Dynamics of Machines. Part 1: Getting started (1) Introduction SPACAR. Part 1: Getting started (2) Mass-spring model

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1 Part : Getting started () Lecturer: Dnaics of Machines Introduction SPACAR Universit of Twente / acult of Engineering Technolog (CTW) Mechanical Autoation (Wa) Horstring (building ) Z 9 Phone: (053) / 50 Eail: R.G.K.M.Aarts@utwente.nl WWW: Link on htts:// Download the software and eales fro our web site. Two arts: The SPACAR GUI to create odels (Microsoft Windows). Unack ZIP-file anwhere on our PC s harddisk. On-line hel. Built-in udate anager. License until March 3, 07. The SPACAR toolbo for MATLAB/SIMULINK (3-bit/64-bit MS Windows onl, MacOS in rearation). ollow the installation instructions. Launch MATLAB and etend the ath (referabl eranentl). Additional hel and deo s available. License until Seteber 30, 06. Note the inforation about ossibl issing files. DoM-Sacar/0/ DoM-Sacar// Part : Getting started () Mass-sring odel Chater Using the SPACAR GUI: Run sacar0.ee Build our odel (*.sa file), create *.dat file and launch MATLAB. The SPACAR toolbo for MATLAB/SIMULINK: Inut is *.dat file, either fro the GUI or created anuall with our favourite tet editor. Run the sacar(...) coand. Outut are MATLAB variables (also stored in data files) to be investigated with our own tools or e.g. savisual. The anual section on the web site includes: A guide on Protote odelling of echanical sstes The anual of the SPACAR toolbo. Model sste as a sile one degree of freedo ass-sring sste. All ass is lued in a single euivalent ass e and the euivalent stiffness k e reresents all elastic coonents. Inut VCM force and outut osition : G(s) = = e s. + k e Suort colun k e e, v DoM-Sacar// DoM-Sacar//

2 irst SPACAR odel: degree of freedo The ass-sring sste can be odelled as a one-diensional SPACAR odel: Identif eleents with their coordinates and deforations. Sect.. Planar truss eleent (PLTRUSS) β l (k) Two translational nodal oints: and. One eleent for the sring: a (two-diensional) truss eleent. Two translational [ nodal ] oints and with two coordinates each: (k) truss = = [,,, ] T. A single deforation ode e of this eleent reresents the elongation: = l (k) l (k) 0, with l(k) = ( ) +( ). DoM-Sacar// DoM-Sacar//3 Building odel fdtr.sa Creating the truss Create the truss. The end nodes are New secif the (initial) coordinates (0.0, 0.0) and (0., 0.0). Edit the truss to set its diensions. Building odel fdtr.sa Constraint coordinates and released deforations Nodal oint coordinates are b default Deendent, i.e. can ove freel deending on the rest of the sste. Set suort coordinates to ied: Both X and Y in and Y in. Eleent deforations are rescribed Zero unless defined otherwise. Set the truss elongation, i.e. deforation e,toreleased. DoM-Sacar//4 DoM-Sacar//5

3 Building odel fdtr.sa Counting the DO s SPACAR coutes the nuber of degrees of freedo NDO in the sste fro NDO = NX NXO NEO, NX =4is the nuber of nodal coordinates, NXO =3the nuber of absolute constraints: ied coordinates, NEO =0the nuber of relative constraints: The reaining unreleased eleent deforation araeters. So NDO =4 3 0=degree of freedo has to be defined. Define e.g. an absolute degree of freedo: Set the X coordinate of node to be a Dnaic DO. A necessar (though not sufficient condition for a valid sste definition is that the couted nuber of DO s euals the nuber of DO s elicitl defined b the user. Building odel fdtr.sa Adding ass Masses and inertial roerties can be secified as a lued ass attached at a nodal oint or as a distributed ass resent along an eleent. Lued ass and inertia are defined b Editing the nodal oint. Set the Mass of node to 0.06 kg. Distributed ass and inertia (er unit length) are defined b Editing the Mass & Inertia roerties of the eleent. Set the Mass er Length of the truss eleent to 0.43 kg/ (reresenting both leaf srings). DoM-Sacar//6 DoM-Sacar//7 Building odel fdtr.sa Adding an eternal force Building odel fdtr.sa Adding stiffness and daing Edit a node to define an eternal force, e.g. a horiontal force of N in node. E d A Edit the Stiffness & Daing roerties of the truss to set the aial rigidit and daing E d A of the truss eleent. Knowing the euivalent longitudinal stiffness k e = 945 N/: () e = k e l 0. or the considered sring leaves the daing is couted assuing a relative daing ζ in the range of 0.0 to 0.00 and knowing that the daing d e =ζ k e e. Net (E d A) e = d e l 0. Building odel fdtr.sa Secification of inut and outut Click the Edit additional Dnaic coands button to enter INPUT OUTX The kewords INPUT and OUTX define a single inut force and a single outut coordinate X, resectivel. The first araeter of both kewords is the inut or outut nuber that corresonds with its osition in the inut vector u or outut vector. The other two araeters define the inut s or outut s nodal oint nuber and the corresonding Cartesian nodal coordinate. DoM-Sacar//8 DoM-Sacar//9

4 Model fdtr.sa SPACAR results 9: Create dat file and launch MATLAB. Start SPACAR fro the MATLAB coand rot in ode 7 : Static analsis and the coutation of the natural freuencies and ode shaes of the sste: >> sacar(7, fdtr ) Outut: MATLAB lot window with the sste in its euilibriu configuration. Log file fdtr.log. Binar data files fdtr.sbd and fdtr.sb. MATLAB variables with results read fro these files. Model fdtr.sa SPACAR results () Horiontal dislaceent of node : >> (ln(,)) = 0.0 The noral force in the truss eleent: >> sig(le(,)) =.0000 Mass, stiffness, and (first and onl) natural freuenc: >> 0 = 0.07 >> k0 = 945 >> srt(eig(k0,0))//i = Visualisation: >> savisual( fdtr ); DoM-Sacar//0 DoM-Sacar// Model fdtr.dat SPACAR results (3) >> sacar(9, fdtr ) With a ode 9 run the state sace atrices A, B, C, and D are couted and stored in binar data file fdtr.ltv. The can be read with the getss coand. >> [A,B,C,D] = getss(fdtr); >> G=tf(getss(fdtr)) Transfer function: sˆ s >> bode(g) The latter coand is used to create a Bode lot. Magnitude (db) Phase (deg) Bode Diagra reuenc (rad/sec) Second SPACAR odel: e degree of freedo Alternativel, a relative degree of freedo can be defined b setting a deforation to be a Dnaic DO. Of course NDO = NX NXO NEO has to be satisfied. Sect.. Change Coorinate te of the X coordinate of nodal oint into its default te Deendent. Change the Deforation te of the Elongation of the truss eleent into a Dnaic DO. Outcoe as before... DoM-Sacar// DoM-Sacar//3

5 Two-diensional odel Chater b, l b s, E s, l s 3 Planar bea eleent (PLBM) R n R n n φ n R n β(k) R n φ Bending of the leaf srings odelled ore accuratel using lanar bea eleents. Translational and rotational lanar nodal oints. our Cartesian coordinates (, ), (, ) describing the osition of the bea in the (, )-coordinate sste. Two rotation angles φ and φ reresenting the angular orientation of the triads (R n, R n ) and (R n, R n ) at the nodes and resectivel. DoM-Sacar/3/ DoM-Sacar/3/ Planar bea eleent (PLBM) () Nodal coordinates: (k) bea = Deforation araeters: elongation: bending: φ φ = l (k) l (k) 0, = (R n, l (k) ), = [,,φ,,,φ ] T. R n 3 = (R n, l (k) ), 3 The deforations are indeendent for rigid bod oveents of the eleent and have a clear hsical eaning. l (k) R n -DO odel b, l b s, E s, l s 3 Three beas: Coon nodes rigid connection. ied suort: Translational nodes and 7, rotational nodes and 8. Solid bar: All deforations Zero (default). In both leaf srings: Release bending odes e and e 3. NDO = NX NXO NEO = as NX =is the nuber of nodal coordinates (4 +4 ), NXO =6the nuber of absolute constraints (ied + ), NEO =5the nuber of relative constraints, so So NDO =: E.g. horiontal osition of node 3. DoM-Sacar/3/3 DoM-Sacar/3/4

6 -DO odel () Distributed inertia of the solid bar in a lued ass reresentation:. The ass of the eleent should be eual to the su of the lued asses.. The center of ass of the eleent and that of the discrete ass odel should coincide. 3. The rotational inertia of the eleent and that of the lued sste should be eual. l c l = / = / J c = J = l J = l Solid bar Discrete ass odel Solid bar with ass and length l: the rotational inertia relative to the centre of ass J c = l. Euivalent lued asses and rotational inertias are then: -DO odel (3) Define lued and/or distributed asses and inertia roerties. or the leaf srings it is convenient to use the Calculate Inertia button, rovided the bea diensions and densit are defined correctl. Define the aial rigidit (related to longitudinal stiffness /l) and fleural rigidit EI (related to bending stiffness EI/l 3 ). The Calculate button offers a user-friendl autoatic calculation of the stiffness roerties. Daing roerties need to be defined anuall, see revious odel with relative daing ζ. = = and J = J = l. DoM-Sacar/3/5 DoM-Sacar/3/6 -DO odel (4) SPACAR analsis ode 7 and 9 as before: Still natural freuenc! SPACAR analsis ode 8 for buckling analsis: The critical loading araeter λ i is couted, such that the buckling load f i = λ i f 0, where f 0 reresents a static reference loading vector of nodal forces/torues. Edit node 3 to al a vertical force with a agnitude of N in the negative direction. Deterine the euilibriu configuration and the critical loading araeter λ : >> sacar(8, fdb ) >> eig(-k0,g0) = >> savisual( fdb ) 3-DO odel Release elongations e of both leaf srings as well. NDO = NX NXO NEO =3 as b, l b 3 s, E s, l s NX =is the nuber of nodal coordinates (4 +4 ), NXO =6the nuber of absolute constraints (ied + ), NEO =3the nuber of relative constraints, so onl the solid bar So NDO =3: E.g. X and Y coordinates of node 3 lus rotation of node 4. Sect.3. Net to the first natural freuenc (0.6 H), now also higher natural freuencies are found (9 H and 3585 H), justifing the revious -DO aroiation. DoM-Sacar/3/7 DoM-Sacar/3/8

7 Three-diensional odel Chater 4 Satial bea eleent (BM) n R n n R n b 3 s, E s, l s, A s, I s n R n n R n R n Modelled with satial eleents. n R n Translational and rotational satial nodal oints. n DoM-Sacar/4/ DoM-Sacar/4/ Satial bea eleent (BM) () Si translational coordinates are fro two osition vectors and describing the osition of the bea in the fied inertial coordinate sste. Si indeendent rotational coordinates as the orientation of each rotational node in three diensions is given b three indeendent rotation coordinates collected in the vectors λ and λ resectivel. orientation of the triads (n, n, n ) at the nodes and. Satial bea: (indeendent) nodal coordinates. As a rigid bod: 6 degrees of freedo. So 6=6indeendent deforation araeters: elongation: = l (k) l (k) 0, torsion: = [ l(k) 0 (R n, R n ) (R n, R n ) ], bending: 3 = (R n, l (k) ), 4 = (R n, l (k) ), 5 = (R n, l (k) ), 6 = (R n, l (k) ). -DO odel NDO = NX NXO NEO = 4 =, Note: Internall in SPACAR rotational nodal oints are described b so-called Euler araeters (4 in each node) NX and NEO are counted differentl. b s, E s, l s, A s, I s NX =4is the nuber of nodal coordinates ( ), NXO =the nuber of absolute constraints (ied 3+ 3), NEO =the nuber of relative constraints, so 3 6 = 7 deforation araeters have to be released, NDO =: Horiontal osition of node 3. DoM-Sacar/4/3 DoM-Sacar/4/4

8 -DO odel (3) DO odel () Create a 3D odel with 3 satial beas. The beas rincile aes ust 3 be secified. Set the coordinates of the suorts to ied. Release the bending odes e 5 and e 6 of both leaf srings. b s, E s, l s, A s, I s In addition the torsion and reaining bending odes of eleent 3 are released in accordance with the local cut in the leaf sring. Define lued and/or distributed asses and inertia roerties. Calculate Inertia button. b 3 s, Es, ls, As, Is Define the aial rigidit (related to longitudinal stiffness /l), torsional rigidit GI t (related to the (euivalent) torsional stiffness GI t /l 3 ), and fleural rigidities EI and EI /l 3 (related to bending stiffnesses EI /l 3 and EI /l 3 ). Calculate button. Daing roerties. Analsis in MATLAB as before... DoM-Sacar/4/5 DoM-Sacar/4/6 Sect.4. 9-DO odel Sect.4. Von Mises stresses Modelling the leaf sring ore accuratel with five beas and releasing 5 deforations: SPAVISUAL can also visualise the stress distribution in fleible satial beas, e.g. in the 9-DO odel with a horiontal force of N: NDO = NX NXO NEO = 7 3 (6 + 5) = 9, NX =4is the nuber of nodal coordinates ( ), NXO =the nuber of absolute constraints (IX 3+ 3), NEO =the nuber of relative constraints (onl solid bar and elongations), NDO =9: Translation of solid bar lus additional deforations. DoM-Sacar/4/7 DoM-Sacar/4/8

9 Modules in SPACAR/MATLAB Manual. SPACAR in MATLAB and SIMULINK Manual.4 echanis toolog geoetr DO s = (, e ) dnaic roerties forces (f, σ ) trajector ath velocit rofile ( o, ẋ o) Rigid or rigidified link odel rescribed trajector μ h leible link odel actual trajector KIN DYN INVDYN LINR ode ode ode 3 ode 4 DO s (, ) joint variables (e 0, ė, σ 0 ) noinal inuts (u 0 ) reference oututs ( 0 ) State sace atrices Linearied euation 0 INVDYN read setoints u 0 0 siulation STATIO LINR ode 7 ode 8 ode 9 Eigen freuencies Buckling loads State sace atrices LINR MATLAB analses read coeff. atrices SIMULINK siulation M 0,C 0,K 0 control araeters DoM-Sacar/5/ DoM-Sacar/5/ Rigid satial aniulator echanis I 4 5,,, 3, 4 3 5, 6, 7 8, 9 Satial aniulator with two (rigid) links (BM) 4, 5 and three rotational joints (HINGE), 3., II v.76 /s Prescribed trajector: Ti otion along straight line with desired velocit rofile. t Manual 3. Procedure (reliinar MATLAB analsis): invdn coutation sacar(, robotinv ): Define the correct degrees of freedo and deforation odes for the inverse dnaics in the kineatics art. Define the trajector. Define the noinal inuts and reference oututs to be used in a future siulation. linear coutation sacar(3, robotinvlin ) using the data fro the revious invdn run: Define the degrees of freedo for the forward dnaics. Define the inuts and oututs to be used in a future siulation. Coute e.g. the reuired (noinal) inut torues and sste atrices (to be discussed later). DoM-Sacar/5/3 DoM-Sacar/5/4

10 Siulation (SIMULINK) oen-loo: Siulation (SIMULINK) closed-loo: robotinvlin robotinvlin Setoint U0 robotsi SPASIM U U(E) Selector E Scoe E Setoint U0 U U(E) Selector Yti Scoe dyti eedforward robotinvlin Reference Y0 U U(E) Selector Yti Scoe dyti sasi rovides a SIMULINK interface siilar to a ode= (forward dnaic) SPACAR analsis. The sste s inuts and oututs have to be secified elicitl in the inut file: INPUTS, OUTE, OUTEP, OUTX. Read noinal inuts u 0 and connect to echanis inuts. Coare echanis oututs to reference oututs 0. Note that this oen-loo aniulator is unstable. robotinvlin Reference Y0 U U(E) Selector E + Ed Oega = 8 beta = 0.85 K Kv control robotinvlin ties M0 eedback robotsi SPASIM Noinal inuts u 0 can be used as a feed-forward signal. Coare echanis oututs to reference oututs 0 and use the difference for feed-back control. The ties M0 block ultilies the aniulator with its reduced ass atri to change it into a MIMO /s sste. The closed-loo sste with PD-like control is stable. DoM-Sacar/5/5 DoM-Sacar/5/6 leible satial aniulator echanis Redo invdn and linear anales with odified inut files including deforations reresenting the fleibilit of the beas. Redo the SIMULINK siulation with this echanis. A non-linear siulation with SPASIM can be ver tie consuing! Reoving degrees of freedo with onl high-freuent vibrations while still retaining correct low-freuent behaviour can reduce the siulation tie considerabl. DoM-Sacar/5/7