Modeling of Elastomeric Engine Isolators for Prediction of Aircraft Cabin Noise

I. Hoenigsberg Lord Corporation

Abstract

The paper describes a unique dynamic characterization modeling process of Lord Corporation’s isolation system for an executive jet aircraft. It was performed as part of a Joint Plan of three companies to build a global acoustic model of the aircraft, engines and engine isolators, which would allow for prediction of the cabin noise. The objective of the isolator modeling process was to calculate the dynamic stiffness matrices of the three isolators, which are made of metal and rubber materials, as functions of frequency for the working point conditions. The Modal analyses of the metal components and rubber packs showed that their dynamics would not affect the dynamic behaviors of the isolators in the frequency range of interest. Isolator Model C, which is a hybrid model, had an analytical part and an experimental part. The analytical part was based on nonlinear hyperelastic FE models of the rubber packs. The outputs of these models were six stiffnesses per rubber pack for the working point static loads. These stiffnesses served as input to model B, which is a spring FE model that allowed for the calculation of the full 6X6 isolator static stiffness matrices. The experimental part of Model C was based on dynamic shear tests of rubber samples, which addressed the effects of the working point temperature, static and dynamic strains, as well as, frequency on the rubber properties. The test results were used to scale the six stiffnesses per rubber pack, which were obtained from the hyperelastic models. The scaled stiffnesses served as input to Model B, which was used as an engine to generate the dynamic stiffness matrices of the isolators as functions of frequency for the working point conditions.

Introduction The dynamic characterization was performed as part of a Joint Plan between an aircraft manufacturer, an engine manufacturer and Lord Corporation, which develops the elastomeric isolators that connect the engine to the aircraft. The purpose of the Joint Plan was to build an acoustic model of the entire system: aircraft, engine and engine isolators, which would allow for prediction of the cabin noise. Each one of these sub-systems was represented by a super element.

The Interface Between the Aircraft and the Engine Each engine is connected to the aircraft by three elastomeric isolators (see Figure 1): two front isolators – a top one and a bottom one - and one aft isolator. The yoke, which is part of the aircraft super element, connects the aircraft pylon to the forward isolators (See Figure 2)

Figure 1. The Interfaces Between the Aircraft and the Engines

Figure 2. The Schematic Interfaces Between the Aircraft and the Engines

The Mechanical Design of the Elastomeric Isolators

The Forward Isolator Figure 3 shows an isometric view of the forward isolator, where a quarter has been removed for visualization purposes. The metallic housing is connected to the engine using four through bolts. The base plate acts as a shear boss, transferring shear loads from the engine to the isolator. The metallic inner member is connected to the airframe. Between the housing and the inner member, there is a pair of radial

rubber packs. Each rubber pack is made of rubber layers with thin metallic shims in between. The rubber geometry shown will be referred to later on as the “in-housing geometry”. The radial rubber packs are manufactured in a molding and adhesion process, where the inner member is part of the mold. The out-of-the-mold rubber thickness is higher than the in-housing one. Therefore, the installation of the rubber packs within the housing requires pre-compression. The out-of-the-mold rubber geometry will be referred later on as the “as-bonded geometry”. The two axial rubber packs carry the thrust and the reverse thrust. The sleeve bolt holds them together. An attachment bolt, which runs through the sleeve bolt, connects the inner member to the airframe’s yoke (shown schematically on Figure 2).

Figure 3. The Forward Isolator

The Aft Isolator Figure 4 is an isometric view of the aft isolator. The housing and the inner member are connected to the engine and the airframe, respectively. Two pairs of pre-compressed rubber packs are located between the housing and the inner member: a radial pair and a tangential one.

Figure 4. The Aft Isolator

The Modeling Process The objective of the modeling process was to provide the dynamic stiffness matrices of the three isolators over a 0 – 300 Hz frequency range for the working point conditions and along 6 DOF.

Figure 5 provides an overview of the elastomeric isolators modeling process. One of the major challenges of this process was that the stiffness and the damping of the elastomeric isolators are affected by the working point conditions: the static loads applied to them (static strains) and even more so than that, the dynamic displacement amplitudes applied to the isolators (dynamic strains), the frequency (loading rate) as well as the temperature.

Modal analyses of the metal components and rubber packs showed that their dynamics would not affect the dynamic behavior of the isolators within the frequency range of interest.

The results of these modal analyses allowed for focusing on the rubber packs. Model B is a simple FE spring model. Each rubber pack is represented in this model by six springs. The output of this model is a full 6 X 6 stiffness matrix of the isolator. This model was also used to demonstrate that the off-diagonal terms would not significantly contribute to the coupling in the isolator systems.

Model A is environmental dynamic tests of the full isolators. It was used for a preliminary prediction of the isolator dynamic behavior in six DOF. It was also used for dynamic validation of the modeling process.

Model C is a hybrid model. It is named ‘hybrid’ because it has an analytical part and an experimental part. The analytical part of it is based on ANSYS 3-D nonlinear hyperelastic FE models of the rubber packs. It allowed for the calculations of six stiffnesses per rubber pack for a given working point static loads. These stiffnesses were used then as input to model B. The output was the static stiffness matrices of the isolators for the working point static loads. The experimental part of the hybrid model was based on rubber sample shear tests over the frequency range and at the static and dynamic strains, as well as, temperature of the working point. The test results were used to scale the rubber pack stiffnesses for each frequency. The scaled

stiffnesses were used as input to Model B. Model B was used as an engine of a frequency loop, which allowed for the computation of the dynamic stiffness matrices of the isolators (see subsection The Computation Process for additional details regarding the computation process of the dynamic stiffness matrices).

Figure 5. The Modeling Process

Modal Analysis

Modal Analyses of the Metal Components The metal components are the housings and inner members of the forward and aft isolators.

The FE Models The FE models of the metal components are shown on Figure 6.

The four models were run with free-free boundary conditions. The housing models were also run with boundary conditions, which simulate their engine interfaces. The first several natural frequencies and mode shapes of the housings were insensitive to the boundary conditions.

Figure 6. The FE Models of the Metal Components

Mode Shapes and Natural Frequencies The lowest first natural frequency of the metal components is the one of the aft housing (see Figure7). The mode shape associated with it is axial bending. This frequency (908 Hz) is more than 200% higher than the upper end of the frequency range of interest (300 Hz). Therefore, the dynamics of the metal components would not affect the dynamic behavior of the isolators within the frequency range of interest.

Figure 7. The Mode Shapes of the Metal Components

Modal Analyses of the Rubber Packs

The FE Models The finite element models of the forward and aft rubber packs are shown on Figure 8 and on Figure 10, respectively. The boundary conditions, which were applied in these models, are shown on Figure 9 and on Figure 11, respectively.

Since modal analysis is linear analysis by definition, the rubber layers were represented by elastic elements (SOLID45), where Young modulus was taken as three times the shear modulus and Poisson’s ratio was taken as 0.4995.

The geometries of the rubber packs in these models are the in-housing ones. That means that the pre-compression effect is not taken into account in the modal analyses. This is also on the conservative side, since the pre-compression increases the stiffnesses of the rubber packs.

The sleeve bolt, which holds the two axial rubber packs together, was modeled using PIPE16 elements (see Figure 8). Constraint equations were used to rigidly connect the rubber packs to the inner member center (see Figure 9). The degrees of freedom of the inner member were coupled with the ones of the sleeve bolt center with the exception of the axial translation and the rotation about the axial axis. The inner member center was grounded in the modal analysis, since it is connected to the aircraft. Constraint equations were also used to connect the sleeve bolt to the end plates of the axial packs. The interfaces of the rubber packs and the housings were grounded as well.

Constraint equations were used to rigidly connect the aft rubber packs to the aft isolator inner member center (See Figure 11). The inner member center was grounded in the modal analysis, since it is connected to the aircraft. The interfaces of the rubber packs and the housings were grounded as well.

Figure 8. The FE Model of the Forward Rubber Packs

Figure 9. The Boundary Conditions of the Forward Rubber Pack FE Model

Figure 10. The FE Model of the Aft Rubber Packs

Figure 11. The Boundary Conditions of the Aft Rubber Pack FE Model

Mode Shapes and Natural Frequencies The first two mode shapes of the forward and aft rubber packs are shown on Figure12.

The first two mode shapes of the forward rubber packs are associated with the Radial In (RI) pack, whereas the first two mode shape of the aft packs are associated with the Radial Out (RO) pack. For both the forward and the aft rubber packs, the first mode is torsion one and the second mode is axial shear one. The lowest frequency (506 Hz) is the first natural frequency of the forward packs. It is higher than 150% of the frequency range of interest (0-300 Hz). Therefore, the dynamics of the rubber packs would not affect the dynamic behavior of the isolators within the frequency range of interest.

Figure 12. The Mode Shapes of the Rubber Packs

Model B The Spring Model The spring model of the aft rubber packs will be used in this section. The spring model of the forward rubber packs is similar.

The Model and the Input Model B of the aft isolator and its input are shown on Figure 13.

Figure 13. Model B of the Aft Isolator

Coordinate System and Schematic Diagram The coordinate system origin is located at the geometric center of the inner member (see Figure 13). X is the axial axis, Y is the radial axis and Z is the tangential one.

The schematic diagram on the bottom left of Figure 13 shows a view of the aft isolator as seen from the axial direction. The green part represents the housing and the yellow one represents the inner member. Between the housing and the inner member there are two pairs of rubber packs. The radial pair consists of a Radial In (RI) and a Radial Out (RO) rubber packs, while the tangential pair consists of a Tangential Top (TT) and a Tangential Bottom (TB) rubber packs. The solid lines within the radial packs represent the metallic shims.

The FE Model The red points in the schematic diagrams of Figure 13 represent the elastic centers of the rubber packs. Model B is actually a simple FE spring model, which allows for the calculation of the isolator static stiffness matrix. It has two coincident nodes at each one of the rubber pack elastic centers (see diagram on the right hand side of Figure 13). Each pair of coincident nodes is connected by six springs with the stiffnesses, which are specified within the input tables above the schematic diagrams on this figure. One set of nodes (one node from each one of the elastic center node pairs), which represents the housing, was grounded. Another set of nodes, which represents the inner member, was rigidly connected to a master node at the origin. Running the model for six unit loads allowed for the calculation of the full static stiffness matrix of the isolators for these example inputs.

Example Input The table, which is shown on the top of Figure 13 shows example input for the aft isolator model. Each line represents a rubber pack. There are four rubber packs: Radial Out (RO), Radial In (RI), Tangential Top (TT) and Tangential Bottom (TB). Each column represents stiffness. There are six stiffnesses: three translation stiffnesses and three rotational ones. The three translation stiffnesses are named Kx, Ky and Kz along the axial (X), radial (Y) and tangential (Z) axes, respectively. The three rotation stiffnesses are named Kxx, Kyy and Kzz around the axial (X-X), radial (Y-Y) and tangential (Z-Z) axes, respectively. These stiffnesses were obtained from closed-form solutions based on Lord Corporation’s design handbook (reference 1) for design loads.

Example Output – Example Static Stiffness Matrix An example static stiffness matrix of the aft isolator, which was calculated for the example input (see Example Input above), is shown on Figure 14.

Figure 14. An Example Static Stiffness Matrix of the Aft Isolator

This 6 X 6 matrix relates the generalized displacement vector to the generalized force vector. The generalized displacement vector contains three displacements and three rotations. The generalized force vector contains three forces and three moments.

Observations 1. The matrix is symmetric. This is expected since the model does not contain any dissipation

elements. The damping will be covered later on with the experimental part of Model C.

2. The matrix can be broken to four 3X3 sub-matrices. The top left quarter of the matrix is the translation sub-matrix and the bottom right quarter is the rotation one. Both sub-matrices are diagonal ones. It means that an application of a displacement along one axis would not generate a force along another axis. Similarly, an application of rotation around one axis would not create a moment around another axis.

3. There are some small off-diagonal terms, which represent coupling. The top right quarter is the Force-Rotation coupling sub-matrix and the Bottom left quarter is the Moment-Displacement coupling one. See the Checks sub-section below for a better understanding of both the diagonal and the off-diagonal terms, while checking the output of Model B.

Checks 1. The three translation diagonal terms (yellow highlighted in the top right matrix of Figure 15) are

the isolator stiffnesses along the axial, radial and tangential axes. Summing up the rubber pack stiffnesses along each axis ends up with the isolator stiffness along that axis (see the top left input table in this figure). Therefore, it is a good sanity check of model B.

2. One may assume that summing up the rotational stiffnesses will end up with the isolator rotation stiffnesses. When we check that around the radial axis (Y), for example, we end up with different numbers (6,385 in-lb/rad from summation of the input table terms vs. 14,902 in-lb/rad calculated

by Model B). When a rotation is applied around the radial axis (ROTy), the radial packs react in torsion and the tangential ones react in cocking (see the bottom left schematic diagram in Figure 15). These are the rotational stiffnesses listed in the input table. However, the tangential packs also react the rotation (ROTy) in shear along the axial axis, which contributes to the rotation stiffness of the isolator around the radial axis. The calculation of the isolator rotation stiffness around the radial axis is shown in the green-bordered box of Figure 15: it starts by summing up the torsion and cocking stiffnesses of the rubber packs (6,385 in-lb/rad). Then, the axial stiffnesses of the tangential packs, multiplied by the square of the distances from their elastic centers to the axis of rotation, are added ((862.2+850.5) X 2.232 = 8,517 in-lb/rad). This calculation ends up with the value calculated by Model B (6,385+8,517 = 14,902 in-lb/rad).

3. The ratio between the actual rotational stiffness of the isolator around the radial axis and the superposition of the rubber pack rotational stiffnesses is 2.33 (see bottom-right purple-bordered boxes on Figure 15). This ratio is relatively high. The reason is the rectangular geometry of the aft isolator and the relatively large distance from the elastic center of each tangential pack to the axis of rotation (2.23”). The second power of this distance affects the translation contribution of these rubber packs to this isolator rotational stiffness. The ratios along the axial and tangential axes are around 1.15, which is more typical.

4. One additional check to model B is running it with very small rubber pack translation stiffnesses. The ratios of the calculated isolator rotation stiffnesses to the sums of the rubber pack rotation stiffnesses are 1.

5. The red-circled off-diagonal term in the top right matrix of Figure 16 couples a rotation around the axial axis (ROTx) with a reaction along the tangential axis (Fz). When a rotation around the axial axis is applied, all the packs react in cocking (see the bottom left schematic diagram in Figure 16). But, in addition to that they also react in shear. The radial packs shear along the tangential axis. The only reason why this off diagonal term is different from zero is that the tangential shear stiffnesses of the two radial packs differ from each other. The calculation of this off-diagonal term is shown in the red-bordered box of Figure 16: it takes the difference of these two stiffnesses (853.3 and 758.4 lb/in), where each one of them is multiplied by the distance from the elastic center of the pack to the axis of rotation (0.9020” and 0.8965”, respectively). The calculation ends up with the isolator off-diagonal term predicted by model B (853.3 X 0.9020 – 758.4 X 0.8965 = 89.8 lb/rad).

Figure 15. Diagonal Term Check of Model B

Figure 16. Off Diagonal Term Check of Model B

Model C The Hyperelastic Model and Working Point Static Stiffness Matrices As a hybrid model, Model C has an analytical part and an experimental part. The analytical part of Model C is based on nonlinear hyperelastic FE models of the rubber packs, which will be presented in this section. The experimental part of Model C is presented in subsection The Experimental Part of Model C below.

The Hyperelastic FE Model of the Rubber Packs The hyperelastic FE model of the forward rubber packs will be used in this subsection since it allows for comparison of the forward top isolator rubber pack stiffnesses to these of the forward bottom isolator. Since the two forward isolators are subjected to different static loads, using the forward isolator in this section allows for understanding of the static load effects on the rubber pack stiffnesses. The hyperelastic FE model of the aft rubber packs is similar.

The FE Model The hyperelastic finite element model of the forward rubber packs is shown on Figure 17. This model is based on the modal analysis one (see subsection Modal Analyses of the Rubber Packs) with several important differences as follows:

a) The rubber layers were simulated by hyperelastic elements HYPER58 (as opposed to elastic SOLID45elements in the modal analysis models). The material model used was a neo-Hookean model.

b) The initial geometries of the rubber packs were the as-bonded ones as opposed to the in-housing geometries, which were used in the modal analysis models (for explanations of “as-bonded geometry” and “in-housing geometry” please refer to subsection The Mechanical Design of the Elastomeric Isolators). That means that the pre-compression effect was accounted for in the hyperelastic analyses.

c) A non-linear pre-tension element was added in the sleeve bolt element in order to apply the pre-tension force.

The boundary conditions of this model consist of pre-compression, which was applied inwards on the radial packs and outwards on the axial ones, and also of a pre-tension force, which was applied to the sleeve bolt (see Figure 18).

Figure 17. The Hyperelastic FE Model of the Forward Rubber Packs

Figure 18. Boundary Conditions in the Forward Rubber Pack Hyperelastic FE Model

The Hyperelastic Analyses The hyperelastic analyses consisted of two sets of nonlinear computations: for the forward top isolator and for the forward bottom one.

Each one of these nonlinear computation sets consisted of seven runs: a run for the working point loads plus six runs for additional incremental unit load runs. The additional incremental unit load runs were required in order to calculate the six tangent stiffnesses per rubber pack for the working point loads.

Each working point run consisted of two load steps. The first load step was used for the application of the pre-compression, which served as boundary conditions for the second load step. The second load step was used for the application of the working point loads.

The Working Point Static Stiffness Matrices

The Working Point Definition The working point was defined as a set of axial, radial and tangential static loads and dynamic displacement amplitude acting over a 0 - 300 Hz frequency range and at a low temperature.

The Working Point Loads Figure 19 shows the static loads that the aircraft applies on the forward top and the forward bottom isolators at the working point conditions. The bottom isolator carries most of the thrust (about 1,100 lb.) and also about 1,900 lb. of radial force acting inwards (with respect to the engine). The main load on the top isolator is about 1,500 lb. of radial force acting outwards (with respect to the engine).

Figure 19. The Working Point Loads of the Forward Isolators

The Rubber Pack Stiffnesses Figure 20 shows the room temperature rubber pack stiffnesses of the forward top and forward bottom isolators at the working point static loads specified in subsection The Working Point Loads above. These stiffnesses are for room temperature. The experimental part of Model C addresses the effect of the working point low temperature (see section The Modeling Process above and subsection The Experimental Part of Model C below). There are some significant differences in the stiffnesses of the top isolator rubber packs compared to the bottom one. For example, the axial stiffnesses of the axial packs of the top isolator (33,212 lb/in vs. 21,317 lb/in) are closer to each other than the ones for the bottom isolator (39,138 lb/in vs. 17,607 lb/in). The reason for that is that the bottom isolator carries most of the thrust load (see subsection The Working Point Loads above). However, the total axial stiffness of the bottom isolator (57,191 lb/in) is less

than 4% higher than that of the top isolator (55,148 lb/in). The maximum isolator stiffness difference (less than 15%) is along the radial axis.

Figure 20. The RT Rubber Pack Stiffnesses of the Forward Isolators for the Working Point Loads

The Static Stiffness Matrices The room temperature static stiffness matrices of the two forward isolators for the working point loads are shown on Figure 21. The two matrices are similar. The significance of the red circled off diagonal term compared to the corresponding green-circled diagonal one is discussed in subsection Low Temperature and High Frequency Effects on Off Diagonal Terms below.

Figure 21. The RT Static Stiffness Matrices of the Forward Isolators for the Working Point Loads

The Dynamic Stiffness Matrices

The Experimental Part of Model C The working point static loads translate to different static strains in each one of the four rubber packs of each isolator. These static strains somewhat affect the dynamic stiffnesses of the isolators.

The working point dynamic displacement amplitudes, which are caused due to imbalance of the engines, translate to different dynamic strains in each one of the rubber packs. Even more so than the static strains, the dynamic strains as well as the frequency (loading rate) and temperature affect the dynamic stiffnesses of the isolator.

In order to account for these effects, shear tests of rubber samples were performed (see Figure 22). These tests, which are the experimental part of Model C, were performed at various temperatures for various static and dynamic strains and for a frequency sweep of 0-300 Hz. The test results were reduced to scale up factors for each rubber pack, which are monotonic ascending functions of frequency. The stiffer the rubber material the more sensitive it is to frequency and to the working point conditions (temperature and strains).

Figure 22. Rubber Shear Test Specimen

The Computation Process A flowchart of the computation process for the isolator dynamic stiffness matrices is shown on Figure 23. The yellow background parallelogram in this flowchart is the output of Model B, which is the 6 X 6 static stiffness matrix of the isolator for the working point static loads (see section Model B The Spring Model). For each frequency, the rubber material test results (see subsection The Experimental Part of Model C) were used to scale the rubber pack stiffnesses (see green background box in the flowchart). The test results were picked according to static and dynamic strains calculated by the analytical portion of model C for the working point conditions. The scaled stiffnesses were used then as input to Model B. Model B served as an engine, which generated a dynamic stiffness matrix for each frequency. This loop was repeated 300 times per isolator for each frequency (see the loop within the blue dashed box in the flowchart). The output for each isolator was a 3-D complex array. The real portion of this array is the stiffness matrix and the imaginary portion is the hysteretic damping one. The dimension of each portion is 6 X 6 X 300 Hz.

Figure 23. Flowchart of the Dynamic Stiffness Matrix Computation Process

Observations and Checks

Low Temperature and High Frequency Effects on Diagonal Terms a) The top matrix in Figure 24 is the room temperature (RT) static stiffness matrix of the forward top

isolator for the working point static loads, whereas the bottom matrix is the low temp 300 Hz stiffness matrix of this isolator for the working point conditions. As can be seen in this Figure (magenta colored terms) the low temp 300 Hz to RT Static ratio of the axial stiffness is 4.7. The radial stiffness ratio is 1.9 (blue colored terms) and the tangential stiffness ratio is 4.2 (green colored terms). The low temp 300 Hz to RT static ratios are more than 4.0 for both the axial and tangential axes versus less than 2.0 for the radial one. Therefore, the low temperature and high frequency affect the stiffnesses along different axes in different fashion.

b) The top table in Figure 25 shows the RT static stiffnesses of the forward top isolator rubber packs along the three axes. The isolator stiffness along each direction is obtained by summing the stiffnesses of the rubber packs up. The yellow highlighted total stiffnesses match the translation diagonal terms in the top matrix of Figure 24. This is another good sanity check of the modeling process. It can also be seen from this table that the contribution of the axial packs (orange background rows) to the isolator axial stiffness is 99%. Also, the axial packs are responsible for 81% of the isolator tangential stiffness. On the other hand side, the radial packs (light blue background rows) contribute 89% of the isolator radial stiffness. Therefore, the axial packs are dominant along both the axial and the tangential axes, while the radial packs are dominant along the radial axis.

c) The bottom table of Figure 25 lists the low temp 300 Hz to RT static stiffness scale factors for the four packs, which were obtained from result reduction of Model C’s experimental part (see subsection The Experimental Part of Model C). A simple average of the axial packs is 4.3. It is not accurate to calculate a simple average because the scale-up factors are multiplied by different stiffnesses, but it gives a good qualitative indication (see quantitative check below). This average is larger than 4.0, which matches the axial and tangential ratios in Figure 24. The simple average

of the radial pack ratios is 1.5, which is lower than 2.0. That also agrees with the tangential ratio in Figure 24.

d) The table in Figure 26 shows the RT static stiffnesses of each pack along each axis side-by-side with the low temp 300-Hz to RT static experimental scale factors. The isolator low temp 300-Hz stiffnesses are obtained from a summation of the scaled-up RT static stiffnesses along each one of the axes. The totals match the translation diagonal terms in the bottom matrix of Figure 24.

Figure 24. Static RT and 300Hz Low Temp Stiffness Matrices of Forward Top Isolator

Figure 25. Low Temperature and High Frequency Effects on Diagonal Terms

Figure 26. Check Diagonal Terms of Forward Top Isolator at Low Temp and 300 Hz

Low Temperature and High Frequency Effects on Off Diagonal Terms a) The matrices in Figure 27 are identical to the ones in Figure 24. For the RT static case, the ratio

between the red-circled off-diagonal term, which is located at the 3-5 place of the matrix, to the corresponding green-circled diagonal term, which is located at the 3-3 place of the matrix, is roughly 0.4. The ratio between these terms at low temp 300 Hz is almost 1.0. Therefore, the low temperature and high frequency affect the contribution of this off-diagonal term compared to the corresponding diagonal one.

b) This off-diagonal term couples a rotation around the radial axis (ROTy) with a reaction along the tangential one (Fz). Looking at the top right diagram in Figure 28, it can be seen that when a rotation is applied around the radial axis (ROTy), the axial rubber packs respond not just by coking, but also by shearing along the tangential axis. The reason that this off-diagonal term is different than zero is that the axial packs have different stiffnesses along the tangential axis. For the RT static case, the off-diagonal term is obtained from the difference between these two stiffnesses multiplied by the distance from their shear centers to the axis of rotation ((1,631.7 – 2,890.6) X 1.56 = -1,963.9 lb/in as can be seen in the top red-bordered box of this figure). This value is identical to the red-circled value in the top matrix of Figure 27.

c) For the low temp 300 Hz case, the static stiffnesses are first multiplied by the experimental scale-up factors (see bottom left table in Figure 28). Then, the difference between the scaled-up tangential stiffnesses of the axial packs is multiplied by the distance to the axis of rotation ((3,672.3 – 18,239.7) X 1.56 = -22,725.2 lb/in as can be seen in the bottom red-bordered box of this figure). This value is identical to the red-circled value in the bottom matrix of Figure 27.

d) The low temp 300 Hz off-diagonal term is larger than the RT static one since the Axial Forward (AF) pack scale-up factor (6.31) is almost three times higher than the Axial Aft (AA) pack one (2.25). The reason for that is that the AF pack, which carries most of the thrust, is made of a stiffer rubber material than the one used for the AA pack. A stiffer rubber material is more sensitive to low frequency and high temperature.

e) However, just comparing an off-diagonal term with units of pound per radian (-22,725.2 lb/rad) to a diagonal term (23,555.4 lb/in) with different units of pound per inch might be somewhat misleading. If we take, for example, a tangential displacement of one thousandth of an inch, the contribution of the diagonal term is 23.6 pounds (23,556 X 0.001 = 23.6 lb. see left table on Figure 29). Due to the about 3 inch distance between the axial packs, the one thousands of an inch displacement translates to a rotation of a third of a thousand of a radian. The contribution of the off-diagonal term is therefore 7.6 pounds (22,725 X 0.001/3 = 7.6 lb. see right table on Figure 29). Therefore, the off-diagonal term contribution is about 32% of the diagonal term contribution (7.6 / 23.6 = 32%) and about 24% of the total response (7.6 / (23.6+7.6) = 24%). Although 24% is not negligible, it is still much smaller than the close-to-1.0 ratio may suggest.

Figure 27. Low Temperature and High Frequency Effects on Off Diagonal Terms

Figure 28. Check Off Diagonal Terms of Forward Top Isolator at Low Temp and 300 Hz

Figure 29. Significance of Dynamic Off Diagonal Terms

Conclusion 1. A unique modeling process, which is based on combination of analysis and experiments, was

developed for the computation of elastomeric isolator dynamic stiffness matrices as function of frequency for given conditions (static and dynamic strains as well as temperature).

2. The modal analysis showed that the dynamics of the metal components and the rubber packs would not affect the dynamic behavior of the isolators within the frequency range of interest.

3. The stiffer the rubber material is, the more sensitive it is to low temperature and high frequency effects.

4. The low temperature and high frequency have different quantitative effects on the dynamic stiffnesses along different axes.

5. The off diagonal term contributions are affected by low temperature and high frequency, but are still relatively small.

6. The significance of the off-diagonal terms for the cabin noise will be evaluated using the global acoustic model of the aircraft, engine and engine isolators.

References 1) Lord Corporation Design Handbook, Volume I, November 24, 1971.