1 ME 344 REVIEW FOR FINAL EXAM LOCATION: CPE 2.204 M. D. BRYANT DATE: Wednesday, May 7, 2008 9-noon Finals week office hours: May 6, 4-7 pm Permitted at final exam: 1 sheet of formulas & calculator I. Physical systems to bond graph Mechanical translational & rotational free body diagrams series elements: 0 parallel elements: 1 electrical series elements: 1 parallel elements: 0 fluidic II. Power P = ef III. Energy E = P dt = ef dt IV. Bond graph elements resistance: R energy storage: C and I links between domains: GY: effort to flow, flow to effort TF: effort to effort, flow to flow sources: effort: Se flow: Sf junctions 0: common effort, balance of flows 1: common flow, balance of efforts power flows: arrows indicate positive direction causality assignment via rules of cookbook approach integral (desired for energy storage elements) independent energy storage (IESE) # IESE determines system order derivative (may be forced on some ESE) dependent or linked devices state equations # St Eqns = # IESE extract from bond graph start with IESE usually revolves around junctions 0 => sum flows and/or common effort 1 => sum efforts and/or common flow input bond contribution ALWAYS on LHS causal stroke /arrow touching only applies to balances
agree w/input bond => + sign disagree w/input bond => - sign 2 V. System response Definition of solution: function that renders the differential equation(s) an identity devoid of derivatives Also satisfies initial conditions linear system x = A x + u homogeneous postulate x i homogeneous = a e λ i t substitute into eqn, get & solve characteristic equation [λ i I - A]a = 0 get eigenvalues λ i back substitute λ i [λ i I - A]a i = 0 solve eigenvectors a i = [a ii ] particular soln: u(t) = Uo us(t) step function: us(t) Note: u(t) = U o e st us(t) step, s = 0 growing exponential, s = σ > 0 decaying exponential, s = σ < 0 oscillatory, s = jω sin ωt, take Im parts cos ωt, take Re parts exp.grow/decay oscillation, s = σ + j ω sin ωt, take Im parts cos ωt, take Re parts variation of parameters integration technique n complete x complete = x particular + x i homogeneous i=1
initial conditions apply to complete solution x complete determine arbitrary constants 1st order system " x + x = h(t) general approach (integrate equations) time constant τ 2nd order system natural frequency ω n damping ratio ζ x + 2"# n x + # 2 n x = f (t) damped natural frequency ω d = ω n 1 - ζ 2 ONLY IF 0 ζ < 1 higher order system 3 VI. Superposition for multiple input excitation functions linear system multi-inputs fi(t) multiple outputs x i p Ν complete particular xp = x i p i=1 VII. Stability eigenvalues IF EVEN ONE Re(λ i ) > 0, e λt becomes large => UNSTABLE all Re(λ i ) < 0, stable stable but oscillatory: Re(λ i ) < 0 but Im(λ i ) 0 VIII. Transfer functions ratio: (specified) output to (specified) input basis: f(t) = F(s) e st input, x(t) = X(s) e st output derive transfer function G(s) = X(s)/F(s) substitute x(t) = X(s) e st, f(t) = F(s) e st ordinary diff eqn, solve for X(s)/F(s) state equations, Cramer's rule for X(s) Laplace transform F(s) = L{ f (t);t " s} = $ % f (t)e #st dt, Apply to Diff. Eq. with input f(t), output x(t) Solve for function G(s) = X(s)/F(s) Inverse Laplace (tables used backwards) => x(t) differential equation(s) from transfer: s = d/dt t= 0
characteristic equation from denominator 4 poles = eigenvalues (s poles transfer ) zeroes from numerator (s zeroes transfer 0) Block Diagrams n Blocks Cascaded, total = product: G = G 1 G 2 G n Summing junctions Total transfer function from block diagram IX. Bode plots Directly from measured sinusoidal data Transfer function G(s) G(jω) = G(s) s = jω Bode (vs. log 10 ω) magnitude: MdB = 20 log 10 M(ω), M(ω) = G(jω) 1st breaks 20 db/dec @ ω = 1/τ 2nd breaks 40 db/dec @ ω = ω n zeroes move MdB up/ poles move MdB down phase: φ(ω) = arg G(jω) 1st through 90, 45 @ ω = 1/τ 2nd through 180, 90 @ ω = ω n zeroes => positive, poles => negative steady state response to A sin ωοt: A M(ωο) sin[ωοt + φ(ωο)] read M(ωο) and φ(ωο) from Bode plot X. Multiport storage elements C device I device energy E = E(q) = V q (q), e i = energy E = E(p) = T p (p), f i = V q q i T p p i XI. Control Systems Goal: output tracks input Feedback: reduces effects of disturbances PID controllers definitions plant controller feedback closed loop transfer function
G cl (s) = X(s) R(s) = G(s) 1+ G(s)H (s) 5 system response / step, sinusoid, etc. stability from denominator s charac. eqn. P, D, I action and effects disturbances steady state error