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Modeling of Physical Systems: HW 5–due 10/9/12 Page 1
For problems 1 through 3 below, do the following: a) develop a bond graph model, a)
assign power convention, number and label the bonds, c) assign causality, d) use causality
to identify the number of independent states and write the state vector, x, e) derive state
equations in state space form. Assume model elements have linear constitutive relations only
if a parameter is explicitly indicated (e.g., bearing friction, b, means bearing imparts torque,
Tb = bω).
Problem 1: Model the system from problem 4-3(f) from Karnopp, et al. (as shown below)
Problem 2: Model the system from problem 4-5(d) from Karnopp, et al. Note: it has been
modified from the original text so that To(t) is applied on the ‘coupling’ disk not on the end
of the shaft.
Problem 3: Model the system from problem 4-2(a) in Karnopp, et al. (as shown below)
Problem 4: Refer to the schematic below of the silent air pump system. Develop a complete
bond graph, assign causality, and indicate states of the system. Assume that all mass in the
slider-crank-piston is referred to the ‘crank’ and lump with motor rotor inertia, Jm. Show
why it is preferred to have causality on the transformer so that ‘flow’ is specified into the
piston. Assume pmdc motor with torque constant, rm. Assume no compliance of air in
the piston chamber for this model. Derive state equations for this system, using unknown
functions for constitutive relations as needed. Assume linear constitutive relations where it
can be justified to do so.
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
Modeling of Physical Systems: HW 5–due 10/9/12 Page 2
Problem 5: Develop a bond graph model for the electric circuit shown below (studied
in HW 2), and use the bond graph to identify the system states and to derive the state
equations.
Problem 6: Study the three systems shown below from Problem 5-13 in Karnopp, et al.
Use causality to identify any difficulties and describe any problems. If possible, derive the
state equations. If not possible, explain how the system should be modified to correct the
problem (but do not derive any equations for corrected systems).
R.G. Longoria, Fall 2012 ME 383Q, UT-Austin
Open valve:
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 2 4 Flow Velocity in Tunnel 0 200 400 600 800 1000 1200 1400 1600 1800 2000
- 0 5 x 10 (^5) Inertial Pressure in Tunnel 0 200 400 600 800 1000 1200 1400 1600 1800 2000 3100 3150 3200 Water Level in Surge Tank (m)
Close valve:
0 500 1000 1500 2000 2500 3000
- 0 5 Flow Velocity in Tunnel 0 500 1000 1500 2000 2500 3000
- 0 5 x 10 (^5) Inertial Pressure in Tunnel 0 500 1000 1500 2000 2500 3000 3100 3200 3300 Water Level in Surge Tank (m)
APPENDIX
Surge Tank Simulation
Page 1 of 4
This appendix contains the listings used for a MATLAB m-file simulation of the surge tank
problem.
File: surge_tank_setup.m (run this file to initialize all variables)
% This file defines the tabular constitutive % laws for functions used in the surge tank % simulation. % % Make variables global. clear all global Vc Pc topen sopen tclose sclose sw global It Qp Pin Gamma_t Rl Rt % % Surge tank: V=V(P) % If called P=table1(volume,V), will return % linearly interpolated pressure. % Vc=[0 200 800 1500 2700 3300 4100 4550 4750]; Pc=[0 98100 196200 294300 588600 686700 784800 882900 1079100]; z=zeros(9,2); z(:,1)=[3100;3110;3120;3130;3160;3170;3180;3190;3210]; z(:,2)=Vc; % % S(t) functions for opening or closing of valve. % tclose=[0 5 20 5000.0]; sclose=[1 0.15 0 0]; % topen=[0 5 15 20 30 35 40 5000]; sopen=[0 0.4 0.4 0.8 0.8 1.0 1.0 1.0]; % % Define variables % It = fluid inertance It = 1000.013580.0/12.0; % Qp = magnitude of flow rate modulated by S(t). Qp = 0.6sqrt(2.09.807(3168-3072)); % Pin = source of pressure at inlet to tunnel. Pin = 1000.09.807(3168-3100); % Gamma_t = Critical value of fluid momentum where flow is turbulent. Gamma_t=2000.010001e-613580/sqrt(412/3.1415927); % Rl = coefficient for friction in tunnel if laminar. Rl = 8.01.0e-613580.0/(9.807144); % Rt = coefficient for friction in tunnel if turbulent. Rt = 4.110009.807/(144It*It);
APPENDIX
Surge Tank Simulation
Page 3 of 4
File: sim_surge_close.m (this closes the valve – needs final values form ‘close’)
% Simulation of the differential equations defined in surgetank % from 0.0 < t < Nstepstintrval. Using the routine ODE45. sw = 0; Nsteps = 600; tinterval = 5.0; t0 = -tinterval; tfinal = Nstepstinterval; t(Nsteps)=tfinal; t(1)=0.0; % note: these initial conditions come from running the % surge_tank_open.m routine, and finding final values x(1,1) = 2247.395202707482; x(1,2) = 2.946369274944220e+07; tsfinal = t(1); xs0 = [x(1,1) x(1,2)]; % Define initial conditions. Pinertial(1) = Pin-fluidresist(x(1,2))-interp1(Vc,Pc,x(1,1)); Wlevel(1) = interp1(Vc,Pc,x(1,1))/9807 + 3100.0; % % for i=2:Nsteps, ts0 = tsfinal; tsfinal = ts0 + tinterval; tol = 1.e-3; % Accuracy trace = 0; [ts,xs] = ode15s(@surgetank,[ts0 tsfinal],xs0); intdim=size(ts); t(i) = ts(intdim(1)); x(i,1) = xs(intdim(1),1); x(i,2) = xs(intdim(1),2); Pinertial(i) = Pin-fluidresist(x(i,2))-interp1(Vc,Pc,x(i,1)); Wlevel(i) = interp1(Vc,Pc,x(i,1))/9807 + 3100.0; xs0 = [x(i,1) x(i,2)]; % Define initial conditions. end vt = x(:,2)/(It*12); figure(2) subplot(3,1,1), plot(t,vt), title('Flow Velocity in Tunnel') subplot(3,1,2), plot(t,Pinertial), title('Inertial Pressure in Tunnel') subplot(3,1,3), plot(t,Wlevel), title('Water Level in Surge Tank (m)')
APPENDIX
Surge Tank Simulation
Page 4 of 4
File: surgetank.m (there are the ODEs to be called by solver)
function xprime = surgetank(t,x); % surgetank(t,x) returns the state derivatives of the surge tank % system. Used by sim_surge. % Equation 1: volume of surge tank % Equation 2: fluid momentum in pressure tunnel % global It Qp Pin Gamma_t Rl Rt tclose sclose topen sopen Vc Pc sw if sw == 1 sfunc = interp1(topen,sopen,t); else sfunc = interp1(tclose,sclose,t); end xprime = [x(2)/It-Qp*sfunc;Pin-fluidresist(x(2))-interp1(Vc,Pc,x(1))];
File: fluidresist.m (this routine switches the fluid resistance function)
function resistance = fluidresist(fmom) global Gamma_t Rt It Rl if fmom > Gamma_t resistance = Rtfmomabs(fmom); else resistance = Rl*fmom/It; end
