Modelling of a hydraulic system coupled with lumped masses

ABSTRACT A coupled hydraulic-mechanical system with a lumped parametric mechanical part has been set up, measured and mathematically modelled in the frequency domain. The main focus of this article is the identification of unknown system parameters, which depends on the models of coupling and dissipation. The set-up under investigation can be excited hydraulically, by flow rate, or mechanically, by force. The responding pressures of the hydraulic subsystem and the accelerations of the mechanical subsystem are measured, from which transfer functions between excitation and system states can be calculated. The property of reciprocity is used for the processing of measurement data. With a suitable two-step strategy and non-linear optimization unknown system parameters can be identified from measurements. Additionally, the agreement of model and measurement and the physical meaningfulness of these parameters are examined. The proposed model succeeds in predicting measured transfer functions, whose data weren't used for the identification of model parameters.


Introduction
The presented article deals with mathematical modelling and model updating of a coupled hydraulic-mechanical system. Model updating is an active area of research whose goal is to calculate unknown parameters of a dynamic model in such a way that the best possible agreement between modelled and measured system responses is achieved. Various applications have been reported for mechanical systems, where the equation of motion is usually discretized by finite element (FE) techniques.
In model updating, frequency response function (FRF) data from measurements can be linked to the modelled FRFs by a suitable optimization criterion. The best known method to update FRF in connection with FEM is the response function method (RFM) [1], which uses analytical sensitivities to solve the inverse problem. The advantage of this method is that it does not require a complete modal analysis of the system at hand. Since in model updating the convergence of the parameters to the correct values cannot be guaranteed if there are too large model deviations, a function weighted sensitivity method was presented in [2]. With this approach, faster convergence and better robustness to measurement noise were shown. In [3] updating of the equations of motion, assurance criterion (POTMAC and KINMAC [22]) the model and measurement data matched much better than in the comparison of transfer functions, which proved to be more sensitive to the choice of model parameters.
An important aspect of model-updating is the way of updating and the strategy behind it. If there is enough measurement data to perform a complete modal analysis of a given mechanical structure, one can use the modal assurance criterion (MAC) [23], the coordinate modal assurance criterion (COMAC [24]) and the coordinate strain assurance criterion (COSMAC) as shown in [25] for parameter identification. If there is not enough data available, i.e. only certain FRFs, the frequency response assurance criterion (FRAC [26]) can be used and an appropriate optimization criterion can be set up. In addition [25], also describes that the location of the analytically calculated resonance frequencies can be compared with the corresponding measured ones and linked into a suitable quality functional. Another approach is offered by [27]. Here, the Response Surface Method (RSM [28]) and Derringer's function method [29] are combined, using MAC values and FRF data for the formulation of an optimization criterion. A two-step procedure is described that updates mass and stiffness matrices in the first step and only the damping parameters in the second step. This is done since these techniques produce large errors in prediction of FRFs of damped systems near resonance and anti-resonance points if one tries to update them in one step. In a later publication [30] this procedure is successfully applied to detect damage of a cantilever beam. An approach using a classical non-linear optimization problem with constraints, solved via Matlab, is offered by [31]. Here, the analytically calculated FRFs are evaluated at certain frequency points and compared with measurement data. Parameters for the mass matrix and the stiffness matrix as well as viscous damping are calculated in one step. This method was also verified by means of an experiment.
Altogether, there are a lot of publications with several methods of model-updating applied to mechanical systems via FE models. In this paper we are dealing with a coupled hydraulic-mechanical system and the mathematical model cannot be expressed by a standard FE approach. Special emphasis is laid on dissipation. In the course of experiments, it became more and more evident that viscous mechanical damping and linear resistances in the hydraulic subsystem are not sufficient to adequately model the measured dissipation effects. A damping model is considered, that couples the dissipation effects of the mechanical and hydraulic subsystems. Furthermore, three different damping models are compared to validate the necessity of the new form of modelling dissipation in case of FSI. In addition, it becomes apparent that the plates closing the rigid-walled hydraulic volumes are themselves flexible, and thus FSI must be taken into account.
In [32], the hydraulic part of the test-bench shown here, the hydraulicchain-oscillator, was studied separately. First, taking into account the coupling effects of individual closing plates as well as an associated dissipation due to their movement, a more detailed mathematical model than in [18] was created. The used model updating strategy proceeds in two stages, similar to that offered in [27,30], but the optimization criterion is formed from measured FRF data and the analytically calculated ones similar to the method shown in [31]. The difference is that only at the location of resonances and antiresonances the analytically calculated FRFs are compared with the measured ones, i.e not every sampling step for which measured data are available is included in the optimization criterion. Additionally, the resulting optimization task with constraints is transformed into an unconstrained task by incorporating a suitable differentiable function. The physical parameters of the chain oscillator were thus calculated and checked for reasonableness. With this approach, a good agreement between modelled and measured transfer functions could be achieved with realistic system parameters.
In this article, the findings and methods from [32] will now be used and applied to the entire test-bench. The property of reciprocity is used to correct the hard-to-measure flow rate with respect to its amplitude. The new physical models as well as the model-updating strategies will be applied to a coupled hydraulic-mechanical system for the first time. The calculated parameters of the system are checked for their physical meaningfulness to validate the presented model approach. In addition, it is also shown that the model has some predictive power. FRF measurement data not used for model updating are compared with the FRFs calculated by the model and their agreement is quantified by means of FRAC.

Experimental set-up
The photograph in Figure 1 displays the test-bench, some of its important components are marked with numbers and described in Table 1. The hydraulic subsystem consists of four cavities which are connected by short bores. The bores can be regarded as rigid walled pipelines. Cavities and pipelines are milled in a massive steel block and closed by steel plates. It can be seen in Figure 1 that the first cavity is closed by a circular steel plate.

Figure 1. Test-bench
Two plates of different thicknesses are available to close this chamber, a thicker rigid one (like in the photograph), which is subsequently referred to as the rigid plate, and another one, half as thick, which is subsequently referred to as the coupling plate. So it is possible to investigate the system's behaviour depending on which closing plate is used for the first chamber. The cavities in the middle of the block are both closed with one thick plate. The last chamber is always closed by a coupling plate. The vertically arranged mechanical two-mass-oscillator is attached on this plate by a screw in the centre. It consists of two masses of different weight, which are connected with leaf springs. Pressure pulsations of the fluid are transmitted via the coupling plate to the mechanical oscillator, and mechanical vibrations vice versa into the fluid. Trapped air can be removed with minimess connections and an additional outlet, which can be seen on the very right of the photograph. During an experiment, the additional outlet is closed with a ball valve.   Figure 2 shows the inside of the first chamber, also in this figure some details are labelled with numbers and the corresponding description can be read in Table 1. The holes inside the chamber are the ends of the pipelines, which are drilled into the steel block. All chambers are sealed with an O-ring as shown in this photograph. This rubber ring lies in a groove milled for it. By screwing on the plate, this ring is pressed into the groove and thus seals the chamber.

Measuring equipment
As can be seen in the schematic of the hydraulic circuit illustrated in Figure 3, there are two alternative ways of excitation. Hydraulic excitation is performed by the injected flow rate Q V , which is controlled by an ultra fast MOOG D760-995A servo valve, whose spool position is measured. A mechanical excitation is exerted by the force F H indicated on top of the two-mass-oscillator. Figure 1 shows that there is a small circular plate on top of the oscillator, where an impact hammer can excite the system. The force of the hammer blow is measured. The pressure in each chamber as well as the pressure p V of the inlet device are measured by pressure sensors. Small asymmetries of the mechanical construction can lead to tilting of the masses. To take this into account, four accelerometers are attached to each mass and the vertical acceleration of a mass is calculated from the mean value of these four signals. The outlet of the system is adjustable by a throttle. The signals of the pressure sensors are acquired with a 24 bit analog input card, the signals of the acceleration sensors and the valve spool position with a 16 bit analog input card. The spool position of the MOOG servo valve is controlled by the National Intruments software tool LabView. The desired output signal is passed through a 16 bit analog output card and then fed to a signal amplifier, from which it is forwarded to the servo valve.

Preparing measurement data
As opposed to pressure, acceleration and force, the flow rate cannot be measured directly. There is an approximation for the determination of the flow rate according to the orifice equation if the oil is flowing through an orifice, where Q N is the nominal flow rate and Δp p N is the ratio of pressure drop across the valve and the nominal pressure. ξ denotes the measured valve spool position relative to full opening. However, the nominal quantities in Eq. (1) are not exactly known. The property of reciprocity can provide a remedy for this issue, as explained in the following.

Reciprocity
At first glance the model appears to have mechano-acoustical properties. Reciprocity relations of a mechano-acoustical system were described in [33] and in more detail including the two-port theory in [34]. In [35], a formal proof of the reciprocity of mechano-acoustical systems was given. It was also demonstrated that the reciprocal property is always valid if the differential equations of motion are symmetrical in the spatial variables. Any asymmetries would become noticeable as a sound absorbing effect. In [34], it is also pointed out that caution should be exercised with respect to reciprocity if there are poor connections in the mechanical system under consideration which may cause non-linearities, or if other nonlinear effects are included.
A vibroacoustical system can be regarded as two-port or as a power-gate. For the test-bench under consideration, Figure 4 shows the two external inputs, the flow rate Q V injected by the valve, and the force F H applied by the impact hammer, as well as the state variables in close proximity to the external inputs, the pressure p V in the inlet device, and the velocity v 10 of the upper mass. Together they build a two-port, respectively, a power-gate. A system is considered reciprocal if there is a special relationship between the transfer functions of the two gates, so that they are coupling-balanced or transmission-balanced. This means that they have the same transmission behaviour in both directions. For a reciprocal model of the test-bench, the transmission behaviour is expressed as If one input is applied separately, the relationships can be established. Depending on the sign conventions of the in-and outputs, there is now a simple relationship between two measured frequency response functions that can be used to calibrate the flow rate.

Reciprocity of the test-bench
The constants in Eq. (1) are now set according to Tab. 2. Q N and p N were taken from [36], where the same servo valve was used, and Δp is the mean pressure drop between supply pressure and the pressure p V in the inlet device.
In case of reciprocity, Eq. (1) for the injected flow rate is adapted for all measurements and now replaced by Eq. (5), including a correction factor α � ¼ 1:156, which is calculated so that the square error between the magnitudes of the two measured transfer functions  In Figure

Experimental conditions
The following conditions apply to all experiments listed in this publication. The used fluid is the HLP oil Shell Tellus S2 MA 32, whose temperature is in the range of 32-35 °C for all experiments performed. With a supply pressure of 12 MPa, the mean valve spool position and the adjustment of the outlet throttle are selected such that a mean flow rate of approx. 8.3 � 10 À 5 m 3 /s flows permanently through the system, which causes a mean pressure of approx. 8 MPa in each chamber. In case of flow rate excitation, a flow rate with chirp shape, whose amplitude is approx. 1.3·10 À 5 m 3 /s, is superimposed on the mean flow rate. The chirp signal lasts 4 s and reaches from 20 to 1000 Hz. In case of force excitation, the system's vibrations are measured for approx. 3 s. To reduce the influence of noise, each experiment is repeated eight times and an average of these eight individual measurements is used for the identification. To assess the scatter range of these measurements, the frequency response function between measured flow rate Q M V and the pressure in the inlet pipeline p M V is shown in Figure 6. Red represents the plus/minus range of the double standard deviation, calculated from the eight individual transfer functions, black is the corresponding average. It can be seen that the measurement uncertainties are small.

Mathematical modelling
In this section, the individual components of the test-bench are described mathematically and combined together for a complete model of the test-bench. According to Figure 7, the test-bench can be separated in three different parts and the coupling element. The inlet device (I), the hydraulic chain-oscillator (II) and the two-massoscillator (III) can be modelled independently, while the coupling plate (IV) links the two mass-oscillator with the hydraulic chain-oscillator.

Assumptions for the mathematical model
It is assumed that there is always laminar flow in the pipelines and we are dealing with a linear elastic (compressible) fluid, with viscosity to account for internal friction. Any temperature-dependent properties of the fluid are neglected, since the temperature of the fluid does not change during measurements. The pipelines considered can neither stretch nor move. As can be seen in Figure 7, the pipelines have elbows, which are considered with discrete linear resistances (see sec. 3.2.2). Furthermore, the pressure in the chambers is assumed to cause small elastic bending of the closing plates and the coupling plate, which is treated in sec. 3.4. The clamping situation of the plates is not exactly known, but in sec. 3.4 two different ideal situations will be considered. This expected motion of the plates is allowed to involve special dissipative effects, which are discussed in section 3.6.
The clamping between the lower mass (see m 20 in Figure 7) and the coupling plate is assumed to be ideal and excitations of the oscillator from its housing via the top springs are neglected. In addition, the movement of the mechanical two-mass-oscillator is assumed to cause a point load in the centre of the coupling plate ((IV) in Figure 7). For the mathematical model, the mechanical two-mass oscillator can only move in vertical direction. The masses are rigid, the leaf springs between them can be modelled with linear stiffnesses and it is assumed that the occurring dissipation effect is viscous and linear, which can be seen in detail in section 3.5.

Modelling straight pipelines
A relation between pressures and flow rates in the end points of a pipeline can be derived from the dissipative pipeline model given in [37]. An overview of different models and solutions from the fundamental equations of fluid dynamics can be found in [15]. Neglecting convective terms, the partial differential equations of the pressure and flow rate dynamics are transformed into the Laplacian domain, leaving a system of ordinary differential equations which can be analytically solved. The remaining unknown system functions are defined by the assumed fluid properties. In this article, we to deal with a viscous compressible fluid whose temperature-dependent properties can be neglected. Additionally, it is shown in [38] that the relationship of the solutions of pressure and flow rate dynamics between two points of a pipeline can be described by a power gate. The characteristics in the end points α and β of a pipeline section can therefore be written as where Q αβ;α is flowing into the pipeline section and Q αβ;β is flowing out. Equations (6) and (7) are including the fluid parameters E, ρ, and ν which denote bulk modulus, density, and kinematic viscosity, respectively, as well as the radius r and the length l αβ of the pipeline section; ω denotes the angular frequency and J 0 and J 2 are Bessel functions of the first kind. Equations (7) are valid in case of a viscous compressible fluid without temperature-dependent properties. If other characteristics are to be considered, these functions will have a different formulation.

Modelling pipelines with resistances
As can be seen in Figure 7, the pipelines in the block have elbows. A possible solution to take them into account would be to divide the pipelines into sections and consider their boundary conditions separately, which would increase the modelling effort. Therefore, a simpler solution like the one presented in [39] was used. In order to take into account elbows, as well as possible deviations from a circular cross-section, laminar resistances at the ends of the pipeline can be modelled. This model is schematically illustrated in Figure 8.
The corresponding equations read from which a relationship between the flow rates and the external pressures can be derived in the form

The inlet device
First introduced and modelled in [32], the inlet device (I) in Figure 7 consists of an inlet resistance as well as capacity and an inlet pipeline. Because of a jump in cross-section, the inlet pipeline has to be split into two sections. The relationship between the flow rates Q V and Q 01 is given by Using Equations (6) or (9), the inlet pipeline with the jump in cross-section can be described by In case of using the pipeline model according to Eq. (6) the entries read where Z V2 , γ V2 and Z V1 , γ V1 are defined in the Equations (7), with the corresponding radii r V2 and r V1 . Eliminating the unknown pressure p a , two relevant equations remain, whereby the model structure is invariant of the pipeline model used.

The hydraulic chain-oscillator
Part (II) in Figure 7 shows the hydraulic chain-oscillator, a massive steel block in which four chambers are milled, connected by bores and closed by steel plates. Like described in [32], it can be considered as a cascaded arrangement of capacities and pipelines. Figure 9 shows the conditions in a chamber β and Eq. (14) describes the situation mathematically sC βp β ¼Q αβ;β ÀQ βγ;β À sV β;χ (14) where the capacity due to the oil volume V β in the chamber reads As can be seen in Figure 9, the pressure in the chamber lifts the plate. This effect causes a displacement volume V β;χ depending on the pressure, which can be calculated from Kirchhoff's plate theory [40]. Pressure is applied to the plate within the radius r p , the radius of the sealing O-ring. The radius to clamping is denoted by r L . The boundary conditions are not exactly known, it is assumed that they are between fixed and pinjoined. According to Equations (16), the two volumes are which is denoted as the plate's flexural rigidity, depending on Poisson's ratio ν S , Young's modulus E S and the plate's thickness h. The rigid plate and the middle plate have the thickness h r which is twice the coupling plate's thickness h c . Substituting V β;χ by C β;χpβ , Eq. (14) becomes The pipelines in the block are modelled according to Equations (6), which leads to for the connection between chambers α and β. Alternatively, the pipeline model (9) can be used in the same manner.

The two-mass-oscillator
Part (III) in Figure 7 shows the mechanical part of the construction. Modelled in the Laplacian domain, the following equation describes the subsystem's dynamics and the state vector z m ¼ a 10 ; a 20 ½ �`. M m is the mass matrix of the oscillator with the masses m 10 and m 20 of approximately 10 and 20 kg, respectively. D m denotes viscous damping effects of the leaf springs and K m is the oscillator's stiffness matrix including the leaf spring's stiffnesses k 10 and k 20 .

The coupling element
Part (IV) in Figure 7 constitutes the coupling element. It is realized by a steel plate that links the dynamics of the hydraulic chain-oscillator with the mechanical two-massoscillator. In order to develop a suitable mathematical model of this component, its loading situation is shown in Figure 10. The pressure in the chamber lifts the plate as for the chambers of the hydraulic chainoscillator (see section 3.4). Additionally, the force from the mechanical two-massoscillator acts on the centre of this plate. Again from Kirchhoff's plate-theory [40] a displacement line w χ , depending on the geometry, the boundary conditions and the acting forces can be calculated. The displacement in the centre of the plate can be written in the form where x d ðtÞ is a displacement through dissipative effects, which are not covered by plate theory. Integrating the displacement w χ ðr; tÞ over the entire surface according to the resulting displacement volume reads For fixed and pin jointed boundary conditions, the parameters ζ p , ζ F , ξ p and ξ F can be seen in the Equations (A1) and (A2) in appendix A. Equations (20) and (22) can be rewritten as where k P denotes the plate's stiffness, C P the capacity due to the coupling plate and A K the coupling surface. Furthermore, ζ p ¼ ξ F is valid. This is related to the circular geometry of the plate and the applied Kirchhoff-theory and occurs if the load due to the force F is assumed to be a point load. It can be seen that A K links the states of the mechanical to the hydraulic system. Vice versa, with negative sign, the hydraulic state is linked to the mechanical subsystem. It is assumed that there are three forms of dissipation. A viscous mechanical damping of the plate can be considered with the damping ratio d P . Hydraulic leakage is modelled by the hydraulic resistance R P . Based on the available measurement data coupled damping must be considered in addition. With V d ðtÞ ¼Ṽ d ðtÞ À A K x d ðtÞ and a coupling dissipation area A d a possible dissipation model for the coupling plate reads Inserting Equations (24) in Equations (23) and transforming to the Laplacian domain, the coupling element's dynamics is obtained as |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl }

Proof of dissipativity
To prove that the suggested dissipation model extracts energy from the system, a closer look on the system's power gates is necessary. Rewriting the model (25) with a dissipation force F d and flow rate _ V d it leads to If the matrix D χ according to Eq. (25) is positive definite, F d and flow rate _ V d build a positive power with the system's state variables _ x and p. Sylvester's criterion [41] says that a symmetric Hermitian matrix is positive definite if, and only if, all its leading principal minors are positive. For D χ the leading principal minors are det

The dissipation model for closing plates
As can be seen in Figure 7, there is no mechanical system attached on the first two plates. In these cases the dissipation model (24) can be simplified, because no external force acts on the plate.
Combining this relation with Eq. (24b), the dissipation through the plate for a chamber α is given in the form

The model of the test-bench
All the previous partial modelling can now be combined together to obtain a mathematical description for the entire set-up. With the chosen state vector z ¼ a 10 ; a 20 ; p 4 ; p 3 ; p 2 ; p 1 ; p V ½ �` and an input vector u ¼ F H ; Q V ½ �`, the system's dynamics can be written in matrix notation as which can be interpreted as the model's mass matrix, including the masses of the mechanical oscillator, the capacities due to the volumes and closing plates and the end capacity of the inlet device. Additionally, the coupling surface A K and the dissipation surface A d between hydraulic and mechanical subsystems can be seen. A dissipation matrix considering the leaf springs' stiffnesses k 10 and k 20 as well as the stiffness k P of the coupling plate. The coupling between mechanical and hydraulic subsystem is also considered in K.
The matrix T is chosen in the form T 1::2;1::2 ¼ I 2 T 3::7;3::7 ¼ sI 5 (35) to link the state-and input-vector by the given system form (30). Finally, the input matrix B links the input-vector with the system's dynamics. Model (30) can be rewritten as whereby one has computed any interesting transfer function G αβ ðsÞ ¼ẑ α =û β , because in each experiment only one excitation quantity of the input vector u is active at a time.

Reciprocity of the mathematical model
In section 2.3.2 the test bench's reciprocal characteristics were shown by measured transfer functions. The reciprocity properties of the model (30) shall now be investigated. The transfer functions between the flow rate and the velocity of the upper mass, and the force of the hammer blow and the pressure in the inlet device are calculated as This shows that the reciprocal property is lost, because the dissipation area A d is always greater than zero. In the following section 4.2.2, unknown dissipative system parameters are identified using measurement data. Since the measured data is calculated with flow rate excitation according to reciprocity, a corresponding correction must be made here. The requirement for the correctly parametrized model is according to the reciprocity theorem (4), but since the correlation between the measured FRFs and the modelled FRFs is different, must now apply, assuming that only the amplitude level of the flow rate was not measured exactly.

Parameter identification
If a mathematical model is available, two problems have to be solved to identify unknown parameters. First, a suitable strategy is presented to find these parameters, then a proposal is made to deal with numerical issues during this process.

Preface: discussion of the modelling of dissipative effects
In the previous section, a model for the test-bench was developed that contains special dissipative parameters that do not occur in standard modelling of hydraulic systems. These include the sealing gap resistances R H;α according to Eq. (29), R P in Eq. (24) and the inlet resistance R L of the inlet device, as well as the resistances for the pipelines according to pipeline model (9). This section shows why these parameters are indispensable. The aim was to develop a model that approximates measurement data sufficiently well to enable the determination of possible damping parameters. Standard modelling of hydraulic systems, usually, only includes dissipative parameters like the kinematic viscosity and resistances. The resistances are usually supposed to model throttles if such exist in the set-up under consideration. In preliminary experiments, however, it became clear that the kinematic viscosity alone is not sufficient to approximate the dissipation seen in the measured data.
The resistance R L of the inlet device is indispensable to ensure the correct damping of the fifth mode at about 780 Hz of the FRFs , which can be seen in the comparison of model and measurement in the sections 5.1 and 5.2 in Figures 16 and 25. If this parameter would not be present, the kinematic viscosity would have to be chosen unrealistically high. Additionally, without the resistances of the pipelines it is also not possible to keep the kinematic viscosity in a reasonable range. The coupling dissipation according to (25) with the dissipation surface A d is not only necessary to link mechanical and hydraulic dissipations, it also serves to adjust the amplitude level of a modelled transfer function. Together with the coupling surface A K it acts like a gain on the transfer functions. The optimization strategy presented in the following section would fail if the amplitude levels of model and measurement differed too much.
In the upcoming section 4.5 three different cases with different dissipation models are therefore compared. This is done to prove that these non-standard modellings are necessary and also useful.

Strategy
Model parameters can be determined in a two-step procedure. First, the model is considered completely frictionless. For this purpose, all dissipative variables are set to zero and the remaining system is in use. The remaining parameters are optimized so that the pole and zero locations of the model match those of the measurement as closely as possible. When these parameters have been determined, the model is considered with dissipation. The magnitudes of the measurement at the pole and zero locations are used as a criterion to determine the dissipative parameters. The process is first explained on a simple example before it is applied to the model of the test-bench.

Step one
In Figure 11, the transfer function between the force of the impact hammer F H and the acceleration of the upper mass a 10 is shown for the case that there is no oil in the hydraulic chain-oscillator, i.e. without coupling. In this case, the transfer function GðsÞ is used as model which satisfies the necessary number of poles and zeros. GðsÞ contains unknown system parameters x d i , which are responsible for the dissipation, as well as unknown parameters x PZ i , which are mainly responsible for the position of poles and zeros. In Figure 11 GðsÞ, evaluated with initial values, is shown in black. Let G ðsÞ be the frictionless version of GðsÞ, and feed it to an optimization task of the form where ω P n ¼ À js P n are the angular frequencies at a pole location and ω Z m ¼ À js Z m are the angular frequencies at a zero location. The optimized quantities x �;PZ i will be calculated. In Figure 11 G ðsÞj x PZ i ¼x �;PZ i is illustrated in green, where poles and zeros of the model can be seen to agree with the measurement.

Step two
In the second step, the unknown parameters x d i are determined. For this purpose, the transfer function GðsÞ is evaluated with the previously calculated values x �;PZ i , and the amplitude heights at the poles and zeros are adjusted to those of the measurement by optimizing the parameters x d i . Let G M ðsÞ be the transfer function of the measurement, then the following optimization task calculates the optimal parameters x �;d i . In Figure 11 GðsÞ is evaluated with the optimal parameters x �;PZ i and x �;d i shown in red. It can be seen that the model response now agrees well with the measurement.

Numerical issues
For a practical application of the presented strategy, numerical problems have to be treated additionally. The unknown parameters of the vibroacoustical test-bench are spread across a wide range of order of magnitudes. For example, the bulk modulus is expected to be in the range of 1.5-1.7�10 9 N/m 2 , and an unknown capacity due to the rigid plate ranges between 5 and 30 � 10 À 14 m 5 =N.
Many works in the field of numerical mathematics, such as [42], deal with the theoretical concepts of scaling in numerical analysis. Mainly, scaling analysis deals with the invariance of a given mathematical model with respect to its scaling group. A practical application to different mathematical problems is shown by [43]. Now it is possible to move parameters into an equivalent numerical range with the help of scaling, but the problem can arise that the functions, which depend on the parameters, drift apart numerically as a result of this, which in turn has a bad effect on the problem posed. This can be counteracted by additionally scaling the dependent functions. For the solution of an optimization problem with the help of a gradient-based method, it is also necessary to consider the effect of both scalings on the gradients. This could lead to a lengthy search for the best possible set of scaling factors. Moreover, the set of scaling factors is bound to the specific model and cannot simply be applied to another one, for which the calculation of these factors would start all over again. F H An additional aspect is that an unconstrained optimization task is a simpler task than a constrained one. In order to circumvent the problems associated with such a wide numerical range, to counteract possible scaling problems and to obtain an unconstrained optimization task, a function is introduced in a similar way as shown in [44]. Designed in that way, f k is a smooth and monotonically increasing function which maps the infinite interval À 1; 1 � ½ of ξ k onto the interval κ k;min x I;k ; κ k;max x I;k � � and f k ð0; x I;k Þ ¼ x I;k applies. x I;k are the initial values of the unknown physical parameters with their percentage limits κ k;min and κ k;max , between which the optimal value is likely to lie. Figure 12 shows an example of how the function f k works. We want an initial value x I ¼ 5 and f should always be between 2 and 10.
In a constrained optimization task, the functions f k replace the unknown physical parameters x k and the dimensionless parameters ξ k can then be used as optimization variables in the resulting unconstrained task. Applied, for example, on the constrained task (42) the new unconstrained task reads |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } The resulting optimal parameters ξ �;PZ k can then be used to calculate the optimal physical quantities x �;PZ k according to by using Def. (44).

Algorithm
The task (45) is a summation of different partial minimization functions that can now be fed to a numerical solver. Since, the individual partial minimization functions ϕ Z m resp. ϕ P n sometimes have very different magnitude values, they are normalized to one for the initial points. After this, it is possible to give each function a scaling factor α for tuning. All optimization tasks performed in this publication were solved with the help of the software tool Matlab by a solver for non-linear least-squares problems using a trust-region algorithm. This solver finds a minimum of the fitness function k λ μ ð Þk 2 2 ; of a given vector-valued function λ by the parameters μ. Considering now task (45), the single functions ϕ Z m and ϕ P n , which are summed up, can be regarded as elements of a vector valued function λ. The corresponding computed task would then read In analogous way task (43) is processed, which is used for optimization of damping parameters. Stopping criteria are the function-and the iteration (step) tolerance, which are suitably chosen, but equal for every optimization task. In addition, the case of a very slow convergence is covered by stopping the calculation after a maximum number of iterations.

Results of step one
The strategy and the function introduced above are now used to identify the unknown system parameters of the test-bench model (30). In the first step, the searched parameters are the stiffnesses of the leaf springs k 10 and k 20 , as well as the coupling plate's stiffness k P . Furthermore, the coupling area A K and the capacities due to the plates, as well as the oil parameters E and ρ have to be determined. The kinematic viscosity ν, the dissipative coupling area A d , all viscous damping parameters of the leaf springs and the closing plates as well as any hydraulic resistances are set to zero. A frictionless model is used for the calculation. The optimization is done for the case of a hydraulic excitation, including all non-trivial measured pole points (which means that the poles at the frequency 0 Hz are not considered) of all possible transfer functions. In addition, five selected zero points are considered, since in a calculation without them, the model result would deviate from the measurement in the range of these zero points. Furthermore, the two possible plate constellations are covered in one task. Figure 13 shows the evolution of the fitness function for the first 1000 iterations. The fitness function in this case consists of 40 partial functions that are each normalized to one in the initial point, as described in sec. 4.4. In the first 50 iterations a faster convergence can be observed than in the following ones. The first 1000 iterations take the solver about 70.5 s and the rather slow convergence of this calculation after the first 50 iterations justifies a stop after these steps. Table 3 shows the results of the physical parameters calculated by this task.  In the course of the previous work [32], values for the bulk modulus, the fluid's density and the capacities could be calculated. These values serve as a guideline for the choice of the initial values of this optimization task. The initial values of the individual parameters lie in their expected range. The bandwidths of the optimization parameters were set according to Def. (44) with the respective percentage deviations κ k;min and κ k;max so that they do not exceed the expected range excessively.
It can be seen that the parameter values except for the coupling plate's capacity C P are in the expected range. The expected ranges for the capacities, the coupling area and the plate stiffness were determined using Kirchoff's plate theory [40] with the two assumed boundary conditions. Since the central plate as shown in Figure 1 is not a circular plate, the expected range of the two central capacities C m cannot coincide with that of the capacity of the circular rigid plate C r , even if it has the same thickness. However, the way it is screwed on suggests that the capacity values must be smaller than those according to the circular plate.
Parameter C end has not been discussed up to here and it is also not part of the optimization. It is only responsible for the 5th pole (at approx. 770 Hz), which one can see in Figures 16 and 25 in sections 5.1 and 5.2. The oil volume of the inlet device without the inlet pipeline was measured and amounts to 10-15 ml. With the calculated bulk modulus, it is possible to give an assessment range for the end capacity within 6.07-9.09 mm^5/N. The best fit with measurement data is given by an end capacity of 7.130 mm 5 N , which confirms the expected range.
The expected ranges for the leaf spring stiffnesses were determined using static beam theory [45], distinguishing between two assumed boundary conditions. In one case, one edge is fixed and the other is free, with a force applied at the free edge. In the other case, one edge is fixed and the other, where a force is applied, is guided. The expected range for the bulk modulus was derived from [46] where experiments on the HLP oil Shell Tellus S32 were done. This oil has similar properties to the oil Shell Tellus S2 MA 32 which is used in the experiments published in this article. Since the chirp excitation causes rapid pressure changes here, the measurement data of the quasi-adiabatic test-cycle in [46], in which the focus was also on rapid pressure changes, was considered. Finally the density should be smaller than 872 kg=m 3 because this value is given in the data sheet of Shell Tellus S2 MA 32 for a temperature of 15 °C and the experiments were done at a measured oil temperature of 32-35 °C.

Results of step two
The results of Table 3 are now used for the second step. Three different types of dissipation models are analysed and their results are compared in Table 4. The evolution of each fitness function is illustrated in Figure 14. Dissipation model (I) is without any laminar resistances at the ends of a pipeline. So the pipeline model according to Equations (6) is in use. Case (II) is a model without hydraulic dissipative effects due to the plate motion, so there is no hydraulic resistance considered for the plates, but hydraulic resistances of the pipelines are modelled like in Equations (9). The third model (III) is considering both effects. In order to allow a qualitative comparison of the optimization process, the same initial values were used for models (II) and (III), only for model (I) an initial value for the kinematic viscosity 36% higher was applied. For all three cases, the same pole and zero points were used for the optimization task.
It can be seen in Figure 14 that the fitness functions are starting at 50 because in this case all three entire fitness functions consists of 38 partial functions, which are all normalized to one in the initial point and four of them are scaled up by a scaling factor α of two as task (47) shows. This choice leads to better results. The fitness function according to model (I) converges fastest, the calculation is completed after 186 iterations and takes about 54.3 s, whereby in this case 10 parameters have to be identified. The fitness function according to model (II) converges slowest, after 200 iterations it no longer decreases strongly per iteration step. For the calculation of the first 500 iteration steps the solver takes about 400 s, while in this case 14 parameters are optimized. This long calculation time and the slow convergence after 200 steps lead to the fact that the solver can be stopped after these 500 iterations without missing a significantly better result. The convergence of the fitness function according to model (III) is better compared to case (II), but also in this case it becomes slower after the first 200-300 iterations. For the first 500 steps, the solver takes about 510 s, which is longer than in case (II), but in case of model (III), 18 parameters are optimized. The optimization is stopped after 500 steps, because of the slow convergence and the duration of the calculation. Table 4 shows the results of the physical parameters, determined by the optimization process for each dissipation model. The kinematic viscosity ν of the oil is in the expected range for the dissipation models (II) and (III). The range was taken from the Shell Tellus hydraulic oil data sheet. As can be seen, the hydraulic resistances at the beginning and end of a pipeline are necessary to keep the viscosity in the realistic range. The hydraulic resistances of the pipelines are in a similar size range for both model approaches. The hydraulic resistances according to the plates are much larger in comparison. For the testbench model (30), this can be interpreted to mean that more friction is generated by the presumed gap effects of the plates than by the elbows of the pipelines.
A d is needed for all dissipative models, because it is independent of the hydraulic resistances due to the plate motion and because it has a dissipative gain effect together with the coupling surface A K on all transfer functions in case of flow rate excitation.
Considering the viscous damping values of the leaf-springs and the coupling plate, the physical reasonableness of the three model approaches can be interpreted. First, it is assumed that the more rigid a mechanical element is, the lower the expected dissipation. It follows that the dissipative damping of the upper leaf-springs has to be the largest, that of the leaf springs in the middle smaller and that of the coupling plate the smallest. This matches for models (I) and (III).
Another interpretation is also possible. Building a stand alone mechanical model with the viscous damping ratios, the stiffnesses of Table 3 and the masses (see Tab. A1), one can calculate the eigenvalues and from these the dimensionless damping ratios. Similar to the model (19) of the mechanical two-mass-oscillator, the autonomous model without excitation reads |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl fflffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl fflffl } which can be written in the form |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } with the eigenvalues It is possible to substitute the above system (48b) with decoupled equations of motion similar to those treated in [47] or [48]. Let the decoupled system be |ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl {zffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl } whose eigenvalues are eigðΛ S Þ ¼ À ξ 1 � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi The dynamics of system (49a) can substitute the dynamics of the physical system (48b) if their eigenvalues are equal. With Equations (48c) and (49b), it is possible to calculate the dimensionless damping ratios ξ i and the system's natural frequencies ω i from the complex eigenvalues of the physical model: ffi ffi ffi ffi ffi ffi ffi ffi ffi ; ω ¼ β ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The results for the damping ratios ξ i are shown in Table 5, defined as percentages of critical damping.
The expected range of the damping ratios was taken from [49], covering the cases of a continuous metal structure and a metal structure with joints. As can be seen, one damping ratio of model (II) is far too large. This can be taken as a proof that a coupling dissipation effect is existing. Only model (III) allows realistic kinematic viscosity and damping ratios.

Comparison of measurement and model results
In this section the focus is on the match between the transfer functions of the model (30) and those of the measurements. In case of flow rate excitation, there are seven possible transfer functions, with two different closing plate situations and three different dissipation models. This would cause 42 different FRFs, whose presentation would go beyond the scope of this article. Therefore, three selected transfer functions are shown for each of the different dissipation models and the different closing plate constellations. Like Figure 15 shows, the FRFs between the flow rate and the acceleration of the upper mass, as well as the pressure in the inlet device and the pressure in chamber three will be analysed.
For this purpose the test-bench's model (30) is evaluated with the identified parameters of the Tables 3 and 4, as well as the fixed parameters of Table A1.   A reason for this can be the torsion mode of the mechanical two-mass oscillator at about 279 Hz. The mechanical part was measured without hydraulic coupling in an earlier experiment. It is obvious that this mode now couples with the hydraulic system and the two-mass-oscillator loses more energy at this frequency than expected, which is due to the participating torsional vibration. An effect that was not considered in the model (19).
In the FRFs p 3 Q V and â 10 Q V a peak can be seen at 225 Hz. Here, the piston pump that pumps the oil to the test bench appears with its excitation frequency.  Also in this case the match between model and measurement for each dissipation case is quite good. Like in the previous case, model (II) and (III) give very similar results, which are both better than model (I).
Comparing these results, measurement as well as modelling, with the previous case, one can see that a different closing plate situation leads to a different FRF shape. Especially, the poles and zeros of FRFs are now more pronounced than before. Furthermore, it should be emphasized that some of the resonances have shifted. In this case, they are around 160, 290 and 390 Hz. The resonance at about 325 has not shifted in the same manner, and the second mechanical resonance at about 700 Hz (see in the FRFs â 10 Q V ) has not shifted. The coupling torsional mode of the two-mass-oscillator at a frequency of approx. 279 Hz again leads to the fact that around the second mode at 290 Hz model and measurement do not correlate that well (see in Figures 27, 30 and 33).
The effect of the piston pump at about 225 Hz can be seen again. The problem of unwanted noise only occurs above 500 Hz for both plate constellations, which only affects the resonance at 700 Hz.

Excitation with an impact hammer
In this section, the focus is on the predictive power of the model. Like Figure 1 shows, the test-bench is excited with an impact hammer. The calculated parameters of Tables 3 and 4 will be used to evaluate the model, but these parameters are determined using measurement data from flow rate excitation. In the following sections 6.2 and 6.3 the FRFs between the force of the hammer blow and the acceleration of the lower mass, as well as force and the pressure in chamber one will be shown, like it is illustrated schematically in Figure 34.

Discussion
As can be seen in Figures 35-46, the prediction of the poles worked well in both plate constellation cases. Of course, the model-measurement match is not as good as in the previous section where the parameters were optimized for it, but the model reproduces the basic shape of the measurement data well. Similar to the case of flow rate excitation, models (II) and (III) give very similar and better results than model (I).
It should be emphasized in this case, that the resonance at about 270 Hz (coupling plate in use) and 290 Hz (rigid plate in use), is worse reproduced in terms of its amplitude height than in case of flow rate excitation. The flow rate excites the two-mass-oscillator via the well-defined centre of the coupling plate. The torsion mode is therefore not excited as much as the vertical one, which leads to less dissipation and better model/measurement match. With force excitation at the top of the two-mass-oscillator, the torsional mode is excited more strongly, which leads to this larger deviation between model and measurement.
Noise is a problem especially in the FRFs from about 400 Hz (this is true for all FRFs between pressure and force). It is not possible to detect the second mechanical resonance from these transfer functions. In addition, the model also indicates that the amplitude level of the second mechanical resonance will be very small.   7. An additional quality review of the graphic results in sections 5 and 6 In order to qualitatively assess the agreement between measured and modelled FRFs, several possibilities are available. In publications dealing with modal analysis, the modal assurence criterion (MAC), as shown in [23], or the coordinate modal assurance criterion (COMAC) [24] is usually used as a quality measure between measured and analytically calculated mode shapes. For this criterion one needs eigenvectors of the measured system, as well as calculated ones from the designed model. Now two inconveniences arise. On the one hand, in order to be able to construct eigenvectors from a test system, one must make a comprehensive investigation of numerous FRFs from this system. On the other hand, it has to be possible to extract eigenvectors at natural frequencies from the mathematical model, which severely limits the model type. From the model (30), for example, eigenvectors cannot be assigned to certain frequencies in a classical sense. However, it is also possible to use experimental and analytical FRFs directly for a correlation study. One can implement the idea of COMAC based on FRF data, which results in the frequency response assurance criterion (FRAC) according to [26]. This criterion is best suited for a calculation of quality measures for the results shown in sections 5 and 6, because with the FRAC two FRFs with the same input-output relationship can be compared. The FRAC between a calculated FRF H and the corresponding measured one H M in a frequency range of f α to f β ¼ f s N þ f α with a constant sampling frequency f s can be defined by which can be interpreted as the normalized quadratic scalar product of the frequency responses over the considered range. The FRAC value is always in the interval [0,1], where 1 means perfect correlation between the two FRFs in the considered frequency range and 0 means none at all. Table 6 shows the FRAC values for the results of section 5. It can be seen that the FRAC values are all clearly above 0.9, with the exception of the FRF p 3 Q V in case of a coupling plate closing chamber one. But also in this case the FRAC value is above 0.85 which means that there is a good correlation. It can be seen that the dissipation models (II) and (III) (see sec. 4.5.2) lead to quite similar FRAC values for each FRF, which are always better than those calculated if dissipation model (I) is in use. Table 7 shows the FRAC values for the results of section 6. It is pointed out again that the measured FRFs from section 6 were not used for the determination of system parameters, which can be a reason for the fact that most of the calculated FRAC values are below 0.9. However, most of them are above 0.85, which indicates a good correlation between model and measurement and confirms the model's good predictive power. Also in this case it can be seen that the dissipation models (II) and (III) lead to very similar FRAC values, but always higher than those calculated when dissipation model (I) is in use. The results from Table 7 also show, compared to the results from Table 6, that the closing plate situation has a stronger effect on the FRAC value.

Conclusion and further investigations
It has been shown that the presented model in combination with the proposed parameter identification strategy leads to a good model-measurement match.
Bending of the outer wall due to oil pressure, as occurs with the closing plates in the test bench, was taken into account with hydraulic capacities. In section 4, it was shown that the necessary capacities agree with the displacement volume calculated by the circular plate theory. Additionally, it was shown that modelling of dissipative effects in the presented way leads to realistic model parameters. This was not possible with a simpler model structure. These results suggest that there are nontrivial dissipative effects in a mechanical hydraulic interaction and that they are coupled as well. Furthermore, it has been shown that elbows in the pipeline have to be taken into account. In case of a suitable dissipation model of the coupling effect it is not only possible to identify system parameters in a physically reasonable range, but it is also possible to make predictive statements with the model as shown in section 6.
In order to quantify the quality of the graphical results from sections 5 and 6, the FRAC values for the corresponding FRFs were calculated, giving an average FRAC of 0.947 for the results of section 5 and one of 0.887 for the results of section 6. This means that the model results fit well to the measured ones. FRFs that are not used to calculate system parameters can be predicted well, under the additional requirement that the model parameters are physically reasonable.  A reciprocal system property can help to adjust hard-to-measure quantities such as the flow rate. However, further investigation is needed to reconcile the dissipative coupling effect with the theory of reciprocity, in order to find out what exactly is going on in the process that causes this property to be lost. Potentially, hydraulic resistances due to the closing plate movements approximate a nonlinear effect.
Moreover, it should be proven whether the strategy shown here for the identification of unknown system parameters can also be transferred to other systems. Important quality criteria of the procedure are the duration of the parameter search, the physical meaningfulness of parameter values and the accuracy of the calculated solutions. Of course, the method reaches its limits if an unsuitable mathematical model is proposed for a given set-up. Apart from this, there are still many possibilities to extend the presented updating method so that it converges faster and becomes more robust against numerical problems and measurement noise.

Appendix A. Tables and mathematical expressions
The parameters of the Equations (20) and (22) differ depending on the assumed boundary condition. In case of a fixed boundary condition, they read and in case of a pin-joined boundary condition they read with K P according to Eq. (16c).