Generalized study of crank-rocker mechanisms driven by a d.c. motor Part I. Mathematical model

Mechanism and Machine TheoryVol. 15. pp. 435-445 © Pergamon Press Ltd., 1980. Printed in Great Bdtain

0094.-114Xl8011201-04351502.0010

Generalized Study of Crank-Rocker Mechanisms Driven by a d.c. Motor Part I. Mathematical Model J. P. Sadlert R. W. Mayne$ and K. C. Fan§ Received 9 January 1980; received for publication 16 May 1980 Abstract A mathematical model is developed for analyzing a 4-bar linkage driven through a speed reducer by a separately-excited d.c. motor. A set of nonlinear differential equations, governing the dynamic response of this system, is derived and converted to dimensionless form. This dimensionless form is described by newly defined linkage and drive parameters. It is believed that these parameters may find convenient use in estimating the overall behavior of motor-mechanism systems from performance graphs and guidelines which can be established by numerical study of typical cases. The performance characteristics which are considered in this analysis are the steadystate speed fluctuation of the mechanism and its steady-state energy consumption as well as the time required for mechanism "start-up" and the energy consumption during start-up. Expressions are derived in a dimensionless form to represent these performance characteristics. In Part II of this paper, specific numerical examples are presented which relate the performance characteristics to the linkage and drive parameters for two different mechanisms. Introduction A MECHANISMis an arrangement of machine elements which converts motion of one type to that of another type, usually accompanied by the performance of mechanical work. The power to accomplish this physical effect generally comes from an external source called an actuator. An actuator is an energy conversion device whose output performs a characteristic motion compatible with the mechanical part to be driven. Often, designers treat the mechanism and the drive as two separate and distinct steps in the design process. Likewise, research efforts have often treated these components independently; for example, much has been done to improve the output of an electric motor under very specific drive conditions, while a number of studies devoted to improving dynamics of mechanisms have been based on restrictive input motion assumptions such as constant velocity. In fact, the performance of these two system components is interrelated. It is necessary to investigate the total system in order to find that arrangement of the design parameters for optimum conditions, such as minimum speed fluctuation, which can lead to the reduction of vibration, noise and stress, and to longer machine life. tDepartment of MechanicalEngineering,Universityof North Dakota,Grand Forks, North Dakota,U.S.A. *Departmentof MechanicalEngineering,State Universityof New Yorkat Buffalo,Amherst,New York,U.S.A. §Departmentof MechanicalEngineering,FengChia Collegeof Engineeringand Business,Taichung,Taiwan. 435

436 The work presented here deals with the mathematical modeling of a class of actuatormechanism systems, consisting of a 4-bar linkage driven by a separately-excited direct current motor. As mentioned previously, although the actuator is an integral part of the system, relatively little attention has been devoted to the design of mechanisms including the role of the actuator. A few investigations dealing with the dynamic response of mechanisms have incorporated actuator characteristics, including a pneumatic drive[I] and electric drives[2-5]. In those cases dealing with electric drives, the actuator was modeled by means of an assumed torque-speed relationship, such as a motor torque which varies linearly with speed[3, 4] or a quadratic relationship used by Liniecki[5] to represent a three-phase a.c. motor driving a slider-crank mechanism. These ideas are extended in the present study, where the motor is represented by a first-order differential equation in the armature current which is coupled to the mechanism via the input angular velocity. In addition, the actuator model includes viscous friction loss and constant mechanical drag effects. This leads to a set of three first-order state equations describing the motor-mechanism system which must normally be solved numerically to understand the behavior of a particular system. The purpose of this paper is to present an approach to generalizing the study of motormechanism systems. Through this approach, it may be possible to learn about the behavior of a particular system without deriving or solving the governing differential equations for that particular system but by applying graphs and guidelines available from previous numerical solutions for similar systems. The use of dimensional analysis concepts is an important aspect of this approach. Dimensionless groups are developed which define linkage and drive parameters and dimensionless performance characteristics are also presented. Relationships between the performance and system parameters are established in Part II of this paper for two particular mechanisms. The results show the potential of this approach in dealing generally with the behavior of motor-mechanism systems. This part of the paper consists of four sections. In the first, the dynamic equations describing the motor and mechanism are developed. In the second section, the equations are inspected to establish the steady-state behavior of the system which, of course, is an important aspect of performance and is also necessary for our dimensional considerations. The nondimensional linkage and drive parameters are developed in the third section, and in the final section, the system performance characteristics are derived in dimensionless form.

SystemEquations In this section, the differential equations are derived which govern the behavior of a crank-rocker mechanism driven by a separately-excited d.c. motor. We begin with the mechanism equations. Consider the 4-bar linkage of Fig. 1, where the symbols • designate the centers of mass of the assumed-rigid linkage members. For arbitrary link i, the location of the center of mass is defined by coordinates ri and 0i, Mi is the mass, Ji is the moment of inertia with respect to the centroid, and l~ is the link length. In order to generalize the mechanism model, a torsional spring of stiffness k and a torsional damper having coefficient c are attached to the follower of the

IY

x

Figure1. Schematic diagram of a general 4-bar linkage.

437 linkage, as shown in Fig. 1. These elements represent possible potential energy and dissipation energy effects within the mechanism. It will be assumed throughout this paper that the mechanism is such that its input link or crank can make a complete 360° rotation. Using input angle ~b2as the generalized coordinate describing the motion of the single degree of freedom mechanism, Lagrange's equation can be written as

d (OK] c~K+OP+OD

(1)

where K is the kinetic energy, P is the potential energy, D is the dissipation energy, and T is the external input torque on link 2, and where dot denotes differentiation with respect to time. The kinetic energy can be expressed as 1

2

i=2

where ~ is the angular velocity of link i and/)ix and/)iy are the x and y velocity components of its center of mass. Using well-known techniques, ~b3 and ~b4 can be expressed as functions of ~b:, and the velocities can be written as /)ix = O~i~2

i=2,3,4

/)iy = fli¢~2

i = 2, 3, 4

(3)

i=2,3,4 where ai, ~,. and y~ are functions of 4~2 and the linkage dimensions (see the Appendix). Substitution into eqn (2) leads to

1

K = ~ A(~b2) 4~22

(4)

where 4

A(~b2) = 2 [Mi(°t? + fl/2) + jiTi2]. i=2

(5)

The potential energy stored in the spring is P = ~ k(~b, - 4~*)2

(6)

where 4~* is the position of the follower corresponding to zero deflection of the spring. The damper produces dissipation energy given by 1

"2

D = [ c~b4

(7)

Differentiating eqns (4), (6) and (7) and substituting into eqn (1), yields •. I d A "2 A4~ + ~ ~ ~b2 + ky,(~b, - ~b*)+ c y 2 ~ = T

(8)

where the velocity relations (3) have been used. This is a second-order, nonlinear differential equation describing the mechanism motion as represented by angle ~b2.The torque T is the load applied to the motor. This torque is time-varying, depending on the mechanism motion and also, because of the mechanical coupling, on the motor characteristics which are described below.

438 Figure 2 shows a d.c. separately-excited motor, including a geared speed-reducer having a gear ratio n

Tb

w,,

7",,

wb

(9)

where the subscripts refer to shafts a and b, shaft a being the motor shaft and shaft b being the input shaft of the mechanism. In Fig. 2, the input is the constant applied voltage V and the output is the torque Tb = T which is supplied to the mechanism. The variables iI, Rr and Lr are the field current, resistance and inductance, respectively, while i, R and L refer to the armature current, resistance and inductance. J is the rotor moment of inertia, B is a damping coefficient due to viscous bearing friction, and Tt is a constant mechanical load, due, for example, to brush friction, gear friction, or dry bearing friction. The Kirchoff equation for the armature circuit is (10)

V = Ri + L[+ e

where e is the generated emf of the rotor. The Newtonian equation for the mechanical load is T = n(Tm - T l - B o J a -Jd)o)

(il)

where Tm is the magnetic motor torque. For a constant field current if, the magnetic torque and generated emf are given by [6] Tm = Kmi

(12)

EgO) a

(13)

e

:

where Km and Ke are constants, characteristic of the d.c. motor. Observing that

(14)

toa = mob = n~2 and substituting eqns (12)--(14) into eqns (10) and (11), we find

V R,(0 nKsr~2(t ) /(t)=~- L - L

(15)

T = nKmi(t) - nTt - n2Brk2(t) - n2J~2(t).

(16)

These are the modeling equations for the motor. If eqn (16) is substituted for T in eqn (8), then eqns (8) and (15) can be solved simultaneously for ~b~(t)and i(t). In the following two sections, these equations will be converted to a more general non-dimensional form. Notice that gravity effects have been neglected in the development of the equations. If so desired, this additional feature can be accounted for by the inclusion of gravitational energy terms in the potential energy function of eqn (6). R

V

L

t"

,

Figure 2. Schematic diagram of a d.c. separately-excited motor.

439

Steady State Considerations In this section, the system model and equations developed in the previous section are specialized to consider the steady-state behavior of the motor-mechanism system. One of the results of this analysis is a pair of algebraic equations which can be solved for the average motor current and average mechanism input speed under steady-state operation. The final result is a recasting of eqns (15) and (16) so that the motor equation and the input torque to the mechanism are expressed in terms of the deviations of current and speed from their average steady-state values. This is an important step because the average values can then be used conveniently to obtain dimensionless system equations. It is assumed that the operation of the motor-mechanism system will consist of a transient response, eventually leading to a steady-state response where the input energy is equal to the total dissipated energy. At steady-state, the current i(t) and the mechanism input angular velocity ~2(t) will be periodic functions of time. Let i0 and ~Oorepresent the average steadystate current and mechanism input velocity, respectively. From eqns (15) and (16), ..

V

l°= L

Rio

nKswo

L

L

(17) (18)

To = nK,,io - n T~ - n2 Btoo - n2 j ~

where, if X represents any time-varying quantity, then Xo is the average value of X per steady-state cycle. Obviously, [o -- ~o = 0. Furthermore, an expression for the average steadystate torque To can be obtained from the Lagrange formulation, eqns (1) and (8). In eqn (1), the first three terms involving kinetic energy and potential energy are conservative torques and will have a zero steady-state average. The torque due to dissipation energy appearing in the fourth term is a nonconservative torque which will result in a net energy loss. Thus, external torque T can be expressed as the sum of conservative and nonconservative parts T = Tc + T.~

(19)

and

T.~o

To =

(20)

because Tco = 0. Therefore, To is the time average value of T,c (the last term in eqn (8)) taken over a steady state cycle

fo

t~T.¢dt

To =

"

tp

fo"

c~42~ d t

2~'/OJo

y4~d~2

c =

"tO

2~r/oJ0

= cfo30.

(21)

Here, tp is the period of the mechanism cycle and f is the average value of y42 over the cycle

f=.__~.l f2" 2zr J0 y42d~b2"

(22)

Examination of this integral shows that f depends on the link lengths only, and hence, is constant for a given linkage btlt varies from linkage to linkage. Because an exact closed-form integration of eqn (22) cannot be achieved, the following approximation will be used based on dividing one cycle into N equal segments

440 Returning to eqns (17) and (18), in steady-state

Rio+ ng~too

--L

L

V

=Z

(24) (25)

n2B)too = nTt

nKmio - ( c f +

Subtracting eqns (24) and (25) from (15) and (16), . i = - T (i - io) -

_?((~ - coo)

(26)

T - cfioo = n K m ( i - i o ) - n2B(cb2 - COo)- n 2 j ~ 2

(27)

Combining eqns (8) and (27), 2 '" I d A . : (A + n J)~b2+ ~ ~ ~b2 + ky4(t~4 ~b*)+ C'y42~/)2 ~-~ CftO 0 "~- n K m ( i - io) - n2B(c~2 - Wo). -

-

(28) Notice that eqns (24) and (25) are equations which can be solved for the average steady-state armature current i0 and the average steady-state mechanism velocity too when given the applied voltage V and the various quantities describing the friction, load, motor and gears. Eqns (26) and (27) define the motor behavior in terms of current and speed variations about their average steady-state values. These two equations form the basis for the generalized dynamic equations of the motor-mechanism system developed in the next section.

NondimensionalSystem Equations The purpose of this section is to describe dimensionless, variables, groups and equations which govern the behavior of the motor driven crank-rocker mechanism. The first step in the process is to define the most basic dimensionless variables as follows I = i , fl = ~2, r = tOot, f" = to

too

T

M l 2 too 2"

(29)

The average steady-state current i0 has been used to nondimensionalize the armature current. The average steady-state input speed tOois used to nondimensionalize the input speed and also is used to define the dimensionless time variable r. To establish the dimensionless torque 7", tOo has been combined with the total mechanism mass M = M2 + M3 + M4 and the length of the fixed mechanism link l = It. From the basic definitions of eqn (29) it may then be shown that .. di

dI

l =~=

io~ ~--~T

~=~/= 2dn

too "~==~•

Using the above relations and eqn (25), eqns (26) and (28) can be converted into the following set of three first-order state equations for unknowns I(¢), O(¢) and I~(¢). Note that q~ is simply a new representation for input position angle and that • = ~ . dI

d~ =

CR(I I ' +

dtI)=~ dr

-

)

CRCK

f C o + G + CB (I - ~ )

(3O) (31)

441 dO d-7 = (A~Cj)

[ / C o t + C,(1 - 1) +

G(t

- n)

-

l dA-2 CPY4(JP4-~I)-C°'y42[]]" u (32)

In the process of developing these state equations, the following dimensionless linkage and drive parameters occur quite naturally R

CR = LoJo CK : n2KsKm/ R

Ml2 o.)o nT~ G = Ml2ojo2 n2B CB = MF ~oo

G -

n2j MF

k G, - Ml2~Oo~

C Co = Ml2oJo•

Also, the function/i is defined as /~=A MF As shown CR is the reciprocal of the nondimensional time constant for the armature circuit. Cr is another nondimensional motor parameter which reflects the electro-mechanical characteristics of the motor. Ct is the equivalent dimensionless load torque including the effect of the speed reducer. Ca and Co are dimensionless viscous damping coefficients, Cj is the equivalent dimensionless motor inertia, and Cp is the nondimensionai spring stiffness. These dimensionless groups have been established so that each characteristic of the system is being compared to the overall mass, size and/or speed of the mechanism. This has been done since mass, size and speed are often specified early in a design because of kinematic and operational requirements of the application. The other quantities would be considered afterward as a motor is selected and the effects of a flywheel and friction are evaluated. The parameters CR and CK are under control of the designer as he selects a motor. The parameters C, Cs and Cj represent a load torque, viscous friction and inertia on the motor shaft but can be easily shifted to the input shaft of the mechanism by dealing appropriately with the gear ratio n. These parameters may be somewhat controlled by the designer, particularly the inertia parameter C: which can be rather easily increased by addition of a flywheel. Spring stiffness and viscous friction at the output link of the mechanism are now represented by the parameters Cp and Ca The designer may exert some control over these parameters, especially Cp, but the friction term, modeled here as damping, may be inherent in the mechanism or bearing configuration. In any case, considering the motor-mechanism parameters in a dimensionless form gives them special meaning for the application at hand. For example, stating the electrical time constant L/R of a motor may describe a characteristic of the motor. However, stating CR = R[l-~o describes the electrical time constant in relation to the rotational speed of the mechanism it will drive. By studying motor-mechanism systems operating with various values of CR we can establish general

442

quantitative conclusions about the influence of CR on system performance. Part II of this paper begins to explore the relationships between motor-mechanism performance and the dimensionless parameters just defined. System P e # o t m a n e e In the preceding sections, differential equations have been derived to describe the behavior of the motor driven crank-rocker mechanism. These equations have been expressed in terms of deviations from the average speed and current in steady-state operation of the system and the equations have been cast into a nondimensional form. Dimensionless parameters have been defined to describe the motor-mechanism system. The remaining step in this analysis is to properly define the variables which measure the behavior or performance of the system. That is the purpose of this section where we have focused on some significant aspects of mechanism design including the transient or start-up time, the steady-state speed fluctuation, and energy requirements both during start-up and in steady-state. The determination of these quantities, except for steady-state energy, requires solution of the governing differential eqns (30)-(32). These are, of course, nonlinear equations and must be solved numerically through digital computation by generating the motor-mechanism state variables at discrete points in time. Start-up time is defined as the time required to accelerate the mechanism from rest to steady-state. The system is considered to be at steady state when either or both of two conditions are satisfied: (1) the difference between the actual period of mechanism rotation and the ideal period (i.e. (27r/to0)) is less than a certain small value, or (2) the difference in the periods of two successive cycles is less than some small amount. In nondimensional form the ideal period is of duration 27r and the following nondimensional equations may be written to describe the criteria for steady state mechanism operation

(1) It* -2rr I -< el

(33)

(2) It* - r*-,1 <- e2.

(34)

and

Here r* is the dimensionless time required for the jth cycle of the mechanism. For the examples of this paper, the error limits used were el = 0.01 and e2 = 0.00002. When the mechanism has reached steady-state speed and has satisfied these criteria, the following definition of start-up or transient time may be made r, = ~ r*. i=1

(35)

Here r* is the time needed for the ith rotation of the input crank and steady-state is achieved on the jth rotation. The steady-state speed fluctuation for the motor-mechanism system is based on 360 degrees rotation of the crank and is defined as AI) = f~max-- l'lmi.

(36)

~'~ave

where f~maxand Ilmi, represent the maximum and minimum dimensionless angular velocities of the crank during a steady-state cycle. II~veis the average value of fl for the same cycle and is expressed approximately as 1 N

llave = -= '~ lli Ni=l

(37)

where N is the number of divisions of the cycle and Ill is the dimensionless speed in the ith division. Clearly, our definition of dimensionless speed requires that llave have a value of one in steady-state.

443 The problem of energy consumption in machinery is becoming increasingly important and can no longer be overlooked by designers. For this reason, the total energy requirements of the system will be examined. First, consider steady state operation. The total input energy per cycle is the integral of the power: Ess --

(38)

Vidt.

The input voltage V is constant for each fixed set of parameters, and the integral of the current is related to the average steady-state current, . -_ ~too f0 tp idt. to

(39)

The energy is then E,, = 21rVio

(40)

tOo

Solving eqns (24) and (25) for V and io, V =~ •

[nTIR + (n2BR + cfR + n 2KgKm )too] 1

to = - ~ m [nTt + (n2B + cf)too].

(41)

(42)

For a given set of parameters and a specified average steady-state speed too,eqns (41) and (42) give the required input voltage V and the resulting average steady-state current io.On the other hand, eqn (41) can be used to compute the resulting average speed for a given applied voltage. Substituting eqns (41) and (42) into eqn (40), 2~r ( uztoo+ U2 +u3'~ E,,=n-rg~. too/

(43)

where ul = R(n2B + cf) ~+ n2KgKm(n2B + cf) u2 = nTt[2R(n2B + cf)+ n2KsK,,] U3 = n2Tt2R.

It might be noted that eqn (43) indicates that the energy level is high for both high and low average operating speeds, with an optimum speed somewhere in between. Differentiating eqn (43) with respect to too and setting the result to zero yields the following value of too for minimum energy comsumption

The resulting minimum steady-state energy per cycle is (Ess)min

= ~

21r

(2X/(/'/lU3) + US).

444

Returning to eqn (43), the dimensionless steady-state energy is defined as ~S ~

ES$

Ml2 too~.

(44)

In ideal electromechanical conversion, which is reasonably approximated by real d.c. motors, the constants Kg and Km are equal when compatible units are considered [6]. In this case, eqns (43) and (44) can be combined to form the following energy expression 27T

~. = ~

( c , + / c , + G + c,,)(c, + / c o + G).

(45)

This equation expresses the steady-state energy requirement as a function of the actuator and mechanism parameters. It is particularly useful, because it is a closed-form algebraic expression which does not require solution of the differential equations. The energy consumption during the transient can be obtained by summing the energy during the transient time interval E, = y v~i~atj = v2ijat~

(46)

where the voltage V is constant. Nondimensionalizing eqn (46) yields =

M---V~g= Q ~ a ~

(47)

where Q = ~

V/o _ ~ s

- ~--~-.

(48)

As used here, Q might be viewed as a nondimensional voltage. In eqn (46) and (47), the time increments At~ are much less than the mechanism cycle time.

Conclusion The state equations have been derived for an electromechanical system consisting of a separately-excited d.c. motor driving a crank-rocker 4-bar mechanism. These eqns (30)-(32) have been expressed in a dimensionless form and new dimensionless parameters have been suggested to describe the motor-mechanism system. Nondimensional performance measures have also been developed including the speed fluctuation and start-up time as well as the energy consumption in steady-state and during start-up. These are shown in eqns (35), (36), (45) and (47). Part II of this paper presents the results of numerical computations solving the nonlinear dynamic equations for two different crank-rocker mechanisms. The results show the influence of the dimensionless linkage-drive parameters on the various performance measures and begin to show the generalized behavior of motor-mechanism systems. It is believed that this basic approach can be applied to other system configurations. Possibilities include analysis with other types of actuators including other d.c. motors and a.c. motors as well as hydraulic and pneumatic motors. Different classes of mechanisms might be considered such as other linkage types and cams. References I. M. Skreiner and P. Barkan, On a model of a pneumatically actuated mechanical system. J. Engng Ind. pp. 211-220 February (1971). 2. M. Skreiner, Dynamic analysis used to complete the design of a mechanism. J. Meck $, 105-119 (1970). 3. M. J. Gardocki and W. L. Carson, Force system synthesis using a weighted velocity squared error criteria. Proc. 3rd Appl. Mech. Conf., pp. 41.1-41.19. Oklahoma State University, November 0973). 4. W. E. Bonham, Calculating the response of a four-bar linkage. ASME Paper No. 70-Mech-69, Presented at the ASME Mechanisms Conf., Columbus, Ohio, November (1970).

~5

5. A. Liniecki, Synthesis of a slider-crank mechanism with consideration of dynamic effects. J. Mech. 5, 337-349(1970). 6. S. A. Nasal Electromagnetic Energy Conversion Devices and Systems. Prentice-Hall, New York (1970). 7. F. L. Conte, G. R. George, R. W. Mayne and J. P. Sadler, Optimum mechanism design combining kinematic and dynamic force considerations. J. Engng Ind. 97(2), 662-670 May (1975).

Appendix Determination o/ velocity influence coe~cients The angles 6s and 64 are determined as functions of 62 by solving the followingcomplex equation 12e~ + 13e~ = Ii e ~' + 14d ca

where 6~ is the fixed-frame orientation angle (6, = 0 in Fig. 1), and ] is the complex operator defined by 1"2= - 1. The velocity influence coefficients are then ')12=1 l~sin(64 -

~)

~3 = 13sin(63 - 64) 12sin(63 -

~)

Y4 = / 4 s i n ( 6 3 - 64)

a2 = - r2sin(d~2+ 02) as = - 12sin~

- y 3 r 3 s i n ( 6 s + Os)

a 4 = - y 4 r 4 s i n ( 6 4 + 04) /32 = r 2 c o s ( d ~ + 02)

f13 = hcosd~ + y3rscos(6s + 03) B4 = Y4r4COS(64 + 04).

ETUDE D~tqqA/,IIOUI~ GRNI~Ih~I.,T-qRE D'IJN IKI~C~M~I'I'Sltd~ENTRA'I'N~. PAR MOTE'tJR A COURANT CONTTNU

J. P. Sadlor, 8. W. Mayne ot K. C. Fan R~sum6 - On d~rive un syat~me d'~quations

diff~rentielle

afin d'obtenir

la r~ponse dynamique

d'un m~caniame entrafn6 par un moteur A courant continu et une bo~te d'engrenages. normalisation

dea ~quations on arrive ~ d~finir quelques nouveaux paramltrea

sent l'asaemblage.

Lea param~tres

normalis~s

conviennent parfaitement

Avec la

qui caract~ri-

pour la description

de la r~ponse et du rendement du syst~me entier. A l'aide du modile math~matique et la conaommation

on analyae lea variations

d'~nergie pendant une marche continue,

de vitease du m~canisme

ainsi que le temps de d~marrage

et la conso~anation d'6nergie pendant le d~marrage. Deux ~tudes de simulation ont ~t~ ex6cut6es

utiliaant deux m6canismes

diff~rents.

L'analyae de la r~ponse dynamique a ~t~ effectu~e par la solution num~rique dea ~quations d~tat

~ l'aide d'un ordinateur.

Lea rSles des param~tres

pr~sent6s ~ 1'aide des courbea normalia~es. r~canisme

et le moteur ~ courant continu.

On d6montre

du moteur et du m~canisme

sont

lea effets de couplage entre le