FINITE ELEMENT ANALYSIS OF A HYDRAULIC SNUBBER WITH RESPECT TO HISTORICAL TEST DATA AND AMERICAN SOCIETY OF MECHANICAL ENGINEERS REQUIREMENTS

Hydraulic snubbers are either acceleration or velocity limiting seismic restraints designed to restrict movement of piping or equipment during dynamic events or operational transients. In a piping analysis, snubbers are modeled as linear elastic spring elements governed by Hooke’s law, F = kx. Snubbers are widely used in nuclear power plants, and as such their qualification testing to verify the spring constant k is governed by American Society of Mechanical Engineers (ASME) codes. ASME mandates experimental determination of spring constant k where practical, or a combination of testing and analysis when not practical. This qualification test includes testing at full load under a sinusoidal forcing function at a maximum frequency of 33 Hz. There remains a practical upper limit to dynamic testing that is imposed by the availability of test equipment. This upper limit is governed by the size of the hydraulic pumps and servo actuators supplying fluid to the actuating test cylinder. At current writing, this limit is approximately 200 kips @ 33 Hz. New reactor designs have applications for snubbers with load capacities up to 1,900 kips. Functional testing can be conducted on these large units to verify the lockup and bleed parameters are correct, as well as the load carrying capacity. However, the dynamic spring rate of the snubber will not be experimentally verified at full rated load. This study developed an FEA model of a hydraulic snubber that was compared to existing ASME qualification test data performed by Anvil Engineered Pipe Supports (EPS). If an accurate model can be developed for smaller snubber sizes, it can be used to determine the spring rate of units that exceed the capacity of existing test equipment. The experimental test data shows a spring constant that decreases at an approximately linear rate between 3 Hz and 33 Hz, with a reduction at higher frequencies of approximately 30%. The simulation models linear elastic behavior, and shows up to a 6% reduction between 3 Hz and 33 Hz. Part of the discrepancy can be explained by the load controlled nature of the test negating inertial effects, and increased deflections due to lost motion caused by assembly clearances and manufacturing tolerances. Further testing should be conducted, measuring load through pressure transducers in the cylinder fill ports to overcome these test setup limitations. This testing should be done on at reduced load so that a model can be developed that agrees with both the reduced load testing and rated/emergency load testing.


Definition and Principles of Operation of Hydraulic Snubbers
A pipe support is an integral part of any nuclear power plant. Pipe supports are load bearing components that support the weight and guide the thermal growth of safety and non-safety related piping systems and equipment. Pipe supports consist of nearly infinite configurations, but all have the same recipe: 1) Structural Attachment -Typically consists of a bracket or lug welded to the building structural steel or bolted to the building foundation or containment structure 2) Pipe/Equipment attachment -A clamp designed to fit around a pipe or lug welded to equipment (pump, steam lines, pressurizer, etc) 3) The "middle" -The pipe support component that functions as required by analysis. Will allow thermal growth, apply a load to the system, or restrain pipe movement as needed. Figure 1 shows typical pipe support assemblies on 34" hot reheat lines at a natural gas combined cycle power plant. When in operation, these lines will operate at a temperature of approximately 1050° F, causing several inches of thermal movement. Figure 2 is a typical snubber assembly per Figure NF-1132-1(c) [1]. There are many other types of pipe support components, but for the purposes of this paper, the discussion will be limited to shock and sway suppressors, also known as snubbers. A snubber is a type of seismic restraint whose design was first conceived in the 1960's and began to see large scale use in the nuclear power industry in the 1970's [2,3]. In a typical installation, it is attached to the building structure by a rear bracket and the piping system or equipment by a clamp, as in Figure 2. When the piping or equipment heats up during normal operation, a snubber is designed to allow thermal growth while applying minimal or no load to the system. It is operating in a passive mode. When subject to a velocity or acceleration greater than the pre-set limit, the restraint activates and applies resistance to any motion. Figure 3 shows a snubber installed on main steam piping at Diablo Canyon Nuclear Power Plant These passive and active modes of operation are common to the two types of snubbers present in industry today: Mechanical and hydraulic. Mechanical snubbers restrain motion through a ratcheting mechanism or an inertial mass limited to a pre-set angular acceleration. Hydraulic snubbers utilize valves to restrict the flow of fluid between the tension and compression sides of a hydraulic cylinder. Both mechanical and hydraulic snubbers can be velocity or acceleration limiting. This study is limited to velocity limiting hydraulic snubbers, specifically those manufactured by the Anvil Engineered Pipe Supports located in North Kingstown, RI (Formerly known as Grinnell Corp.).

Principles of Operation of Hydraulic Snubbers
Hydraulic snubbers are very similar in operation to any hydraulic actuator found commonly in industrial equipment, such as a forklift, hydraulic press, excavator etc… The primary difference is in the valve configuration. Where hydraulic actuators create motion of a piston rod by pumping fluid into either the tension or compression side of the cylinder, hydraulic snubbers resist motion by restricting fluid flow out of the tension or compression side. This fluid resistance through the valves prevents free movement of the piston, and a load is applied to the piping or equipment in the opposite direction of system motion. All snubbers are double acting, and will provide a resisting force in either tension or compression.

Functional Characteristics
Functional characteristics are so called because they can be field tested to verify that the snubber will perform its function as a dynamic restraint. The valve is a crucial component for the operability of a hydraulic snubber. On Anvil hydraulic snubbers, it consists of a check and bleed valve set in parallel on both the tension and compression sides. The check or lockup valve, is of the spring loaded poppet type. The bleed valve is a flow restrictor which provides resistance to flow as function of pressure. A typical control valve arrangement is shown in Figure 4. In passive mode, the check valve is held open with a spring, and as fluid starts to flow over the poppet, a pressure differential develops between the reservoir and cylinder side. As fluid flow velocity increases, the pressure difference becomes large enough to overcome the force of the spring holding the valve open. At this point, the snubber goes from passive to active, and the poppet closes. Fluid must now flow through the flow restrictor. Enough resistance is provided that the pressure inside the cylinder begins to increase, and a load is applied through the piston and piston rod.
A typical snubber pre-service test plot is shown in Figure 5. The test plot shows the two functional characteristics of a snubber that must be within specification to be installed in a nuclear power plant. These characteristics are defined in [5] as follows: Activation -The change of condition from passive to active, in which the snubber resists rapid displacement of the attached pipe or component Drag force is defined in [5] as "The force that will sustain low-velocity snubber movement without activation throughout the working range of the snubber stroke". It is not of interest in most functional tests of hydraulic snubbers. Typical values for hydraulic snubbers are 2% or less of rated load. A snubber with a rated load of 10 kips should see a drag force of no more than 200 lbf. This resistance to motion is primarily caused by friction between the energized elastomeric seals and cylinder/piston rod. In most piping analysis, this drag force is neglected.

Dynamic Characteristics
Plant operations and maintenance personnel are concerned with the functional characteristics of a snubber; Plant design engineers are more concerned about the dynamic characteristics. These engineers analyze piping systems and equipment for their response to a ground excitation or operational transient, including water hammer, turbine trip, or pipe rupture. In their models, snubbers are placed at key node points to restrain motion of the system and reduce piping stresses to within American Society of Mechanical Engineers (ASME) code allowables. To simplify analysis, an assumption is made that the seismic or transient velocities of piping and equipment are much greater than the pre-set activation velocity of a snubber. It can then be assumed that the snubber is a linear elastic spring element subject to Hooke's law, F = kx.
When subjected to a sinusoidal load, hydraulic snubbers exhibit a load/displacement relationship that follows a hysteresis loop, as shown in Figure 6.

Figure 6 -Typical oscilloscope dynamic test plots [8]
From the plot in Fig. 6, the numerical value of k, or the dynamic spring rate of a snubber can be derived. The vertical axis represents load, and the horizontal axis displacement. The load and displacement of each cycle are plotted against one another, and the peak to peak loads and displacements are measured. Spring rate is defined in [9] as the "Applied load in tension and compression divided by the recorded displacements in tension and compression". This corresponds to the equation . The result k is usually given in units of kips/in.
The region of non-linear motion is known as "dead band". This is free-motion of the hydraulic snubber while activated with no resistance being applied to the piping or equipment. This is a consequence of clearances required for installation of load pins in the rod eye and pivot mount or mechanical gaps in the assembly due to manufacturing tolerances. At low frequencies ( < 3 Hz), it is also includes the time The number of analysis load cases would grow exponentially.
To circumvent this complexity, the nuclear power industry wrote plant design specifications to include a Minimum Spring Rate parameter. In first generation design of large scale commercial reactors, overly conservative assumptions of accident conditions led to very stiff piping systems [12]. This then required manufacturers to supply snubbers that will always be at least as stiff as the minimum spring rate value.
This made analysis of large piping systems practical. For very critical equipment applications, such as steam generator, pressurizer, or reactor coolant pump snubbers, the engineer may specify a spring rate and require that the manufacturer test each production unit to verify the dynamic performance is within spec. This presents its own set of challenges and is the motivation for this study.
History of Dynamic Qualification Testing and Development of ASME QME/QDR Early snubber populations in the 1970's and early 1980's were prone to failure on a very large scale. Depending on the number of snubbers in service at each plant, failure rates hovered between 4% -30% [2]. Fluid and seal compatibility issues with radiation and temperature caused many snubbers to leak or clog the valves, impairing their safety related function. Minimal or non-existent codes and requirements for snubber design qualification testing resulted in a low degree of confidence of their operability during a seismic event or transient. Compounding the problem is the fact that many early plant designs had upwards of 1000+ snubbers on their piping systems [2]. Optimizing seismic support locations in a piping analysis is an iterative process, and during the design phase of many plants, it was cheaper to buy more snubbers than it was to run hundreds of analysis iterations on mainframe computers. In some cases, entire plant populations were tested and/or replaced each refueling cycle at great expense in radiation exposure to plant workers, time and money.
The NRC began to address the issue of snubber operability as early as 1978, and incorporated it into its standard review of all licensee safety analysis by 1980 [11].
This put the onus on the licensee to provide proper assurance that safety related hydraulic snubbers would be operable during postulated seismic events or transients.
Another consequence of the high failure rate of early snubbers was the development of GE Specification 21A3502. This purchasing specification [13]  hydraulic snubbers installed in nuclear power plants were supplied by Grinnell [4]. As a result of this industry dominance, many of the design requirements from the GE specification became the accepted industry standard.
By the early 1980's, the issue of snubber operability had been solved from a regulatory standpoint [11]. Licensees began snubber reduction programs, more rigorous inspection regimens, and adoption of improved hydraulic snubber designs.
However, it was also apparent that most licensees had two distinct snubber populations that required different approaches to qualification of the design and inservice inspection. Licensees and snubber vendors differentiated snubbers as largebore and small-bore, sometimes also referred to as equipment and piping snubbers.
The common definition of a large bore snubber is a snubber with a rated load of 50 kips or greater, and as of 1992, large bore snubbers made up approximately 10% of the US hydraulic snubber population [4]. In practice, functional testing and snubber repair/replacement activities involve removal of the snubber from service and installation onto a test bench. This can present real issues inside the containment structure, as space is limited and any time spent performing maintenance activities exposes plant personal to radiation. For small

Figure 7 -Large bore hydraulic snubber (LBHS) in assembled in a functional test bench. [4]
bore piping snubbers, many designs are light enough that they can be carried by hand or maneuvered through tight spaces with minimal trouble.
Large bore equipment snubbers are much larger in size. Some cylinders have piston diameters as large as 20" and can weigh several thousand pounds. In many cases, it was deemed impractical to do any functional testing and prior to 1980, most large bore hydraulic snubbers were classified as inaccessible and were not subject to the same in-service functional testing of piping snubbers [4]. To rectify this, the NRC issued Generic Letters 80-99 and 84-13 [19,20] requiring that all licensees add large bore hydraulic snubbers to their snubber surveillance populations. The first in-service functional tests of large bore hydraulics occurred at 13 plants between 1982-84. Each of these 13 plants had large equipment snubbers that failed to activate during testing, and would have been inoperable during a seismic event. In fact, all 14 steam generator snubbers at Palisades were found to be inoperable since initial reactor startup [10].
As a result of these serious safety deficiencies, the NRC created Generic Issue 113 which culminated in the technical evaluation NUREG/CR-5416, published in 1992 [4]. The technical evaluation determined that one of the main causes of large bore snubber failures was the lack of qualification testing for early designs. In many cases, design requirements for large bore snubbers were not properly specified by the licensee. Test equipment limitations in the 1960's and 1970's also prevented full scale static or dynamic testing of larger sizes. In fact, some designs did not see a proof-load test to verify that they were indeed capable of operating at full rated load. However, a survey of five large bore snubber manufacturers yielded that by 1992, qualification tests that were very similar to the GE 21A3502 requirements had been conducted for many models with a rated loads of under 100 kips.
One of the recommendations made in NUREG/CR-5416 was to implement a uniform set of qualification standards for large bore hydraulic snubbers, either through a regulatory guide issued by the NRC or the addition of snubber qualification requirements to the existing ASME Qualification of Mechanical Equipment (QME) standard. ASME created a subgroup of the QME code to develop this standard. Given that the test requirements of GE 21A3502 had been adopted by several manufactures and were well understood by much of the industry, this specification became the basis for the development of the new testing standard. This standard became known as

"Qualification of Active Mechanical Equipment Used in Nuclear Facilities Section
Qualification of Dynamic Restraints", also known as QME/QDR [9]. QME-  was the first QME standard to be endorsed by the NRC for rulemaking in 2009 [14].

Current Snubber Qualification Testing
Nuclear power plants designed to ASME Section III 2007 or later must have their dynamic restraints qualified per ASME QME/QDR. The minimum requirements for functional and dynamic testing are outlined in in QME/QDR-6223. 1     The following description assumes a movement that causes the cylinder piston rod to retract. The rod eye is connected by a pin to the piping or equipment to be restrained. Movement is transmitted to the piston from the rod eye via the front piston rod. The pivot mount is pinned to a rear bracket, and the piston moves relative to the cylinder. This is known as the compression side. The piston displaces fluid on the compression side, and it flows through the control valves into the tension side. The cylinder and piston dimensions are such that the rod eye can move a magnitude of 6" from the fully retracted position to the fully extended position. This is referred to as cylinder stroke. A piston at mid-stroke can retract or extend 3" relative to the fixed pivot mount.
The rear piston rod is not load bearing but does provide key advantages. It

Materials of Construction
Many different materials go into the construction of a Fig. 3306 The pressure differential developed over the poppet valve is partly a function of viscosity, thus snubber activation is also viscosity dependent. The viscosity of the working fluid must be relatively stable through a variety of environmental conditions.
The fluid is a silicone polymer known as polydimethyldiphenylsiloxane, and is widely used in many brands of hydraulic snubbers. This silicone based fluid was originally manufactured by General Electric under the brand name GE SF-1154. A critical characteristic of this fluid is its excellent viscosity stability when exposed to radiation.
It has been tested by Anvil EPS up to 1 10 rads of gamma exposure with no significant gelation. Its viscosity is also fairly stable at elevated temperatures. This provides more consistent functional test results, as field testing is not always done in a temperature controlled setting.
For the load/displacement relationship of interest in this study, fluid bulk modulus is of great importance. As mentioned previously, fluid bulk modulus can vary from batch to batch. At the time Anvil's testing was carried out, Anvil was in the process of qualifying a new fluid vendor, and the lot of fluid used in the qualification tests underwent laboratory analysis to determine the bulk modulus. This laboratory testing was done at 70° F and will be used for this study. For simulations at 200° F, experimental bulk modulus data is not available and historical Anvil data will be used.

Fig. 3306 Qualification Testing to ASME QME/QDR Requirements
The Fig. 3306  Each snubber was pinned at the pivot mount to a fixed bracket and to an actuating hydraulic cylinder at the rod eye. A linear transducer was mounted on the rod eye and cylinder head to measure deflection of the rod eye relative to the cylinder ( Figure 11). This configuration does not measure cylinder deflection, and an assumption was made that cylinder axial deflection was minor compared to fluid column deflection. It is worth noting that in this test configuration, in addition to cylinder deflection in tension, the clearance between the pins, spherical bearings, rear bracket holes, and the pivot mount deflections, are not taken into account.
A load cell was placed in the load path between the actuating cylinder and snubber rod eye. In addition to measuring the load applied to the snubber, the load cell provides feedback for actuator load control. Load control is utilized to prevent the snubber from generating high working pressures inside the cylinder. As the loaded fluid column deflects, the unloaded column will draw fluid from the secondary reservoir to prevent cavitation in the unloaded side. Under displacement control, the additional fluid volume creates a higher pressure and over the course of many cycles, internal pressure can become high enough to cause seal failure. By using load control, this can be avoided.
The rated load of each cylinder is applied by the actuating cylinder with a sinusoidal waveform at frequencies between 1 Hz -33 Hz. Cylinder load and displacement were sampled at a time increment of .0002 seconds. During testing, it was determined that there was electronic "noise" in the system, and 150 Hz filter was placed on both the load and deflection outputs to filter out high-frequency signals.
Data was recorded into LabView software, and made available in a format readable to Microsoft Excel.  displacements was zeroed at mid-stroke. All displacement values are given relative to this zeroed value. A positive value indicates the piston has extended from mid-stroke; a negative value indicates the piston has retracted.

Test Sequence and Data Reduction by Anvil EPS (PE 9851-6)
The full test sequence conducted by Anvil EPS includes all functional and dynamic testing mandated by QME/QDR. After testing was conducted, the data was reduced by Anvil EPS to come up with numerical values for snubber spring rate. To reduce the data, the following method was employed:  Spring Rate Values as Reported by Anvil EPS [15,16]

Data Reduction Through Use of Computer Code
One of the challenges Anvil faced in reducing the data was the sheer volume of information. As an example, Figure 16 shows 5 cycles of a test that lasted for 12 seconds at 10 Hz, for a total of 120 cycles. The spring rate value was determined by using only 4% of the available data. To some degree, the reduction of the test data is subject to a small bias by selection of different cycles, or to interpretation of the analyst, or where the line of best fit is drawn on a particular day.
To work with a larger set of data and remove graphical interpretation from the equation, code was written in Microsoft Excel Visual Basic for Applications, or VBA to return the extrema for each cycle in the selected time period. The user must still choose a start and end time of the cycles in question; the recorder was on before a load was applied to the system, and the actuating cylinder takes several cycles to ramp up to the desired load. This can be seen in Figure 15, where it takes approximately 12 seconds to achieve the minimum test load of 100 kips peak to peak, and the last 5 seconds have no meaningful data. Once the start and end times are specified, the start and end of each cycle is periodic as a function of the test frequency. By iterating through each cycle, the maximums and minimums can be extracted for the entire test.
One behavior that is evident in the dynamic test plots is drift of the piston position over the course of the test. In Fig. 15 An understanding of the drift mechanism is not required for determining spring rate. Fig. 17 plots the displacement extrema of 5 cycles at 1 Hz. Upon inspection, it is apparent that the maximums and minimums of each cycle drift by approximately -.025" each cycle. This "drift" will skew the peak to peak displacements depending on how they are measured. Using the 2 nd cycle as an example, measuring the distance A will result in a smaller displacement than measuring peak to peak distance B. Because snubber "drift" happens at a rate that is approximately linear, the peak to peak displacements A and B can be averaged to find the true peak to peak distance.

Summary of Results as Reduced by Computer Code
The spring rates for the Size 35 and 100 at room temperature and 200° F are given in tables 2 through 5 as a function of frequency.   The test data supports that the opening and closing of these valves only has an impact for frequencies lower than 3 Hz, and the frequencies of interest here are between 3 Hz -33 Hz. This assumption was made to reduce the number of elements and stay away from the realm of CFD simulations.

2) "Walking" does not occur
A consequence of Assumption 1 is that snubber drift does not occur.
The test data shows that this drift adjusts the deflection extrema at a linear rate, and this has been taken into account in the experimental test data reduction. It will not have any impact on the cycle to cycle values of deflection in the simulation.

3) Fluid bulk modulus is not variable w/ pressure
During dynamic cycling, the internal pressure of a snubber will increase above the value due to the applied load. For example, if the snubber cycles first in tension, the compression fluid column will deflect, causing fluid to flow into the tension side from the reservoir. As the snubber moves to a neutral position, the pressure on the compression and tension sides balances at a non-zero value. With no bleed valves, this value asymptotically approaches the pressure that would be present at full load with each cycle. When the snubber is now in a neutral position of zero deflection, it is pressurized in the tension and compression sides at 1 X full load. When a load is applied, the loaded side is at a pressure that is 2 X full load, and the unloaded side is at zero.
In actuality, some of this pressure is bled off through the bleed valves at a pressure dependent rate. There is no test data that shows what actual pressure inside the cylinder is during a dynamic event, and it is assumed that ΔP between the tension and compression sides is 2 X full load. In the simulation, with Assumption 1, ΔP will still be 2 X full load, although the loaded side will be pressurized to 1 X rated load and the unloaded side will in a vacuum at -1 X rated load.

4) Fluid deflection governs the response to load
In the dynamic model, it is assumed that steel deflections are orders of magnitude smaller than the deflection of the fluid. This assumption is valid, as Young's modulus for steel is approximately 150 X the bulk modulus of the hydraulic fluid.

Fluid Cavity Material Models
Two material behaviors were developed for the fluid cavity:

1) Fluid Cavity Interaction
This behavior models a fluid cavity as a linear elastic volumetric spring [18], with properties of density and bulk modulus. A fluid cavity is defined by an enclosed surface with normal pointing inward. The elements are surface elements, but they are coupled to a cavity reference node that has properties of pressure and volume. As the surface elements deform due to an external load or internal pressure, the pressure and volume properties of the reference node are updated to the new conditions. These properties are then applied to the next step or increment. This interaction can be used with both implicit and explicit static and dynamic procedures.  [18].

2) Hydrodynamic Equation of State
Energy is dissipated either through volumetric or shear deformation. The user must specify density, viscosity, and the speed of sound through the material, which is a function of bulk modulus and density. For the dynamic model, the explicit/dynamic procedure is used. This procedure satisfies equations of motion at increment n, and uses the acceleration of that increment to advance the model to the next n + 1 increment. The explicit dynamics procedure allows for a time history of linear response to be modeled. It has been verified through testing that the natural frequency of the size 35 and size 100 snubbers does not exist in the frequencies of interest, therefore all load/deflection response is expected be linear.
A consequence of using this procedure is the required time increment can be very small compared to the static simulation and can lead to long run times for the same model. The minimum time increment is the time required for the dilatation wave to pass through the smallest element in the model [18]. Complicated geometries that cause large elements to distort or require small elements for an accurate model can decrease the time increment by orders of magnitude for the rest of the model, greatly increasing run times. This drove some differences in the static and dynamic models to keep run times on the order minutes. It is worth noting that implicit dynamic and modal based procedures were also tried, with similar results.

Static Model Geometry
The first simplifying assumption in the model is recognizing that the hydraulic cylinder is round, and the number elements can be reduced by using symmetry. The pivot mount and rod eye are both rectangular in shape, and because of this ¼ symmetry was used in the XY and YZ planes. The cylinder is oriented so the Y axis is parallel with the axis of the snubber cylinder. +Y is cylinder extension, and -Y is cylinder retraction. The Fig. 3306 consists of hundreds of individual parts. In order to reduce the number of contact interactions and elements, many subassemblies were merged together to form continuous pieces. This model consists of 6 parts: 1) Cylinder -Modeled as a 3D solid. For the Size 35, the cylinder ID is 5" and the wall thickness is .5", and with a thickness to ID ratio of .1, a thinwalled cylinder model could be used. However, the Size 100 has a ratio of .125, and the assumption of a thin walled cylinder cannot be made. In order to achieve consistency between the 35 and 100 simulations, they were modeled as solids. Partitions to the internal diameter surface had to be made to accurately describe the fluid cavities. There are also bolt holes at the cylinder ends (Fig. 19).
2) Cylinder Head -Modeled as 3D solid. This part is part of the cavity surface boundary and includes bolt holes (Fig. 20) 3) Cylinder End Cap -Modeled as a 3D solid. This piece is only modeled so as to include bolts and the threaded connection of the pivot mount to the cylinder. Included in the model are the bolt holes (Fig. 21) 4) Pivot Mount -Modeled as 3D solid. This piece is used for the boundary conditions at the pin hole (Fig. 22).

5) Piston and rod assembly -Modeled as a 3D solid. A partition was made in
the rod eye to apply a load, and in the piston rods to describe the fluid cavity (Fig. 23) 6) Bolts -Modeled as 2D wire features. These bolts were modeled as wires to reduce the number of elements in the model. They are partitioned to include a node for a bolt pre-load force.

Static Model Interactions and Constraints
For the static model, the master/slave surface contact definition was used for all interactions of solid parts that come into contact. In this contact definition, pressure is transmitted between the two surfaces. The slave surface is not allowed to penetrate the master surface, although both surfaces are allowed to deform. Friction is not expected to have a significant effect in the model, so frictionless behavior was used.
The fluid cavity interaction was used for both the tension and compression cavities.
The cavity reference nodes lie on the axis of the cylinder. There is also a tie constraint between the pivot mount and the cylinder end cap. This constraint ties the movement and rotation of the nodes on both surfaces together via the master/slave relationship.
To simulate the bolted connections of the cylinder head and endcap to the cylinder, multi-point beam constraints were used. This constraint ties the displacement and rotation of a single control point to the nodes that lie on a slave surface. They are connected by an idealized rigid beam. The wire bolt features had two multipoint constraints; One from the bolt "head" to the counter-bored surface, and a second from the threaded end to the internal threads of the cylinder.

Static Model Boundary Conditions
There are four boundary conditions applied to the model 1) Symmetry in the XY plane 2) Symmetry in the ZY plane 3) Fixed surface at the pivot mount pin 4) Fixed X and Z translation at the rod eye exterior flat plane. This is to prevent any moment developing around the fixed pivot mount surface. In actuality, these nodes would be part of a pinned connection and no moment would develop.

1) Bolt preload
This load is applied to an internal node of the wire features used for bolts. This load is applied during the first step after the initial step to determine the deformed lengths of the bolts under the pre-load. This deformed length is then applied to step 2.

2) Primary Loading of the Cylinder
This load is the applied load to the rod eye and simulates a pin applying a distributed load over 45° of the rod eye ID (Fig. 25). Due to ¼ symmetry, the magnitude of the load will be ¼ the experimental test value. Because it is a pressure distribution acting over 45°, the input magnitude will be greater than the actual value according to the equation

• cos /
This amounts to an increase of 1.111 for the simulation input from the actual test load. The portion of the load not directed down the axis of the snubber will cancel due to symmetry. There is no input waveform for the static model. It is applied instantaneously in compression in one step, and in tension in a second step.

Static Model Mesh
ABAQUS has a mesh generation algorithm that has a parameter known as seed size. The algorithm places nodes on the model edges, and these nodes are the seed nodes. Seed size refers to the spacing between these nodes, and when possible, node spacing maintains this value throughout the model. ABAQUS suggests a default seed size based on part geometry and element type. Contact models work best with elements of similar size, so the smallest default seed size was used for the entire model. This is governed by the cylinder head and endcap, which have a default seed size of .42 for the size 35 model, and .71 for the size 100. General purpose continuum elements were utilized. Due to the complex geometries, quadratic tetrahedral C3D10 elements were selected. Various seed sizes were tried, both larger and smaller, and the default provided results that were good enough at a reasonable computational cost.
( Fig. 26). For the bolts, B31beam elements with a seed size of .062 and .069 were used.

Dynamic Model Geometry
To isolate the fluid behavior, a very simple axisymmetric model was created.
Because of Assumption 4, the rod eye and cylinder portion between the compression cavity and the pivot mount could be neglected. This model consisted of three parts, all modeled as planar shells:

1) Cylinder
Models the cylinder and head as one piece. The interior edge was partitioned to specify cavity surfaces.

2) Piston and Rod Assembly
Models the piston and piston rods as one piece. To more accurately model the mass of the part, the interior portion of the rear piston rod was removed for the secondary reservoir cavity.

Dynamic Model Materials
For simplicity, only one steel material was modeled. It has the following properties: Poisson's Ratio: . The only boundary condition needed to prevent rigid body motion is a fixed cylinder base where it would extend to the pivot mount. (Figure 28). The load is applied as a pressure on the main piston rod. It has a sinusoidal amplitude of the form

• sin
This load was applied for .5 seconds for 3 Hz to 5 Hz and .25 seconds for frequencies between 10 Hz and 33 Hz.

Dynamic Model Mesh
The dynamic model is more sensitive to changes in mesh density. The fluid is a much softer material than the steel, and must be a much finer mesh.  In the Fig. 3306 qualification testing, there was not much variation cycle to cycle of the input load. This is a consequence of the load controlled nature of the test.
Therefore, to determine spring rate, the maximum magnitude of the input load can be doubled to obtain the peak to peak load. This is divided by the peak to peak deflection.
The reaction forces of the nodes fixed in the axial direction relative to the snubber can be summed. The difference in the summed reaction forces and the input load show the inertia effects of the piston/piston rod mass moving with high accelerations.

Static Results
The static spring rates from the ABAQUS models are shown in Table 6  It was assumed in the experimental data that at 3 Hz, the activation valves were closed and had no effect on piston deflection. The spring rate at 3 Hz is as close to static as the experimental data can get. When measured against the static model, it appears that the values are not in agreement. However, this discrepancy can be explained by the configuration of the reservoir and compression cavity. In the ABAQUS model, the seal gland disks were modeled rigidly connected to the cylinder, and not floating between two retaining rings. When two gaps are added to the peak to peak static deflection, the resulting static spring rate is shown in Table 7

Dynamic Simulation Results
The dynamic simulation results utilizing the equation of state material model and a dynamic explicit procedure are as follows in Tables 8and 9 Mechanical gaps were not included in the peak to peak deflections.
Assumption 4 states that deformation of the steel cylinder is minor compared to the larger fluid deflections, and can be ignored. This means that variations in geometry between the axisymmetric dynamic model and the experimental test data can be ignored as long as the fluid behavior is modeled accurately. Using 3 Hz as the frequency where spring rate is near static, Table 10 shows that the values for "static" spring rate are the correct order of magnitude and the axisymmetric model is realistic.
Arbitrarily adding mechanical gaps or altering geometries of the steel components will not make the dynamic results any more or less valid.

Discussion of Dynamic Test Results
The frequency dependence of spring rate in the experimental data is much greater than the simulation. Experimentally, the difference in spring rate between 3 Hz and 33 Hz is approximately 30%, and in the simulation is less than 6%. The 6% deviation in the simulation is due to the inertia of the piston and piston rod assembly.
This inertia can be seen by measuring the reaction forces of the fixed nodes at the cylinder wall. When the reaction forces in the direction of the applied load are summed, the value is larger than the maximum value of the sinusoidal input.
Inertia should be seen in the experimental data. Under load control, the applied force ramps up at a higher rate for each cycle, causing larger accelerations of the actuator ram and thus larger inertial forces. The simulation data confirms this; Maximum accelerations at 3 Hz for the size 35 room temperature simulation is 250 , and 2870 at 33 Hz. However, the experimental data shows no discernable increase in input load as a function of frequency. Table 13 shows dimensionless load as calculated from the reaction forces of the fixed nodes in the simulation, and as measured from the load cell in the experimental data. Figure 34 shows this graphically. There is a linear increase in reaction forces in the simulation, but no pattern in the experimental data. Consideration was also given to the possibility that the increase in deflection as a function of frequency in the experimental data could be explained by the sinusoidal input approaching a natural frequency. Examining the experimental plot in Fig. 18, reduction in spring rate is approximately linear. If the snubber was approaching natural frequencies, values of deflection would increase exponentially with frequency.
A key difference between the dynamic simulations and the experimental test setup that has not yet been addressed is dead band, or lost motion. In the simulation, there is no free movement due to manufacturing tolerances or clearances required for assembly. In the experimental test setup, the linear transducer was mounted to measure deflections between the cylinder and rod eye, and thus dead band was not measured. This does not mean it did not exist. This alters the input wave form, from a smooth sinusoid into the pumps controlling pressure in the actuator, to a sharp sawtoothed wave form seen at the connection of the snubber rod eye to the actuating bracket. The input wave form is now , where P is the maximum test load and A(t) is a periodic coefficient that is a function of time. This wave form would look similar to the one seen in Fig. 35.

Figure 35 -Modified Waveform to Simulate Lost Motion
In a sinusoidal waveform, load increases from 0 to , where T is the period. From to , load is decreasing to the minimum, and at t = T, load is zero. In Figure 35, from 0 to , it as assume there is no load applied to the snubber due do lost motion. Deflections are on the magnitude of .04", and dead band is analytically determined to be .02". Thus, it is a reasonable assumption that ½ of the loading time interval is spent in free motion.
During the time when the snubber is unloaded due to deadband, pressure in the actuating cylinder is increasing. By the time the actuating ram makes contact with the snubber, the load applied is not zero, but a larger value. This impact load waveform now looks like Figure 36 In Fig. 36, it is assumed that the load applied at is ½ the maximum magnitude. This wave form can be applied to the dynamic model to test the effects of impact loading on dynamic response. The experimental value of the magnitude of the applied load at impact cannot be known from the data given, but an assumption will be made that it increases linearly with frequency, from 0 to .4. It will increase with frequency because of larger actuator accelerations at higher frequencies. Table 14 and Fig. 37 show simulated dimensionless spring rate plotted against actual dimensionless spring rate.

Figure 37 -Dimensionless Spring Rate w/ Simulated Deadband
The simulated dimensionless data trends downward, as does the actual dimensionless spring rate. While this looks promising, the simulated data is not predictable enough to say for certain that this is a trend or poor choice of modeling parameters. A drawback of using tie constraints to simulate the Newtonian fluid noslip condition at media boundaries is that the internal fluid nodes can still drastically deform. There is no interaction or constraint defined for the internal fluid elements with respect to the metallic cavity surface. These elements are then allowed to deform beyond the tie constraints, locating some of their nodes "outside" the fluid cavity. This creates instabilities in the simulation. Adaptive meshing, a finer mesh, or a different element choice may fix these instabilities and give more uniform data.
For comparison, the acoustic medium fluid model was used to check the effects of simulated deadband on deflection. The interior nodes not subject to tie constraints of the cavity walls are subject to hydrostatic behavior only, and will not see the same excessive deformations due to shear forces that develop in the equation of state material model. Deadband was increased linearly with frequency, up to a maximum value of .4 of the initial load. The results are given in Table 15 and Figure   38. The simulation results for the Size 35 and 100 do show a more pronounced downward trend with this material model.  Large bore equipment snubbers are more heavily scrutinized during the design stage of the supported equipment. Their application is thoroughly studied and environmental variables are specified with a higher degree of certainty. On the manufacturer's side, large bore snubbers are quite often made for a specific application, so if the production unit is carefully studied all the dimensional parameters can be known. Fluid bulk modulus can still be problematic, as the fluid is usually replaced several times over a snubber's life cycle, but can be mitigated somewhat through quality control testing of each batch.
In conclusion, the FEA model showed only a 6% deviation from the static spring rate due to inertia of the piston assembly inside the cylinder. This is in contrast to the 30% deviation found in the experimental test values. However, the load control nature of the dynamic test understates the actual load seen by the snubber, decreasing the spring rate. It is also theorized that while the input of the hydraulic pumps into the actuating cylinder is sinusoidal, the actual waveform seen by the snubber is a combination of a step-function and sinusoid. This is a consequence of dead band, and as the initial load seen by the snubber during the loading intervals of each cycle increases, so too does deflection.
Further testing and study is required to say with certainty that an analytical method has been found for determining snubber spring rate with no testing.
Specifically, 1) Conduct a similar analysis for the rest of the Fig. 3306 snubber product line. These cylinders are of a single piston rod configuration and have the same set of qualification test data available. There are 5 sizes of snubbers, which is a larger sample size than the scope of this study.
2) Performing further dynamic testing on the Size 35 and 100 will be cost prohibitive. If, after an analysis of the smaller sizes of Fig. 3306, the same result is shown, dynamic testing should be conducted on a smaller size.
This time, instead of measuring load through a load cell in the load path, pressure transducers should be placed in the tension and compression fill ports. This way, the load seen by the snubber can be exactly measured independent of lost motion or inertia effects. Valuable insight will also be gained into when exactly the check valves close each cycle, and what the working pressure of the snubber truly is.
3) Conduct dynamic testing at loads less than rated load on smaller sizes. In the simulations, applying the same simulated deadband to both sizes will not yield the same result. Testing at loads less than rated load could be conducted to find the correct input waveform for the simulations at those loads. It is possible that a trend can be found that can be found that can be applied to the full rated load of the snubber. For example, if deadband is accurately modeled against experimental tests conducted at 4% to 20% of rated load, perhaps the result can be applied to a simulation at 100% load.

4)
Refine the finite element model to remove instabilities due to simulated dead band. The relationship between dead band and deflection of the snubber under an impact load can be better explored, and applied to the study of the smaller sizes.
Load controlled dynamic testing of hydraulic snubbers has always recorded the values of load with a load cell in the load path of a hydraulic actuator to the snubber, and has always been subject to the same limitations as discussed above. It may be that load controlled testing in this fashion does indeed show some frequency dependence that is overstated, and measuring loads through pressure transducers could show different results. Were this to happen, it would not invalidate a piping analysis done using the Anvil supplied stiffness as a minimum value as intended. In fact, if it could be demonstrated that frequency dependent effects are smaller than earlier reported,