Multibody System Dynamics: MBDyn Hydraulics Modeling

Transcription

1 Multibody System Dynamics: MBDyn Hydraulics Modeling Pierangelo Masarati Politecnico di Milano Dipartimento di Scienze e Tecnologie Aerospaziali

2 Outline 2 Introduction Modeling Scales Multibody/Multiphysics Dynamics: MBDyn Software Modeling Approach Hydraulic Library Overview Examples

3 Introduction 3 Multibody Dynamics: unconstrained mechanical systems... M x ẍ=f x, ẋ, t plus kinematic constraints: constrained mechanical systems M x ẍ T / x =f x, ẋ, t x, t =0 System of differential-algebraic equations (DAE) Infinitely fast dynamics (time scale 0) a=m 1 f ẍ=a a c a c =M 1 f c f c = / x T Requires unconditionally stable integration, algorithmic dissipation implicit A/L-stable schemes

4 Introduction 4 Hydraulic system dynamics: approximate or neglect local dynamics spatial resolution: 0D & 1D circuit theory: node pressures, branch flows Flow balance at nodes q 3 1 q b = d dt m n q 2 1 d dt m 1 q4 1 Constitutive properties of branches (e.g. pressure loss) q b, q b,p n1,ṗ n1,p n2,ṗ n2, t =0 d dt m 1 q b =q 1 2 d dt m 2

5 Introduction 5 Constitutive properties of fluid: linearization about reference condition d dt m=d dt V = V V = p ṗ T Ṫ= 0 1 ṗ Ṫ volume change (e.g. actuator) compressibility can be neglected if: m small bulk modulus large pressure rate small thermal expansion can be neglected if: m small coefficient small temperature rate small d dt m=m β ṗ mα Ṫ+ ρ V assume temperature rate small compared to time scale of hydro-mechanical processes

6 Modeling Scales 6 Multiphysics problem: interaction between different domains (e.g. mechanical and hydraulic) Interaction described in terms of: Frequency, bandwidth (how rapid phenomena are) Power (how much work is transferred between domains) Determine how interaction can be simplified: Truncation Steady approximation Quasi-steady approximation Complete dynamics coupling

7 Modeling Scales 7 Example: actuator p A M F truncated model: commanded force independed from dynamics p A M F (quasi-)static model: commanded force depends on dynamics approx. as mass-spring-damper

8 Approximations 8 Coupled dynamic problem: state-space representation (e.g. linear) ẋ=a x Bu y=c x Du States partitioned based on frequency separation ( slow vs. fast ): slow fast {ẋ s ẋ f } = [ A ss A fs A sf ]{ A ff x s x f } [ B ] s B u f y=[c s C f ] { x } s x Du f Approximations consist in reducing the system to the slow dynamics while preserving information about the fast dynamics

9 Approximations 9 Truncation: only consider slow states slow {ẋ s } 0 = [ A ss A sf ]{ A ff A fs ] B u f x } s 0 [ B s y=[c s C f ] { x s 0 } Du Reduced system: ẋ s =A ss x s B s u y=c s x s Du

10 Approximations 10 Steady approximation: statically approximate fast states {ẋs A fs A ff ]{ 0 } = [ A ss A sf x s x f } [ B ] s B u f y=[c s C f ] { x } s x Du f Reduced system: ẋ s = A ss A sf A 1 ff A fs x s B s A sf A 1 ff B f u x f = A 1 ff A fs x s A 1 ff B f u y= C s C f A 1 ff A fs x s D C f A 1 ff B f u Note: the original system becomes differential-algebraic (DAE) of index 1, thus reducible to ODE by direct substitution

11 Approximations 11 Quasi-steady approximation: use low-order dynamics In Laplace's domain: Taylor y s = C s I A 1 B D u s =H s u s y s H 0 s dh s2 d2 H sn dn H ds s=0 2 ds 2 n! ds n H 0 = C A 1 B D H' 0 = C A 2 B H'' 0 = 2C A 3 B H n 0 = n! C A n 1 B Back to time domain: s=0 y t =H 0 u t H' 0 u t 1 2 H'' 0 ü t s=0 u s Note: time derivative of input! (Only makes sense when u is state of interacting system)

12 Approximations Example: actuator; equations: p 1 p 2 12 Flow balance in chamber 1 p s V 1 /βṗ 1 = A p ẋ C eli A eli (p 1 p 2 ) C ele A ele p 1 C els A els (p s p 1 ) p r Flow balance in chamber 2 V 2 /βṗ 2 =+A p ẋ C eli A eli (p 2 p 1 ) C ele A ele p 2 C elr A elr (p 2 p r ) Equilibrium of piston mẍ=a p (p 1 p 2 ) r ẋ+f State-space realization {ṗ1 ṗ 2 ẋ v }=[ Ce11 / V1 Ce12 / V1 0 Ap / V1 C e12 V 2 C e22 / V 2 0 A p / V 2 p x ]{p1 A p /m A p /m 0 r /m v } {Ce10 / V1ps C e20 / V 2 p r } 0 F/m

13 Approximations 13 Truncation approximation: Neglect hydraulics (velocities small, infinite power available) m ẍ=a p p s p r r ẋ F Directly control the force exerted by the fluid {ẋ v} = [ r /m]{ x v} { 0 A p /m p s p r F/m} Static approximation: Neglect compressibility (volume is small, bulk modulus is high) 0 A p ẋ C eli A eli p 1 p 2 C ele A ele p 1 C els A els p s p 1 0 A p ẋ C eli A eli p 2 p 1 C ele A ele p 2 C elr A elr p 2 p r m ẍ=a p p 1 p 2 r ẋ F Pressure depends on velocity: equivalent damping {ẋ v} = [ r/m c eq /m]{ x v} + { 0 A p /m(p s p r )+F/m} p 1, p 2 ẋ

14 Software 14 AMESim: ADAMS: Monolithic software (can be interfaced as dynamic module to other solvers) Models hydraulic networks in detail Module for basic solver Introduces modeling capabilities of hydraulic components Modelica (Dymola, MathModelica, OpenModelica?) Modeling language, based on open library of elements Broad library for general-purpose components Many fields, including hydraulics and multibody Needs specific (closed) solver MBDyn

15 Software 15 MBDyn: Developed at Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano Monolithic multibody/multiphysics general-purpose software It's free: released under GPL (GNU General Public License) Includes integrated hydraulic library

16 Software 16 Web site: Distributed as source code Developed for Linux Needs Un*x-like build environment Linux Linux in virtual machine MSYS/MinGW in windows Cygwin in windows

17 Software 17 Command-line software Prepare an input file using your favourite text editor Execute: # mbdyn -f input_file -o output_file_prefix Output in files with specific extensions Load output files in math environment (octave, scilab, matlab, ) for plotting and further manipulation

18 Software 18 Output can be reformatted for some post-processing tools built-in: EasyAnim... Ongoing third-party project ( Blender & MBDyn ) about using Blender for pre/post-processing:

19 Software 19 The model and the analysis are defined in a textual input file Use your preferred editor to prepare the input file The structure and the syntax of the statements are described here Download (pick the manual for the version in use, or follow instructions) A set of tutorials and other documentation is presented here Documentation including example input files

20 Modeling Approach 20 Nodes instantiate degrees of freedom and the corresponding balance equations Static structural nodes only instantiate equilibrium equations 0= f 0= m Dynamic structural nodes also instantiate momentum and momenta moment definitions M ẋ= J = = = f m Hydraulic nodes (pressure) instantiate flow balance equations

21 Modeling Approach 21 Elements write contributions to equations instantiated by nodes Elements represent connectivity and constitutive properties Elements can add further, private equations and variables (e.g. algebraic constraints and Lagrange multipliers) Mechanical elements typically add forces and moments to equilibrium equations Mechanical constraints ( joints ) may add algebraic relationships between kinematic degrees of freedom Hydraulic elements typically add flow contributions to flow balance equations in nodes

22 Modeling Approach 22 Multibody vs. FEM: nodes at bodies vs. nodes at frontier (node body; element hinge) hinge body 2 dof node body 1 connectivity element multibody FEM

23 Modeling Approach 23 Hydraulic modeling: network outer pressure chamber 1 leaks actuator motion x supply line return line chamber 2 pressure generator reservoir

24 Hydraulic Element Library: Pressure Generator 24 Not specifically implemented; use a clamp genel instead This element adds a scalar algebraic equation that enforces a specific value (possibly time-dependent) on a scalar node p=p 0 t This algebraic equation implies a Lagrange multiplier on the flow balance equation related to the pressure node q =0 The Lagrange multiplier represents the flow required to grant the imposed pressure value

25 Hydraulic Element Library: Imposed Flow 25 Not specifically implemented, use an abstract force instead This element adds a contribution, possibly time-dependent, to an arbitrary scalar equation q=q t Note: in MBDyn, a negative flow enters the circuit at the given node, a positive flow leaves the circuit at the given node

26 Hydraulic Element Library: Dynamic Pipeline 26 Dynamic pipes are formulated using a finite-volume approach Degenerate into static when pressure time derivatives neglected Differential mass balance: D D t dm=0 q / x A / t =0 Differential momentum balance: D Dt dq=df q q2 / t A A p =f v / x Discretization: q(x, t)=[ 1 ξ 2 p(x, t)=[ 1 ξ 2 1+ ξ 2 ]{ q 1 (t) q 2 (t)} 1+ ξ 2 ]{ p 1(t) p 2 (t)} x=x(ξ)

27 Hydraulic Element Library: Dynamic Pipeline 27 Piecewise-constant weight functions: Weight the mass and momentum balance equations Four constitutive equations in pressure and flow at nodes w 1 x w 2 x 1 2 q 1 2

28 Example: Pressure Wave in Pipeline 28 From: R. Piché, A. Ellman, A Fluid Transmission Line Model for Use with ODE Simulators, 8 th Bath International Fluid Power Workshop, Sep , University of Bath, UK Length: Radius: Density: Viscosity: Sound celerity: Impulsive flow: Duration: m 6.17e-3 m 870 kg/m^3 8.e-5 m^2/s 1.4e3 m/s m^3/s 0.1e-3 s Impulsive dynamics q t imposed flow pressure nodes pipe elements p t imposed pressure

29 Example: Pressure Wave in Pipeline 29 Dynamic pipe (10 dynamic pipe elements) compared to static pipe, impulsive flow input: pressure, Pa pressure, Pa length, m time, s length, m time, s

30 Example: Pressure Wave in Pipeline 30 Dynamic pipe (10 dynamic pipe elements) compared to static pipe, smooth flow input: pressure, Pa pressure, Pa length, m time, s length, m time, s

31 Example: Hydraulically Actuated Beam 31 From: J. MäkinenM kinen,, A. Ellman, R. Piché, Dynamic Simulations of Flexible Hydraulic-Driven Multibody Systems using Finite Strain Beam Theory, 5 th Scandinavian International Conference on Fluid Power, Linköping, 1997, Sweden revolute hinge beam rigid body actuator prismatic imposed flow orifice pipe Input file:

32 Example: Hydraulically Actuated Beam node beam elements imposed flow time, s vertical position, m actuator horizontal position, m

33 Example: Hydraulically Actuated Beam 33 Tip vertical displacement tip vertical position, m time, s

34 Example: Hydraulically Actuated Beam 34 Internal bending moment close to actuator connection moment, Nm time, s

35 Example: Hydraulically Actuated Beam 35 Pressure at imposed flow node pressure, Pa time, s

36 Consclusions 36 Multibody: natural environment for integration of multiphysics Need for further development: complete library, build test suite, validate components The software is free: try it, and feed back!

37 37 Questions?