The shock absorber from LABEIN

                                                         mesfet [1]
Sara Casado (LABEIN)

The system under consideration has 27 equations and 27 unknowns with several parameters whose value is only known approximately. Two kind of unknowns are considered:

• the pressures: {P_{i} : 1 ≤ i ≤ 14}.

• the flows: {q_{j} : 1 ≤ j ≤ 15}.

The total degree of the equations is equal to one or two. These come from two different physical conditions: flow equilibrium in the different components of the electric-hydraulic circuit and pressure value in every node of such a diagram. The flow equilibrium equations are:

\begin{aligned}q_{1} & = & q_{2} \\ q_{2} & = & q_{3} + q_{7} \\ q_{3} & = & q_{4} \\ q_{4} & = & q_{5} \\ q_{5} & = & q_{6} \\ q_{7} & = & q_{8} \\ q_{8} & = & q_{9} + q_{13} \\ q_{13} & = & q_{14} \\ q_{14} & = & q_{15} \\ q_{9} & = & q_{10} \\ q_{6} + q_{10} & = & q_{11} \\ q_{11} & = & q_{12} \end{aligned}

and the pressure equations are:

P_{2} − P_{1} = q_{1}^{2} \frac{ρ}{2}k_{cont} \frac{S_{1} − S_{2}}{S_{1}^{3}}
P_{3} − P_{2} = q_{2}^{2} \frac{ρ}{2}[k_{exp} \frac{(S_{3} − S_{2})^{3}}{S_{2}^{2}S_{3}^{2}} + k_{gr} \frac{1}{S_{2}^{2}} ]
P_{4} − P_{3} = q_{3}^{2} \frac{ρ}{2}k_{cont} \frac{S_{3} − S_{4}}{S_{3}^{3}}
P_{5} − P_{4} = q_{4}^{2} \frac{ρ}{2}k_{exp} \frac{(S_{5} − S_{4})^{2}}{S_{5}^{2}S_{4}^{2}}
P_{6} − P_{5} = q_{5}A\frac{νρℓ}{d^{4}}
P_{10} − P_{6} = q_{6}^{2} \frac{ρ}{2}[k_{exp} \frac{(S_{10} − S_{6})^{3}}{S_{6}^{2}S_{10}^{2}} + k_{gr} \frac{1}{S_{6}^{2}} ]
P_{7} − P_{3} = q_{7}^{2} \frac{ρ}{2}[k_{cont} \frac{S_{3} − S_{7}}{S_{3}^{3}} + k_{gr} \frac{1}{S_{3}^{2}} ]

P_{8} − P_{7} = q_{8}A\frac{νρℓ}{d^{4}}
P_{9} − P_{8} = q_{9}^{2} \frac{ρ}{2}[k_{cont} \frac{S_{8} − S_{9}}{S_{8}^{3}} + k_{gr} \frac{1}{S_{8}^{2}} ]

P_{10} − P_{9} = q_{10}^{2} \frac{ρ}{2}[k_{exp} \frac{(S_{10} − S_{9})^{2}}{S_{9}^{2}S_{10}^{2}} + k_{gr} \frac{1}{S_{9}^{2}} ]

P_{11} − P_{10} = q_{11}^{2} \frac{ρ}{2}[k_{cont} \frac{S_{10} − S_{11}}{S_{10}^{3}} + k_{gr} \frac{1}{S_{10}^{2}} ]

P_{14} − P_{11} = q_{12}^{2} \frac{ρ}{2}k_{exp} \frac{(S_{14} − S_{11})^{2}}{S_{14}^{2}S_{11}^{2}}

P_{12} − P_{8} = q_{13}^{2} \frac{ρ}{2}[k_{cont} \frac{S_{8} − S_{12}}{S_{8}^{3}} + k_{gr} \frac{1}{S_{8}^{2}} ]

P_{13} − P_{12} = q_{14}^{2} \frac{ρ}{2}[k_{exp} \frac{(S_{13} − S_{12})^{2}}{S_{12}^{2}S_{13}^{2}} + k_{gr} \frac{1}{S_{12}^{2}} ]

P_{14} − P_{13} = q_{15}^{2} \frac{ρ}{2}k_{exp} \frac{(S_{14} − S_{13})^{2}}{S_{14}^{2}S_{13}^{2}}

The initial data is given by the knowledge of the pressure and flow at the entry node:

q_{1} = Q, P_{1} = 0

In this case, only real solutions are required.

Characteristics:

• Number of variables: 27.

• Bound on the number of complex solutions: 2^{12}

Example 1:

params := [kcont, kexp, kgr,r,Anrd,
           S1 S2,S3,S4,S5,S6,S7,S8,S9,S10,S11,S12,S13,S14, Q];
[q1-q2, q2-(q3+q7), q3-q4, q4-q5, q5-q6, q7-q8, q8-(q9+q13), q13-q14,
q14-q15, q9-q10, q6+q10-q11, q11-q12,
P2-P1-q1^2*(r/2)*kcont*(S1-S2)/(S1^3),
P3-P2-q2^2*(r/2)*(kexp*(((S3-S2)^3)/(S2^2*S3^2))+kgr*(1/(S2^2))),
P4-P3-q3^2*(r/2)*kcont*((S3-S4)/(S3^3)),
P5-P4-q4^2*(r/2)*kexp*(((S5-S4)^2)/(S5^2*S4^2)),
P6-P5-q5*Anrd,
P10-P6-q6^2*(r/2)*(kexp*(((S10-S6)^3)/(S6^2*S10^2))+kgr*(1/(S6^2))),
P7-P3-q7^2*(r/2)*(kcont*((S3-S7)/(S3^3))+kgr*(1/(S3^2))),
P8-P7-q8*Anrd,
P9-P8-q9^2*(r/2)*(kcont*((S8-S9)/(S8^3))+kgr*(1/(S8^2))),
P10-P9-q10^2*(r/2)*(kexp*(((S10-S9)^2)/(S9^2*S10^2))+kgr*(1/(S9^2))),

P11-P10-q11^2*(r/2)*(kcont*((S10-S11)/(S10^3))+kgr*(1/(S10^2))),
P14-P11-q12^2*(r/2)*kexp*(((S14-S11)^2)/(S14^2*S11^2)),
P12-P8 -q13^2*(r/2)*(kcont*((S8-S12)/(S8^3))+kgr*(1/(S8^2))),
P13-P12-q14^2*(r/2)*(kexp*(((S13-S12)^2)/(S12^2*S13^2))+kgr*(1/(S12^2))),
P14-P13-q15^2*(r/2)*kexp*(((S14-S13)^2)/(S14^2*S13^2)),
q1-Q,P1
];

References

[1] D. Bini and B. Mourrain. Polynomial test suite. 1996.