Intro Wb3303 Kineto-statics and Dynamics

Transcription

1 Wb333 Kineto-statis and Dynais 8 ineto-statis and Dynais Introdution Virtual wor and equilibriu equations (statis) General priniple Driving fores Support fores Connetion fores Mass fores riniple of d Alebert Mass entre and bea eleent Mass atri of the binary eleent Mass distribution of a ehanis (eaple) Internal fore distribution Mass distribution of joints Other eternal fores Springs Daping fores Gravity fores Other eternal fores (roess fores) Equation of otion General priniple Redution of fores to the driving shaft Driving fore and otor odel Integration of the driving equation of otion Stiffness of the eleents and deforations The onstitutive equations Bending of the bea eleent osition auray of the ehanis Dynai siulation with fleible eleents Eaple with RUNMEC...8.3

2 Wb333 Kineto-statis and Dynais 8. Wb333 / KB ig. 8.. Ebodient properties in a o-planar ehanis odel etra nodes (ass distribution) etra eleents (fore distribution) Wb333 / KB etra node (deforation/vibration) in g g bending oent balaning ass ig. 8.. Typial adaptation of topology for dynais

3 Wb333 Kineto-statis and Dynais 8. 8 ineto-statis and Dynais 8. Introdution Dynais = branh of theoretial ehanis dealing with the otion and equilibriu of bodies and ehanial systes under the ation of fores (IToMM [.]). Kineto-statis: this ter will be used here onsidering the ehanial syste as a rigid body syste with given tie behaviour of its otion. Dynais is the ore general ter. Kineto-statis deals with a liited field of dynais: the theory that an be used as if the ehanial syste is, in a ertain position, a stati onstrution. Tied otion, and thereby aeleration, plays anyhow a role. The fores inlude ertainly the ass fores, both in ineto-statis and dynais. The intention of this hapter should be regarded in the ontet of the design proess of ehaniss, see fig..3.. It will be assued that the ineatis part of the ehanis design has been opleted: the ehanis type has been hosen and the ineati diensions are given quantities. The fores ating at the individual parts are to be alulated now. The results are needed for the design of bearings, the hoie of a driving otor, the thiness diensions of the lins and so on. Typially the dynais part has iterative loops. When thiness diensions and the onstrution aterial have been hosen, the asses of the parts are nown. But the asses are needed to alulate the ass fores, so usually the stresses and deforations an only be verified properly after having ade this hoie. A lot of easures deserve attention for better dynai behaviour, lie adding etra ass or springs (balaning of fores), shape of the parts and so on. inally the ehanial odel should be regarded as a three-diensional odel, even when it is a o-planar ehanis (fig. 8..). During the various design stages it will frequently be needed to adapt the odel (with respet to its physial properties) to the design purpose. Soe eaples of adaptation are shown in fig The dynai odel will oasionally be uh ore etended opared with a pure ineati odel, but the ineati otion will be the sae. It will be lear that the inite Eleent approah is very suitable to handle suh odel definitions and etensions towards dynais and ineto-statis. The eaples show that any of the design deisions in dynais don t need a three-diensional ehanis odel (for a o-planar ehanis). The ore ople 3D-odel an thus be avoided during a great part of the oneptual design of the ehanis. This boo onentrates therefor on the theory that an be used with a D-odel of a planar ehanis, whih is ineatially deterined (the atri D p is regular). It will be assued further: The parts will have only sall deforations due to the fores. It eans that the parts are fairly rigid, but it is the intention to find out whih fleibility is allowed. The priniple of superposition an be used: The total otion of the ehanis (o-ordinates) is the su of rigid otion and displaeent due to elastiity of the lins.

4 Wb333 Kineto-statis and Dynais 8. Wb333 / KB o-ordinates T = y y for paraeters T = ont. equations (and ) see binary eleent = - = - st order ontinuity equation os sin os sin os sin sin os sin sin os os y y ig Bea eleent, speial for dynais

5 Wb333 Kineto-statis and Dynais 8.3 rition will, in the oneptual design phase, ainly be iportant for ehaniss that are supposed to ja. In the ajority of ases it will be tried to eep the frition low, allowing to neglet frition in the ehanial syste. It is beyond the sope of this boo to inlude frition in the theory. To be better prepared for dynais a new eleent will be introdued here: the planar bea eleent (see fig. 8..3). It has the harateristi property that, opared with the binary eleent, bending deforation has been added. Naturally the two for paraeters desribing the bending ( and ) should be understood as presribed for paraeters. The ajor advantage of this eleent is that the internal fore distribution (and deforation odes) an, in speifi ases where bending in a bar is signifiant, better be odelled. In a later hapter (8.6.) it will be shown that the two bending deforations are oupled by the bending stiffness of the bea. This property annot be obined with the idea that a deforation should be understood as a large deforation, as required to odel input otion. When a bea is to be used as a driven ran, one of the angular o-ordinates would be a proper quantity to use as the ran angle. In this boo however only for paraeters are used as driving quantities. Adding a half bea eleent (etra eleent) an solve this proble, sharing the angular o-ordinate with the bea. The half bea has the proper for paraeter to drive the ehanis. 8. Virtual wor and equilibriu equations (statis) 8.. General priniple The ehanis an be onsidered as a fleible onstrution with the input otion frozen (rigid, but fleible). At suh a onstrution fores an be applied, just lie in statis. In this oneptual ehanis odel the driving otion quantities (onerning the degrees of freedo) will be treated lie all other presribed for paraeters. In statis the priniple of virtual wor is the sae as the ondition for equilibriu: If the virtual wor of the applied (eternal) fores is equal to the virtual wor of the internal fores of the ehanis, then there is equilibriu. The reverse stateent is also true: If there is equilibriu in the ehanial syste, then the virtual wor of applied fores and internal fores are equal. The virtual wor of the applied fores ust be onsidered with the displaeents of the oordinates (of the nodes and other types of o-ordinates), as they an be oved ineatially. This iplies, for instane, that a fied point ( o = ) annot ontribute to the aount of virtual wor: W e f, f, (8.) The inde indiates the oving o-ordinates of the ehanis. The <> braets indiate the inner (salar) produt of two vetors fro different physial vetor spaes. Co-ordinates are ineatially ovable due to allowed deforations (aused by the fores) of the presribed for paraeters. Atually this is a hoie that an be ade for eah ehanis odel. Suppose for instane that all presribed for paraeters are onsidered as undeforable (really rigid), then the aount of virtual wor of the applied fores an only be zero. In general however one or ore presribed for paraeters will be onsidered as deforable, allowing the to absorb the virtual wor. Assue now that one partiular for paraeter p is deforable: p p W in,, (8.)

6 Wb333 Kineto-statis and Dynais 8.4 Wb333 / KB G A A B input y G f y / A A p p N N N Virtual Wor:,, f W N f y G y G T / A A A A ig. 8.. Driving fore =N as an internal fore (noral fore) Eaple Wb333 / KB A A B f y B A A p p N M N 3 3 os sin os os sin sin os D ] [D Virtual Wor:.94 M M f M T / / ig. 8.. Driving oent =M as a generalized internal fore (eaple )

7 Wb333 Kineto-statis and Dynais 8.5 The oeffiient of vetor p ontains the (internal) fore that, when ultiplied with the orresponding deforation, yields the virtual wor. In the equilibriu situation it ust hold that: p p f,, (8.3) How the o-ordinates an ove ineatially is epressed by the first order transfer funtion (5.6), noted here as: p p (8.4) After substituting (8.4) in (8.3) the following vetor equation an be obtained: T p pt p f p Dividing at both sides by p, whih is by definition non-zero, it follows that T pt T f f p p (8.5) Assuing that all presribed for paraeters p are independent variables, the result (8.5) an be written in one equation syste for all possible deforations p : p T f p pt D f (8.6) It an be onluded: the internal fores p and the applied (eternal) fores f are related by the first order ineati transfer funtions. Note that the desription of fores, both the internal and the eternal fores, is to be understood in the EM in the generalized way. It depends on the odelled definition of the orresponding o-ordinate (or for paraeter respetively) whih physial diension the fore has. The ultipliation of eternal fore and o-ordinate displaeent, or internal fore and deforation ust yield (virtual) wor. 8.. Driving fores A speifi appliation of (8.5) onerns the driving fores of the ehanis. In the EMapproah they are onsidered as internal fores (one or ore oeffiients of the p vetor). Writing these driving fores epliitly: / T f (8.7) it an be reognized easily that the applied fores ust be ultiplied with the vetor of first order ineati transfer funtions to obtain the (stati) driving fore to ae equilibriu with the eternal fores. The eaple of figure 8.. onerns a siple ehanis with two lins, in a situation that they are perpendiular. The vertial lin is a driving ylinder (for paraeter ). There is one oving node A at whih fores +G and - eert: f = +G - T

8 Wb333 Kineto-statis and Dynais 8.6 Wb333 / KB D A 4 E 3 A 5 B 3,4,5 : support eleents v p T 5 D f Ao yao A ya B yb f p 3 4 v5 p N M N N3 N4 N5 driving fore reation fores ig Reation fores, ipliit ethod Wb333 / KB A R A B R R3 p pt D f o pot p f D A ya B o Ao yao yb p o f R R R3 reation fores ig Reation fores, epliit ethod

9 Wb333 Kineto-statis and Dynais 8.7 In this speial position the first order transfer funtion an be found intuitively as: /T = and the vetor produt (8.) yields that the driving fore (noral fore N ) equals (pressure). The eaple of figure 8.. shows a slider-ran ehanis with eternal fore at node A and at node B. To find the vetor /T the atri D p an be written out and filled with its nuerial values (using here =45 and =). The result of anual atri inversion is presented in the figure. Beause the driving quantity is the seond for paraeter (seond row of atri D p ), the vetor of first order transfer funtions / is the seond olun in the inverse of atri D p. Applying (8.7) the driving fore, to be understood here as an internal driving oent =M, an be alulated (M = -.94 ). The inus sign indiates that the internal fore is to be understood as a opressive fore, sine it tries to derease the orresponding for paraeter (driving angle, opare the ylinder in the previous eaple) Support fores Two ethods to find the support fores (reation fores eerting at the frae) will be onsidered. Ipliit alulation. The odel an be etended with etra eleents, suh that the fied nodes will be onneted to the frae by presribed for paraeters (oneptually the sae tri as in ineati optiization). The sae ehanis as in the previous hapter will be used as an eaple, see fig oint A has been onneted by binary lins (eleents 3 and 4) to the frae. The noral fores in these bars (N 3 and N 4 ) will be oeffiients of the vetor of internal fores p, when applying the general rule (8.6). These noral fores are then the support fores of node A (but ind the diretion: a positive value eans a tensile fore, a negative value eans a opressive fore). Epliit alulation. The reation fores need to be inluded diretly in the ethod. Now the displaeent of fied o-ordinates o ust be understood as nonzero, otherwise the reation fores annot ontribute to virtual wor. It will be suffiient to onsider all for paraeters as undeforable ( p =). Equilibriu of the eternal fores requires then that W e f, f, f Referring to eq.(5.4) it follows that p p po o, D D and [D p ] D po o o o ust be applied instead of eq. (8.4). The result beoes: f o pot pt po T p D D f D In other words: the supporting fores an be alulated diretly after that the internal fores are nown. A speifi part of the D-atri, referring to the fied o-ordinates and presribed for paraeters [D po ], is needed here. In figure 8..4 the odel of the saple ehanis, with whih the reation fores an be found epliitly, has been depited. In the present version of RUNMEC the epliit ethod is not supported. (8.8)

10 Wb333 Kineto-statis and Dynais 8.8 Wb333 / KB Ipliit ethod A A A v u 3 3: bearing eleent (TR) resribed for paraeters: u 3 = v 3 = 3 = Epliit ethod A y y f : equilibriu fores of single eleent Bearing fores as internal fores inluded in p f D T ig Connetion fores (bearing fores) in a hinge Wb333 / KB R y ain diretion of bearing fore t 5 R y t 5 t 4 t 6 t 6 t 3 t 4 R t 3 t R t 7 t t t 7 t8 t ig Typial behaviour of bearing fore (polar diagra)

11 Wb333 Kineto-statis and Dynais Connetion fores Coparable with the support fores the onnetion fores (bearing fores) an be deterined by two different approahes. Ipliit ethod. An etra eleent is used to odel the bearing fores as internal fores. or a revolute joint the ternary eleent- is typially suited for this tas, see fig (left part). The two neighbouring eleents ust not share a node, but get a node eah (A and A ) that will be shared with the onneting eleent. The two points A and A an oinide by presribing the for paraeters of the onneting eleent (u = v = ). The angular o-ordinate of the ternary eleent is to be understood as fied and will usually hosen to be zero. An advantage of this ethod is that the ass of the bearing house an properly be divided over the two nodes involved with the onnetion. Disadvantage is that the odel (the diension of the D -atri) beoes bigger. Epliit ethod. Assue that the internal fores have already been alulated (the p -vetor is nown). One of the eleents having the onnetion is to be disassebled, see fig (right part). The onnetion fores of this eleent are to be understood now as (part of) the total eternal fores (equilibriu fores f eq ) of that eleent nuber. Naturally the equilibriu fores ust ause the nown internal fores of that eleent: eq f,, (8.9) Applying the ontinuity equation =D it follows that f eq = DT (8.) The onlusion is that the ontinuity atri D of just that eleent is required to alulate the equilibriu fores of that eleent. In these values the onnetion fores of the bearing under onsideration are oprised. The epliit ethod loos siple enough to perfor oasionally a anual alulation or verifiation. It ust be notied however that the eternal fores f, eerting at the eleent oordinates (inluding ass fores, see net hapter), ontribute to the equilibriu fores. The onnetion fores (f on ) are then: f on = f eq - f (8.) An abiguity arises when the ass distribution of the bearing house oes into aount. (Mass) fores an only be obined with displaeents of o-ordinates. Using the shared node onnetion of the two lins, a further speifiation is needed whih part of the bearing house ass belongs to the disassebled eleent. This speifiation needs to be introdued here anually in the vetor f. In any design ases the final shape of the eleents will be doinated highly by the onstrution of the bearings. The alulation of the onnetion fores, usually for worst ase assuption of eternal fores, is ertainly required for the bearing design (or to selet bearings fro a atalogue). ig shows soe eaples of typial bearing fore behaviour in a polar diagra. Suh inforation is a basis for the priary deisions about the bearing design.

12 Wb333 Kineto-statis and Dynais 8. Wb333 / KB ass partile y inertia fores d Alebert reation fores y y - y - y M,J y M y y J p - -M y - y ig Replaeent of ass by eternal fores (d Alebert priniple) Wb333 / KB z z,j z equilibriu equivalent ass distribution for bea eleent z = +.z = ( -z) J + J +.z + ( -z) =J z = z ( - z/) = z.z/ J p = J =.5 {J z - z.z( -z)}, J, J d Alebert fore vetor f J J y y ig Equivalent ass distribution for a bea eleent, without using a entre of ass

13 Wb333 Kineto-statis and Dynais Mass fores 8.3. riniple of d Alebert Aording to Newton the fore oponents to give a partile (a point ass ) aeleration are: y y These fore oponents ust be delivered by the eleent the partile belongs to. The partile eerts thus a reation fore at the eleent (d Alebert priniple, see fig. 8.3.): y y (8.) These fores will be onsidered thus as eternal fores, whih replae the ass. In any ases a wheel-lie part an be odelled as a point ass with finite rotational (polar) inertia J p. The orresponding reation oent aording to d Alebert is then: M (8.3) J In the EM-approah eternal fores ust be related to a (oving) o-ordinate. The d Alebert fores have only eaning when the orresponding o-ordinate is available in the ehanis odel Mass entre and bea eleent In any ases an eleent with length (binary eleent, bea) has, ore or less, a unifor ass distribution. To desribe a ass distribution in the EM approah a proper set of eleent o-ordinates is required to onnet the d Alebert fores. A distributed ass needs thus to be replaed by soe disrete ass distribution. Two onepts are useful, see fig. 8.3.: With a ass entre. An etra node for the ass entre will be used. Now the ass z of the bar an be onentrated as a point ass in the ass entre. Usually the bar has also a polar inertia, so the odel needs also a rotational o-ordinate at the etra node. It loos natural to use two bea eleents rigidly onneted in the ass entre. There are now only ass fores ating at the ass entre point, whih are equivalent (with respet to the equilibriu of fores) with the real ass fores. Without a ass entre. No etra point will be introdued, so the equivalent ass fores an only at at the eisting o-ordinates of the bar or bea. The total ass z ould be divided over the two end-points and of the bar, proportional to their distane of the (virtual) ass entre (see fig lower part). But this affets also the polar inertia of the lin, whih will beoe now: z ( z) The really eisting polar inertia J z (= for a unifor distribution) an be orreted however by opensating rotational inertia at the end points, whih ust have then a rotational o-ordinate. The bea eleent is very well suited for this tas: it has a rotational o-ordinate at both end points. A hoie an be ade now where to opensate: either at or at (or half at both points, as suggested in fig. 8.3.).

14 Wb333 Kineto-statis and Dynais 8. y. M d d( ) d =.d d M = total ass = ass per length ig y Uniforly distributed ass of a binary eleent, ontribution of a partile d Wb333 / KB Linear displaeent field d d( ) dy dy( ) d d dy Linear aeleration field ( ) y y( ) y D Alebert fores: ( ) y d y ( ) y Wb333 / KB d M d T T W T T T M [ ] [ ] d = vetor of equivalent node fores M = total ass M = ass atri y y 3 M y y ig Mass atri of a binary eleent (uniforly distributed ass), using equivalent node fores

15 Wb333 Kineto-statis and Dynais 8.3 Both onepts speify a solution for equivalent ass fores, whih eans they are orret with respet to the equilibriu of the fores. The (internal) fore distribution is possibly only an approiation (disussion later in hapter 8.3.5). Both onepts ouple diretly d Alebert fores to o-ordinates. Therefor the (vetor of) ass fores an be epressed siply using a diagonal ass atri, see fig Mass atri of the binary eleent The bar, see fig has a unifor ass distribution. The total ass is M and the speifi ass (ass per length unit) is denoted. The intention is to find the equivalent ass fores (vetor, its oponents ating at the four o-ordinates of this eleent). The bar ould be divided into any partiles with ass d at line. A partile at position along the bar will be displaed dependent on the displaeents of the nodes and. Sine the displaeent field is linear, a linear atri an be onstruted to epress the displaeent (d, dy) of the ass partile: d d( ) dy d dy( ) d (8.4) dy Observing that the aeleration field is also linear (see hapter 3.3.) the sae atri an be used to epress the aeleration of the partile, dependent on the aeleration of and (vetor ). The d Alebert fores at the partile d M d [ ] (8.5) ust give the sae aount of virtual wor as the equivalent node fores : T T W Substitution of (8.4) and (8.5) leads to (see also fig ) T T M [ ] [ ] d T in whih the integral at the interval yields the atri This atri, ultiplied with the total ass M of the eleent, is nown as the ass atri M. The final result, the equivalent ass fores, an be epressed then as: 3 6 y 3 6 y M (8.6) 6 3 y y 6 3 Note that this ass atri has also ters off the diagonal. Considering the internal fores, this ass odel will not desribe the bending effet due to the unifor ass.

16 Wb333 Kineto-statis and Dynais 8.4 C A A unifor ass: eleents,,3 luped ass at nodes A,A,B,C Wb333 / KB 6 M 6 M A 3 M A 6 B 6 B C C J M B rotation inertia at eleent 4 (o-ord. 4 ) T = Ao y Ao A y A B y B C y C 4 ig Mass atri of a ehanis (eaple) Case: bar rotating at onstant speed, ass distribution: unifor Wb333 / KB Distribution odels M eat M.5M.5M M M M M M 6 3 M 3 ig Internal fore distribution (eaple)

17 Wb333 Kineto-statis and Dynais Mass distribution of a ehanis (eaple) The ehanis of fig will be onsidered as an eaple. The three lins have a unifor ass distribution. In addition the points A, A, B and C have been given point asses (to indiate the ass of the bearing houses). The polar inertia J M at the input shaft (driving otor, gear-bo, oupling, fly-wheel) needs an angular o-ordinate, so the half bea eleent 4 has been onneted to the ran A A. Now the assebled ass atri of the whole ehanis an be written out, see the figure The ontributions of the ass speifiation, referring to the sae atri oeffiient, an siply be added Internal fore distribution Mass fores an only be onneted to o-ordinates of the ehanis odel. Espeially when a unifor ass distribution of an eleent oes into aount, the effet to the internal fore distribution should be onsidered. The eaple of figure shows a ehanial syste onsisting of one bar rotating with onstant speed around one of the end-points (rotating ran). The noral fore in the bar will be onsidered. The eat internal fore distribution is soe (quadrati) funtion of the radius: the aiu is M at the entre, the iniu at the end. In a ass distribution with the ass divided equally over the two end-points, the noral fore would be onstant at M. The ass onentrated in the ass entre would give a onstant noral fore of in the inner half of the bea, and zero fore in the outer half of the bea. M A ass distribution aording to the ass atri of eq. (8.6) would speify a onstant noral fore at 3 M. It will be lear that none of these ass distributions is orret. It depends on the user s hoie whih ass distribution will be preferred. In a oneptual phase of the design however all odels are qualitatively orret regarding the aiu value of the fore. A oparable note an be ade with respet to the deforations of the eleent, as aused by the fores. They will ertainly not be eat, but ight be useful as an indiation in the oneptual design phase (ore about deforations in hapter 8.6).

18 Wb333 Kineto-statis and Dynais 8.6 Wb333 / KB BIN luped T ot y y TR R unifor T y ot vr y R yr TR R S T y y R ot RS vr v yr S ys ig at at RS Soe ass distribution options with a slider pair Wb333 / KB A + Bearing eleent (Ternary TR) A A v u 3,J,J ig Mass distribution of a revolute joint (eaple)

19 Wb333 Kineto-statis and Dynais Mass distribution of joints A sliding joint an in the EM-approah easily be odelled with a binary eleent, by assuing the length non-presribed. Suppose now that a unifor ass distribution will be taen. In that ase the speifi ass (ass per length unit) would vary and this is physially not realisti. Sliding joint odels, whih are ore realisti with respet to the onstrution, are depited in fig In any ases one of the sliding eleents (the guide) is bea-lie and an be given a unifor ass, see the iddle part of the figure. The other sliding eleent is short and an be regarded as a point ass. The ternary eleent() is very well suitable to speify suh a ass distribution. Using this eleent the internal fore referring to for paraeter v is the bearing fore (perpendiular to the guiding eleent). When the slider pair onsists of two telesopi ylinders, a unifor ass distribution ould be proposed using two ternary() eleents. Mutually they share a third point and an end-point, as indiated in the figure (lower part). Notie that the distane R is not allowed to be zero (but in the real onstrution this is also not allowed). The two internal fores, referring to the for paraeters v and v R, are the bearing fores. A revolute joint deserves attention with respet to the ass of the bearing house. In addition to the way a (stati) bearing fore an be odelled ipliitly using a help eleent (preferably a ternary- eleent, see fig. 8..5), the ass of the bearing house an be split up as indiated in fig When polar inertia is signifiant, an angular o-ordinate ust be present at the node. In that ase a ternary eleent() or a bea eleent ould be used. Oasionally the ideas epressed in the figures and need to be obined. or instane: when the revolute joint speifiation is required at the sliding point it an be neessary then to add etra (duy) eleents, in order to reate an angular o-ordinate. 8.4 Other eternal fores 8.4. Springs Charateristi for a spring is, that the (ineati) degree of freedo of the ehanis reains unhanged when a spring has been added. By its nature a spring an have an internal fore and a large deforation. In the EM-approah these harateristis of the spring an be aintained while the spring fores are onverted to eternal (reation) fores. Sine suh a fore is dependent on the geoetry of the ehanis, it needs to be oupled to a dependent for paraeter. A typial spring has been depited in fig This (oil) spring an be represented by a binary eleent, but with non-presribed length. The idea is, that the relation between the internal fore (here the noral fore N ) and the for paraeter (here the length of the lin), usually indiated with the spring harateristi, is given. The spring eleent will be disassebled to find the equilibriu fores f using eq.(8.). The reation fores f are then the (eternal) fores eerted by the spring at the o-ordinates of the ehanis. This idea is general for all inds of dependent for paraeters. All inds of springs an thus be odelled. Eaple: a flat spiral an be oupled to the angle between two bars (the differene angle needs to be oposed by subtration of the two angular for paraeters). Although in ost ases springs will be assued to have no ass, all ass speifiations onerning the spring eleent an norally be used.

20 Wb333 Kineto-statis and Dynais 8.8 Wb333 / KB Equilibriu fores: Reation fores: - f f onerns a (ineatially) non-presribed for paraeter D T y = N y = N - N -N os y sin f N os y sin ig Spring fore as an eternal (reation) fore at onneting nodes Wb333 / KB etension oil proper woring range opression oil + + = tan = - / triple spring definition ( ) ( ) ; ; ; + + ig Linear spring and woring range (generalized definition)

21 Wb333 Kineto-statis and Dynais 8.9 Usually a linear spring harateristi oes into aount, lie depited in fig (upper left). The spring harateristi is then typially, with as the spring onstant (stiffness): ) ( ) (8.7) ( or the oil, oupled to a binary eleent, this spring harateristi an be interpreted as: ) ( ) (8.8) ( Notie the typial differene between an etension oil and a opression oil with respet to the woring rang of the internal fore. It loos a good idea to define the (linear) spring harateristi in a speial way: by the value of (spring onstant) and the value of (). The advantage of this ethod is that it is also possible now to speify a onstant fore with variable diretion. The spring eleent is borrowed for this purpose, speifying = and () as the onstant fore. Realisti springs usually have a woring range. A opression oil for instane has in its released state (and for greater length) a zero fore. In the state of iniu length all windings touh and the stiffness inreases enorously. An etension oil will be sla below a ertain iniu length and be peranently defored beyond its aiu allowable length. A triple range definition of a (linear) spring, see figure 8.4. (right and lower part), ay define ore realisti spring fores. The proper woring range, the iddle part of the spring harateristi having stiffness between the values and, an be defined as above. The state outside this range needs a orretion of the stiffness, as proposed here with values and. In the Rune progra this triple spring harateristi has been ipleented Daping fores Visous daping (internal fore proportional to veloity of any dependent for paraeter) an be taen into aount by (8.9) in whih is the visous daping oeffiient. The equilibriu fores of the eleent aording (8.), with negative sign (reation fores), an be used as eternal fores ating at the o-ordinates of the eleent Gravity fores Atually these fores onern ass fores in the aeleration field of the gravity. A useful idea is to define a gravity vetor g onerning all o-ordinates of the ehanis odel. or the usual gravity in the y diretion suh a vetor would have oeffiients g in all y-oordinates. Suh a vetor will typially loo lie (g without underline is the gravity onstant): T g g g g (8.) The (eternal) gravity fores an be alulated then using the ass atri M of the ehanis: f z [M] g (8.) Notie that the gravity fores at at all o-ordinates, even the fied ones. The latter fores ontribute to bearing fores, but only in an unabiguous way when the bearing has been odelled properly (see 8..4).

22 Wb333 Kineto-statis and Dynais 8.

23 Wb333 Kineto-statis and Dynais Other eternal fores (roess fores) Mehaniss will be designed to transfer otion and fores. Usually soe proess is to be arried out in whih fores are needed. Soe eaples an be found in hapter, lie in the utting of steel blade, the digging of iron ore, the gas opression in a obustion engine et. Suh fores will be onsidered here as reaining eternal fores. They need a user speifiation. 8.5 Equation of otion 8.5. General priniple The equilibriu equation (8.6) an be used again, but now with the ass fores and other eternal fores: p pt D f M ( g ) (8.) The atri M is the part of the total ass atri dealing with the oving o-ordinates (shaded part in fig ). Vetors and g onern the oving o-ordinates of the aeleration vetor and the gravity vetor (eq. 8.) respetively. In vetor f all other eternal fores (also fro springs and dapers) are ontained here. The atri D p onerns the first order ontinuity equations, the ineati relation between presribed for paraeters and oving o-ordinates of the ehanis. The aelerations as epressed in (5.3) with transfer funtions and driving funtions (of tie) T //.. /. (8.3) an be substituted in the equilibriu equation. Now this equation (8.) is a (differential) equation in the driving quantities. This equation will be alled further the equation of otion of the ehanis or pratial use the part of the equation of otion, whih onerns the input of the ehanis, deserves speial attention. This part an be written separated as / T f M ( g ) (8.4) Together with (8.3) this part is alled further the equation of driving otion. The equation of otion an be used in the following two ases. All input otions (t) are given funtions of tie, and so are the input veloities and aelerations. In any ehanis position the aelerations of all o-ordinates an be alulated then aording (8.3). The equation of otion provides the internal fores diretly, inluding the driving fores. This type of alulation will be alled here dynai analysis. The driving fores are given. Now the equation of driving otion is a differential equation in, whih ust be solved first. One (t) and its tie derivatives are nown, the first ase an be followed. There is no objetion to speify the driving fore as a funtion of the input otion: (,, ) (8.5) Suh a speifiation is alled further a otor odel. This type of alulation is nown also as (dynai) siulation.

24 Wb333 Kineto-statis and Dynais 8.

25 Wb333 Kineto-statis and Dynais Redution of fores to the driving shaft Assue here a ehanis with one degree of freedo. Substitution of (8.3) in (8.4) leads to / T / / T // M M In the literature a oparable equation an be found: / T (f M g ) (8.6) dj red T J red f red (8.7) d J red is alled the inertia oent of the ehanis redued to the input shaft. red ontains the redued eternal (other) fores, inluding gravity fores. The oparable ters in the EMapproah an be reognized easily. The ass atri is usually a onstant atri, but the first and seond order transfer funtions are typially not onstant. The redued inertia is thus dependent on the ehanis position. Notie that (8.6) shows very nie how ineatis plays a role in dynais. To find the stati oent (brae oent) to hold the ehanis iovable eq.(8.7) defaults to / T f (8.6) whih has been found earlier in hapter 8.., see eq.(8.7). This result is also valid for ulti- DO ehaniss ( is a vetor then and / a atri). Writing out the equation (8.6) for a ulti-do ehanis will be oitted here, both eq. (8.3) and (8.4) together epress the equation orretly Driving fore and otor odel Eaples of otor odels: A onstant fore (for instane a braing fore to stop a running ehanis). A zero fore (for instane the ehanis will be set free at a ertain tie and starts oving due to the eternal fores). A spring otor (linear relation between and ), opare eq. (8.7). An asynhronous AC-otor with a steady-state harateristi as depited in fig..4. (left part). Here the driving oent M = - is a funtion of the angular speed. In the literature a atheatial epression for suh a otor odel an be found: s M M a s a ;s a ; (8.9) In design pratie the hoie of a driving otor ay need both appliations of the otion equation. The seletion of the driving otor (out of a atalogue) will typially be based on worst-ase assuptions, lie aiu required power, aiu required torque at a ertain (aiu) speed et. In a ahine onept with a entral driving shaft it will be aeptable then to suppose a onstant speed of the driving shaft. In a ulti-do roboti ehanis the given otion funtions and proess fores will be applied at the wrist of the robot (otion tas) and need to be transfored first to the input otion to find the aiu required driving torques (using the inverted ineati odel). One a driving otor has been seleted, the properties of the otor are nown. The siulation an be used for instane to verify whether or not the driving speed is onstant enough or the robot tas will be perfored aurate enough.

26 Wb333 Kineto-statis and Dynais 8.4 Wb333 / KB integration error t t t dt t orretion predition of eat value of t t t t ig Integration error in the equation of otion for the ran angle

27 Wb333 Kineto-statis and Dynais Integration of the driving equation of otion or dynai siulation the equations of driving otion need to be solved (integrated). Sine the oeffiients depend on the ehanis position (the variables of the equation to solve) they are non-onstant. Only nuerial integration ethods oe into aount. Standard proedures for nuerial integration are well nown (lie ethod of Heun, ethod of Runge/Kutta, ethod of Gordon and Shapine and others). To give an ipression of the required alulations and soe probles of the integration proedures, an eaple will be presented here (Heun). Assue a one-do ehanis with no driving otor attahed at the input shaft (the ehanis will ove freely when released at start tie). The driving fore is thus zero. At start (tie t ) the input position is and the veloity. In this position the aeleration of the input shaft an be alulated with (8.6), using the ineati first order transfer funtions, the eternal fores and the ass atri: f J red red / T / T f M / It will be lear that the ehanis will start to ove only when at least one of the eternal fores is non-zero. After soe sall tie step t, at tie t = t + t, the new position and veloity an be predited: t t t and (8.6) an be used to alulate the aeleration of the input shaft at t : / T / T / T // / (8.3) f M (8.3) M Now it should be verified how aurate the integration is. This an be done by realulating and, reading for i the position : i i i i ( ( i i ) t i i i ) t i (8.3) The differene between (8.3) and (8.3) is the integration error on and respetively, see fig for the error on. The orretion yle an be repeated until this error is sall enough. The preise aount is dependent on the auray deands of a user. When the error reains too big, a saller tie step ay iprove the solution. urther rears: The integration requires that the denoinator in eq. (8.3) is non-zero. The ehanis ust have soe oving ass! It is well nown that suh nuerial integration proedures ay beoe divergent (unstable). One reason ould be that an eternal fore is not ontinuous. In that ase the predited aeleration (8.9) will also have the jup. Bad onvergene an be the result. Typially the use of a triple spring as proposed in fig is risy in siulation.

28 Wb333 Kineto-statis and Dynais 8.6 Wb333 / KB EA S EI S ** G I S torsion EA S ** ig Soe stiffness interpretations of eleents Wb333 / KB deforations internal fores eternal fores D N M M M M N D N D M N y 3 D M y 3EI EI D M EI EI ig Coupled bending stiffness of the bea eleent

29 Wb333 Kineto-statis and Dynais Stiffness of the eleents and deforations 8.6. The onstitutive equations It will be assued that the deforations are sall and proportional to the fores. The deforations (due to fores) in the EM approah onern the presribed for paraeters. In ineatis they are rigid (presribed zero), in dynais they an be given a finite stiffness by eans of a so-alled onstitutive equation. Stiffness (S) is a property of an eleent, so the equation an be desribed for eah eleent separately: S (8.33) The stiffness ust be understood as generalized. It depends on the type of the for paraeter how the stiffness is defined. Soe typial eaples are depited in fig Soeties the user an give his own interpretation to the stiffness, with respet to the onstrutive realisation of the lins. In ost eleents presented so far, the deforations within one eleent are deoupled, that eans there is one equation for eah presribed for paraeter. This is even true when two eleents are assebled, lie in the fied angle (by eans of a torsion bar) between two lins, see fig left below. The deforation an then, for given internal fore, be epressed as: S (8.34) There is however one eeption: the bea eleent, see the net hapter Bending of the bea eleent The two bending deforations at the end-points should be oupled to define proper bending deforation, see fig To find the deforations of the whole eleent (assue also the length is presribed and has noral stiffness) the eleent an be plaed in an arbitrary but suitable position (horizontal in fig. 8.6.). In this position the deforations are related to the displaeents of the o-ordinated by the ontinuity equations (as used in ineatis): D y (8.35) Standard forulas are available to epress the displaeents of the o-ordinates (only for point, sine point is ept fied): * S f EA y 3 3EI EI EI EI N D M Assue now the internal fores ( N, M and M ) are given. Eq.(8.36) and the equilibriu equations (8. ) substituted in (8.35) yields: * T D S D S (8.36) (8.37)

30 Wb333 Kineto-statis and Dynais 8.8

31 Wb333 Kineto-statis and Dynais 8.9 The ter in braets is by definition the (inverse) stiffness atri. Although the D atri is dependent on the position of the eleent, the result is of ourse a onstant atri. Notie that it is not neessary to do the atri inversion, it an be obtained diretly by atri ultipliation as indiated. The final result is: S EA 3EI 6EI 6EI 3EI (8.38) in whih the bending stiffnesses are oupled. It is a good idea to all this inverted stiffness atri the fleibility atri of the bea eleent. All stiffnesses are atually regarded as fleibilities. ratially it eans that rigidity (infinite stiffness) should be speified as zero fleibility. This is a better idea regarding nuerial values in oputer progras osition auray of the ehanis or a given internal fore distribution and given fleibility of the lins the deforations of the presribed for paraeters an be alulated using (8.34) or (8.36). The displaeents of the o-ordinates due to these deforations an be onsidered with respet to position auray. They an be alulated with the ontinuity equations (atri D ) as used in ineatis for position update (5.8). Atually the inverse of atri D an be understood as the sensivity atri for deforations. When ultiplied with the deforation vetor, the (linearized) o-ordinate displaeents are the result. or preise alulation of the auray the iterative proedure ould be used as desribed in hapter Rear: the deforations due to the ass fores an be less aurate (see hapter 8.3.5) Typially suh auray alulation will be used to find out the aiu deviations of rigid body otion. The linearized estiation of o-ordinate deviation provides usually suffiient inforation whether or not the eleent is stiff enough. It is a good idea to tae here all presribed for paraeters as deforable, even if the fleibility is very low Dynai siulation with fleible eleents To find out how the deforations of a ehanis behave as a funtion of tie, a dynai siulation is needed. There is no new theory required. The ehanis odel an be adapted to do this tas. The basi idea is that any elastiity ust be regarded (odelled) as a degree of freedo (drive) at whih a driving harateristi an be added. Here the spring harateristi oes into aount as the otor odel. To avoid onfusion suh a fleibility will be alled a dynai degree of freedo, and the original ones the ineati degrees of freedo. Rear: do not onfuse with a spring (a spring does not hange the degree of freedo) Dynai degrees of freedo need not to desribe large deforations. Bending deforation of the bea eleent is typially suited as a dynai degree of freedo. With respet to the integration of the equations of otion, in addition to hapter it an be reared: The use of fleibility, without daping, will lead to undaped osillating otion. When the eternal fores vary with about the sae frequeny, the osillations ay beoe very large. A high frequeny of the osillation (high stiffness or low ass) needs a sall stepping tie in the integration proedure, whih will beoe slow then.

32 Wb333 Kineto-statis and Dynais 8.3 DYN ANA ;test eaple dynais (for Rune version 3.) TOOLOGY ele BIN 8 9 ; support of point A BIN ; support of point A 3 BIN TR ; etra ass point on the oupler 5 BIN TR ; bearing in point B 7 BIN ; spring for lenl lenl 3 lenl lenl lenu lenv lenl lenu lenv 6 9 angb 3 lenl 7 - ; dependent for paraeter (spring) nrdof GEOMETRY fi Table ov para DYNAMICS inertia unifor spring - -5 ; and siga at length zero gravity STORAGE bufinp ; ran angle buf 6 ; oupler point buf 7 bufd 6 5 ; aeleration of oupler point bufd 7 6 bufe bufsiga - 3 ; length of the spring ; noral fore in lin (vertial support) ; noral fore in lin (horizontal support 6 6 ; ass fore at oupler 8 8 ; horizontal bearing fore in B 9 9 ; vertial bearing fore ; driving fore RINTBU ANIMATE fil BEGINIC ploturv ; oupler urve draweh ENDIC BEGINIC ploturv ; polar urve of support fore in A ploturv 8 9 ; ide of bearing fores in point B ENDIC MOVEMENT tstepe ; eatly one yle END ;(auto. saling, true distane)

33 Wb333 Kineto-statis and Dynais 8.3 It will be lear that the aount of alulations rapidly inreases with the nuber of dynai DO s. It is reoended then to arefully ae use of fleibilities in siulation alulation. Typially only few fleibilities should be onsidered, and only in ase the fleibility is relatively high. 8.7 Eaple with RUNMEC The ehanis of fig has been analysed dynaially (ineto-statis). The ehanis has been given several dynai properties, lie: A spring (eleent 7) A oupler with bending fore (eleent 4) Support eleent of the fied point A Bearing fores at point B The input file for the RUNMEC progra is listed in table 8.. The output plot of the oupler urve is presented in fig The polar plots of the bearing fores (in A and B) have been depited in fig Wb333 / KB ig Saple ehanis for dynai analysis

34 Wb333 Kineto-statis and Dynais 8.3 Wb333 / KB B C spring A A B ig Mehanis and oupler urve Wb333 / KB at B at A ig olar plots of bearing fores