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A physics based methodology for modeling of hydraulic devices within multibody-based comprehensive models of rotorcraft systems is developed. The proposed models are developed in two stages. At first, models are developed for three basic hydraulic elements: the hydraulic chamber, the hydraulic orifice and the pressure relief valve. These models con...

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... the second approach, the hydrodynamic behavior of the device is linearized to obtain one or more ordinary differential equations relating control inputs to the forces generated by the device; typical equations are given in text books such as those of Viersma (Ref. 1) or Canon (Ref. 2). While this approach is physics based and captures some basic aspects of hydraulic devices, the linearization process is clearly too restrictive. In fact, rotorcraft lead–lag dampers are often purposely designed to behave in a nonlinear manner. Indeed, a linear device would generate high damping forces under high stroking rates; these high forces must be reacted at the hub and at the root of the blade, creating high stresses and decreasing fatigue life. A possible remedy to this situation is to use pressure relief valves that act as force limiters, implying a nonlinearity essential to the design and behavior of the device. In the last approach, a physics based, fully nonlinear representation of hydraulic devices is implemented. This enables the determination of the complex interaction phenomena between the structural and actuator dynamics: pressure levels in the hydraulic chambers are now coupled with the dynamic response of the system. This paper describes such an approach in detail, and its predictions are validated against bench test measurements and flight test data for Sikorsky’s UH-60 aircraft. The modeling of hydraulic devices has been the subject of detailed studies. In Ref. 3, Welsh proposed a detailed model for predicting the dynamic response of helicopter air–oil landing gear that included several degrees of freedom representing the tire, floating piston, orifice piston, and simple fluid and adiabatic gas models. In a later effort (Ref. 4) the same author addressed the problem of modeling the lubrication system of a helicopter using a similar approach. In both cases, detailed models of the hydraulic systems were developed, but these were not coupled with the dynamic response of the vehicle. A variety of hydraulic devices are used in the rotorcraft industry: hydraulic actuators are crucial components of many main rotor control systems, hydraulic lead–lag dampers are used in many rotor designs, and landing gear often involve hydraulic or pneumatic elements. In the case of lead–lag dampers, the hydraulic device tightly interacts with the rotor response; in fact, blade root edgewise moments depend to a large extent on damper response characteristics. To deal with this variety of devices, a modular approach is taken. At first, models are developed for three basic hydraulic elements: the hydraulic chamber, the hydraulic orifice, and the pressure relief valve. Models for entire hydraulic devices are then constructed by assembling the models of a number these hydraulic elements. In this work, models for hydraulic actuators, simple hydraulic dampers, and hydraulic dampers with pressure relief valves are discussed. Once a model of the hydraulic device is in hand, it is to be coupled with a comprehensive rotorcraft simulation code. In this effort, hydraulic device models are coupled to a finite element based multibody formulation of a helicopter rotor system within a comprehensive analysis (Ref. 5). Within the framework of flexible mechanism analysis codes, the modeling of hydraulic devices has attracted limited attention; in Ref. 6, models were proposed for a hydraulic jack and for the actuator of an aircraft retractable landing gear. Conceptually, the coupling of a hydraulic device model with a comprehensive rotorcraft modeling code is straightforward. First, the hydraulic device model predicts the instantaneous force the device applies on the supporting structure. In turn, this force is applied to the dynamic model of the vehicle to predict displacements and velocities. Finally, these kine- matic quantities change the stroke of the hydraulic device, and hence, its force output. In a finite element formulation, this is readily achieved by connecting the end points of the hydraulic device to two nodes of the finite element discretization. The present paper has two main goals. First, a comprehensive modeling approach will be presented for hydraulic devices, such as hydraulic actuators and dampers. Second, these models will be coupled to a com- prehensive rotorcraft model using a finite element based multibody formulation. The paper is organized in the following manner. The first section presents models for the basic hydraulic elements, and the second section shows how these basic models can be combined to deal with various hydraulic devices. Next, issues associated with the coupling of hydraulic devices models with finite element based multibody formulations of rotorcraft dynamic simulation are discussed, with special focus on the integration of the hydraulic equations. The modeling approach is then validated using a number of numerical examples. Finally, conclusions of this work are offered. Hydraulic devices can be seen as an assembly of simple hydraulic elements; in this work, three basic hydraulic elements will be presented: hydraulic chambers, orifices, and pressure relief valves. These basic elements are described in the following sections. Of course, a variety of other elements could be developed, such as hydraulic accumulators or check valves. The hydraulic chamber, shown in Fig. 1, is probably the most common hydraulic component. The chamber, of volume V and cross-sectional area A , is filled with a hydraulic fluid of bulk modulus B under pressure p . Often, due to the presence of a piston, the length of the chamber can vary. The change in length of the chamber, due to piston motion, is denoted d . Finally, hydraulic fluid can flow into the chamber; Q denotes the net volumetric flow rate into the chamber. The evolution of the pressure in the chamber is governed by the following first order differential ...

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## Citations

... As an alternative to the monolithic schemes, in a second approach, known as multirate integration, there exist different subsystems that are integrated separately, exchanging information between them in predetermined time intervals. This can be carried out by using a single environment where the different problems are integrated separately (co-integration) [18,19] or resorting to a different software for each problem (co-simulation) [20][21][22]. Some relevant aspects have been addressed in the literature, such as co-simulation configuration [23], energy-based coupling error minimization [23,24] or the multirate cosimulation [25]. ...

... Simplifying by using Eq. (19) and retaining up to firstorder terms in Eq. (25): ...

Hydraulics is often used to actuate mechanisms in the applications of heavy machinery. In this work, a linearization approach for hydraulically driven multibody systems is presented. The approach allows linearizing the equations of motion of general multibody systems with holonomic and nonholonomic constraints, augmented with the hydraulic equations of the hydraulic subsystem. The derivation of this linearization approach is of interest in many applications, such as the performance of linear stability analyses. The procedure is tested with a three-dimensional multibody model of a hydraulically actuated four-bar mechanism. The validation of the approach is performed by means of the forward dynamics simulation of the linear and nonlinear systems. The results show the power of the approach, obtaining the linearized equations of motion around the equilibrium position of the four-bar mechanism multibody model in terms of the mechanical and hydraulic parameters. A comparison of the proposed procedure with a conventional counterpart approach is included, demonstrating the great accuracy and computational efficiency of the approach developed in this work.

... When considering a machinery device as a combination of mechanical mechanisms and fluid power systems, one finds that the mechanical mechanisms can be modelled in a straightforward manner in terms of multi-body system dynamics [3], whereas a fluid power system is often modelled via the theory of lumped fluids, in which the system gets divided into hydraulic volumes with evenly distributed pressure [4]. The integration time step applied for a fluid power system is usually smaller than that in a multi-body solver [5]- [7], so managing fast response for the fluid power system is crucial to real-time simulation for vehicles [8]. There are two popular approaches to dealing with assemblies comprising a mechanical mechanism and fluid power system, known as the unified approach and the multi-rate integration approach [8]. ...

Machinery devices often consist of mechanical mechanisms that are actuated by fluid power
systems. In many applications, the mechanical system can be modelled and analysed in terms of the multibody
system dynamics. Fluid power systems, in turn, can be analysed via the lumped-fluid theory, with
which simulation of fluid power systems requires smaller integration time steps than needed by multi-body
solvers. This leaves simulation of the entire machinery device beyond reach for a real-time framework,
with the main reason for the very small time steps in modelling of fluid power systems being the presence
of a small hydraulic volume, which creates a numerical stiffness problem. The stiffness issue may arise
from numerical singularity emerging in the fluid power system, which implies that solving the governing
equations involves different time scales – small and large. To resolve the numerical singularity in hydraulic
circuits, the authors developed a perturbed model to alleviate the stiffness problem demonstrated that it can
increase the integration time step by an order of magnitude. Since the perturbed model does necessitate
a correction factor for the volumetric flow rate, the method of multiple scales is applied to compute the
pressure within the small volume to second-order accuracy, O("2), in comparison with the perturbed
model’s O(\epsilon). The results reveal that if the correction parameter is not set, the perturbed model’s cumulative
error leads to considerable deviation in piston position with respect to the reference model, whereas the
multiple-scale model eliminates the issue of cumulative error without demanding any flow-rate correction
factor.

... The control system for the articulation, minutely presented by the authors in [1], was developed on the basis of studies reported in [3][4][5][6][7]. Performance of the algorithm had to be tested with respect to its sensitivity to delays that occur in the control loop. ...

An algorithm for controlling the hydraulic damping of an innovative articulation for an articulated vehicle was tested with respect to the stability of the vehicle movement while avoiding a road obstacle at 50 km/h by a double change of the lane. Related maneuvers were simulated using LabVIEW software. Two testing methods were employed - with the steering wheel motion being dependent or independent on delays introduced into the control loop. Simulation results according to the first method proved that the mean absolute slip angle of all the tires and the mean absolute lateral acceleration of the rear car centre of mass did not exceed, respectively, 0.65° and 0.55 m/s², whereas for the second method these values were lower than 0.85° and 0.75 m/s². In both methods the results depended on the delay in signal transmission. It was concluded that the control loop delay should be kept lower than 0.1 s. Moreover, this delay should be monitored and used as an input to the control algorithm, as in certain conditions deactivating the articulation damping in presence of excessive delays can be confusing to the driver.

... Bauchau and Liu [17] used a multi-integration algorithm to solve a finite element based mechanical system coupled with hydraulic equations. They used a time-step of 0.1 ms for the structural dynamics analysis and a four-step Runge-Kutta integrator with a time-step 48 times smaller than the structural solver in the hydraulic integration. ...

... where ε is determined by the small volume divided by the effective bulk modulus, which is lowered by the entrained air [17], [28]. It is often observed when a hydraulic system is turned-ON after a period of shutdown, which allows air to collect in the system. ...

... Based on the boundary-layer stability analysis of the system, it has been obtained that (17) and (19) have no challenges in applying the perturbation theory to the control signal U > C. It should be noted that the boundary-layer stability analysis is performed if the control signal is less than C, i.e., U < −C. By taking a similar approach as mentioned previously, it can be found that the boundary layers of (17) are not unconditionally stable; thus, the perturbation theory cannot be applied. ...

... The hydraulic setup of the articulation consists of two cylinders, each paired with an electrically-driven proportional valve. The piston motion is modelled with the following equations [5][6]: ...

Operation of an articulated vehicle is dependent on an appropriate damping action taking place in its rotary articulation. In order to analyse an impact of the control of the articulation on the motion of the vehicle a model of the vehicle with a controllable hydraulic damping system has been developed. A 90 degree turn and lane change manoeuvres were simulated using LabVIEW software. Modification of the damping parameters of the articulation, according to the velocity and articulation angle of the vehicle, proved to have a significant impact on the vehicle motion stability. Moreover, the sensor layer necessary for the control algorithm as well as the diagnostic system is described.

... The same modeling approach has also been used by Eyres for modeling hydromechanical dampers to investigate vibration reduction of helicopters with lag dampers [9]. More recently, Bauchau and Liu developed a hydromechanical damper model using a similar approach for modeling lead-lag dampers in a rotorcraft comprehensive analysis, DYMORE [10]. Similarly, a hydraulic lag damper model was formulated and integrated in an established industrial aeroelastic simulation code by Titurus and Lieven [11]. ...

... Initially, the model was validated by comparing to test data from [10] and was found to match peak values within 6% [13]. However, the available data were presented as force versus time rather than forcedisplacement or force-velocity relationships, which are more useful for validation. ...

A physics-based, fully nonlinear hydraulic damper model was developed within the framework of the Rotorcraft Comprehensive Analysis System (RCAS) in order to facilitate integrated damper and rotorcraft analysis and design. The hydraulic damper governing equations of motion were formulated using flow continuity equations for each hydraulic device within the damper. The developed damper model was evaluated for an integrated rotor-damper solution and was verified against test data from the NASA-Army UH-60A Airloads Program. For these evaluations, the accuracy of blade load predictions using the developed damper was compared against conventional linear and nonlinear (table lookup) damper models. From these comparisons, the developed damper model is shown to better capture the isolated damper dynamics. The assessment help address the previously identified issues and uncertainties and identify trends or dependencies.

... The same modeling approach has also been used by Eyres for modeling hydromechanical dampers to investigate vibration reduction of helicopters with lag dampers [9]. More recently, Bauchau and Liu developed a hydromechanical damper model using a similar approach for modeling lead-lag dampers in a rotorcraft comprehensive analysis, DYMORE [10]. Similarly, a hydraulic lag damper model was formulated and integrated in an established industrial aeroelastic simulation code by Titurus and Lieven [11]. ...

... Initially, the model was validated by comparing to test data from [10] and was found to match peak values within 6% [13]. However, the available data were presented as force versus time rather than forcedisplacement or force-velocity relationships, which are more useful for validation. ...

The present study focuses on the optimal trim states of a compound helicopter with an articulated main rotor, at a high-speed of 225 kt. In addition to conventional helicopter trim variables, variation in main rotor revolutions per minute, auxiliary thrust, stabilator pitch and aileron deflection are considered. Simulations are based on a compound derivative of a modified UH-60A helicopter with a 20,110 lb gross weight operating at standard sea level conditions. The results show that to achieve a reduced power requirement the stabilator should be used to bring the aircraft to a nearly nose-level pitch attitude, resulting in a significantly high wing lift share (approaching70% in this study). Using the ailerons to introduce a roll right moment allows the rotor to operate at a higher lift-offset and generate its share of the lift with greater aerodynamic efficiency on the advancing side. In addition to the use of stabilator and ailerons, reduction in main rotor revolutions per minute is favored for low power, and the combination of these control surface and revolutions per minute settings reduces the rotor drag thereby requiring lower auxiliary thrust from the propulsor. The paper provides detailed discussions on the physical mechanisms producing changes to the trim state and the associated power benefits realized.

... The damper arm and damper horn are modeled as rigid bodies. The leadlag damper is modeled as a hydraulic damper (Bauchau and Liu, 2006); its physical properties are described in Welsh (1988). In addition, 68 equally spaced airstations along with C81-type airfoil tables are used to calculate the aerodynamic loads on the blade. ...

Purpose ‐ This paper aims to correlate the flexible multibody analysis for the performance, blade airloads, rotor pitch control angles, and blade structural loads of a full-scale utility helicopter rotor in low-speed forward flight with wind tunnel test and flight test data. Design/methodology/approach ‐ A nonlinear flexible multibody dynamics analysis code, DYMORE, is used to analyze the performance and aeromechanics of a utility helicopter rotor in low-speed forward flight. The main rotor system is modeled using various multibody elements such as rigid bodies, nonlinear elastic beams, mechanical joints, and elastic springs/dampers. The freewake model is used to capture rotor wakes more elaborately in low-speed forward flight. Findings ‐ Fair to good correlations of rotor performance such as figure of merit in hover, rotor power, propulsive force, and lift in low-speed forward flight are achieved with sweeps of the thrust, rotor shaft tilting angle, and advance ratio, against wind tunnel test data. The blade section normal forces from the mid-span to outboard are fairly or well correlated with flight test data, but the normal force at the inboard blade station is under-predicted. The trimmed pitch control angles are reasonably predicted; however, the lateral cyclic pitch control angle is moderately under-predicted. The flap bending moments are compared fairly with measurements; however, the oscillations of the lead-lag bending and torsion moments are not captured well. Practical implications ‐ Reasonable predictions of the performance and aeromechanics of the rotor in low-speed forward flight will allow the flexible multibody dynamics to be used for the rotorcraft comprehensive analysis, in place of expensive flight and wind tunnel tests of the rotor. Originality/value ‐ Up to now, the stand-alone flexible multibody dynamics without the aid of external aerodynamic analysis has not been widely used for the analyses of rotor performance and aeromechanics in low-speed forward flight. However, the present flexible multibody dynamics analysis directly integrated with the freewake model gives fair to good correlation of the rotor performance and aeromechanics predictions in low-speed forward flight.

... The force generated by the damper is given by equation (8), as proposed in Bauchau [14]. Equations (2)-(4) and (7)- (8) constitute the full damper model. ...

In this paper a dynamic optimization methodology for designing a passive automotive damper is proposed. The methodology proposes to state the design problem as a dynamic optimization one by considering the nonlinear dynamic interactions between the damper and the other elements of the suspension system, emphasizing geometry, dimensional and movement constraints. In order to obtain realistic simulations of the suspension, a link between a Computer-Aided Engineering Model (CAEM) and a multi-objective dynamic optimization algorithm is developed. As design objectives we consider the vehicle safety and the passenger comfort which are represented by the contact area of the tire and the vibrations of the cockpit respectively. The damper is optimized by stating a set of physical variables that determine the stiffness and damping coefficients as independent variables for the dynamic optimization problem, they include the spring helix diameter, the spring wire diameter, the oil physical characteristics and the bleed orifice diameters among others. The optimization algorithm that we use to solve the problem at hand is a multi-objective evolutive optimization algorithm. For this purpose we developed a parameterized model of the damper which is used to link the CAE tools and the optimization software, thus enabling fitness evaluations during the dynamic optimization process. By selecting the physical characteristics of the damper as design variables instead of the typical stiffness and damping coefficients, it is possible to consider important design constrains as the damper size, movement limitations and anchor points. As result of the proposed methodology a set of blueprints of non dominated Pareto configurations of the damper are provided to the decision maker.

... The hydraulic semi-active damper and its model were developed and validated using laboratory tests conducted in periodic working conditions[14]. The model uses concepts adopted from hydraulic system theory and hydraulic actuator modeling[30],[31]. Research presented in[20]and[30]focuses on aspects of the damper and coupled rotor-damper modeling. ...

... The model uses concepts adopted from hydraulic system theory and hydraulic actuator modeling[30],[31]. Research presented in[20]and[30]focuses on aspects of the damper and coupled rotor-damper modeling. Stability questions for helicopters with semi-active lag dampers are addressed in[26]and[27]. ...

... Stability questions for helicopters with semi-active lag dampers are addressed in[26]and[27]. Studies on load alleviation potential in a hypothetical semi-active hydraulic lag damper[28]and passive hydraulic lag damper coupling studies[30]are examples of similar research with a different focus. The research presented in this paper uses a damper model adopted from[14]which represents a refined and validated version of the earlier model discussed in[25]. ...

This paper describes the research on the use of semi-active hydraulic lag dampers for vibration control in helicopters. A semi-active vibration control of the hydraulic lag damper is investigated in the context of a midsized production helicopter in steady level flight. The semi-active operation of the damper is achieved via periodic flow restrictor modulation. The influence of the modulations and constraints on the nonrotating hub loads of a five-bladed main rotor is evaluated. The selected harmonic modulation frequencies are determined by the combination of the hydromechanical coupling in the damper and the mechanical filtering effect of the symmetric rotor. The modulation frequencies of 3, 5, and 7 per revolution are used to modify the 5-per-revolution lateral and longitudinal hub loads, whereas the frequencies of 4 and 6 per revolution are efficient at modifying the 5-per-revolution hub yaw moment. The harmonic modulations larger than seven per revolution are not used due to limited damper responses at these frequencies. The out-of-plane hub loads and moments cannot be significantly influenced by the rotor-damper configuration considered. A maximum load reduction of 72% on the lateral hub load component decreases to 58% when considering a limited damper response authority due to the relief valve activation.