Let s examine some of the items from the product gallery in Chapter 1 and see what kinds of motion we might need to produce.

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1 Chapte 2 Mechanics Fom the mechanical point of view, the goal of the electomechanical design pocess outlined in Chapte 1 is to cause an object o objects (the load) to pefom a specific, contolled motion. In this chapte we will examine how the load, the mechanism, and the electically poduced foces inteact to poduce motion. 2.1 Motion Examples of Motion Let s examine some of the items fom the poduct galley in Chapte 1 and see what kinds of motion we might need to poduce. Dill. In the economy model dill, the chuck tuns at about 1000 pm when the tigge is pulled, and will slow down when dilling, the actual speed depending on the size of the dill and the toughness of the mateial being dilled. As we pay moe money, we get additional featues such as selectable speed, constant speed unde load, evesible diection, and a toque limite fo diving scews. Elevato. Hee we wish to contol a position athe than a velocity. Moe specifically we want to be able to select one of a fixed set of vetical positions fo the ca. Although ou goal is the steady state position following a move, we must also contol the tansitions between stops (i.e. the velocity and acceleation) so that the ide is safe and comfotable. 2-1

2 CD Playe. We could think of the desied action hee as a single motion: moving the focal point of the lens along the tack at a fixed tangential velocity. O we could look at it as thee coodinated motions: spinning the disk, following the tack, and staying in focus. In any case, the motion is poduced by thee sepeate actuatos. Robot. So fa, the motions we ve seen have eithe been one dimensional o have been a set of independently contolled one dimensional motions. In a obot, although made up of a combination of single dimensional motions, the desied end poduct is a completely abitay six degee of feedom (6 DOF) motion. This flexibility allows a obot to assume any position and any oientation (within its wok envelope) and to move between them along any path. Depending on the application, we may equie accuacy only of the end points, o of all points along the path. Fuby. The Fuby has a lage vocabulay of motions: opening and closing his eyes and mouth, ocking back and foth, and wiggling his eas. These ae combined in a numbe of pattens depending on whethe he is eating, sleeping, dancing, o chatteing. It is significant that each of the individual motions is cyclic. Washing Machine. The typical top loading washing machine has fou moving pats: the tub, the agitato, the pump, and the wate valve. Duing the wash and inse cycles, the tub is stationay and the agitato oscillates back and foth. At the beginning of these cycles, the wate valve opens until the tub is filled, then closes. Duing the spin cycles, both the tub and agitato otate at high speed, and the pump uns to empty the wate fom the tub Types of Motion Fom these examples and othes, we can collect a numbe of qualitative chaacteizations about the types of motion a system might be equied to poduce. These ae summaized in Table

3 Regulation of fixed steady state value Washing machine spin cycle Automobile cuise contol Tacking a slowly vaying value CD focus Heliostat (sola collecto) Continuously adjustable steady state value Dill CD otation Multiple discete fixed values Washing machine wate valve Elevato Following an abitay path CD tack following Numeically contolled lathe Multiple coodinated motions Fuby Assembly obot Table 2.1: Types of Motion We can descibe these motions by saying that we want to be able to set the value of m(t) whee m is a mechanical quantity (position o velocity) and t may be a specific value o set of values, o in the case of steady state, t = Chaacteistics of Motion Hee is a catalog of some of the quantitative chaacteistics of motion that we will need to be able to achieve. The Contolled Quantity The pimay physical vaiable being contolled, e.g. the vetical position in the elevato o the otational velocity in the dill. In a linea system this may be a position, velocity, o foce, o in a otational system an angula position, angula velocity, o toque. Numbe of Desied States This may be a set of discete values (elevato, wate valve) o a continuum (obot, CD playe). Range The motion may be equied ove a limited ange, o may be unlimited. In the washing machine, the agitato otates ove a ±90 ange in the wash and inse cycles, while in the spin cycle the agitato and tub otate ove an unlimited ange. Load If the desied quantity is a position o velocity, how much foce will be equied to sustain it? 2-3

4 Accuacy How closely must the actual value match the desied value? Constaints on Path Must the actual value be within the accuacy equiements at all times (CD playe tacking) o only at the endpoints (elevato)? Ae thee equiements fo smoothness of motion such as maximum values fo velocity, acceleation, o jek? Mechanisms Responsibility fo poducing the desied motion is shaed between the mechanism and the contolle. In many cases the geomety of the path is constained by the mechanism (often to one dimension) and the tavesal of the path is diected by the contolle. Some components of the mechanism ae esponsible fo implementing constaints on the motion (fo example, the ails in the elevato o the spindle beaing in the dill), othes povide suppot fo the load (the elevato cage o the dill chuck), some tansfom motion (windlass, geabox) o move it fom one place to anothe (cable, shafts), and some ae esponsible fo holding eveything else togethe in the pope oientation. We could egad the mechanism as a black box which attaches to the load on one side and to the electomechanical devices on the othe. The motion pesented at the electomechanical side of this box has usually been decomposed and tansfomed into a set of single dimensional linea and otational motions. Fo example, a obot am takes an abitay 6 DOF motion and decomposes it into a set of six otations which must be coodinated by the contolle Electomechanical Devices Having constained and decomposed the desied motion down to a set of simple, one dimensional motions we can now animate ou mechanism by attaching electically contolled actuatos to these pots. Ideally, an actuato would convet an electical signal (e.g. a voltage) in a popotional manne to the desied mechanical quantity (e.g. a displacement o a velocity). Similaly, to monito motion a senso should convet the mechanical quantity being measued to a coesponding electical quantity. We will efe to actuatos and sensos jointly as electomechanical devices. 2.2 Mechanical Foces If we attempt to change the position o velocity of a pat of ou mechanism it esponds with a foce. These foces must be matched o ovecome if we ae to achieve the desied motion Linea Inetia 2-4

5 Newton s Laws Conside the simple mechanical system in Figue 2.1. Accoding to Ideal Rolles M F Figue 2.1: One-dimensional, Fictionless Mechanical System Newton s Fist Law, the box will emain stationay in the absence of any applied foces. To move the box to a diffeent position, it is necessay to apply appopiate foces. Newton s Second Law tells us what the esult of these foces will be: F ma whee F is the foce acting on the object, m is its mass, and a is the esulting acceleation. If an appopiate system of units is chosen, the constant of popotionality becomes one. We will utilize SI (Système Intenational) units fo the most pat, as they minimize the numbe of extaneous constants in vaious fomulas. In this case, fo F in Newtons, m in kilogams, and a in metes/second 2, we have the familia: F = ma (2.1) Rotational Inetia Many of the devices we will be consideing poduce otay, athe than staight line motion. Figue 2.2 is the otational equivalent of Figue 2.1. If a tangential foce is applied to a cylinde of adius, suppoted by its axis, it will expeience a toque of T = F If it is fee to otate, the toque will poduce an angula acceleation T = J θ whee J is the moment of inetia of the cylinde about its axis. We may also wite whee ω is the angula velocity in adians/second. T = J ω (2.2) 2-5

6 T Figue 2.2: One-dimensional, Fictionless Rotational System F Moment of Inetia Moment of inetia is the otational countepat to mass, and epesents the ease (o difficulty) with which a toque can acceleate an object. Fo a solid cylinde, the moment of inetia about its axis is J = 1 2 m2 Fo a hollow cylinde with inne adius 1 and oute adius 2 J = 1 2 m( ) Gavity If Figue 2.1 was taken on Eath, thee is always at least one foce acting on the box. Newton s Law of Gavitation tells us that thee is a foce: F = G m boxm eath eath 2 F = mg (2.3) acting vetically downwad on the box. Newton s Thid Law tells us that thee is an equal eaction foce fom the table acting upwad on the box. Hence the net foce is zeo and in the absence of any additional foces, the box emains stationay. If a otational system is statically balanced, gavitational foces will cancel and thee will be no net gavitationally induced toque, egadless of oientation Fiction The idealized systems in Figues 2.1 and 2.2 will continue to move foeve if undistubed. Any eal system will contain fiction which will eventually aest the motion. 2-6

7 Fiction comes in two basic flavos. If we eplace the olles in Figue 2.1 with an ideal oil film, we have what is called viscous fiction which causes a foce which is popotional to the velocity and opposite the diection of motion. F = cu = cẋ Note that we will use u = ẋ to denote linea velocity to avoid confusion with the use of v to denote voltage. The othe common fom of fiction is dy o Coulomb fiction. It is simila in that it acts in the diection opposite to the motion (o impending motion) of the objects, but the magnitude is popotional to the nomal foce between the two objects. F = µn whee µ is called the coefficient of fiction. Section A.4.1 gives the appoximate coefficient of fiction fo seveal common mateial pais. Coulomb fiction is also dependent on velocity in the following sense: the value of µ just pio to the onset of motion, called the coefficient of static o limiting fiction, is usually diffeent fom the value afte motion has begun, called the coefficient of sliding o kinetic fiction. Typically sliding fiction is slightly less than static fiction. This chaacteistic causes the phenomenon known as stick-slip which is esponsible fo the sound of the violin and squealing bakes, and can cause instability in contol systems when executing small motions Elastic Foce Just as the beaings of a eal mechanical system ae not ideal (fictionless), neithe ae the stuctual elements pefectly igid. If a foce is applied to a membe which is constained at some othe point it will deflect. In most cases, fo easonably small foces, the amount of deflection is popotional to the amount of applied foce. This phenomenon is known as elastic defomation and its linea egime is descibed by Hooke s Law: F = kx (2.4) whee k is called the sping stiffness. In a otational system, a tosional sping poduces the following elation between angula displacement and the esulting toque: T = kθ 2-7

8 2.3 Wok and Enegy Wok is pefomed when a foce is applied though a distance: W = F x (2.5) fo a constant foce, o moe geneally: dw = F dx W = F dx (2.6) Wok done on a system inceases the enegy of the system. Convesely, wok may be done by a system, theeby deceasing its enegy. Kinetic Enegy In acceleating a mass m with a foce F: F = ma = m u W = = F dx m u dx = m u u dt = 1 2 mu2 (2.7) Fo a otating system: W = 1 2 Jω2 (2.8) Stain Enegy Fo a sping of stiffness k compessed though a distance x: W = F = kx F dx = kx dx = 1 2 kx2 (2.9) Gavitational Potential Enegy In moving a mass m vetically in a gavitational field: W = F = mg F dx = mgdx = mgx (2.10) It is impotant to emembe that wok entails both foce and distance, and that abitaily lage foces may be applied o abitaily lage distances tavesed without the expenditue of any enegy, if the coesponding distances o foces ae zeo. 2-8

9 2.3.1 Consevation of Enegy Consevative Foces The enegy that these foces (inetial, elastic, gavitational) impat to a system is stoed in a fom (kinetic, stain, gavitational potential) which is ecoveable. If the pocess which put the enegy into the system is evesed (by deceleating the mass, eleasing the sping, o loweing the weight) the enegy may be taken back without loss. The foces involved in ecoveable enegy tansfe ae called consevative foces. Anothe definition of a consevative foce is that the total wok pefomed in tavesing a closed cycle is zeo. The law of consevation of enegy states that in a system involving consevative foces, the sum of all enegies in the system is constant. Non-consevative Foces The wok expended in ovecoming a fictional foce may not be ecoveed by evesing the pocess. Foces such as fiction ae temed non-consevative o dissipative foces. Mechanical enegy in a system involving such foces is not conseved but is changed into anothe fom such a heat which is dissipated and is not ecoveable. Consevation of enegy encompasses dissipative foces as well, if themal enegy is included. The total enegy of a system emains constant, but that pat of it involving dissipative foces is conveted to heat (o some othe fom of enegy) and is not diectly ecoveable. 2.4 Dynamics Equations of Motion Fee Response Fo the system in Figue 2.3 the foces acting on the mass ae the estoing foce due to the extension o compession of the sping, and the viscous dag foce fom the dampe. We may wite: F = ma = mẍ = kx cẋ o mẍ + cẋ + kx = 0 (2.11) If we define the undamped natual fequency k ω n = m the damping coefficient α = c 2m (2.12) (2.13) 2-9

10 k x m c Figue 2.3: Second-ode, Unfoced Mechanical System and the damping atio ζ = α/ω n (2.14) then solutions to this equation will have one of thee foms: Ovedamped (ζ > 1) x = k 1 e s1t + k 2 e s 2t whee s 1 = α + α 2 ω 2 n s 2 = α α 2 ω 2 n Citically Damped (ζ = 1) x = (k 1 + k 2 t)e αt Undedamped (ζ < 1) x = ke αt cos(ω d t + θ) whee ω d = ω 2 n α 2 is the damped natual fequency Foced Response If an extenal foce F(t) is applied to the mass, as in Figue 2.4, (2.11) becomes mẍ + cẋ + kx = F(t) (2.15) 2-10

11 k x m F c Figue 2.4: Second-ode, Foced Mechanical System The solution of (2.15) is of the fom x(t) = x c (t) + x p (t) i.e. it consists of the sum of a tansient solution, x c (the complementay function), and a steady state solution, x p (the paticula integal). The complementay function is a solution of the homogeneous system (2.11). The paticula integal depends on both the dynamics of the system and the focing function. Two special cases fo the applied foce ae of inteest: Step Response If F(t) = 0 fo t < 0 and F(t) = F 0 fo t > 0, then x p (t) = F 0 /k. Figue 2.5 shows a typical step esponse fo each of the classes of damping. x ζ<1 ζ=1 ζ>1 t Figue 2.5: Step Response. 2-11

12 Sinusoidal Response When the excitation is sinusoidal, the steady state esponse is also sinusoidal at the same fequency, with amplitude depending on both the damping atio and the atio between the diving fequency and the undamped natual fequency. Thee is also a phase shift between the input and the esponse which vaies fom zeo at low fequencies, though 90 deg at the undamped natual fequency, to 180 deg at high fequencies. Fo an excitation of F(t) = F 0 cos(ωt), the steady state esponse is x(t) = Xcos(ωt φ) whee X = X 0 [ ( ) ] 2 2 ) 2 1 ω + (2ζ ω ωn ωn tanφ = 2ζ ω ω n ( ) 2 1 ω ωn and X 0 = F 0 /k Figue 2.6 shows the magnification atio X/X 0 fo vaious damping atios. ;; R ] ] ] ] ] ] Z Z Q Figue 2.6: Fequency Response vs. Damping Ratio. 2.5 Contol As we ll see in Section 3.1, the only mechanical quantity which can be made to diectly coespond to an electical voltage o cuent is foce (o toque). What if we want to contol a position o velocity? We have a numbe of altenatives. 2-12

13 1. We will develop a numbe of actuatos fo which the steady state position o velocity is detemined by a voltage o cuent (o fequency). 2. We can combine an electically contolled foce with an elastic element whose foce vaies with displacement o a viscous element whose foce vaies with velocity. In this case the equilibium position o velocity will be detemined by the electical vaiable. 3. We can ceate a closed loop contol system whee the electically poduced foce is applied in such a way as to poduce motion which will educe the diffeence between the desied position o velocity and its actual value Open Loop Contol Figue 2.7 shows how we might use method 2 to contol the position of the elevato. Assume the moto poduces a toque which is popotional to the applied cuent: T = K T i. This will poduce an upwad foce of F up = T = K T i on the cage. If m is the mass of the cage, k the sping constant, and c is the viscous fiction of the oil on the ails, then the total downwad foce is F down = mẍ + cẋ + kx + mg. Since these must be equal, we have mẍ + cẋ + kx + mg = K T i (2.16) In the steady state we get kx + mg = K T i which gives x(i) = K T k i mg k Figue 2.7: Open loop elevato contol Solving this fo i tells us how to set the cuent to achieve a desied position i(x) = K T (kx + mg) Note that while this cuent will detemine the steady state position of the ca, the actual position as a function of time when moving between floos will be given by the solution of Also, this contol law is only valid fo an empty ca. If any passenges get into the ca, the total weight will incease and the ca will sink. 2-13

14 2.5.2 Closed Loop Contol Suppose that we have a senso whose output is popotional to the vetical position of the ca (x) o altenately, to the angula position of the dum (θ) such that v x = K x x. Let v des = K x x des be a contolling signal which epesents the desied position of the ca (x des ). Then v e = v des v x is an eo signal which will be zeo when the desied and actual positions ae the same. If we amplify this signal and use it to dive the moto in the pevious example, i.e. i = G(v des v x ) = GK x (x des x) then mẍ + cẋ + kx + mg = K T A(x des x) (2.17) whee A = GK x. In the steady state we get kx + mg = K T A (x des x) which gives x(v des ) = K TA/ k + K T A/ x mg des k + K T A/ Note that lim G x(v des) = x des. This says that if we make the amplifie gain G sufficiently lage the eo caused by changing the load in the ca can be made abitaily small. Also as we incease G the effect of the sping (k) becomes negligible and we can emove it. Howeve, these benefits don t come fo fee. Let s examine the dynamic behavio of the closed loop system. mẍ + cẋ + kx = mẍ + cẋ + (k + K TA K TA (x des x) mg )x = K TA x des mg Define w = mg to be the weight of the ca and its contents and take the Laplace tansfom. ms 2 X(s) + csx(s) + (k + K TA Let k 1 = k + K T A. We can define two tansfe functions H c (s) = X(s) X des (s) = )X(s) = K TA X des (s) W(s) K T A/ ms 2 + cs + k 1 gives the esponse of the ca position X to the contolling input X des and H d (s) = X(s) W(s) = 1 ms 2 + cs + k 1 epesents the distubance caused by vaiations in the weight of the ca. 2-14

15 The natual esponse of the system is given by the oots of the chaacteistic equation ms 2 + cs + k 1 = 0 which gives ω n = α = ζ = k 1 m c 2m α = c/2m ω n k1 /m = cm 2 k 1 As we incease A to impove the accuacy, the system will eventually become undedamped and oscillate with changes in position o load. To pevent this, we must incease the damping by inceasing eithe c o m, neithe of which is desiable. Howeve, we can incease the effective damping by incopoating additional feedback. Note that adding position feedback inceased the stiffness of the system, just as if we had inceased the stiffness of the sping. If we incopoate velocity feedback, we can incease the damping of the system. Suppose that we attach a senso whose output is popotional to the velocity of the ca: v 2 = K v ẋ and feed it into the contol cuent along with the position eo Taking Laplace tansfoms and ms 2 X(s) + (c + K TA v i = G(K x (x des x) K v ẋ) = A x x des A x x A v ẋ I(s) = A x X des (s) A x X(x) A v sx(s) )sx(s) + (k + K TA x Defining k 1 as befoe and c 1 = (c + K T A v ) we get k 1 ω n = m α = c 1 2m ζ = c 1 2 mk 1 )X(s) = K TA x X des (s) W(s) so that we can now contol both the stiffness and damping of the system by adjusting A x and A v. 2-15

16 The idea of feeding back measuements of the state of a system, compaing them with the desied state, and geneating appopiate signals to dive the actuatos can be genealized to any system whee we want a mechanical output to coespond to an electical input, as shown in Figue 2.8. Such systems ae sometimes called sevomechanisms. With a little imagination this can be seen as an instance of the geneic electomechanical system shown in Figue 1.2. v des desied esponse contolle v ctl dive v a actuato F mechanism x,u load actual esponse v x,v u sensos Figue 2.8: Genealized closed-loop contol system 2-16