modeling and experimentation: compound...

Modeling Simulation Block Diagrams Experimentation Summary

Modeling and Experimentation:Compound Pendulum

Prof. R.G. Longoria

Department of Mechanical EngineeringThe University of Texas at Austin

Fall 2014

ME 144L Dynamic Systems and Controls Lab (Longoria)


Overview

This lab focuses on developing a mathematical model of the compoundpendulum system studied in lab experiments.

This lecture reviews the model, then shows how to develop themathematical model both as a second order ODE and in state space form.

A key goal is to learn about how numerical simulations of math models canbe used to predict response of a system. A model simulation can be thoughtof as another way to experiment with a system.

Block diagrams are also introduced as a way to describe systems withinsimulation environments, and LabVIEW examples are provided.

The model simulation results from the pendulum system can be comparedto measured response data in order to improve the model, especially fordetermining the dominant type of friction causing the pendulum motion todecay.



Lab Objectives

1 Practice system modeling methods, including constitutive modeling forfriction effects

2 Become familiar with the National Instruments Control and Simulationmodule

3 Use a simulation model to compare with measured data and tune systemparameters



Use modeling to build insight and derive a proper dynamicsystem model

Understand the type of model that is needed. To predict thependulum response observed in the lab, we need a dynamic systemmodel in the form of ODEs

Recognize any special considerations. There is significant friction inthe pivot/bearing causing the pendulum oscillation to decay, so themodel needs to include torques induced by friction.

Make sure the model can be used for the intended purpose. Given aproper model, we need to be able to solve the equations. Since theODEs cannot be solved in closed-form, we need to put the model in aform suitable for simulation.



In introductory dynamics courses, you often solved forforces, accelerations, or velocities of bodies at specificinstants in time. Consider the example below.



Example continued...



We seek system dynamics models as ordinary differentialequations (ODEs) for key variables of interest in order tostudy how a system behaves over all time.

The measured decay of the pendulum oscillations in the lab indicates lossof energy over time due to dissipative effects.



Example: the simple pendulum model

A simple pendulum modelrepresents dynamics of a pointmass at the end of fixed length,massless, rod or string, constrainedto move in a plane as shown below.Gravity acts downward.

If there are no other forces, the nettorque for rotation about the pivotis, ∑

T = −Tθ = −mgl sin θ

Now, apply Newton’s law,

dh

dt= h = Jθ =

∑T

where J = ml2, so a 2nd orderODE in θ is found,

ml2θ +mgl sin θ = 0



Contrast the simple pendulum with the compoundpendulum

Simple pendulum:

J = ml2

m is the total mass, l is distance fromthe pivot to the mass


θ +g

lsin θ = 0

Compound pendulum:

J0 = J +ml2C

m is the total mass, lC is distancefrom the pivot to the CG

J0θ +mglC sin θ = 0

θ +mglCJ0

sin θ = 0

Modeling a compound pendulum as a simple pendulum would incur error thatdepends on how widely mass is distributed. In some cases, a simple pendulummight be a reasonable approximation.



Additional torques in the pendulum model

Aside from proper modeling of the inertia effects, the model also needs toconsider other torques that cause changes in the angular momentum. Primarily,we are concerned here with torques due to dissipative effects (friction at thepivot, air friction, etc.), or any applied torques or forces on the pendulum body.Both can be included in the Newton’s law equation,

h = −Tθ − Tf − Ta

where Tf is the net torques due to friction/losses and Ta are applied or actuatortorques.

It is clear that the pendulum studied in the lab is not lossless since it does notoscillate indefinitely, and there are no other applied forces or actuator torques.



Modeling frictional torques

Frictional effects can be modeled as frictional torques that act about the pivot.These torques are modeled as functions of angular velocity, T = f(ω). Considertwo types:

These models represent different types of physical processes. For example, if abearing has a (viscous) fluid film, induced forces tend to vary linearly withvelocity (left). On the other hand, bearings with dry contact result in forces moresimilar to Coulomb-type friction, being relatively constant with velocity (right).



Case 1: Compound pendulum with only linear damping

Consider a linear damping torque in the pendulum equation,

h =∑

T = −Tθ − Tb

whereTb = bω = linear damping torque

where ω = θ. Now,J0θ + bθ +mglC sin θ = 0

This equation can be made linear by assuming small motion, so sin θ would be

approximately θ and the equation becomes a linear 2nd order ODE with a

closed-form solution.



Case 2: Compound pendulum with linear damping andnonlinear friction

Now include a nonlinear friction torque,

h =∑

T = −Tθ − Tb − Tc

whereTc = Tosgn(ω) = Coulombic (nonlinear) frictional torque

Linearizing for small motion as done for Case 1 is problematic, because of the‘sgn’ (signum) function.

The signum function has a very large change in value when its argument is closeto zero: sgn(a) = +1 if a > 0 and sgn(a) = −1 if a < 0.

Closed form solutions for motion cannot be found as readily as for Case 1. It turns

out that this type of friction is more prevalent in many practical systems. Solving

this ODE requires numerical integration schemes for simulating the system.



Basic idea behind ODE system simulation

When we talk about system simulation in DSC, this typically refers to solving anODE initial value problem. This means that we have a set of ODEs that modelour system of interest1.

Consider the simplest type of dynamic system, a first order ODE,

dx

dt= f(x, u, t)

which we’ll call a state equation for the state variable, x. The right hand side is afunction of x itself, an input to the system, u, and time, t.

The modeling process provides f(x, u, t).

Numerical integration requires an initial value at initial time, t0, x(t0) = x0.Then, the solution at the next step, t = t0 + ∆t, is to be approximated.

1Some systems may involve both differential and algebraic equations (DAEs), whichwon’t be considered in this course.ME 144L Dynamic Systems and Controls Lab (Longoria)


Euler ODE Solver

You may recall that numerically solving ODEs for initial value problems is simplymarching forward in time. The solution is found at discrete time steps, given theinitial value at t0. Given x0, the value at initial time, an estimate of the value att1 is,

x1 = x0 + ∆x0

The job of the solver is to estimate ∆x0. The simplest algorithm is the Eulersolver, which is basically a Taylor series approximation.

The Euler solve estimates ∆x0 by,

∆x0 = f(x0, u0, t0) · ∆t

using an approximation from the ODE∆x/∆t ≈ f(x, u, t), and given allinitial values. There is always error inthe estimate of the true value, x1.



Selection of ODE Solvers

The Euler method is the simplest ODE solver, and it usually uses a fixedtime step. To get good solutions (more accurate, stable), usually you needto make the time step very small.

The Euler method is a 1st order Runge-Kutta (RK) method. You may havelearned about Runge-Kutta methods in a computational methods course.The most commonly used RK algorithm is 4th order, which uses fourevaluations of the ODEs to estimate the next value of x, as opposed to thesingle evaluation made by the Euler routine. The RK4 algorithm allows youto take larger time steps than Euler, is more stable, but as for Euler the errormust be managed by the user. More sophisticated algorithms use variabletime step to control numerical errors.

Many commercial and open-source software packages have built-in fixed-stepand variable step algorithms that can be used for ODE simulation. Thiscourse will provide practice in using the algorithms available in LabVIEW.



ODE solvers require that the equations be put into 1storder form

The Euler solver was described using the simple 1st order ODE in the variable x.This is the form required by ODE solvers.

Consider the single x variable generalized as a vector x formed by n statevariables of a system.

ODE algorithms are designed to accept descriptions of the system ODEs as avector formed by the model equations. In general, the 1st order vector of stateequations is,

x = f(x,u, t)

where u is now a vector of r inputs.

Writing the equations in this form can either be done by converting the nth orderODE to n 1st order ODEs, or the n 1st order ODEs can be derived directly. Thelatter method is the way equations are directly derived when the bond graphmethod is used.



Example: Convert the simple pendulum model from 2ndorder to 1st order form

Recall the simple pendulum model


To convert this 2nd order ODE to 1storder, first define n = 2 state variablesas x1 = θ and x2 = θ. We now wantto write two 1st order ODEs in termsof these new state variables. The firstone is found by taking the derivativeof the first variable, x1 = θ, which isrecognized as x2. Therefore, the firstequation is,

x1 = x2

Now, take derivative of secondvariable, x2 = θ. To find thisequation, we must use the original 2ndorder ODE, which has θ as well as θ.Solve for θ,

θ = −(g/l) sin θ = −(g/l) sinx1

This gives us the second stateequation,

x2 = −(g/l) sinx1

These two equations are the statespace equations for the simplependulum.



State space form

Here are the simple pendulum state space equations in matrix form:

x =

[x1x2

]= f(x,u, t) =

[x2

−gl sin(x1)

]

The rightmost term is a vector of the n nonlinear, state equations.These are in the form required by ODE solvers.

The state equations are generally coupled; i.e., each equation can bea function of any and all system states.

Each equation quantifies how each state changes over time (the slopeat each time step).

In a numerical algorithm, each equation is used to estimate the valueof the state at each time step, as illustrated for the simple Eulerroutine.



Using solvers with script files

Commercial software packages such as Matlab and LabVIEW have solversthat will numerically integrate ODEs. The ODEs are usually formatted ina script function file.

For example, for the simple pendulum, a script file may take the form:

function filefunction f = PendulumEqs(t,x)

global g l

f1 = x(2);

f2 = -g*sin(x(1))/l;

f = [f1;f2];

Here, the values returned by this function are sent to the ‘solver’ (e.g., ode45).The use of script files is very effective. In this course, we will be learning adifferent approach that uses block diagrams to graphically describe the ODEs.Using this approach allows use of LabVIEW for simulation.



Block diagram representations of systems

Block diagrams are used to graphically represent signal flow andfunctional relationships between signals.

A directed line (with arrowhead) indicates a ‘signal’, which canrepresent a system variable or a control input.

Nodes, shown as blocks, represent input-output relationships betweensignal variables..

Basic functions are gains, summers, integrators and differentiators,and as such block diagrams are effective in representing differentialequation models.

Modern software programs use block diagrams as a way tocommunicate system representations to computer-aided analysis tools.



Block diagram programming environments have becomevery popular as part of computer-aided engineeringpackages

Why?

They allow us to describe systems using a graphical form, which canbe useful for communicating how different components interact.

Block diagrams are functional and not just schematics (as we willsee).

There is a rich history not only in describing control systems but alsoin how (analog) computational algorithms were originally designed.



Block diagram algebra

Summing point

Branching point

Let x(t) input and y(t) output. For

general nonlinear systems, the output

is simply, y = g(x).

If the system is linear, y = G · x

G (gain) may be a constant or asystem transfer function, G(s).

The ‘s’ represents the ‘Laplace

operator’.



Block diagram calculus

Other blocks especially useful for

representing physical system models

include the integrator:

There is also a derivative

These are common blocks, but they

are sometimes shown in software

products using the s’ Laplace operator.

For example, the integrator is:

and the derivative block is

Important: these are just symbols - the blocks performnumerical integration/differentiation (not Laplacetransforms).



How do you build a block diagram of system stateequations?

For a given complete set of state equations (as you will learn to derive in ME344), the following steps are taken:

1 Identify an integrator for each first order equation, so each integratorgenerates a state

2 Form the terms of each equation using system parameters, gain blocks, andfunctional blocks

3 Use summing blocks form the state derivatives (i.e., add terms as needed toform the right hand side of each 1st order equation)

4 Specify initial conditions, x(0), and system inputs, u

NOTE: There are some systems that result in equations where the ODEs do nottake this desirable form. You may learn about algebraic loops and derivativecausality in your DSC course.



Block diagram of nth order state space system

The system equations generate the dx/dt for each state, and the nintegrators produce the states, which are then passed back into the systemequations.

Note each integrator requires an initial condition, x(0).



Example: block diagram for linear 1st order ODE

Re-write the 1st order ODE in the form:

dx

dt=

1

τ[−x+ u(t)]

Then we can use basic block elements to describe the algebra and thecalculus in the equations, as shown below.

Remember that to find x the integrator needs to have an initial condition,x(0).



Two popular software products that use block diagramsare LabVIEW and Matlab/Simulink.

These programs feature:

combined structured and block diagram programming

a capacity to communicate with physical hardware

efficient means for designing a human-user interface

modeling and simulation tools for physical and engineering systems

We have adopted LabVIEW for use in this course.



LabVIEW Control and Simulation Module



LabVIEW Control and Simulation Module

Build your block diagrams within theC&S Loop:

Basic block diagram VIs are foundunder Signal Arithmetic:

and Continuous Linear Systems:



The integrator block is a key element for integrating yourstate equations

Remember that the input to theintegrator block will be the derivativeof your state:

and the output is the state.

By double-clicking the integrator VIblock, you can access settings:

The settings allow you to configurehow you will set key integrationparameters and whether you will setthem in the dialog box or wire to aterminal.



Example: LV simulation diagram for standard 1st ordersystem



Example: LV simulation diagram for the simple pendulum

In this example VI for the simple pendulum, a formula node used as analternate way to code the state equations rather than with block diagrams.



Experimentation

The experimentation is now to be conducted using a numerical model.

Here are some suggestions for using your simulation model:

determine if your estimates of the pendulum moment of inertia allowspredictions to compare well with measured response data; show howthe simulation can be tuned to improve the model

show that the simulation gives the same type of responsecharacteristics, especially with proper frictional torque models; weighthe effect of linear versus nonlinear type friction

calculate stored energy, and improve prediction on how energydecreases with each cycle



Summary

Build experience formulating system models from physical laws

Use a known physical problem for learning about constitutive models,especially linear vs. nonlinear types

More opportunity to experiment with LabVIEW VIs for simulation,learning about Control and Simulation Module

Use LabVIEW for analyzing data from experiments, relating tomodeling results


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