Integrated Modeling of Physical System Dynamics© Neville Hogan 1994 page 1 Hydraulic Ram Figure 5.32 shows a sketch of a hydraulic actuator of the kind commonly used in heavy earth-moving machinery. The difference in fluid pressure acting on the two sides of the piston exerts aforce on the rod. As pressure is defined as force per unit area, the power transduction phenomenonconnecting the hydraulic and translational domains is that of a transformer. In general, the piston area in the two chambers need not be the same1, and the rate of increase of the volume of onechamber, Qc1, need not equal the rate of decrease of the volume of the other chamber, Qc2, so two transformers may be needed to describe the piston. They are connected to a one-junction on thetranslational side (representing the unique piston velocity) as shown in figure 5.33(a). cylinder rod fluid lines (a) F v Q pistono-ring seals supply line return line P PP Q c1 c2 1 2 1 P2 (b) Figure 5.32: Sketch of a hydraulic actuator (a). A cross-sectional view is shown in (b). 1 For example, the sketch shows a rod protruding from each side of the piston, but single-rodcylinders are also common; with a single rod, the area on one side of the piston exceeds that on theother side by the area of the rod.Integrated Modeling of Physical System Dynamics© Neville Hogan 1994 page 2 If the piston area in each chamber is the same, then the two transformers may be replaced by asingle equivalent transformer as shown in figure 5.33(b). Any increase in volume of one chamberequals the decrease in volume of the other and the transformer is connected to a one junction in thefluid domain (representing the common rate of volume change, Qc). Constitutive equations are then as follows. F = A (Pc1 - Pc2) (5.69) Qc = A v (5.70) TF 1 1 TF 2 Qc1 1 1 Qc2 A A v TF1 1 Qc A v (a) (b) Figure 5.33: Bond graph fragments representing power transduction in the hydraulic actuator.(a): general model for unequal piston areas. (b): special case of equal piston areas. Parasitic Dynamics Like other power transduction devices, a hydraulic ram also embodies energy storage anddissipation effects which may need to be modeled. For simplicity, we will assume the hydraulicfluid to be incompressible (though that is often one of the prominent "parasitic" dynamicphenomena) and model only the inertial effects due to acceleration of the fluid and the powerdissipation due to its motion through the lines and orifices. On the mechanical side we will model the inertia of the piston and rods (considered to be a single rigid body) and the friction due to thesliding seals, described for simplicity as a single linear friction element (though in reality, asignificant component of the dissipative behavior is likely to resemble dry friction). Friction due to flow in a pipe was considered in Chapter 4. A crude model of the inertial effects due to acceleration of the fluid may be derived as follows. Consider an incompressible fluidmoving in a pipe of uniform cross-sectional area, Ap. The mass, m, of the "slug" of fluid between two cross sections separated by a distance l is determined by the dimensions of the pipe and the density, ρ. m = ρlAp (5.71) Considerable simplification is obtained by assuming that the fluid flow velocity has the same value,v, at all points. That velocity is determined by the volumetric flow rate, Q, divided by the crosssectional area. v = Q/Ap (5.72) Given these assumptions, the kinetic co-energy of the moving fluid is:Integrated Modeling of Physical System Dynamics© Neville Hogan 1994 page 3 *m ρlAp Q2 ρl Ek = 2 v2= 2 Ap2 = Ap Q2 (5.73) Thus, by analogy with a rigid body in translation, the energy stored in the moving fluid may bedescribed by an ideal inertia, characterized by a linear relation between a flow, Q, and a generalized momentum, Γ, termed pressure momentum. Γ Q = (5.74)I The parameter I = ρl/Ap is the inertance of the fluid, determined by the geometric and material properties of the pipe and the fluid. From the fundamental definition of generalized momentum, thepressure difference accelerating the fluid is the time derivative of the pressure momentum. ∆P = dΓ/dt (5.75) Thus pressure difference is proportional to the rate of change of fluid flow rate. ρl dQ∆P = Ap dt (5.76) A bond graph of the complete system is shown in figure 5.34(a). For the purpose of this example,it has been assumed that the supply line is connected to a power source at constant pressure, P1, the return line to a sump at constant pressure, P2, (e.g. atmospheric pressure) and that the mechanical load is a constant force, F. We may analyze this system by assigning causality. After assigning the required causality to thesource elements, we are free to assign integral causality to one of the energy storage elements(arbitrarily chosen in figure 5.34(b) to be the translational inertia). Propagating the causalconsequences of that choice through the graph completes the causality assignment. Causality assignment reveals that two of the three inertias are dependent energy-storage elements.The velocity of the piston determines the flow rate in the fluid lines and a single variable issufficient to characterize the kinetic energy stored in both the translating masses and the movingfluid. In effect, the piston and rods and the fluid in the lines have been described as a single "rigidbody" even though the fluid in the lines would hardly be considered rigid in the usual sense of theword.Integrated Modeling of Physical System Dynamics© Neville Hogan 1994 page 4 P:Se1 TF1 1 Qc A v 1 1 Q Q 2 1 R I R I R : b : m : 1I R: 2 R: 1 Se :F P:Se2 I : I2 Hydraulic domain Translational domain (a) I Se TF1 1 1 1 R R I R Se Se I (b) Figure 5.34: (a) Bond graph of a model of the hydraulic actuator including parasitic effects. (b)Bond graph with causality assigned. The parasitic fluid elements may be transformed into the translational domain. The fluid inertias are equivalent to translational masses. Combining equations 5.69, 5.70 and 5.76 the effective mass dueto each fluid line is meff = A2I = A2ρl/Ap (5.77) The fluid resistors are equivalent to linear translational friction elements. The effective friction coefficient due to each fluid line is beff = A2R (5.78)Integrated Modeling of Physical System Dynamics© Neville Hogan 1994 page 5 ISe I I 1 TF 1 Se Se R R R (a) S eP1: I : m + A I + A I 21 22 TF1 1 RS e S e F: P2 A : : b + A R + A R1 2 22 (b) Figure