HYDRODYNAMICS OF PUMPS

by Christopher Earls Brennen     © Concepts NREC 1994

CHAPTER 9.
UNSTEADY FLOW IN HYDRAULIC SYSTEMS

 

9.1 INTRODUCTION

This chapter is devoted to a description of the methods available for the analysis of unsteady flows in pumps and their associated hydraulic systems. There are two basic approaches to the solution of unsteady internal flows: solution in the time domain or in the frequency domain. The traditional time domain methods for hydraulic systems are treated in depth elsewhere (for example, Streeter and Wylie 1967, 1974), and will only be touched upon here. They have the great advantage that they can incorporate the nonlinear convective inertial terms in the equations of fluid flow, and are best suited to evaluating the transient response of flows in long pipes in which the equations of the flow and the structure are fairly well established. However, they encounter great difficulties when either the geometry is complex (for example inside a pump), or the fluid is complex (for example in the presence of cavitation). Under these circumstances, frequency domain methods have distinct advantages, both analytically and experimentally. On the other hand, the nonlinear convective inertial terms cannot readily be included in the frequency-domain methodology and, consequently, these methods are only accurate for small perturbations from the mean flow. This does permit evaluation of stability limits, but not the evaluation of the amplitude of large unstable motions.

It should be stressed that many unsteady hydraulic system problems can and should be treated by the traditional time domain or ``water-hammer'' methods. However, since the focus of this monograph is on pumps and cavitation, we place an emphasis here on frequency domain methods. Sections 9.5 through 9.10 constitute an introduction to these frequency domain methods. This is followed by a summary of the transfer functions for simple components and for pumps, both noncavitating and cavitating. Up to the beginning of section 9.15, it is assumed that the hydraulic system is at rest in some inertial or nonaccelerating frame. However, as indicated in section 8.13, there is an important class of problems in which the hydraulic system itself is oscillating in space. In section 9.15, we present a brief introduction to the treatment of this class of problems.

 

9.2 TIME DOMAIN METHODS

The application of time domain methods to one-dimensional fluid flow normally consists of the following three components. First, one establishes conditions for the conservation of mass and momentum in the fluid. These may be differential equations (as in the example in the next section) or they may be jump conditions (as in the analysis of a shock). Second, one must establish appropriate thermodynamic constraints governing the changes of state of the fluid. In almost all practical cases of single-phase flow, it is appropriate to assume that these changes are adiabatic. However, in multiphase flows the constraints can be much more complicated. Third, one must determine the response of the containing structure to the pressure changes in the fluid.

The analysis is made a great deal simpler in those circumstances in which it is accurate to assume that both the fluid and the structure behave barotropically. By definition, this implies that the change of state of the fluid is such that some thermodynamic quantity (such as the entropy) remains constant, and therefore the fluid density, ρ(p), is a simple algebraic function of just one thermodynamic variable, for example the pressure. In the case of the structure, the assumption is that it deforms quasistatically, so that, for example, the cross-sectional area of a pipe, A(p), is a simple, algebraic function of the fluid pressure, p. Note that this neglects any inertial or damping effects in the structure.

The importance of the assumption of a barotropic fluid and structure lies in the fact that it allows the calculation of a single, unambiguous speed of sound for waves traveling through the piping system. The sonic speed in the fluid alone is given by c∞ where
......(9.1)
In a liquid, this is usually calculated from the bulk modulus, κ = ρ/(dρ/dp), since
......(9.2)
However the sonic speed, c, for one-dimensional waves in a fluid-filled duct is influenced by the compressibility of both the liquid and the structure:
......(9.3)
or, alternatively,
......(9.4)
The left-hand side is the acoustic impedance of the system, and the equation reveals that this is the sum of the acoustic impedance of the fluid alone, 1/ρc2∞, plus an ``acoustic impedance'' of the structure given by (dA/dp)/A. For example, for a thin-walled pipe made of an elastic material of Young's modulus, E, the acoustic impedance of the structure is 2a/Eδ, where a and δ are the radius and the wall thickness of the pipe (δ « a). The resulting form of equation 9.4,
......(9.5)
is known as the Joukowsky water hammer equation. It leads, for example, to values of c of about 1000m/s for water in standard steel pipes compared with c∞ of about 1400m/s. Other common expressions for c are those used for thick-walled tubes, for concrete tunnels, or for reinforced concrete pipes (Streeter and Wylie 1967).

 

9.3 WAVE PROPAGATION IN DUCTS

In order to solve unsteady flows in ducts, an expression for the sonic speed is combined with the differential form of the equation for conservation of mass (the continuity equation),
......(9.6)
where u(s,t) is the cross-sectionally averaged or volumetric velocity, s is a coordinate measured along the duct, and t is time. The appropriate differential form of the momentum equation is
......(9.7)
where gS is the component of the acceleration due to gravity in the s direction, f is the friction factor, and a is the radius of the duct.

Now the barotropic assumption 9.3 allows the terms in equation 9.6 to be written as
......(9.8)
so the continuity equation becomes
......(9.9)
Equations 9.7 and 9.9 are two simultaneous, first order, differential equations for the two unknown functions, p(s,t) and u(s,t). They can be solved given the barotropic relation for the fluid, ρ(p), the friction factor, f, the normal cross-sectional area of the pipe, A0(s), and boundary conditions which will be discussed later. Normally the last term in equation 9.9 can be approximated by ρ u(dA0/ds)/A0. Note that c may be a function of s.

In the time domain methodology, equations 9.7 and 9.9 are normally solved using the method of characteristics (see, for example, Abbott 1966). This involves finding moving coordinate systems in which the equations may be written as ordinary rather than partial differential equations. Consider the relation that results when we multiply equation 9.9 by λ and add it to equation 9.7:
......(9.10)
If the coefficients of ∂u/∂s and ∂p/∂s inside the square brackets were identical, in other words if λ = ± c, then the expressions in the square brackets could be written as
......(9.11)
and these are the derivatives du/dt and dp/dt on ds/dt= u±c. These lines ds/dt= u±c are the characteristics, and on them we may write:

  1. In a frame of reference moving with velocity u+c or on ds/dt=u+c:
    ......(9.12)
  2. In a frame of reference moving with velocity u-c or on ds/dt=u-c:
    ......(9.13)
A simpler set of equations result if the piezometric head, h*, defined as
......(9.14)
is used instead of the pressure, p, in equations 9.12 and 9.13. In almost all hydraulic problems of practical interest p/ρL c2 « 1 and, therefore, the term ρ-1 dp/dt in equations 9.12 and 9.13 may be approximated by d(p/ρ)/dt. It follows that on the two characteristics
......(9.15)
and equations 9.12 and 9.13 become
  1. On ds/dt = u+c
    ......(9.16)
  2. On ds/dt = u-c
    ......(9.17)

Figure 9.1 Method of characteristics.

These are the forms of the equations conventionally used in unsteady hydraulic water-hammer problems (Streeter and Wylie 1967). They are typically solved by relating the values at a time t+δ t (for example point C of figure 9.1) to known values at the points A and B at time t. The lines AC and BC are characteristics, so the following finite difference forms of equations 9.16 and 9.17 apply:
......(9.18)
and
......(9.19)
If cA=cB=c, and the pipe is uniform, so that dA0/ds=0 and fA=fB=f, then these reduce to the following expressions for uC and h*C:

......(9.20)
......(9.21)

 

9.4 METHOD OF CHARACTERISTICS

The typical numerical solution by the method of characteristics is depicted graphically in figure 9.2. The time interval, δt, and the spatial increment, δs, are specified. Then, given all values of the two dependent variables (say u and h*) at one instant in time, one proceeds as follows to find all the values at points such as C at a time δt later. The intersection points, A and B, of the characteristics through C are first determined. Then interpolation between the known values at points such as R, S and T are used to determine the values of the dependent variables at A and B. The values at C follow from equations such as 9.20 and 9.21 or some alternative version. Repeating this for all points at time t+δt allows one to march forward in time.

Figure 9.2 Example of numerical solution by method of characteristics.

There is, however, a maximum time interval, δt, that will lead to a stable numerical solution. Typically this requires that δt be less than δx/c. In other words, it requires that the points A and B of figure 9.2 lie inside of the interval RST. The reason for this condition can be demonstrated in the following way. Assume for the sake of simplicity that the slopes of the characteristics are ±c; then the distances AS=SB=c δt. Using linear interpolation to find uA and uB from uR, uS and uT leads to
......(9.22)
Consequently, an error in uR of, say, δu would lead to an error in uA of c δu δt/δs (and similarly for uT and uB). Thus the error would be magnified with each time step unless c δt/δs < 1 and, therefore, the numerical integration is only stable if δt < δx/c. In many hydraulic system analyses this places a quite severe restriction on the time interval δt, and often necessitates a large number of time steps.

A procedure like the above will also require boundary conditions to be specified at any mesh point which lies either, at the end of a pipe or, at a junction of the pipe with a pipe of different size (or a pump or any other component). If the points S and C in figure 9.2 were end points, then only one characteristic would lie within the pipe and only one relation, 9.18 or 9.19, can be used. Therefore, the boundary condition must provide a second relation involving uC or h*C (or both). An example is an open-ended pipe for which the pressure and, therefore, h* is known. Alternatively, at a junction between two sizes of pipe, the two required relations will come from one characteristic in each of the two pipes, plus a continuity equation at the junction ensuring that the values of uA0 in both pipes are the same at the junction. For this reason it is sometimes convenient to rewrite equations 9.16 and 9.17 in terms of the volume flow rate Q=uA0 instead of u so that

  1. On ds/dt=u+c
    ......(9.23)
  2. On ds/dt=u-c
    ......(9.24)

Even in simple pipe flow, additional complications arise when the instantaneous pressure falls below vapor pressure and cavitation occurs. In the context of water-hammer analysis, this is known as ``water column separation'', and is of particular concern because the violent collapse of the cavity can cause severe structural damage (see, for example, Martin 1978). Furthermore, the occurrence of water column separation can trigger a series of cavity formations and collapses, resulting in a series of impulsive loads on the structure. The possibility of water column separation can be tracked by following the instantaneous pressure. To proceed beyond this point requires a procedure to incorporate a cavity in the waterhammer calculation using the method of characteristics. A number of authors (for example, Tanahashi and Kasahara 1969, Weyler et al. 1971, Safwat and van der Polder 1973) have shown that this is possible. However the calculated results after the first collapse can deviate substantially from the observations. This is probably due to the fact that the first cavity is often a single, coherent void. This will shatter into a cloud of smaller bubbles as a result of the violence of the first collapse. Subsequently, one is dealing with a bubbly medium whose wave propagation speeds may differ significantly from the acoustic speed assumed in the analytical model. Other studies have shown that qualitatively similar changes in the water-hammer behavior occur when gas bubbles form in the liquid as a result of dissolved gas coming out of solution (see, for example, Wiggert and Sundquist 1979).

In many time domain analyses, turbomachines are treated by assuming that the temporal rates of change are sufficiently slow that the turbomachine responds quasistatically, moving from one steady state operating point to another. Consequently, if points A and B lie at inlet to and discharge from the turbomachine then the equations relating the values at A and B would be
......(9.25)
......(9.26)
where H(Q) is the head rise across the machine at the flow rate, Q. Data presented later will show that the quasistatic assumption is only valid for rates of change less than about one-tenth the frequency of shaft rotation. For frequencies greater than this, the pump dynamics become important (see section 9.13).

For more detailed accounts of the methods of characteristics the reader is referred to Streeter and Wylie (1967), or any modern text on numerical methods. Furthermore, there are a number of standard codes available for time domain analysis of transients in hydraulic systems, such as that developed by Amies, Levek and Struesseld (1977). The methods work well so long as one has confidence in the differential equations and models which are used. In other circumstances, such as occur in two-phase flow, in cavitating flow, or in the complicated geometry of a turbomachine, the time domain methods may be less useful than the alternative frequency domain methods to which we now turn.

 

9.5 FREQUENCY DOMAIN METHODS

When the quasistatic assumption for a device like a pump or turbine becomes questionable, or when the complexity of the fluid or the geometry makes the construction of a set of differential equations impractical or uncertain, then it is clear that experimental information on the dynamic behavior of the device is necessary. In practice, such experimental information is most readily obtained by subjecting the device to fluctuations in the flow rate or head for a range of frequencies, and measuring the fluctuating quantities at inlet and discharge. Such experimental results will be presented later. For present purposes it is sufficient to recognize that one practical advantage of frequency domain methods is the capability of incorporation of experimentally obtained dynamic information and the greater simplicity of the experiments required to obtain the necessary dynamic data. Another advantage, of course, is the core of fundamental knowledge that exists regarding such methodology (see for example, Pipes 1940, Hennyey 1962, Paynter 1961, Brown 1967). As stated earlier, the disadvantage is that the methods are limited to small linear perturbations in the flow rate. When the perturbations are linear, Fourier analysis and synthesis can be used to convert from transient data to individual frequency components and vice versa. All the dependent variables such as the mean velocity, u, mass flow rate, m, pressure, p, or total pressure, pT, are expressed as the sum of a mean component (denoted by an overbar) and a complex fluctuating component (denoted by a tilde) at a frequency, ω, which incorporates the amplitude and phase of the fluctuation:
......(9.27)
......(9.28)
......(9.29)
where j is (-1)½ and Re denotes the real part. Since the perturbations are assumed linear (|| « , | | « , etc.), they can be readily superimposed, so a summation over many frequencies is implied in the above expressions. In general, the perturbation quantities will be functions of the mean flow characteristics as well as position, s, and frequency, ω.

We should note that there do exist a number of codes designed to examine the frequency response of hydraulic systems using frequency domain methods (see, for example, Amies and Greene 1977).

 

9.6 ORDER OF THE SYSTEM

The first step in any unsteady flow analysis is to subdivide the system into components; the points separating two (or more) components will be referred to as system nodes. Typically, there would be nodes at the inlet and discharge flanges of a pump. Having done this, it is necessary to determine the order of the system, N, and this can be accomplished in one of several equivalent ways. The order of the system is the minimum number of independent fluctuating quantities which must be specified at a system node in order to provide a complete description of the unsteady flow at that location. It is also equal to the minimum number of independent, simultaneous first order differential equations needed to describe the fluid motion in, say, a length of pipe. In this summary we shall confine most of our discussion to systems of order two in which the dependent variables are the mass flow rate and either the pressure or the total head. This includes most of the common analyses of hydraulic systems. It is, however, important to recognize that order two systems are confined to

  1. Incompressible flows at the system nodes, definable by pressure (or head), and flow rate.
  2. Barotropic compressible flows in which, ρ(p), so only the pressure (or head) and flow rate need be specified at system nodes. This category also includes those flexible structures for water-hammer analysis in which the local area is a function only of the local pressure. If, on the other hand, the local area depends on the area and the pressure elsewhere, then the system is of order 3 or higher.
  3. Two-phase flows at the system nodes that can be represented by a homogeneous flow model that neglects the relative velocity between the phases. Any of the more accurate models that allow relative motion produce higher order systems.
Note that the order of the system can depend on the choice of system nodes. Consequently, an ideal evaporator or a condenser can be incorporated in an order two system provided the flow at the inlet node is single-phase (of type 2) and the flow at the discharge node also single-phase. A cavitating pump or turbine also falls within this category, provided the flow at both the inlet and discharge is pure liquid.

 

9.7 TRANSFER MATRICES

The transfer matrix for any component or device is the matrix which relates the fluctuating quantities at the discharge node to the fluctuating quantities at the inlet node. The earliest exploration of such a concept in electrical networks appears to be due to Strecker and Feldtkeller (1929) while the utilization of the idea in the context of fluid systems owes much to the pioneering work of Pipes (1940). The concept is the following. If the quantities at inlet and discharge are denoted by subscripts i=1 and i=2, respectively, and, if {}, n=1,2→ N denotes the vector of independent fluctuating quantities at inlet and discharge for a system of order N, then the transfer matrix, [T], is defined as
......(9.30)
It is a square matrix of order N. For example, for an order two system in which the independent fluctuating variables are chosen to be the total pressure, , and the mass flow rate, , then a convenient transfer matrix is
......(9.31)
The words transfer function and transfer matrix are used interchangeably here to refer to the matrix [T]. In general it will be a function of the frequency, ω, of the perturbations and the mean flow conditions in the device.

The most convenient independent fluctuating quantities for a hydraulic system of order two are usually

  1. Either the pressure, , or the instantaneous total pressure, . Note that these are related by
    ......(9.32)
    where is the mean density, is the fluctuating density which is barotropically connected to , and z is the vertical elevation of the system node. Neglecting the terms as is acceptable for incompressible flows
    ......(9.33)
  2. Either the velocity, , the fluctuating volume flow rate or the fluctuating mass flow rate, . Incompressible flow at a system node in a rigid pipe implies
    ......(9.34)
The most convenient choices are {, } or {, }, and, for these two vectors, we will respectively use transfer matrices denoted by [T*] and [T], defined as
......(9.35)
If the flow is incompressible and the cross-section at the nodes is rigid, then the [T*] and [T] matrices are clearly connected by
......(9.36)
and hence one is readily constructed from the other. Note that the determinants of the two matrices, [T] and [T*], are identical.

 

9.8 DISTRIBUTED SYSTEMS

In the case of a distributed system such as a pipe, it is also appropriate to define a matrix [F] (see Brown 1967) so that
......(9.37)
Note that, apart from the frictional term, the equations 9.12 and 9.13 for flow in a pipe will lead to perturbation equations of this form. Furthermore, in many cases the frictional term is small, and can be approximated by a linear term in the perturbation equations; under such circumstances the frictional term will also fit into the form given by equation 9.37.

When the matrix [F] is independent of location, s, the distributed system is called a ``uniform system'' (see section 9.10). For example, in equations 9.12 and 9.13, this would require ρ, c, a, f and A0 to be approximated as constants (in addition to the linearization of the frictional term). Under such circumstances, equation 9.37 can be integrated over a finite length, ℓ , and the transfer matrix [T] of the form 9.35 becomes
......(9.38)
where e[F]ℓ  is known as the ``transmission matrix.'' For a system of order two, the explicit relation between [T] and [F] is
......(9.39)
where λ1, λ2 are the solutions of the equation
......(9.40)
Some special features and properties of these transfer functions will be explored in the sections which follow.

 

9.9 COMBINATIONS OF TRANSFER MATRICES

When components are connected in series, the transfer matrix for the combination is clearly obtained by multiplying the transfer matrices of the individual components in the reverse order in which the flow passes through them. Thus, for example, the combination of a pump with a transfer matrix, [TA], followed by a discharge line with a transfer matrix, [TB], would have a system transfer matrix, [TS], given by
......(9.41)

The parallel combination of two components is more complicated and does not produce such a simple result. Issues arise concerning the relations between the pressures of the inlet streams and the relations between the pressures of the discharge streams. Often it is appropriate to assume that the branching which creates the two inlet streams results in identical fluctuating total pressures at inlet to the two components, . If, in addition, mixing losses at the downstream junction are neglected, so that the fluctuating total pressure, , can be equated with the fluctuating total pressure at discharge from the two components, then the transfer function, [TS], for the combination of two components (order two transfer functions denoted by [TA] and [TB]) becomes
......(9.42)

On the other hand, the circumstances at the junction of the two discharge streams may be such that the fluctuating static pressures (rather than the fluctuating total pressures) are equal. Then, if the inlet static pressures are also equal, the combined transfer matrix, [TS*], is related to those of the two components ([TA*] and [TB*]) by the same relations as given in equations 9.42. Other combinations of choices are possible, but will not be detailed here.

Using the above combination rules, as well as the relations 9.36 between the [T] and [T*] matrices, the transfer functions for very complicated hydraulic networks can be systematically synthesized.

 

9.10 PROPERTIES OF TRANSFER MATRICES

Transfer matrices (and transmission matrices) have some fundamental properties that are valuable to recall when constructing or evaluating the dynamic properties of a component or system.

We first identify a ``uniform'' distributed component as one in which the differential equations (for example, equations 9.12 and 9.13 or 9.37) governing the fluid motion have coefficients which are independent of position, s. Then, for the class of systems represented by the equation 9.37, the matrix [F] is independent of s. For a system of order two, the transfer function [T] would take the explicit form given by equations 9.39.

To determine another property of this class of dynamic systems, consider that the equations 9.37 have been manipulated to eliminate all but one of the unknown fluctuating quantities, say . The resulting equation will take the form
......(9.43)
In general, the coefficients an(s), n=0→ N, will be complex functions of the mean flow and of the frequency. It follows that there are N independent solutions which, for all the independent fluctuating quantities, may be expressed in the form
......(9.44)
where [B(s)] is a matrix of complex solutions and {A} is a vector of arbitrary complex constants to be determined from the boundary conditions. Consequently, the inlet and discharge fluctuations denoted by subscripts 1 and 2, respectively, are given by
......(9.45)
and therefore the transfer function
......(9.46)
Now for a uniform system, the coefficients an and the matrix [B] are independent of s. Hence the equation 9.43 has a solution of the form
......(9.47)
where [C] is a known matrix of constants, and [E] is a diagonal matrix in which
......(9.48)
where γn, n=1 to N, are the solutions of the dispersion relation
......(9.49)
Note that γn are the wavenumbers for the N types of wave of frequency, ω, which can propagate through the uniform system. In general, each of these waves has a distinct wave speed, cn, given by cn=-ω/γn. It follows from equations 9.47, 9.48 and 9.46 that the transfer matrix for a uniform distributed system must take the form
......(9.50)
where [E*] is a diagonal matrix with
......(9.51)
and ℓ =s2-s1.

An important diagnostic property arises from the form of the transfer matrix, 9.50, for a uniform distributed system. The determinant, DT, of the transfer matrix [T] is
......(9.52)
Thus the value of the determinant is related to the sum of the wavenumbers of the N different waves which can propagate through the uniform distributed system. Furthermore, if all the wavenumbers, γn, are purely real, then
......(9.53)
The property that the modulus of the determinant of the transfer function is unity will be termed ``quasi-reciprocity'' and will be discussed further below. Note that this will only be the case in the absence of wave damping when γn and cn are purely real.

Turning now to another property, a system is said to be ``reciprocal'' if, in the matrix [Z] defined by
......(9.54)
the transfer impedances Z12 and Z21 are identical (see Brown 1967 for the generalization of this property in systems of higher order). This is identical to the condition that the determinant, DT, of the transfer matrix [T] be unity:
......(9.55)
We shall see that a number of commonly used components have transfer functions which are reciprocal. In order to broaden the perspective we have introduced the property of ``quasi-reciprocity'' to signify those components in which the modulus of the determinant is unity or
......(9.56)

We have already noted that uniform distributed components with purely real wavenumbers are quasi-reciprocal. Note that a uniform distributed component will only be reciprocal when the wavenumbers tend to zero, as, for example, in incompressible flows in which the wave propagation speeds tend to infinity.

By utilizing the results of section 9.9 we can conclude that any series or parallel combination of reciprocal components will yield a reciprocal system. Also a series combination of quasi-reciprocal components will be quasi-reciprocal. However it is not necessarily true that a parallel combination of quasi-reciprocal components is quasi-reciprocal.

An even more restrictive property than reciprocity is the property of ``symmetry''. A ``symmetric'' component is one that has identical dynamical properties when turned around so that the discharge becomes the inlet, and the directional convention of the flow variables is reversed (Brown 1967). Then, in contrast to the regular transfer matrix, [T], the effective transfer matrix under these reversed circumstances is [TR] where
......(9.57)
and, comparing this with the definition 9.31, we observe that
......(9.58)
Therefore symmetry, [T]=[TR], requires
......(9.59)
Consequently, in addition to the condition, DT=1, required for reciprocity, symmetry requires T11=T22.

As with the properties of reciprocity and quasi-reciprocity, it is useful to consider the property of a system comprised of symmetric components. Note that according to the combination rules of section 9.9, a parallel combination of symmetric components is symmetric, whereas a series combination may not retain this property. In this regard symmetry is in contrast to quasi-reciprocity in which the reverse is true.

In the case of uniform distributed systems, Brown (1967) shows that symmetry requires
......(9.60)
so that the solution of the equation 9.40 for λ is λ=±λ* where λ*=(F21F12)½ is known as the ``propagation operator'' and the transfer function 9.39 becomes
......(9.61)
where ZC=(F12/F21)½=(T12/T21)½ is known as the ``characteristic impedance''.

In addition to the above properties of transfer functions, there are also properties associated with the net flux of fluctuation energy into the component or system. These will be elucidated after we have examined some typical transfer functions for components of hydraulic systems.

 

9.11 SOME SIMPLE TRANSFER MATRICES

The flow of an incompressible fluid in a straight, rigid pipe will be governed by the following versions of equations 9.6 and 9.7:
......(9.62)
......(9.63)
If the velocity fluctuations are small compared with the mean velocity denoted by U (positive in direction from inlet to discharge), and the term u| u| is linearized, then the above equations lead to the transfer function
......(9.64)
where (R+jωL) is the ``impedance'' made up of a ``resistance'', R, and an ``inertance'', L, given by
......(9.65)
where A, a, and ℓ  are the cross-sectional area, radius, and length of the pipe. A number of different pipes in series would then have
......(9.66)
where Q is the mean flow rate. For a duct of non-uniform area
......(9.67)
Note that all such ducts represent reciprocal and symmetric components.

A second, common hydraulic element is a simple ``compliance'', exemplified by an accumulator or a surge tank. It consists of a device installed in a pipeline and storing a volume of fluid, VL, which varies with the local pressure, p, in the pipe. The compliance, C, is defined by
......(9.68)
In the case of a gas accumulator with a mean volume of gas, , which behaves according to the polytropic index, k, it follows that
......(9.69)
where is the mean pressure level. In the case of a surge tank in which the free surface area is AS, it follows that
......(9.70)
The relations across such compliances are
......(9.71)
Therefore, using the definition 9.35, the transfer function [T] becomes
......(9.72)
Again, this component is reciprocal and symmetric, and is equivalent to a capacitor to ground in an electrical circuit.

Systems made up of lumped resistances, R, inertances, L, and compliances, C, will be termed LRC systems. Individually, all three of these components are both reciprocal and symmetric. It follows that any system comprised of these components will also be reciprocal (see section 9.10); hence all LRC systems are reciprocal. Note also that, even though individual components are symmetric, LRC systems are not symmetric since series combinations are not, in general, symmetric (see section 9.10).

An even more restricted class of systems are those consisting only of inertances, L, and compliances, C. These systems are termed ``dissipationless'' and have some special properties (see, for example, Pipes 1963) though these are rarely applicable in hydraulic systems.

As a more complicated example, consider the frictionless (f=0) compressible flow in a straight uniform pipe of mean cross-sectional area, A0. This can readily be shown to have the transfer function
......(9.73)
where is the mean fluid velocity, M=/c is the Mach number, and θ is a reduced frequency given by
......(9.74)
Note that all the usual acoustic responses can be derived quite simply from this transfer function. For example, if the pipe opens into reservoirs at both ends, so that the fluctuating pressures at inlet and discharge are zero then the transfer function, equation 9.35, can only be satisfied with zero inlet fluctuating mass flow if T*12=0. According to equations 9.73, this can only occur if sin θ=0, θ=nπ or
......(9.75)
which are the natural organ-pipe modes for such a pipe. Note also that the determinant of the transfer matrix is
......(9.76)
Since no damping has been included, this component is an undamped distributed system, and is therefore quasi-reciprocal. At low frequencies and Mach numbers, the transfer function 9.73 reduces to
......(9.77)
and so consists of an inertance, ℓ /A0, and a compliance, A0ℓ /c2.

When friction is included (as is necessary in most water-hammer analyses) the transfer function becomes
......(9.78)
in which f*=fℓ  M/2a(1-M2) and k1,k2 are the solutions of
......(9.79)
The determinant of this transfer matrix [T*] is
......(9.80)
Note that this component is only quasi-reciprocal in the undamped limit, f→0.

 

9.12 FLUCTUATION ENERGY FLUX

It is clearly important to be able to establish the net energy flux into or out of a hydraulic system component (see Brennen and Braisted 1980). If the fluid is incompressible, and the order two system is characterized by the mass flow rate, m, and the total pressure, pT, then the instantaneous energy flux through any system node is given by mpT/ρ where the density is assumed constant. Substituting the expansions 9.28, 9.29 for pT and m, it is readily seen that the mean flux of energy due to the fluctuations, E, is given by
......(9.81)
where the overbar denotes a complex conjugate. Superimposed on E are fluctuations in the energy flux whose time-average value is zero, but we shall not be concerned with those fluctuations. The mean fluctuation energy flux, E, is of more consequence in terms, for example, of evaluating stability. It follows that the net flux of fluctuation energy into a component from the fluid is given by
......(9.82)
and when the transfer function form 9.31 is used to write this in terms of the inlet fluctuating quantities
......(9.83)
where
......(9.84)
and
......(9.85)
Using the above relations, we can draw the following conclusions:

  1. A component or system which is ``conservative'' (in the sense that ΔE=0 under all circumstances, whatever the values of the inlet fluctuating total pressure and mass flow rate) requires that
    ......(9.86)
    and these in turn require not only that the system or component be ``quasi-reciprocal'' (| DT| =1) but also that
    ......(9.87)
    Such conditions virtually never occur in real hydraulic systems, though any combination of lumped inertances and compliances does constitute a conservative system. This can be readily demonstrated as follows. An inertance or compliance has DT=1, purely real T11 and T22 so that T11= and T22=, and purely imaginery T21 and T12 so that T21=- and T12=-. Hence individual inertances or compliances satisfy equations 9.86 and 9.87. Furthermore, from the combination rules of section 9.9, it can readily be seen that all combination of components with purely real T11 and T22 and purely imaginery T21 and T12 will retain the same properties. Consequently, any combination of inertance and compliance satisfies equations 9.86 and 9.87 and is conservative.
  2. A component or system will be considered ``completely passive'' if ΔE is positive for all possible values of the inlet fluctuating total pressure and mass flow rate. This implies that a net external supply of energy to the fluid is required to maintain any steady state oscillation. To find the characteristics of the transfer function which imply ``complete passivity'' the expression 9.83 is rewritten in the form
    ......(9.88)
    where x=/ . It follows that the sign of ΔE is determined by the sign of the expression in the square brackets. Moreover, if Γ2 < 0, it is readily seen that this expression has a minimum and is positive for all x if
    ......(9.89)
    which, since Γ2 < 0, implies Γ1 < 0. It follows that necessary and sufficient conditions for a component or system to be completely passive are
    ......(9.90)
    where
    ......(9.91)
    The conditions 9.90 also imply Γ2 < 0. Conversely a ``completely active'' component or system which always has ΔE < 0 occurs if and only if Γ1 > 0 and G < 0 which imply Γ2 > 0. These properties are not, of course, the only possibilities. A component or system which is not completely passive or active could be ``potentially active.'' That is to say, ΔE could be negative for the right combination of inlet fluctuating mass flow and total pressure, which would, in turn, depend on the rest of the system to which the particular component or system is attached. Since Γ1 is almost always negative, it transpires that most components are either completely passive or potentially active, depending on the sign of the quantity, G, which will therefore be termed the ``dynamic activity''. These circumstances can be presented graphically as shown in figure 9.3.

Figure 9.3 Schematic of the conditions for completely active, completely passive and potentially active components or systems.

In practice, of course, both the transfer function, and properties like the dynamic activity, G, will be functions not only of frequency but also of the mean flow conditions. Hence the potential for system instability should be evaluated by tracking the graph of G against frequency, and establishing the mean flow conditions for which the quantity G becomes negative within the range of frequencies for which transfer function information is available.

While the above analysis represents the most general approach to the stability of systems or components, the results are not readily interpreted in terms of commonly employed measures of the system or component characteristics. It is therefore instructive to consider two special subsets of the general case, not only because of the simplicity of the results, but also because of the ubiquity of these special cases. Consider first a system or component that discharges into a large, constant head reservoir, so that fluctuating total pressure at discharge is zero. It follows from the expression 9.82 that
......(9.92)
Note that ΔE is always purely real and that the sign only depends on the real part of the ``input impedance''
......(9.93)
Consequently a component or system with a constant head discharge will be dynamically stable if the ``input resistance'' is positive or
......(9.94)
This relation between the net fluctuation energy flux, the input resistance, and the system stability, is valuable because of the simplicity of its physical interpretation. In practice, the graph of input resistance against frequency can be monitored for changes with mean flow conditions. Instabilities will arise at frequencies for which the input resistance becomes negative.

The second special case is that in which the component or system begins with a constant head reservoir rather than discharging into one. Then
......(9.95)
and the stability depends on the sign of the real part of the ``discharge impedance''
......(9.96)
Thus a constant head inlet component or system will be stable when the ``discharge resistance'' is positive or
......(9.97)
In practice, since T11 and T22 are close to unity for many components and systems, both the condition 9.94 and the condition 9.97 reduce to the approximate condition that the system resistance, Re{-T12}, be positive for system stability. While not always the case, this approximate condition is frequently more convenient and more readily evaluated than the more precise conditions detailed above and given in equations 9.94 and 9.97. Note specifically, that the system resistance can be obtained from steady state operating characteristics; for example, in the case of a pump or turbine, it is directly related to the slope of the head-flow characteristic and instabilities in these devices which result from operation in a regime where the slope of the characteristic is positive and Re{-T12} is negative are well known (Greitzer 1981) and have been described earlier (section 8.6).

It is, however, important to recognize that the approximate stability criterion Re{-T12} > 0, while it may provide a useful guideline in many circumstances, is by no means accurate in all cases. One notable and important case in which this criterion is inaccurate is the auto-oscillation phenomenon described in section 8.7. This is not the result of a positive slope in the head-flow characteristic, but rather occurs where this slope is negative and is caused by cavitation-induced changes in the other elements of the transfer function. This circumstance will be discussed further in section 9.14.

 

9.13 NON-CAVITATING PUMPS

Consider now the questions associated with transfer functions for pumps or other turbomachines. In the simple fluid flows of section 9.11 we were able to utilize the known equations governing the flow in order to construct the transfer functions for those simple components. In the case of more complex fluids or geometries, one cannot necessarily construct appropriate one-dimensional flow equations, and therefore must resort to results derived from more global application of conservation laws or to experimental measurements of transfer matrices. Consider first the transfer matrix, [TP], for incompressible flow through a pump (all pump transfer functions will be of the [T] form defined in equation 9.35) which will clearly be a function not only of the frequency, ω, but also of the mean operating point as represented by the flow coefficient, φ, and the cavitation number, σ. At very low frequencies one can argue that the pump will simply track up and down the performance characteristic, so that, for small amplitude perturbations and in the absence of cavitation, the transfer function becomes
......(9.98)
where d(ΔpT)/dm is the slope of the steady state operating characteristic of total pressure rise versus mass flow rate. Thus we define the pump resistance, RP=-d(ΔpT)/dm, where RP is usually positive under design flow conditions, but may be negative at low flow rates as discussed earlier (section 8.6). At finite frequencies, the elements TP21 and TP22 will continue to be zero and unity respectively, since the instantaneous flow rate into and out of the pump must be identical when the fluid and structure are incompressible and no cavitation occurs. Furthermore, TP11 must continue to be unity since, in an incompressible flow, the total pressure differences must be independent of the level of the pressure. It follows that the transfer function at higher frequencies will become
......(9.99)
where the pump impedence, IP, will, in general, consist of a resistive part, RP, and a reactive part, jω LP. The resistance, RP, and inertance, LP, could be functions of both the frequency, ω, and the mean flow conditions. Such simple impedance models for pumps have been employed, together with transfer functions for the suction and discharge lines (equation 9.73), to model the dynamics of pumping systems. For example, Dussourd (1968) used frequency domain methods to analyse pulsation problems in boiler feed pump systems. More recently, Sano (1983) used transfer functions to obtain natural frequencies for pumping systems that agree well with those observed experimentally.

Figure 9.4 Impedance measurements made by Anderson, Blade and Stevans (1971) on a centrifugal pump (impeller diameter of 18.9cm) operating at a flow coefficient of 0.442 and a speed of 3000rpm. The real or resistive part of (-T12) and the imaginary or reactive part of (T12) are plotted against the frequency of the perturbation.

Figure 9.5 Typical inertance and resistance values from the centrifugal pump data of figure 9.4. Data do not include the diffuser contribution. The lines correspond to analytical values obtained as described in the text.

The first fundamental investigation of the dynamic response of pumps seems to have been carried out by Ohashi (1968) who analyzed the oscillating flow through a cascade, and carried out some preliminary experimental investigations on a centrifugal pump. These studies enabled him to evaluate the frequency at which the response of the pump would cease to be quasistatic (see below). Fanelli (1972) appears to have been the first to explore the nature of the pump transfer function, while the first systematic measurements of the impedance of a noncavitating centrifugal pump are those of Anderson, Blade and Stevans (1971). Typical resistive and reactive component measurements from the work of Anderson, Blade and Stevans are reproduced in figure 9.4. Note that, though the resistance approaches the quasistatic value at low frequencies, it also departs significantly from this value at higher frequencies. Moreover, the reactive part is only roughly linear with frequency. The resistance and inertance are presented again in figure 9.5, where they are compared with the results of a dynamic model proposed by Anderson, Blade and Stevans. In this model, each pump impeller passage is represented by a resistance and an inertance, and the volute by a series of resistances and inertances. Since each impeller passage discharges into the volute at different locations relative to the volute discharge, each impeller passage flow experiences a different impedance on its way to the discharge. This results in an overall pump resistance and inertance that are frequency dependent as shown in figure 9.5. Note that the comparison with the experimental observations (which are also included in figure 9.5) is fair, but not completely satisfactory. Moreover, it should be noted, that the comparison shown is for a flow coefficient of 0.442 (above the design flow coefficient), and that, at higher flow coefficients, the model and experimental results exhibited poorer agreement.

Subsequent measurements of the impedance of non-cavitating axial and mixed flow pumps by Ng and Brennen (1978) exhibit a similar increase in the resistance with frequency (see next section). In both sets of dynamic data, it does appear that significant departure from the quasistatic values can be expected when the reduced frequency, (frequency/rotation frequency) exceeds about 0.02 (see figures 9.5 and 9.6). This is roughly consistent with the criterion suggested by Ohashi (1968) who concluded that non-quasistatic effects would occur above a reduced frequency of 0.05 ZR φ/cos β. For the inducers of Ng and Brennen, Ohashi's criterion yields values for the critical reduced frequency of about 0.015.

 

9.14 CAVITATING INDUCERS

In the presence of cavitation, the transfer function for a pump or inducer will be considerably more complicated than that of equation 9.99. Even at low frequencies, the values of TP11 will become different from unity, because the head rise will change with the inlet total pressure, as manifest by the nonzero value of d(ΔpT)/dpT1 at a given mass flow rate, m1. Furthermore, the volume of cavitation, VC (pT1,m1), will vary with both the inlet total pressure, pT1 (or NPSH or cavitation number), and with the mass flow rate, m1 (or with angle of incidence), so that
......(9.100)
Brennen and Acosta (1973, 1975, 1976) identified this quasistatic or low frequency form for the transfer function of a cavitating pump, and calculated values of the cavitation compliance, -ρL(dVC/dpT1)m1 and the cavitation mass flow gain factor, -ρL(dVC/dm1)pT1, using the cavitating cascade solution discussed in section 7.10. Both the upper limit of frequency at which this quasistatic approach is valid and the form of the transfer function above this limit cannot readily be determined except by experiment. Though it was clear that experimental measurements of the dynamic transfer functions were required, these early investigations of Brennen and Acosta did highlight the importance of both the compliance and the mass flow gain factor in determining the stability of systems with cavitating pumps.

Figure 9.6 Typical transfer functions for a cavitating inducer obtained by Brennen et al. (1982) for a 10.2cm diameter inducer (Impeller VI) operating at 6000rpm and a flow coefficient of φ1 =0.07. Data is shown for four different cavitation numbers, σ= (A) 0.37, (C) 0.10, (D) 0.069, (G) 0.052 and (H) 0.044. Real and imaginary parts are denoted by the solid and dashed lines respectively. The quasistatic pump resistance is indicated by the arrow (adapted from Brennen et al. 1982).

Ng and Brennen (1978) and Brennen et al. (1982) conducted the first experiments to measure the complete transfer function for cavitating inducers. Typical transfer functions are those for the 10.2cm diameter Impeller VI (see section 2.8), whose noncavitating steady state performance was presented in figure 7.15. Transfer matrices for that inducer are presented in figure 9.6 as a function of frequency (up to 32Hz), for a speed of 6000rpm, a flow coefficient φ1=0.07 and for five different cavitation numbers ranging from data set A that was taken

Figure 9.7 Determinant, DTP, of the experimental transfer functions of figure 9.6. The real and imaginary parts are shown by the solid and dashed lines respectively and, as in figure 9.6, the letter code A→H refers to steady state operating points with increasing cavitation (adapted from Brennen et al. 1982).

under noncavitating conditions, to data set C that showed a little cavitation, to data set H that was close to breakdown. The real and imaginary parts are represented by the solid and dashed lines, respectively. Note, first, that, in the absence of cavitation (Case A), the transfer function is fairly close to the anticipated form of equation 9.99 in which TP11=TP22=1, TP21=0. Also, the impedance (TP12) is comprised of an expected inertance (the imaginary part of TP12 is linear in frequency) and a resistance (real part of -TP12) which is consistent with the quasistatic resistance from the slope of the head rise characteristic (shown by the arrow in figure 9.6 at TP12RT1/Ω=1.07). The resistance appears to increase with increasing frequency, a trend which is consistent with the centrifugal pump measurements of Anderson, Blade and Stevans (1971) which were presented in figure 9.5.

Figure 9.8 Polynomial curves fitted to the experimental data of figure 9.6 (adapted from Brennen et al. 1982).

It is also clear from figure 9.6 that, as the cavitation develops, the transfer function departs significantly from the form of equation 9.99. One observes that TP11 and TP22 depart from unity, and develop nonzero imaginary parts that are fairly linear with frequency. Also TP21 becomes nonzero, and, in particular, exhibits a compliance which clearly increases with decreasing cavitation number. All of these changes mean that the determinant, DTP, departs from unity as the cavitation becomes more extensive. This is illustrated in figure 9.7, which shows the determinant corresponding to the data of figure 9.6. Note that DTP is approximately unity for the non-cavitating case A, but that it progressively deviates from unity as the cavitation increases. We can conclude that the presence of cavitation can cause a pump to assume potentially active dynamic characteristics when it would otherwise be dynamically passive.

Polynomials of the form
......(9.101)
were fitted to the experimental transfer function data using values of n* of 3 or 5. To illustrate the result of such curve fitting we include figure 9.8, which depicts the result of curve fitting figure 9.6.

Figure 9.9 The inertance, -A112, non-dimensionalized as -A112 RT1, as a function of cavitation number for two axial inducer pumps (Impellers IV and VI) with the same geometry but different diameters. Data for the 10.2cm diameter Impeller VI is circled and was obtained from the data of figure 9.6. The uncircled points are for the 7.58cm diameter Impeller IV. Adapted from Brennen et al. (1982).

Figure 9.10 The compliance, -A121, nondimensionalized as -A121Ω2/RT1 for the same circumstances as described in figure 9.9.

We now proceed to examine several of the coefficients Anij that are of particular interest (note that A011=A022=1, A021=0 for reasons described earlier). We begin with the inertance, -A112, which is presented nondimensionally in figure 9.9. Though there is significant scatter at the lower cavitation numbers, the two different sizes of inducer pump appear to yield similar inertances. Moreover, the data suggest some decrease in the inertance with decreasing σ. On the other hand, the corresponding data for the compliance, -A121, which is presented in figure 9.10 seems roughly inversely proportional to the cavitation number.

Figure 9.11 The mass flow gain factor, -A122, nondimensionalized as -A122Ω for the same circumstances as described in figure 9.9.

Figure 9.12 The characteristic, A111, nondimensionalized as A111Ω for the same circumstances as described in figure 9.9.

And the same is true for both the mass flow gain factor, -A122, and the coefficient that defines the slope of the imaginary part of TP11, A111; these are presented in figures 9.11 and 9.12, respectively. All of these data appear to conform to the physical scaling implicit in the nondimensionalization of each of the dynamic characteristics.

It is also valuable to consider the results of figures 9.9 to 9.12 in the context of an analytical model for the dynamics of cavitating pumps (Brennen 1978). We present here a brief physical description of that model, the essence of which is depicted schematically in figure 9.13, which shows a developed, cylindrical surface within the inducer. The cavitation is modeled as a bubbly mixture which extends over a fraction, ε, of the length, c, of each blade passage before collapsing at a point where the pressure has risen to a value which causes collapse.

Figure 9.13 Schematic of the bubbly flow model for the dynamics of cavitating pumps (adapted from Brennen 1978).

The mean void fraction of the bubbly mixture is denoted by α0. Thus far we have described a flow which is nominally steady. We must now consider perturbing both the pressure and the flow rate at inlet, since the relation between these perturbations, and those at discharge, determine the transfer function.

Figure 9.14 Transfer functions for Impellers VI and IV at φ1=0.07 calculated from the bubbly flow model using K=1.3 and M=0.8 (adapted from Brennen et al. 1982).

Pressure perturbations at inlet will cause pressure waves to travel through the bubbly mixture and this part of the process is modeled using a mixture compressibility parameter, K, to determine that wave speed. On the other hand, fluctuations in the inlet flow rate produce fluctuations in the angle of incidence which cause fluctuations in the rate of production of cavitation at inlet. These disturbances would then propagate down the blade passage as kinematic or concentration waves which travel at the mean mixture velocity. This process is modeled by a factor of proportionality, M, which relates the fluctuation in the angle of incidence to the fluctuations in the void fraction. Neither of the parameters, K or M, can be readily estimated analytically; they are, however, the two key features in the bubbly flow model. Moreover they respectively determine the cavitation compliance and the mass flow gain factor, two of the most important factors in the transfer function insofar as the prediction of instability is concerned.

The theory yields the following expressions for A111, A112, A121 and A122 at small dimensionless frequencies (Brennen 1978, 1982):
......(9.102)
where ζ=ℓ  ZR/RT1 where ℓ  is the axial length of the inducer, and ZR is the number of blades. Evaluation of the transfer function elements can be effected by noting that the experimental observations suggest ε is approximately equal to 0.02/σ. Consequently, the Anij characteristics from equations 9.102 can be plotted against cavitation number. Typical results are shown in figures 9.9 to 9.12 for various choices of the two undetermined parameters K and M. The inertance, A112, which is shown in figure 9.9, is independent of K and M. The calculated value of the inertance for these impellers is about 9.2; the actual value may be somewhat larger because of three-dimensional geometric effects that were not included in the calculation (Brennen et al. 1982). The parameter M only occurs in A122, and it appears from figure 9.11 as though values of this parameter in the range 0.8→0.95 provide the best agreement with the data. Also, a value of K of about 1.3 seems to generate a good match with the data of figures 9.10, 9.11 and 9.12.

Finally, since K=1.3 and M=0.8 seem appropriate values for these impellers, we reproduce in figure 9.14 the complete theoretical transfer functions for various cavitation numbers. These should be directly compared with the transfer functions of figure 9.8. Note that the general features of the transfer functions, and their variation with cavitation number, are reproduced by the model. The most notable discrepency is in the real part of TP21; this parameter is, however, usually rather unimportant in determining the stability of a hydraulic system. Most important from the point of view of stability predictions, the cavitation compliance and mass flow gain factor components of the transfer function are satisfactorily modeled.

 

9.15 SYSTEM WITH RIGID BODY VIBRATION

All of the preceding analysis has assumed that the structure of the hydraulic system is at rest in some inertial coordinate system. However, there are a number of important problems in which the oscillation of the hydraulic system itself may play a central role. For instance, one might seek to evaluate the unsteady pressures and flow rates in a hydraulic system aboard a vehicle undergoing translational or rotational oscillations. Examples might be oil or water pumping systems aboard a ship, or the fuel and hydraulic systems on an aircraft. In other circumstances, the motion of the vehicle may couple with the propulsion system dynamics to produce instabilities, as in the simplest of the Pogo instabilities of liquid propelled rocket engines (see section 8.13).

In this section we give a brief outline of how rigid body oscillations of the hydraulic system can be included in the frequency domain methodology. For convenience we shall refer to the structure of the hydraulic system as the ``vehicle''. There are, of course, more complex problems in which the deformation of the vehicle is important. Such problems require further refinement of the methods presented here.

In order to include the rigid body oscillation of the vehicle in the analysis, it is first necessary to define a coordinate system, x, which is fixed in the vehicle, and a separate inertial or nonaccelerating coordinate system, xA. The mean location of the origin of the x system is chosen to coincide with the origin of the xA system. The oscillations of the vehicle are then described by stating that the translational and rotational displacements of the x coordinate system in the xA system are respectively given by
......(9.103)
It follows that the oscillatory displacement of any vector point, x, in the vehicle is given by
......(9.104)
and the oscillatory velocity of that point will be
......(9.105)
Then, if the steady and oscillatory velocities of the flow in the hydraulic system, and relative to that system, are given as in the previous sections by and respectively, it follows that the oscillatory velocity of the flow in the nonaccelerating frame, , is given by
......(9.106)
Furthermore, the acceleration of the fluid in the nonaccelerating frame, , is given by
......(9.107)
The last three terms on the right hand side are vehicle-induced accelerations of the fluid in the hydraulic system. It follows that these accelerations will alter the difference in the total pressure between two nodes of the hydraulic system denoted by subscripts 1 and 2. By integration one finds that the total pressure difference, ( - ), is related to that which would pertain in the absence of vehicle oscillation, ( - )0, by
......(9.108)
where x2 and x1 are the locations of the two nodes in the frame of reference of the vehicle.

The inclusion of these acceleration-induced total pressure changes is the first step in the synthesis of models of this class of problems. Their evaluation requires the input of the location vectors, xi, for each of the system nodes, and the values of the system displacement frequency, ω, and vibration amplitudes, and . In an analysis of the response of the hydraulic system, the vibration amplitudes would be included as inputs. In a stability analysis, they would be initially unknown and the system of equations would need to be supplemented by those of the appropriate feedback mechanism. An example would be a set of equations giving the unsteady thrust of an engine in terms of the fluctuating fuel supply rate and pressure and giving the accelerations of the vehicle resulting from that fluctuating thrust. Clearly a complete treatment of such problems would be beyond the scope of this book.

REFERENCES

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Last updated 12/1/00.
Christopher E. Brennen