Finite Segment Model Complexity of an Euler-Bernoulli Beam

8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, Austria 8th Vienna International Conference on Mathematical Available onlineModelling at Vienna, www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, Austria 18 - 20, 2015. Vienna University of Technology, Vienna, February Austria Austria

ScienceDirect

IFAC-PapersOnLine 48-1 (2015) 334–340 Finite Segment Model Complexity of an Euler-Bernoulli Beam Finite Segment Model Complexity of an Euler-Bernoulli Beam Finite Segment Model of an Euler-Bernoulli Beam Finite Segment Model Complexity Complexity of an Euler-Bernoulli Beam Loucas S. Louca

Loucas S. Louca Loucas S. Louca Loucas S. Loucaand Manufacturing Engineering University of Cyprus, Department of Mechanical UniversityCYPRUS of Cyprus,(Tel: Department of Mechanical and Manufacturing Engineering +357-2289 2279; e-mail: [email protected]) University of Cyprus, Department of Mechanical and Manufacturing Engineering +357-2289 2279; e-mail: [email protected]) UniversityCYPRUS of Cyprus,(Tel: Department of Mechanical and Manufacturing Engineering CYPRUS (Tel: +357-2289 2279; e-mail: [email protected]) CYPRUS (Tel: +357-2289 2279; e-mail: [email protected]) Abstract: A common approach for modeling the dynamic behavior of distributed parameter systems is the Abstract: A common approach for modeling the dynamic behavior of distributed parameter systems is the approximation through finite-segment models. models are able to accurately predictsystems the dynamic Abstract: A common approach for modeling the These dynamic behavior of distributed parameter is the approximation through finite-segment models. These models are able to accurately predict the dynamic Abstract: A common approach for modeling the dynamic behavior of distributed parameter systems is the behavior of thethrough systemfinite-segment given that "adequate" segments are included in accurately the model.predict Frequency-based approximation models. These models are able to the dynamic behavior of the system given that "adequate" segments are included in the model. Frequency-based approximation through finite-segment models. These models are able to accurately predict the dynamic methodologies be used to address the complexity of such purpose of theFrequency-based current work is behavior of thecan system given that "adequate" segments are models. includedThe in the model. behavior system given that "adequate" segments arepreviously includedThe in the model. Frequency-based methodologies can be used to address the complexity of such models. purpose of the current work is to addressofthethe complexity using the developed metric.work More methodologies can be usedoftodistributed address theparameter complexity of such models. The purpose activity of the current is methodologies can be usedofto address theparameter complexity of model such models. The purpose of the current is to address the complexity distributed using the previously developed activity metric.work More specifically the complexity of an Euler-Bernoulli beam is considered. Bond graph models of this to address the complexity of distributed parameter using the previously developed activity metric. More to address the usingmodel theidentify previously developed activity metric. specifically thecomplexity complexity ofdistributed an Euler-Bernoulli beam is considered. Bond graph models ofMore this system already in theofliterature and parameter the objective to the necessary complexity (number of specifically the exist complexity of an Euler-Bernoulli beamismodel is considered. Bond graph models of this system already exist in the literature and the objective is to identify the necessary complexity (number of specifically the complexity of an Euler-Bernoulli beam model is considered. Bond graph models of this segments). A new modeling procedure is proposed for this type of systems where the model starts from system already exist in the literature and the objective is to identify the necessary complexity (number of segments). A new modeling procedure is proposed for this type of systems where the model starts from system already exist in the literature and the objective is to identify the necessary complexity (number of simple and Athenew number of segments until activity basedwhere criterion is satisfied. An segments). modeling procedure isis increased proposed for thisantype of systems the model starts from simple andexample number of segments isis increased until activity based criterion is satisfied. An segments). Athenew modeling procedure proposed for thisantype systems where the model starts from illustrative is provided to demonstrate the effectiveness ofofthis methodology. simple and the number of segments is increased until an activity based criterion is satisfied. An illustrative is provided to demonstrate the effectiveness of this based methodology. simple andexample the number of segments is increased until an activity criterion is satisfied. An illustrative example is provided to demonstrate the effectiveness of this methodology. Keywords: Model Complexity, Energy-based Modeling Metric, Bond Finite Segment Model © 2015, IFAC (International Federation of Automatic Control) Hosting byGraphs, Elsevier Ltd. All rights reserved. illustrative is provided to demonstrate the effectiveness of this methodology. Keywords: example Model Complexity, Energy-based Modeling Metric, Bond Graphs, Finite Segment Model Keywords: Model Complexity, Energy-based Modeling Metric, Bond Graphs, Finite Segment Model Keywords: Model Complexity, Energy-based Modeling Metric, Bond Graphs, Finite Segment Model 1. INTRODUCTION resistance) in the model. The contribution of each element in 1. INTRODUCTION resistance) in ranked the model. The contribution of each element in the model is according to the element activity metric 1. INTRODUCTION in the model. The contribution of each element in Modeling and simulation have yet to achieve wide utilization resistance) the model is ranked according to the element activity metric 1. INTRODUCTION resistance) in the model. The contribution of each element in under specific excitation. Elements with small contribution the model is ranked according to the element activity metric Modeling and simulation have yet to achieve wide utilization as commonplace engineering tools, even wide though current the model is ranked according to the element activity metric under specific excitation. Elements with small contribution Modeling and simulation have yet to achieve utilization are eliminated in order to Elements produce awith reduced The Modeling and simulation simulation have yet toreduced achieve wide utilization specific excitation. small model. contribution as commonplace engineering tools, even efficiency. though current modeling and tools have One under under specific Elements small contribution are eliminated in order produce reduced The as commonplace engineering tools, even though current activity metric excitation. was alsoto used as a aawith basis for model. even further as commonplace engineering tools, even though current are eliminated in order to produce reduced model. The modeling and simulation tools have reduced efficiency. One drawback is that they require sophisticated users who One are are eliminated in order to produce a reduced model. The activity metric was also used as a basis for even further modeling and simulation tools have reduced efficiency. reduction, through partitioning a amodel into smaller and activity metric was also used as basis for even further drawback is that they require sophisticated users who are modeling and simulation tools have reduced efficiency. One often not domain experts. Non-domain experts metric was also used as a basis for even further reduction, through partitioning a model into smaller drawback is that they require sophisticated usersfrequently who are activity decoupled submodels (Rideout et aal.,model 2007).into smaller and drawback is that they require sophisticated users who are reduction, through partitioning and often not domain experts. Non-domain experts frequently lack the to experts. effectivelyNon-domain utilize models and simulation through partitioning decoupled submodels (Rideout et aal.,model 2007).into smaller and often notability domain experts frequently reduction, often not domain experts. Non-domain experts frequently decoupled submodels (Rideout et al., 2007). lack the ability to effectively utilize models and simulation tools the in ability order toto uncover keyutilize designmodels trade-offs. Another decoupled Such modeling approaches be able to handle real submodels (Rideoutshould et al., 2007). lack effectively and simulation lack the effectively models simulation tools in ability order toto models uncover keyutilize design trade-offs. Another Such modeling approaches be able to handle real drawback is that large andand complicated, mechanical systems that should typically include distributed tools in order to uncover are keyoften design trade-offs. Another Such modeling approaches should be able to handle real tools in order to uncover key design trade-offs. Another Such modeling approaches should be able to handle real drawback is that models are often large and complicated, mechanical systems that typically include distributed with a large number of parameters, making the physical mechanical parameter (continuous) components. Frequently, modeling drawback is that models are often large and complicated, systems that typically include distributed drawback is that models are often large and complicated, mechanical systems that typically include distributed with a large number of parameters, making the physical parameter (continuous) components. Frequently, modeling interpretation the model outputs, even by domain experts, parameter objectives and assumptions allow the lumping of continuous with a large of number of parameters, making the physical (continuous) components. Frequently, modeling with a large number of parameters, making the physical (continuous) components. Frequently, modeling interpretation the model outputs, even by domain experts, parameter objectives and assumptions allowenergy the lumping of that continuous difficult. Thisof especially true when “unnecessary” properties into ideal elements lead to interpretation ofis the model outputs, even by domain features experts, component objectives and assumptions allow the lumping of continuous interpretation of the model outputs, even by domain experts, objectives and assumptions allow the lumping of continuous difficult. This is especially true when “unnecessary” features component properties into ideal energy elements that lead to are included in the model. It is the premise of this work that a dynamic model described by a set of ordinary differential difficult. This is especially true when “unnecessary” features component properties into ideal energy elements that lead to are included in the model. It is the premise of this work that a dynamic model described by a set of ordinary differential difficult. This is especially true when “unnecessary” features component properties into ideal energy elements that leadnot to more effective use model. of modeling andpremise simulation necessitates However, when lumping is are included in the It is the of this work that equations. a dynamic model described by a property set of ordinary differential more effective use of modeling and simulation necessitates equations. However, when property lumping is not are included in the model. It is the premise of this work that a dynamic model described by a set of ordinary differential the for proper that and is, models withnecessitates physically acceptable, of awhen continuous component moreneed effective use ofmodels, modeling simulation equations. modeling However, property lumpingrequires is nota the need for proper that and is, models physically acceptable, modeling of its awhen continuous component more effective use ofmodels, modeling simulation necessitates However, property lumping is nota meaningful parameters are ofwith necessary but equations. different approach since inertial, compliance andrequires resistive the need forstates properand models, that is,that models with physically acceptable, modeling of a continuous component requires a meaningful states and parameters that are of necessary but different approach since its inertial, compliance and resistive the need for proper models, that is, models with physically acceptable, modeling of a continuous component requires sufficient complexity meet the engineering propertiesapproach are spatially distributed and cannot be and lumped intoa meaningful states andtoparameters that are ofobjective. necessary but different since its inertial, compliance resistive meaningful states andtoparameters that are ofobjective. necessary but different approach since its inertial,and compliance resistive sufficient complexity meet the engineering properties are spatially distributed cannot be and lumped into single equivalent elements. of sufficient to meet the engineering objective. and properties are spatially distributedThe and dynamic cannot be behavior lumped into A varietycomplexity of algorithms have been developed sufficient complexity to meet the engineering objective. properties are spatially distributed and cannot be lumped into single equivalent elements. The dynamic behavior of continuous components is described by partialbehavior differential single equivalent elements. The dynamic of A variety of algorithms have been developed and implemented automate the been production of proper single equivalent elements. The dynamic behavior of components is described by partial differential A variety ofto help algorithms have developed and continuous equations with derivatives in time and space. Another A variety of algorithms have been developed and continuous components is described by partial differential implemented to help automate the production of proper dynamic system Wilsonthe andproduction Stein developed the equations continuous components is described by partial differential with derivatives in time and space. Another implemented to models. help automate of proper approach that is considered workand is thespace. modeling of a implemented to models. help automate of proper with derivativesininthistime Another dynamic system Wilsonthe andproduction Stein developed the Model Order Deduction deduces the equations approach that is considered in thistime workand is the modeling of a dynamic system models.Algorithm Wilson (MODA) and Steinthat developed the equations with derivatives in space. Another continuous component with finite segments that are spatially dynamic system models. Wilson (MODA) and subsystem Steinthat developed that is considered in this work is the modeling of a Model Deduction Algorithm deduces needed Order system model complexity modelsthe of approach continuous component with finite segments that are spatially Model Order Deduction Algorithmfrom (MODA) that deduces the approach that is considered in this work is the modeling of a This is an approximation for which Model Deduction Algorithm (MODA) that deduces the continuous component with finite segments that the are accuracy spatially needed system modelusing complexity from subsystem models of distributed. variableOrder complexity a frequency-based metric (Wilson This is an approximation for which the accuracy needed system model complexity from subsystem models of distributed. continuous component with finite segments that are spatially is a function This of theisnumber of segments. model accuracy needed system model complexity from subsystem models of distributed. an approximation forThe which the accuracy variable complexity using a frequency-based metric (Wilson and Stein, 1995). Additional work on deduction algorithms a function ofthe theisnumber number of model variable complexity using a frequency-based metric (Wilson is distributed. an approximation forThe which the accuracy improves asThis ofsegments. segments increases and the variable complexity using a frequency-based metric (Wilson is a function of the number of segments. The model accuracy and Stein, 1995). Additional work on deduction algorithms for proper models work has been reported in order to improves number of segments increases and the and generating Stein, 1995). Additional on deduction algorithms is a function ofthe the number of segments. The model selection ofas the appropriate number of segments is accuracy typically and Stein, 1995). Additional work on deduction algorithms improves as the number of segments increases and the for generating proper models has been reported in order to extend the applicability of thehas algorithm (Ferris inand Stein, as the number of segments increases and the selection of the appropriate number of segments is typically for generating proper models been reported order to improves done through frequency-based metrics. for generating proper models has been reported in order to selection of the appropriate number of segments is typically extend the applicability of the algorithm (Ferris and Stein, 1995; Walker et al., 1996). These algorithms have been selection of the appropriate number of segments is typically done through frequency-based metrics. extend the applicability of the algorithm (Ferris and Stein, done through frequency-based metrics. 1995; Walker et al., 1996). been Element extend the applicability of the These algorithm (Ferris have and Stein, implemented demonstrated in a algorithms computer automated activity is another metric that has been demonstrated frequency-based metrics. 1995; Walker and et al., 1996). These algorithms have been done through implemented and demonstrated in a computer automated Element activity is another metric that has been demonstrated 1995; Walker et al., 1996). These algorithms have been modeling environment, (Stein and in Louca, 1996). automated to have more flexibility than frequency-based metrics. It is implemented and demonstrated a computer Element activity is another metric that has been demonstrated implemented and demonstrated a computer Element activity is another metric that has modeling environment, (Stein and in Louca, 1996). automated the to have more flexibility than frequency-based metrics. It is purpose offlexibility this paper to frequency-based develop a been new demonstrated methodology modeling environment, (Stein and Louca, 1996). to have more than metrics. It is In an attempt to overcome theand limitations of the frequency- the modeling environment, (Stein Louca, 1996). to have more flexibility than frequency-based metrics. It is purpose of this paper to develop a new methodology usingpurpose the activity metric for to addressing the model complexity the of this paper develop a new methodology In an attempt to overcome the limitations of the frequencybased metrics the author introduced a new model reduction the purpose of this paper to develop a new methodology using the activity metric for addressing the model complexity In an attempt to overcome the limitations of the frequency- of distributed parameter systems. The methodology is In an attempt overcome the proper limitations the frequencythe activity metric for addressing the model complexity based metrics the author introduced a models newofmodel reduction technique that to also generates (Louca et al., using using the activity metricforforan addressing the model of distributed parameter systems. The methodology is based metrics the author introduced a new model reduction specifically developed Euler-Bernoulli beamcomplexity but it can based metrics the author introduced a new model reduction of distributed parameter systems. The methodology is technique that also generates proper models (Louca et al., 1997). This uses an proper energy-based (element developed for an Euler-Bernoulli beam but it can technique thatapproach also generates modelsmetric (Louca et al., specifically of distributed parameter systems. The methodology is be generalized for models of other continuous beam systems. technique that also generates proper models (Louca et al., specifically developed for an Euler-Bernoulli but it can 1997). This approach uses an energy-based metric (element activity)This that approach in general,uses canan beenergy-based applied to nonlinear systems be generalized for models of other continuous beam systems. 1997). metric (element specifically developed for an Euler-Bernoulli but it can 1997). an metric (element for models of other continuous systems. activity) in 2010), general,uses canconsiders beenergy-based applied to importance nonlinear systems (Louca This etthat al.,approach of all be Thegeneralized paper is organized as follows: first, background about the activity) that in general,and can be appliedthe to nonlinear systems be generalized for models of other continuous systems. activity) that in general, can be applied to nonlinear systems (Louca etelements al., 2010), and considers the importance of and all energy-based The paper is organized asprovided. follows: first, background about the energetic (generalized inductance, capacitance metric is The next section presents (Louca et al., 2010), and considers the importance of all The paper is organized as follows: first, background about the (Louca al., 2010), and considers the importance of and all The paper is organized follows: first, background the energeticetelements (generalized inductance, capacitance energy-based metric isasprovided. The next sectionabout presents energetic elements (generalized inductance, capacitance and energy-based metric is provided. The next section presents energetic elements (generalized inductance, capacitance and energy-based metric is provided. The next section presents

Copyright © 2015, IFAC 334 2405-8963 © © 2015, 2015, IFAC Copyright IFAC (International Federation of Automatic Control) 334 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015, responsibility IFAC 334 Control. Peer review© under of International Federation of Automatic Copyright © 2015, IFAC 334 10.1016/j.ifacol.2015.05.155

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340

the finite segment model of the beam along with input activity calculations under steady state conditions. Following that, the new model complexity methodology is presented. An illustrative example of the Euler-Bernoulli beam is then provided in order to demonstrate the efficacy of the modeling methodology. Finally, discussion and conclusions are given.

system has a linear junction structure and constitutive laws, state equations are linear time invariant and for a single input have the following general form:

x! = A ⋅ x + b ⋅u

The original work on the energy-based metric for model reduction is briefly described here for convenience. More details, extensions, and applications are given in previous publications (Louca and Stein, 2002; Louca, et al., 2004; Louca and Stein, 2009; Louca et al., 2010). The main idea behind this model reduction technique is to evaluate the “element activity” of the individual energy elements of a full system model under a stereotypical set of inputs and initial conditions. The activity of each energy element establishes a hierarchy for all elements. Those below a user-defined threshold of acceptable level of activity are eliminated from the model in order to generate a reduced model. The activity metric has been previously formulated for systems with nonlinearities in both the element constitutive laws and junction structure. In this work, the activity metric is applied to linear systems for which analytical expressions for the activity can be derived, and therefore, avoid the use of numerical time integration that could be cumbersome. The analysis is further simplified if, in addition to the linearity assumption, the system is assumed to have a single sinusoidal excitation, and only the steady state response is examined. These assumptions are motivated from Fourier analysis where an arbitrary function can by decomposed into a series of harmonics. Using this frequency decomposition, the activity analysis can be performed as a function of frequency in order to study the frequency dependency of element activity in a dynamic system.

0

y = c⋅x

(3)

where c ∈ !1×m is the output state space matrix. The input power is therefore calculated as the product of the input by its dual as given in (3). Thus the input power is: in

(t) = u ⋅y = u ⋅ c ⋅ x

(4)

The above input power is then used to calculate input activity, as it is shown in details in the following section. 3. EULER-BERNOULLI BEAM MODEL The state space representation used in the previous section assumes that real components exhibit only inertial, compliant, or resistive lumped behavior. This means that the dynamic behavior of a component can be spatially lumped and modeled as single inertial, compliant or resistive energy elements. This can be a valid assumption for some components, however, real system components can simultaneously exhibit all dynamic properties (inertial, compliant, resistive). These properties may also vary or be distributed spatially. In these cases, a lumped parameter modeling approach cannot be used since it will be inaccurate. These components must be considered as continuous and require a different modeling approach. Examples of such components are rods, cantilever beams or plates under dynamic loading.

τ

P (t) ⋅dt = ∫ e(t)⋅ f (t) ⋅dt

x ∈ !m is the state vector, u ∈ ! is the input, and m is the number of independent states. Appropriate outputs are needed for the calculations of power and thus the activity as defined in (1). For the purposes of this work only the activity of the input port is considered thus the output variable is the dual of the input that can be an effort or flow. Effort and flow are dual variables to each other and their product is power. Thus, the system is assumed to have a single input and the output equation is:

P

A measure of the power response of a dynamic system, which has physical meaning and a simple definition, is used to develop the modeling metric, element activity (or simply “activity”). Element activity, A , is defined for each energy element as: τ

(2)

where, A ∈ !m×m , b ∈ !m are the state space matrices,

2. BACKGROUND

A= ∫

335

(1)

Models of continuous systems can be developed using solid mechanics techniques, which lead to Partial Differential Equations (PDE) with derivatives in both space and time (Bauchau and Craig, 2009; Genta, 2009). The continuous cantilever beam used in this work is shown in Fig. 1, where its transverse motion is considered. The motion of a given gross section, w(x,t) , from its undeformed state varies with time and location thus having PDEs describing its motion. One method for solving these PDEs is separation of variables, which produces a modal expansion solution (Meirovitch, 1967). This approach can also be combined with other lumped parameter elements in order to model a real system that consists of both lumped and distributed parameter components (Karnopp et al., 2006). An analysis of the advantages and disadvantages of this approach is beyond the scope of this work, however, it is safe to say that the solution of PDEs is more cumbersome than the solution of

0

where P (t) is the element power, e(t), f (t) are the respective efforts and flows, and τ is the time over which the model has to predict the system behavior. The activity has units of energy, representing the amount of energy that flows in and out of the element over the given time τ . The energy that flows in and out of an element is a measure of how active this element is (how much energy passes through it), and consequently the quantity in (1) is termed activity. The activity can be calculated for each energy element or input based on the system response. In the case that the system is modeled using a bond graph formulation, the state equations are derived using the multi-port bond graph representation (Borutzky, 2004; Brown, 2006; Karnopp et al., 2006; Rosenberg and Karnopp, 1983). In addition, when a 335

MATHMOD 2015 336 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340

ordinary differential equations that describe the behavior of lumped parameters systems.

accurately predicting low frequency dynamics. It is the purpose of this work to systematically determine the “optimum” number of segments in order for the model to accurately predict the system response. For calculating the constitutive law parameters of the energy storage elements, the beam is assumed to have density ρ ,

F(t)

Young's modulus E , length L , cross sectional area A and cross sectional moment of inertia I . Given these physical parameters of the beam, the element parameters in the above model are given by:

w(x,t)

x

Fig. 1. Cantilever beam transverse vibration A different approach for modeling the transverse vibration of a cantilever beam is to divide it into segments of equal length. This approach is motivated by the procedure for deriving the PDEs describing the motion of a beam. Each of these segments has inertial and compliant properties that can be determined from solid mechanics theory. Each segment is also assumed to have energy losses due to structural damping, which is modeled as a linear resistive element connected in parallel with the compliance of the segment. Shear effects and rotational inertial forces are assumed negligible, which is a valid assumption for slender beams. This is known as the Euler-Bernoulli beam model. In the case of a non-slender beam these effects are significant and are considered in order to have accurate model predictions. b1 c1

bi ci

b2 c2

b i+1 c i+1

b i+2 c i+2

... m1

w1

bn cn

ci =

w i-1

wi

mi+1 w i+1

Δx E ⋅I

(5)

where Δx = L n , mi and ci is the length, mass and compliance (inverse of stiffness) of each segment, respectively. Viscous damping constant, bi , is set such that the system exhibits lightly damped behavior. The transverse velocity of each mass, vi , represents the velocity at a given location of the continuous beam and the following relation holds between the discrete and continuous variables:

vi (t) = w! i = w! (i ⋅ Δx,t )

F(t)

(6)

For developing the dynamic equations, the bond graph formulation is used. Bond graphs provide the power topography of the system and it is a natural selection for implementing the power-based activity metric. The bond graph model of the ideal physical model shown in Fig. 2 is developed and given in Fig. 3. The bond graph has 2n independent state variables and its state vector is

... mi-1 mi

w2

b n-1 c n-1

mi = ρ ⋅A ⋅ Δx, i = 1,…,n

mn-2 m mn n-1 w i+2 w n-2 w n-1 w n

Fig. 2. Ideal physical model of an Euler-Bernoulli beam The ideal physical model under these assumptions is shown in Fig. 2 where the beam is divided into n segments. This model approaches the partial differential equations of the continuous system, as the number of segments approaches infinity. However, it is difficult to predict the number of segments required to achieve a given level of accuracy. It is well known that a large number of segments is required for

x = {p1,…, pn ,q1,…,qn } . In addition, for easy calculation T

of the output equations required for calculating power, the state equations are derived using the multi-port approach (Rosenberg, 1971). According to this approach, the state space and input matrices are calculated using the junction structure matrices and they are given by (7).

Fig. 3. Bond graph model 336

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340

−1 ⎛ ⎞ A = ⎜⎜ JSS + JSL ⋅ L ⋅ (I − JLL ⋅ L) ⋅ JLS ⎟⎟⎟ ⋅ S ⎝ ⎠

b = JSU + JSL ⋅ L ⋅ (I − JLL ⋅ L) ⋅ JLU −1

closed form expression can be used to calculate the activity of the input for a given single harmonic excitation. The superscript 'ss' in equation reference goes here denotes the activity under a steady state harmonic response. The input activity depends on excitation frequency but also on the number of segments and this is made clear in (13).

(7)

Similarly, the output matrix that is required for calculating the input power flow, as defined in (3), is given by: −1 ⎛ ⎞ c = ⎜⎜ JUS + JUL ⋅ L ⋅ (I − JLL ⋅ L) ⋅ JLS ⎟⎟⎟ ⋅ S ⎝ ⎠

⎞ ⎞ ⎛π 2 ⋅U 2 ⋅Y (ω ) ⎛⎜ ⎜⎜sin φ + ⎜⎜⎜ − φ⎟⎟⎟ cosφ⎟⎟⎟ ⎟⎟⎠ ⎟⎠ ⎜⎝ ω ⎝⎜ 2 where φ ∈ ⎡⎢0, π ⎤⎥ ⎣ ⎦

Ass (ω,n ) =

(8)

The junction structure matrices in (7) and (8) are derived analytically and given in the Appendix.

Activity was originally developed as a model reduction metric. Given a model that includes all possible complexity, the activity metric identifies unimportant physical phenomena that do not contribute significantly to the overall system behavior. The methodology starts from a complex model and removes elements in order to simplify (reduce) the model. In the case of a continuous system, which is modeled with a finite segment approach, the modeling problem is the reverse. Starting from a rigid model, the minimum number of segments must be identified in order to meet some accuracy requirements. In this modeling paradigm, the model starts from simple and complexity is increased. The question that needs to be answered in this case is, “What is the minimum number of segments needed in order to accurately predict the system behavior?” The proposed work will provide a new methodology for answering this question.

The time response of the output, y(t) , as given in (3) is required in order to complete the calculation of the input power. For nonlinear systems, numerical integration is typically used to calculate the system response; however, in this case linear system analysis can be used to obtain closed form expressions. In addition, for the purposes of this work, the excitation is assumed to be a single harmonic given by: (9)

where U ∈ ! is the amplitude of the excitation and ω is the excitation frequency. The steady state response of the linear system described by (2) and (3), and for the harmonic excitation in (9), is calculated using linear system analysis theory. This gives the following closed form expression:

(

)

y (t, ω ) =U ⋅Y (ω ) ⋅ sin ω ⋅t + ϕ (ω )

A metric that accounts for the overall system behavior is needed for addressing the minimum number of segments. All energy elements contribute to the overall system behavior and their importance must be accounted for in the metric that will be used for making the modeling decisions. Given this requirement, one option for the modeling metric is to use system activity that is the sum of the activity of each energy element, which can be computationally expensive for a large number of segments. An alternative is to use the input activity that also provides a measure of the overall system activity level and it will be used as the metric for addressing the minimum number of segments.

(10)

where Y (ω) and ϕ(ω) are the steady state amplitude and phase shift, respectively. The steady state response of the output (10) along with the input (9) are substituted in (4) in order to calculate input power as given below:

P (t, ω) = u (t ) ⋅y (t, ω) =U ⋅Y (ω ) ⋅ sin (ω ⋅t ) ⋅ sin (ω ⋅t + ϕ) in

2

(11)

Finally, the input activity can be calculated using (1), but the upper bound of this integral must be specified first. For this case, the steady state and periodicity of the response are exploited. A periodic function repeats itself every T seconds, and therefore, a single period of this function contains the required information about the response. Thus, the upper bound of the integral is set to one period of the excitation, τ = T = 2π ω . Therefore, the steady state activity for the input is given by: T

The input activity, as defined in (13), changes as the number of segments varies and the proposition of this new methodology is to increase the number of segments until the input activity reaches a steady state value. This approach is motivated from the principle of the activity metric stating that low activity elements do not contribute to the system behavior and should not be included in the model. In this case the equivalent is that an increase in model complexity producing a “small change” in the input activity is not significant, thus such increase is not necessary for the model accuracy. Again, based on the activity principle, a small change in the input activity is insignificant and thus does not contribute to the overall system behavior.

P (t, ω) ⋅dt

Ass (ω ) = ∫

in

0

T

(13)

4. MODEL COMPLEXITY METHODOLOGY

3.1 Activity for Single Harmonic Excitation

u(t) =U ⋅ sin(ω ⋅t)

337

(12)

=U ⋅Y (ω ) ∫ sin (ω ⋅t ) sin (ω ⋅t + ϕ) ⋅dt 2

This “small change” must be quantified in order for the methodology to be able to produce the anticipated identification of the minimum model complexity. Using an absolute definition of “small change” is not appropriate since

0

The integral in (12) has closed form solution thus the steady state input activity is calculate as shown in (13). This simple 337

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340 338

the input activity is a function of the excitation amplitude and this will result in model complexity that depends on the amplitude. To overcome this unwanted behavior a relative measure of change is used, that is defined as the ratio of the absolute activity change when the number of segments is increased by one, to the current input activity. This defines the relative activity that is mathematically expressed in (14). The relative activity is not a function of the input amplitude due to its definition as the ratio of activities.

δA (ω,n ) =

ss

A

(ω,n + 1) − A (ω,n ) A (ω,n )

identified for a range of frequencies from 10 to 10,000 rad/s that includes the first 4 natural frequencies. The relative activity threshold is set to 5x10-3 and for simplicity the number of segments was not allowed to increase above 200. The methodology is easy and computationally inexpensive to implement due to the simple and closed form expressions used for calculating the state space matrices, frequency response and activity.

ss

ss

(14)

Given the relative activity, the new modeling procedure can be established. The proposed procedure as shown in Fig. 4 is iterative and it starts with the simplest model that is the rigid body model. Then, at each step of the iteration the number of segments is increased by one and the relative activity is calculated. If the relative activity is greater than a user defined engineering threshold, ε , then the number of segments is increased by one and the procedure is repeated. The procedure is repeated until the relative activity becomes smaller than the threshold ε . At the end of the procedure the optimum number of segments, no , is identified. Rigid Body n=1

n = n+1

Relative Activity δΑ(ω,n)

No

δA(ω,n)<ε

Engineering Specifications,ε

Around the second natural frequency (1644 rad/s), a behavior similar to that seen around the first natural frequency region is observed. The optimum number of segments exponentially increases and reaches the maximum just before the natural frequency. At a close neighborhood of the second natural frequency starting at 1213 rad/s there is a drop in the optimum number of segments as it reduces to 163 at 1320 rad/s and then increases again to the maximum of 200 at 1468 rad/s. After the second natural frequency the optimum complexity drops to another minimum of 110 segments that is between the second and third natural frequencies that is about 2800 rad/s. The same trend is repeated between the third and forth natural frequencies with a minimum of 137 segments at a frequency of 6450 rad/s. For higher frequencies more than 200 segments are needed in order to accurately predict the system behavior. Optimum Complexity [# of segments]

Yes

The results for the optimum complexity are shown in Fig. 5. For low frequencies and up to 32 rad/s, 55 segments are necessary to accurately predict the system response. As the input frequency increases the number of segments increases in order to maintain accuracy. It exponentially increases as it approaches the first natural frequency (262 rad/s) and just before 236 rad/s the maximum allowable number of segments is reached. More segments need to be added at frequencies around the fundamental natural frequency. At higher frequencies and after 285 rad/s the number of segments reduces again reaching a minimum of 81 segments at 700 rad/s and then it increases again.

Optimum Complexity, no

Fig. 4. Identification of optimum model complexity The above modeling procedure is based on the steady state harmonic response of the system given a single harmonic excitation. The input and relative activities vary with the frequency of the excitation, and therefore, the optimum complexity depends on the excitation frequency. Thus, the optimum complexity can be identified for a range of frequencies in order to generate the “frequency response” of the optimum complexity. It is expected that the optimum complexity will vary with the input frequency and probably increase around the system's natural frequencies. This will be verified with the illustrative example that follows.

200 180 160 140 120 100 80 60 40 1 10

2

10

3

Input Frequency [rad/s]

10

4

10

Fig. 5. Optimum complexity In order to verify the results of the optimum complexity a comparison of the steady state response over the same range of frequencies is considered. The comparison is made between the 'Exact' response and the response of the system with the optimum number of segments as identified by the proposed methodology. The 'Exact' frequency response is derived from the solution of the PDE describing the vibration of the cantilever beam. The steady state amplitude of two variables is considered in order to demonstrate the global nature of the methodology.

5. ILLUSTRATIVE EXAMPLE The proposed methodology is applied to a cantilever beam with parameters ρ = 7,860 kg/m3, E = 210 GPa, L = 2 m, A = 6 10-3 m2, I = 2 10-5 m4. The optimum complexity is 338

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340

The first variable to compare is the velocity of the free end where the force is applied, which is variable vn on the bond

6. DISCUSSION AND CONCLUSIONS A new methodology is developed for the systematic modeling of continuous systems that are modeled through the finite segment approach. The previously developed activity metric is used as the basis for determining the optimum number of segments that are needed in order for the model to accurately predict the system dynamic behavior. Specifically, the activity that flows through the input is used for identifying the optimum number of segments. The optimum complexity model is shown to be accurately predicting the system behavior over a range of input frequencies.

graph in Fig. 3. The comparison of the amplitude is shown in Fig. 6 where on the top plot the exact frequency response is plotted over the whole frequency range. The response for the optimum complexity is only plotted for the frequencies for which the maximum number of segments is not reached. For example, there is no curve for frequencies above 7379 rad/s since the optimum number of segments is greater than 200. Similarly, there are four more frequency regions, for lower frequencies, where no curve is plotted. The two amplitude responses overlay each other exposing the good level of accuracy of the model with optimum complexity. For a more refined comparison, the ratio between the optimum and exact response is plotted on the lower plot of Fig. 6. The maximum error is 6.5% with local maximum values near the natural frequencies. The error also has local maxima around the zeros of the velocity transfer function.

The activity metric considers the overall system dynamics with all its state variables and not any specific variable. Therefore, the accuracy of all states is guaranteed with some states being more accurate than others. This can be seen in the comparison of the two outputs in Fig. 6 and Fig. 7. For the reaction torque accuracy it is clear from Fig. 7 that a reduced model could be used to accurately predict the torque around 1100 rad/s. However this is not predicted by the activity metric since the input velocity accuracy of the reduced models is low for around the same frequencies. This results in higher number of segments in order to guarantee that all states variables are accurate. A methodology that considers a given variable rather than the complete state vector would be very beneficial and would result in further reduction in the number of segments. However, this is not addressed in this work and remains as a topic for future development.

0

Amplitude [m/s]

10

Exact Optimum

−2

10

−4

10

−6

10

−8

Amplitude Accuracy [−]

10

1

10

2

10

2

10

10

3

10

4

3

10

1.1 1.05 1 0.95 0.9 1 10

10

Input Frequency [rad/s]

When considering the sinusoidal steady state response, analytical expressions for the input activity are derived. It is shown that activity varies with the frequency content of the excitation and the number of segments in the model. This is expected as different input frequencies excite different system dynamics; nevertheless, the activity metric accounts for the contribution of each energy element to the system dynamic response. This allowed the use of the activity as a metric in the proposed methodology.

4

Fig. 6. Velocity comparison at free end The second variable that is compared is the reaction torque, acting on the beam from the support at the fixed end. The same comparison as before is made and it is shown in Fig. 7. The comparison of the amplitude shows similar acceptable agreement over the frequency ranges that the optimum complexity is identified. Again, high error is observed around the poles of the transfer function of the reaction torque with respect to the input force.

Because this work uses an energy-based modeling metric, it is convenient to use a model representation and formulation approach from which power can be easily extracted or calculated. The bond graph approach explicitly presents the power topography of a dynamic system, and therefore, it is used in this work for calculating the necessary variables required for the power calculations. However, the use of this methodology is not limited to systems represented by bond graphs. It can also be applied when the continuous system is modeled using any other modeling methodology, e.g., Lagrange’s equations, Newton’s Law, etc. However, in this case the calculation of power that is required for the proposed methodology might be cumbersome.

3

Amplitude [N*m]

10

Exact Optimum 1

10

−1

Amplitude Accuracy [−]

10

1

10

2

3

10

10

4

10

1.1 1.05

The developed methodology is valid for a single harmonic input, however, this is not a limitation since generic inputs can also be addressed. For generic inputs, the reduced model can be generated using a Fourier expansion of the input into a series of harmonics. Using this decomposition, the proposed modeling methodology is first carried out for each harmonic of the excitation. The model is then assembled as the union

1 0.95 0.9 1 10

2

10

3

Input Frequency [rad/s]

10

339

4

10

Fig. 7. Torque comparison at fixed end 339

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Loucas S. Louca et al. / IFAC-PapersOnLine 48-1 (2015) 334–340 340

of all the individual (single harmonic) optimum complexity models. However, this procedure has to be formalized and remains as an item for future research.

Louca, L.S. and J.L. Stein, 2002. “Ideal Physical Element Representation from Reduced Bond Graphs”. Journal of Systems and Control Engineering, Vol. 216, No. 1, pp. 73-83. Published by the Professional Engineering Publishing, ISSN 0959-6518, Suffolk, United Kingdom. Louca, L.S., D.G. Rideout, J.L. Stein, and G.M. Hulbert, 2004. “Generating proper dynamic models for truck mobility and handling”. International Journal of Heavy Vehicle Systems Vol. 11, No. 3/4 pp. 209-236. Published by Inderscience Enterprises Ltd., ISSN 1744−232X, St. Helier, United Kingdom. Louca, L.S. and J.L Stein, 2009. “Energy-Based Model Reduction of Linear Systems”. Proceedings of the 6th International Symposium on Mathematical Modeling, Vienna, Austria. Published in the series ARGESIMReports no. 35, ISBN 978-3-901608-35-3, Vienna, Austria. Louca, L.S., J.L. Stein, and G.M. Hulbert, 2010. “EnergyBased Model Reduction Methodology for Automated Modeling”. Journal of Dynamic Systems Measurement and Control, Vol. 132, No. 6, 061202 (16 pages). Published by the American Society of Mechanical Engineers, ISSN Print 0022-0434, ISSN Online 1528-9028, New York, NY. Meirovitch, L., 1967. Analytical Methods in Vibrations. Macmillan Publishing Inc., New York, NY, ISBN 0-02-380140-9. Rideout, D.G., J.L. Stein, and L.S. Louca, 2007. “Systematic Identification of Decoupling in Dynamic System Models”. Journal of Dynamic Systems Measurement and Control, Vol. 129, No. 4, pp. 503 – 513. Published by the American Society of Mechanical Engineers, ISSN Print 0022-0434, ISSN Online 1528-9028, New York, NY. Rosenberg, R.C., 1971. “State-Space Formulation for Bond Graph Models of Multiport Systems.” Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 93, pp. 36-40. Published by ASME, New York, NY. Rosenberg, R.C., and D.C. Karnopp, 1983. Introduction to Physical System Dynamics. McGraw-Hill, ISBN 0070539057. Stein, J.L. and L.S. Louca, 1996. “A Template-Based Modeling Approach for System Design: Theory and Implementation.” TRANSACTIONS of the Society for Computer Simulation International. Published by SCS, ISSN 0740-6797/96, San Diego, CA. Walker, D.G., J.L. Stein, and A.G. Ulsoy, 1996. An InputOutput Criterion for Linear Model Deduction. Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 122, No. 3, pp. 507-513. Published by ASME, New York, NY. Wilson, B.H. and J.L. Stein, 1995. “An Algorithm for Obtaining Proper Models of Distributed and Discrete Systems.” Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 117, No. 4, pp. 534-540. Published by ASME, New York, NY.

APPENDIX: JUNCTION STRUCTURE MATRICES

⎡ m ⋅I 0n×n ⎢ S = ⎢ i n×n ⎢ 0n×n ci ⋅ In×n ⎣ L = bi ⋅ In×n

JSS

⎡ 0 ⎢ = ⎢ n×nT ⎢ −J1 ⎢⎣

J1 0n×n

⎤ ⎥ ⎥ ⎥ ⎦

−1

⎤ ⎡ J ⎥ ⎢ 1 , J = ⎥ ⎢ SL ⎥ ⎢ 0n×n ⎥⎦ ⎣

⎧ ⎪ 0(n−1)×1 ⎪ ⎪ ⎤ ⎪ ⎪ ⎥ 1 ⎥ , JSU = ⎨ ⎪ ⎪ ⎥ ⎪ 0n×1 ⎦ ⎪ ⎪ ⎩

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

JLS = −JTSL , JLL = 0n×n , JLU = 0n×1 JUS = JTSU ,

JUL = JTLU ,

JUU = 0

⎡ −1 2 −1 0 … 0 0 0 ⎤⎥ ⎢ ⎢ 0 −1 2 −1 … 0 0 0 ⎥⎥ ⎢ ⎢ 0 0 −1 2 … 0 0 0 ⎥⎥ ⎢ J1 = ⎢ ! ! ! ! " ! ! ! ⎥ ⎢ ⎥ ⎢ 0 0 0 0 # −1 2 −1 ⎥ ⎢ ⎥ 0 0 0 … 0 −1 2 ⎥ ⎢ 0 ⎢ ⎥ 0 0 0 … 0 0 −1 ⎥⎦ ⎢⎣ 0

REFERENCES Bauchau, O.A. and J.I. Craig, 2009. Structural Analysis. Springer, Netherlands. ISBN 978-90-481-2515-9 Borutzky, W., 2004. Bond Graph Methodology: Development and Analysis of Multidisciplinary Dynamic Systems. Springer, ISBN 978-1848828810. Brown, F.T., 2006. Engineering System Dynamics: A Unified Graph-Centered Approach, Second Edition. CRC Press, ISBN 9780849396489. Ferris, J.B., and J.L. Stein, 1995. “Development of Proper Models of Hybrid Systems: A Bond Graph Formulation.” Proceedings of the 1995 International Conference on Bond Graph Modeling, pp. 43-48, January, Las Vegas, NV. Published by SCS, ISBN 1-56555-037-4, San Diego, CA. Genta, G., 2009. Vibration Dynamics and Control. Springer, Netherlands. ISBN: 978-0-387-79579-9. Karnopp, D.C., D.L. Margolis, and R.C. Rosenberg, 2006. System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th Edition. Wiley, ISBN 978-0-471-70965-7. Louca, L.S., J.L. Stein, G.M. Hulbert, and J.K. Sprague, 1997. “Proper Model Generation: An Energy-Based Methodology”. Proceedings of the 1997 International Conference on Bond Graph Modeling, Vol. 29, No.1, pp. 44-49, Phoenix, AZ. Published by SCS, ISBN 1−56555−103−6, San Diego, CA. 340