::

1;1;2;;;1

;

1;1;2;; 1;2;; 1;2;; 11;12; ::::::::::;121;22; ::::::::::;2

1;22;23;;21;32;33;;3

1;21;22;;2;11;31;32;;3;1

::::::::::::::::::::::::: ;01 :::::::::::::::::::::: ;02 :::::::::::::::::::::: ;mi mi

;

1;11;12;;11;21;22;;2

21;22;;231;32;;3

11;12;;131;32;;3

whereDm Xi1

mii a0;A1;A2;;A 52arbitrary collocation points within a period. When requiring this system to be relative to the coefcients D0 and Dk [20], we willobtainki i

is the value of the argument at the collocation point i. The determinant of system (51) is not equal to zero for the selectedwhere

coskD0 11D1 12D2 1D cos a0 Xk1

1kAk 1 a0;A1;A2;;A

D0 21D1 22D2 2D cos a0 Xk1

2kAk

! 2 a0;A1;A2;;A

::

D0 1D1 2D2 D cos a0 Xk1

kAk

! a0;A1;A2;;A

51equations

!oosing the required number of collocation points in the interval 0 kp 2, we will obtain the system of algebraicwhere

A ]

cos a0 Xk1

Ak cosk

! D0

Xk1

Dk cosk 50which depends on two integrals of motion: Ak and . Let us present the nonlinear portion of Eq. (48) in the form given in[1719us write the system of equations in (20) in the following form:

z z cos z B 47

us consider the librational motion of the system in (47). It is desirable to transform Eq. (47) to the standard form given in [13].s purpose, let us consider the conservative system, (generated) described by Eq. (47) with = 0

z cos z B 48

s system the has the general solution

z a0 Xk1

Ak cosk; 49

Nowe wi

Eqexpres

It irst eqcharac

By

where

where

165 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 1531702 2 2

211212

221222

231232 21101

22102 231031a 1 0 arccosB

1A21

!

1 sin arccos

B1A21

!" #vuutA 1 2

4A21ctg arccos

B1A2

!" #5618 9 18 9 2 2

1;11;121;21;221;31;32

;01

11;1221;22

;02 11;1231;32

;03 11;1221;22

;11 1;221;32

;12 1;121;32

13 1;121;22

;21 1;211;31

;22 1;111;31

;23 1;111;21

Let us assume for the rst approximation that a0 0; A1 0; A2= 0. By using these values in place of the cosine functions for threemembers of the Taylors series expansion, let us present the system of equations in (55) in the following formA2 1

4221

cos a0 11A1 12A2 22

cos a0 21A1 22A2 23

cos a0 31A1 32A2

11 cos7

;12 cos7

;21 cos8

;22 cos8

;31 cos

;32 cos2 2 1

A111

cos a0 11A1 12A2 12

cos a0 21A1 22A2 13

cos a0 31A1 32A2 55. (54) in the form obtained is convenient for iteration. Solutions obtained using the iteration method coincide, in most cases, assions for k ranges of Ak harmonics and are inversely proportional to the k22 multiplier.s convenient to start the iterationmethod setting a0 0; A1=A2==A=0. Let us express a0 throughA1with thehelp of theuation in (54) and, subsequently, use this result in the second equation in (54) to obtain the expression for the range-frequencyteristics. The remaining equations specify the form of the system oscillation represented by an abridged trigonometric series.using only three members in Eq. (49), let us present the system of equations in (54) in the following form:

01 cos a0 11A1 12A2 02 cos a0 21A1 22A2

03 cos a0 31A1 32A2 BTherefore, the rst equation determines the permanent member of solution of the Eq. (49), the second equation expresses thevalue of the frequency of the nonlinear system through ranges of the solution harmonics, and the following equation determinesthe ranges of the high harmonics.D0 a0;A1;A2;;A B2 D1 a0;A1;A2;;A

A1

Ak Dk a0;A1;A2;;A

k22k 2;3;;

54Let us present these equations in the following formw, after using expressions (49) and (50) in Eq. (47) and requiring the coefcients to be equal under similar harmonics (cos k),ll obtain + 1 equations

k22 Ak D0 a0;A1;A2;;Ak Dk a0;A1;A2;;Ak Bk 1;2;;

53

Through the properties of the reverse trigonometric function, let us transform expression (56) to appear as the following

a 1 0 arccosB

1A21

!; a 1 0 2

1 1 B1A21

!2" #14

A 1 2 4A21

B1A21 2B2q

57

Let us assume that a0 0; A1 0; A2 0 for the second approximation.Wewill obtain the second iteration step in (55) by using therst approximation A(1) in (55), which is determined using the function in (56) and repeating the reasoning for the system of equa-

Figimatio

The

If tNo

166 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170Fig. 7.Amplitude and frequency characteristics of the oscillationmovements of the driving link. Curves 1 and 2were drawn based on the second and the rst approach.Curve 3 is derived from the application of the subsequent computational method.this assumption, we introduce A1 and = (A1) as new variables.expression in (57) shows that the range of the libration motion is limited by the motion

A1

1B

s

his condition is violated, the librational motion transforms into a rotational motion.w, let us consider the solution of Eq. (47). Let us assume that the values A1 andwill change slowly in the disturbedmotion. ForThe proximity of curves 1 and 2 provides information about the rapid convergence of the iteration processes. 2 1 B

1A21 141211

A1B

1A21 2B2p

" #>>>>>>:

>>>=>>>>;

uuuuut 58b

A 2 2 4A21

B1A21 1

41211

A1B

1A21 2B2p

" #( )2B2

vuuta 2 0 2

58c

. 7 shows the constructed oscillation frequency- range curve. In this gure, curve 1 is constructed based on the second approx-n in (57), and curve 2 represents the rst approximation. Curve 3 is constructed based on the results of the numeric method.4 11 1A212B2: ;

8> 9>2vuua 2 0 arccosB

1A21 1 12 A1

B q264

375

>>>>>>>>>>>>>

>>>>>>=>>>>>>>>

58a2

tions in (55), and we will obtain

8>> 9>>

By considering the presentation of the general solution of the conservative system as a formula to replace the variables, we canwrite the following [15]:

z X A1; ; z A1 X A1;

X A1; 2 X A1; 59

The subscript index indicates differentiationwith respect to this parameter. The replacement in (59) is reversible, and if you candetermine A1() and () in the disturbed system in (47), then z() and () will be completely determined and vice versa. Replace-ment of Eq. (59) while taking into account Eq. (48) leads to an identical equation

2 A1 X cos X B 60

It is necessary to require that

for com

Sol

Dubut th

where

167 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170Fig. 8. Oscillograph charts of the librational movement of the driving link of the mechanism. Lines 1 and 2 present the dependency of offset and amplitude of thedriving link of the mechanism on time .dA1X A1; A1 sinConsider replacing the variables in (59) with the following form

X A1; a0 A1 cos;XA1 da0 cosuation is legitimate:

E 0; EA1 W 62

E is the total energy of the oscillatory motion of the system.It is interesting to note that the determinantW does not depend on . This fact was established in thework in [16], and the follow-ing eqe to the possibility of reversing the replacement, the obtained equations in (59) are equivalent to the initial motion equation,ey may be directly studied using the principle of averaging. A1 W

XXA1ving this equation and taking into account the identical equation in (60) we will nd that

A

1 W

X2 61The two last functions represent a system of linear equations with respect to A

1 and

with the main determinant being

W XA1XXA1

X200 ddA1

A1 Let us use Eq. (59) in Eq. (47) and determine that

X0 XA1

A

1 X cos X BXXA1A

1 X X

patibility of this replacement.

Let us average the right-hand side of Eq. (61) in terms of over the period 0 2 and obtain the a system of the rstapproximation

A

1 A212EA1

1B

1A21

!2" #12

A1

63

Taking into account that EA1 2A1 from (63) we will nd that

A1 A10e12

In Fig. 8, line 1 presents an oscillogramof the librational oscillation, and line 2 presents the progress of the change in the oscillationamplit

Wh

focus 2

Let

168 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170Fig. 9. Graphs of the dependency of the bias angle of the mechanisms driving link on time for various moments of excitement.(47). Let us assume that

z zC us study the stability of the trivial solutions of the system in (47). Let us prepare an equation that is a variation for the system in9. Stability of the librational motions of the engine shaft of the orthogonal mechanismline represents a separatist, and the line represents the path of the motion in the direction from the saddle point 1 to the.i.e., the stationary solutions will be asymptotically stable when and 0.Figs. 9 and 10 show and versus curves for different excitation moments B for the following parameter values:

= 0.1, B = 0.3, B= 0.5, B= 0.8.Areas of initial conditions that correspond to trivial solutions of Eq. (47) are shown in Fig. 11 for parameter values =0.4, B= 0.22.The 0 1

2 0

1B2

pFrom which we will nd that

2

2

1B2

pequations in (63) result in the following:

A1 0

The phase of the stationary process becomes

1B2 1

4 C1

where C1 is the constant of integration.An equation describing the variations is composed to study the stability of the stationary mode A1 . Given that the value A1 has a

small initial increment A1 = (0) and given Eq. (63), we obtain a small deviation () from the stationary value A1 = 0 for 0ude in time.ile studying the stationary modes, previous researchers assumed that A

1 0. If this is the case, the stationary solutions of the

Using this value in (47) and after linearization relative to variations , we will obtain

powerof the

Wiabout

motioTh

were ecoordi

Fig. 10. Graphs of the dependency of the frequency of mechanisms driving link ism on time for various moments of ignition.

169 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170ude of the links librational motion and the dependence of the oscillation frequency on the amplitude of the source parametersstablished. The stationary mode librational motions of the orthogonal mechanisms are stable in quadrants III and IV of thenate system.tion was built, and conditions were dened that apply to the amplitude of the driving links oscillations. The law of variation of theamplite average value of the angular rate depends on the length of the driving link and on features of the power source. The rotationalns of the orthogonal mechanism are demonstrated to be stable over the entire timeframe.e librational motions of the orthogonal mechanism were examined. An amplitude-response curve of the mechanisms oscilla-average value of this rate, and the amplitude is inversely proportional to this value.Thin the presence of a xed load on the horizontal link. Dynamic and mathematic models were built, and the elds of existenceoperating conditions for the mechanism in the terms of the driving power were dened.th stable rotational motion of the mechanism, it was determined that the value of the angular rate of the driving link changesits average value according to the harmonic law. Furthermore, the frequency of the change in the angular rate equals the sinzC 0 64

Let us prepare a characteristic equation

2 sinzC 0

This equations radices are equal to

1=2 2

2

4 sinzC

s

Depending on the values of the functions included in the expression under the radical, the system may be either stable or non-stable. If sinzCj jN 24 , i.e., the value zC is found in quadrant III or IV, then 1/2 will have complex conjugate values with negativereal parts and the trivial solutions of the system in (47) are stable. If sinzCj jb 24 or sin zC N 0, i.e., the value zC is found in quadrantI or II, then 1/2 will be real numbers, and one of the valueswill be positive. This result indicates that the trivial solutions of the systemin quadrant I or II are not stable.

10. Conclusion

This work has examined the features of vibrational motion of an orthogonal mechanism with disturbances, such as restrictedFig. 11. Fields of the initial conditions causing the orthogonal mechanism to equipoise.

The results obtained during the theoretic research can be successfully used to design vibration equipment with orthogonalmechanisms.

The method research can be used both for orthogonal vibration mechanisms with a load on the operating link that changes overtime and in space as well as in other conditions in which load is applied.

This research was nanced by the Ministry of Education and Science of the Republic of Kazakhstan with the grant won by theauthors at the competition.

References

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(37) (2001) 10511071.[8] Yu.. Drakunov, .. Tuleshov, Dynamics of a vibrating table in generalized functions, Proceedings of the international scientific technical conference

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international scientific conference Modern achievements in the science and education, 2012.[10] . Bissembayev, Zh. Iskakov, Mathematic model of the orthogonal mechanism of the press machine vibrating table, Bulletin of The Kazakh National Teachers

Training University named after Abai, 3 (39)2012. 3238.[11] V.O. Kononenko, Oscillation systems with restricted excitation, Nauka, Moscow, 1964.[12] A.A. Alifov, K.V. Frolov, Interaction of non-linear oscillation systems with sources of energy, Nauka, Moscow, 1985.[13] A.M. Krivtsov, Influence of the torque of restricted power on the stability of stationary motions of non-symmetrical spin, Mech. Solid Body 2 (2000) 3343.[14] V.I. Gulyayev, V.. Bazhenov, S.L. Popov, Applied tasks of the theory of non-linear oscillations of mechanical systems, Higher School, Moscow, 1989.[15] N.N. Bogolyubov, Yu.. Mitrpolskiy, Asymptotic methods of the non-linear oscillations theory, Science, Moscow, 1974.[16] N.N. Moyseyev, Asymptotic methods of non-linear mechanics, Science, Moscow, 1969.[17] C. Hayashi, Nonlinear Oscillations in Physical Systems, Chapters 1, 3 6, McGraw Hill, 1964.[18] . Tondl, Automatic oscillations of mechanical systems, World, Moscow, 1979.[19] . Tondl, Nonlinear oscillations of mechanical systems, World, Moscow, 1973.

170 Bissembayev, Z. Iskakov / Mechanism and Machine Theory 92 (2015) 153170[20] . Bissembayev, V.. Pyatetskiy, Research of non-linear oscillations of body on linear guides with aligned surfaces, Bulletin of the Kiev UniversityThe series ofphysics and mathematics, No. 51992.

Oscillations of the orthogonal mechanism with a non-ideal source of energy in the presence of a load on the operating link1. Introduction2. Kinematic relations3. Mechanism motion equation4. Conservative systems4.1. Autonomous conservative systems

5. Dissipative systems6. Oscillations of the orthogonal mechanism subject to rotational motion from the engine rotor7. Stability of the rotational motion of the orthogonal mechanism of the engine shaft8. Oscillations of the orthogonal mechanism under librational motion of the engine shaft9. Stability of the librational motions of the engine shaft of the orthogonal mechanism10. ConclusionReferences