Matlabmodel 2

Chapter

2

Matlab model

2.4 Box attached at one point to the engine . . . 11

2.4.1 Behaviour of the box at different connections point and using different mounts . . . 12

2.4.2 Box attached using hard mount at soft point to the engine 19 2.4.3 Box attached using soft mount at soft point to the engine 22 2.4.4 Box attached using hard mount at hard point to the engine 25 2.4.5 Box attached using soft mount at hard point to the engine 27 2.5 Box attached at two points to the engine . . . 29

2.6 Box attached at four points to the engine with displace-ment control . . . 37

2.7 Check with Ansys model . . . 39

2.8 Discussion . . . 39

2.8.1 Box attached at one point to the engine . . . 40

2.8.2 Box attached at two points to the engine . . . 41

2.8.3 Box attached at four points to the engine with displacement control . . . 42

2.1

Introduction

The chapter compares the dynamic behaviour of a box, a general accessory, in two different configurations. A real configuration, when the box is attached at an aircraft engine, and a test room configuration, when the box is installed on a shaker

as a different input. Different ways of mounting the box on the engine and different ways of testing it are analysed here. The Matlab scripts used in this chapter are reported in Appendix A.1 and Appendix A.2.

2.2

Assumption

The study was carried out starting from a simple model subjected to the following hypothesis:

• masses, stiffness are concentrated • displacement only in one direction • the body rotation is neglected • structural damping, 2 % • harmonic forces are in phase Therefore

• the mass matrix for the system is diagonal

• the stiffness matrix for the single element, shown in Figure 2.1, is:  Ki  = " ki −ki −k_i k_i #

Figure 2.1: Single element

The stiffness matrix for the complete system is obtained using the connective matrix method. The detailed form of the case is shown in Table 2.1. The range of the

Table 2.1: Connective matrix 1st DOF 2nd DOF N◦ _{of the element}

frequencies for the engine is 0 ÷ 1000[Hz]. However attention focuses on a limited working range of frequencies when the displacements are considerable, that happened

2.3

Engine

The parameters for the engine were selected according to the dynamic behaviour of the whole system, engine plus box, and must refer to the behaviour of the engine. This means when the box is added to the engine there are no changes in the dynamic behaviour of the engine, with the exception of the small area where the box is fixed. The parameters set affected the engine in terms of decreasing the stiffness from the constraint up to the opposite side and of decreasing the mass from the heart of the engine up to surface. This enable each DOF of the engine to have different behaviour as in a real engine. 15DOF s were chosen for the engine, one of these was constrained, 0 ÷ 1000[Hz] for the frequency range and 1000 [kg] for the total mass. The disposition is shown in Figure 2.2. The forces are shown in Table 2.2.

Figure 2.2: The engine

The choice of the mass and stiffness parameters were chosen according to the main number of natural frequencies, these were ideally between 0 ÷ 200[Hz]. The values of mass and stiffness adopted for the engine are reported in Table 2.3. The connective matrix is shown in Table 2.4 and the natural frequencies result from the modal analysis are reported in Table 2.5.

Table 2.2: Forces

Force N◦ _{of DOF Value [N ]}

f5 5 1

f6 6 2

f7 7 3

Figure 2.3: Displacement of the mass 15 from the engine

2.4

Box attached at one point to the engine

The behaviour of the box when it is attached at one point to the engine, called real configuration Figure 2.4, is compared to the behaviour of the box when it is attached at one point to the shaker, test room configuration Figure 2.5. When the box is attached to the shaker, this is subjected to the vibration of the mounting point of the engine without the box. 5[kg] was chosen as the total mass for the box and 2DOF s which one of these was constrained on the engine. The disposition of the DOFs is shown in Figure 2.6. In this case, there are two possibilities for stiffness parameters: if the box is supported on a bracket and there is a ’soft’ anti-vibration mount between the metal bracket and the box, the frequency range is 20 ÷ 30[Hz]. Otherwise if the box is on the bracket with no anti-vibration mount, the range is 100 ÷ 150[Hz]. The mass and the stiffness parameters for the box, depending on the bracket with the anti-vibration or without the anti-vibration mounts, are reported in Table 2.6. Table 2.7 shows the connective matrix for the box constrained on the

Figure 2.4: Real configuration, box attached at one point to the engine

Figure 2.5: Test room, box attached at one point to the engine

the DOF of the frame with the DOF of the engine where the box is fixed, it possible to obtain the connective matrix for the whole system, engine plus box.

The different dynamic behaviour of the box in real and test configurations depends on the different mounting points on the engine and on the type of mount. For this box, the displacement of mass 17, called zEB, for the real configuration, box on

the engine, and zB for the test configuration were measured. The vibration of the

connection point on the engine without box, is called xE.

2.4.1 Behaviour of the box at different connections point and using different mounts

The box was fixed using hard and soft mounts to all DOFs of the engine, one by one, the results are shown in Figure 2.7, Figure 2.8 and in Figure 2.9. The response for the first DOFs are not shown because there were no differences between real and test configurations. The different behaviour of the box increased from the stiff part to the soft part of the engine and from the soft to the hard mounts.

Figure 2.6: The box attached at one point

(a) Hard mount 6th DOF of the engine (b) Hard mount 7th DOF of the engine

(c) Hard mount 8th DOF of the engine (d) Hard mount 9th DOF of the engine

Figure 2.7: Behaviour of the box fixed using hard mount at different point to the engine, from 6th DOF up to 9th DOF

Table 2.3: Parameters of the engine N◦ _{of DOF Mass Stiffness N}◦ _{of element}

[kg] [N/m] 1 0 7 · 108 I 2 90 5 · 108 II 3 80 6 · 108 III 4 60 5 · 108 IV 5 130 2 · 108 V 6 140 6 · 108 VI 7 150 1 · 108 VII 8 110 4 · 108 VIII 9 90 8 · 107 IX 10 70 5 · 107 X 11 30 9 · 107 XI 12 50 5 · 106 XII 13 40 1 · 106 XIII 14 5 2 · 106 XIV 15 5 4 · 105 XV 5 · 105 XVI 7 · 105 XVII 6 · 105 XVIII

Table 2.4: Connective matrix data for the engine 1st DOF 2nd DOF N◦ _{of the element}

1 2 I 2 3 II 2 4 III 3 5 IV 3 6 V 2 6 VI 4 6 VII 4 7 VIII 5 8 IX 6 9 X 7 10 XI 8 11 XII 9 12 XIII 10 13 XIV 11 14 XV 12 14 XVI 12 15 XVII 13 15 XVIII

Table 2.5: Natural frequencies of the engine Engine [Hz] 25.7 38 59.5 72.3 83.8 99.3 119.7 133.2 205.6 261.3 353.1 608.4 673.3 924.5

Table 2.6: Parameters of the box

N◦ _{Mass [kg] Stiffness [N/m] Stiffness [N/m] N}◦

of DOF no A/V mount A/V mount of element 16 2 5 · 107 2 · 105 XIX 17 3 2 · 106 2 · 106 XX

Table 2.7: Connective matrix data for the box attached at one point 1st DOF 2nd DOF N◦ _{of the element}

Frame 16 XIX 16 17 XX

Table 2.8: Natural frequencies of the box attached at one point Box no A/V mount Box with A/V mount [Hz]

127.4 31.3 811.9 209.2

(a) Hard mount 10th DOF of the engine (b) Hard mount 11 DOF of the engine

(c) Hard mount 12th DOF of the engine (d) Hard mount 13th DOF of the engine

(e) Hard mount 14th DOF of the engine (f) Hard mount 15th DOF of the engine

Figure 2.8: Behaviour of the box fixed using hard mount at different point to the engine, from 10th DOF up to 15th DOF

(a) Soft mount 10th DOF of the engine (b) Soft mount 11th DOF of the engine

(c) Soft mount 12th DOF of the engine (d) Soft mount 13th DOF of the engine

(e) Soft mount 14th DOF of the engine (f) Soft mount 15th DOF of the engine

Figure 2.9: Behaviour of the box fixed using soft mount at different point to the engine, from 10th DOF up to 15th DOF

To sum up, there are four different configurations for the box on the engine: • soft point and hard mount

• soft point and soft mount • hard point and hard mount • hard point and soft mount

each of these is discussed below. The displacement from the engine where the box is fixed, called xEB, for the real configuration and xE for the test configuration can be

also measured and used as comparison between the real and the test configuration. xE

xEB

assumed = 1, zB zEB

should be = 1

2.4.2 Box attached using hard mount at soft point to the engine

Mass 15 was chosen as a soft mounting point on the engine, a stiffness of 5 · 107

N/mm represented the hard mount. For this configuration the natural fre-quencies of the whole system, engine plus box, are shown in the first column of Table 2.9. The second and third columns represent the natural frequencies of the engine and the box respectively. Figure 2.10 compares the response of the box when it is mounted on the shaker subjected to the vibration of the connection point of the engine (test room configuration zB), to the response of the box mounted on the

engine (real configuration zEB). Figure 2.11 shows the response of the connection

point of the engine without the box (test room configuration xE), and with the box

mounted (real configuration xEB). Figure 2.12 compares the response of the box

mounted on the shaker subjected to the vibration of the connection point of the engine with the box mounted (test room configuration zB as input xEB ), to the

Figure 2.10: Output from the box, hard mount and soft point

Table 2.9: Natural frequencies of the system hard mount soft point Engine plus box [Hz] Engine [Hz] Box [Hz]

25 25.7 127.4 37.6 38 811.9 58.3 59.5 60.4 72.3 72.4 83.8 99.3 99.3 119.7 119.7 133.2 133.2 159.3 205.6 205.6 261.3 261.3 353.1 353.1 608.4 608.4 673.3 673.3 924.5 924.5 952.3

2.4.3 Box attached using soft mount at soft point to the engine

Mass 15 was chosen as a soft mounting point on the engine. A soft mount is represented of a stiffness equal 2 · 106

N/mm. In the first, second and thrid columns of Table 2.10 are shown the natural frequencies of the whole system (engine plus box), the engine and the box, respectively. Figure 2.13 compares the response of the box

Table 2.10: Natural frequencies of the system soft mount soft point Engine plus box [Hz] Engine [Hz] Box [Hz]

23.9 25.7 31.3 30.8 38 209.2 38.5 59.5 59.4 72.3 72.3 83.8 89.7 99.3 99.3 119.7 119.7 133.2 133.2 205.6 205.6 261.3 209.3 353.1 261.3 608.4 353.1 673.3 608.4 924.5 673.3 924.5

when it is mounted on the shaker subjected to the vibration of the connection point of the engine (test room configuration zB), to the response of the box mounted on

the engine (real configuration zEB). Figure 2.14 shows the response of the connection

point of the engine without the box (test room configuration xE), and with the box

mounted (real configuration xEB). Figure 2.15 compares the response of the box

mounted on the shaker and subjected to the vibration of the connection point of the engine with the box mounted (test room configuration zB as input xEB ), to the

2.4.4 Box attached using hard mount at hard point to the engine

Mass 6 was chosen as a hard mounting point on the engine. A stiffness of 5 · 107

N/mm represented the hard mount. For this configuration the natural fre-quencies of the whole system, engine plus box, are shown in the first column of Table 2.11. The other two columns represent the natural frequencies of the engine and the box. Figure 2.16 compares the response of the box mounted on the shaker subjected

Table 2.11: Natural frequencies of the system hard mount hard point Engine plus box [Hz] Engine [Hz] Box [Hz]

25.7 25.7 127.4 38 38 811.9 59.5 59.5 72.3 72.3 83.8 83.8 99.1 99.3 119.7 119.7 127.2 133.2 133.3 205.6 205.6 261.3 261.3 353.1 351.8 608.4 606.6 673.3 673.3 924.5 811.9 924.5

to the vibration of the connection point of the engine (test room configuration zB),

to the response of the box when it is mounted on the engine (real configuration zEB).

Figure 2.17 shows the response of the connection point of the engine without the box (test room configuration xE), and with the box mounted (real configuration xEB).

Figure 2.16: Output from the box, hard mount and hard point

2.4.5 Box attached using soft mount at hard point to the engine

Mass 6 was chosen as a hard mounting point on the engine. A soft mount is represented of a stiffness equal 2 · 106

N/mm. In the columns of Table 2.12 are shown the natural frequencies of the whole system (engine plus box), the engine and the box. Figure 2.18 compares the response of the box when it is mounted on the

Table 2.12: Natural frequencies of the system soft mount hard point Engine plus box [Hz] Engine [Hz] Box [Hz]

24.9 25.7 31.3 30.8 38 209.2 59.5 59.5 72.3 72.3 89.7 83.8 99.1 99.3 119.7 119.7 127.2 133.2 133.3 205.6 205.6 261.3 209.3 353.1 261.3 608.4 353.1 673.3 608.4 924.5 673.3 924.5

shaker and subjected to the vibration of the connection point of the engine (test room configuration zB), to the response of the box mounted on the engine (real

configuration zEB). Figure 2.19 shows the response of the connection point of the

engine without the box (test room configuration xE), and with the box mounted

Figure 2.18: Output from the box, soft mount and hard point

2.5

Box attached at two points to the engine

In this section, the box is attached at two points to the engine as presented in Figure 2.20. The parameters for the engine are the same as the parameters for the engine in the previous section, while those for the box are shown in Table 2.13. Table2.14 represents the connective matrix for the whole system. The last four rows correspond to the connective matrix for the box. The natural frequencies of the whole system, engine and box are shown in the first, second and third column of Table 2.15 respectively.

Ideally, the displacements obtained from the engine can be applied to the box by

Figure 2.20: Whole system, box attached at two points

the shaker as shown in Figure 2.21. Figure 2.24 compares the response of the box when it is mounted on the shaker subjected to the vibration of the two connection points of the engine (test room configuration zB), to the response of the box mounted

Table 2.13: Parameters of the box attached at two points N◦ _{of DOF Mass [kg] Stiffness [N/m] N}◦ _{of element}

16 2 5 · 107 19 17 3 5 · 106 20 18 4 1 · 106 21 2 · 106 22

of the two connection points of the engine without the box (test room configuration xE and yE ), and with the box mounted (real configuration xEB and yEB). When

the two outputs from the engine with the box mounted (xEB and yEB) are applied

to the shaker, the test configuration has the same response as the real configuration (see Figure 2.23). In fact the two outputs can be measured, but clearly they cannot be applied by the shaker to the box. The shaker can just apply one of the two outputs to the box, as shown in Figure 2.22. Figure 2.27 compares the response of the box mounted on the shaker and subjected to the vibration of the first connection point of the system engine plus box, xEB, to the response of the box mounted on

the engine, whereas in Figure 2.28 the shaker is subjected to the vibration of the second connection point of the engine with the box mounted.

Figure 2.21: Test room, box connected at two points, two different input for the shaker

Table 2.14: Connective matrix data for the whole system connected at two points 1st DOF 2nd DOF N◦ _{of the element}

1 2 I 2 3 II 2 4 III 3 5 IV 3 6 V 2 6 VI 4 6 VII 4 7 VIII 5 8 IX 6 9 X 7 10 XI 8 11 XII 9 12 XIII 10 13 XIV 11 14 XV 12 14 XVI 12 15 XVII 13 15 XVIII 14 16 XIX 15 17 XX 16 18 XXI 17 18 XXII

Table 2.15: Natural frequencies of the system connected at two points Engine plus box [Hz] Engine [Hz] Box [Hz]

24.6 25.7 135.5 38.6 38 663 56.3 59.5 803.8 64.4 72.3 89.7 83.8 99.3 99.3 119.7 119.7 133.2 133.2 159 205.6 205.6 261.3 261.3 353.1 353.1 608.4 608.4 673.3 673.3 924.5 924.5 947.1

Figure 2.22: Test room, box connected at two points

Figure 2.23: Comparison output from the box connected at two points, output from the whole system as input for the shaker

Figure 2.24: Comparison output from the box connected at two points, output from the engine as input for the shaker

Figure 2.25: Comparison the first output from the engine with the output from the whole system, x

Figure 2.27: Comparison output from the box connected at two points, first output from the whole system as input for the shaker

Figure 2.28: Comparison output from the box connected at two points, second output from the whole system as input for the shaker

2.6

Box attached at four points to the engine with

dis-placement control

The box here is attached at four points to the engine. Clearly this is the more realistic configuration. As discuss in the previous sections, even if the box is mounted on the engine at four points only one displacement can be applied by the shaker to the box. The focus here is on two specific points of the box, called target point z and monitoring point w. The aim of this section is to find an input for the shaker that enables the target point to have the same response on the test configuration (zB) as on the real configuration (zEB). The monitoring point response for the box

on the engine, wEB, was compared to the monitoring point response for the box on

the shaker, wB. The input for the shaker, which enables the target point z to have

the same behaviour in both configurations, is called xS. Figure 2.29 shows the box

on the engine used to obtain the value of the target point, the box on the shaker used to find the input xS and the box on the shaker as input xS used to compare

w. The disposition for the DOFs used for this configuration is shown in Figure 2.30.

for the box are reported in Table 2.16. The natural frequencies of the whole system and the box are shown in the first and second column of Table 2.18, respectively.

Figure 2.31 and Figure 2.32 compare the displacement of the target point and

Figure 2.30: Whole system, box attached at four points, disposition of DOFs of the monitoring point for the box on the shaker and for the box on the engine. Applying to the shaker the four outputs from the engine with the box mounted (x11,

x13, x14 and x15), there are no difference between real and test configurations, see

Figure 2.33. Figure 2.34 and Figure 2.35 compare the response of the connection points of the system engine plus box with the input for the shaker which enables the target point response to have the same behaviour on both configurations.

Table 2.16: Parameters of the box attached at four points N◦ _{of DOF Mass [kg] Stiffness [N/m] N}◦ _{of element}

16 2 3 · 107 19 17 1 3 · 107 20 18 2 3 · 107 21 19 1 3 · 107 22 20 0.5 5 · 106 23 21 0.2 4 · 106 24 22 0.3 3 · 106 25 23 0.4 2 · 106 26 24 0.1 2 · 105 27 3 · 105 28 1 · 105 29 7 · 104 30

2.7

Check with Ansys model

The main Matlab models were checked with a Ansys model. The following figures present the comparison between the results from these models. Figure 2.36 compares the displacement of the point on the engine where the box is fixed. Fig-ure 2.37 compares the displacement of the box connected at one point to the engine. In Figure 2.38 is compared the displacement of the box connected at one point to the shaker subjected to the vibration of the connection points of the engine. Finally Figure 2.39 compares the target response.

2.8

Discussion

Figure 2.7, Figure 2.8 and Figure 2.9 show that the box can have a different behaviour when it is fixed on stiff DOFs of the engine close to the constraint, or on flexible DOFs on the opposite side. This proves the validity of the engine model, as in a real aircraft engine there are stiff and flexible parts.

Figure 2.31: Target response

excitation problems while the last one applies to fatigue issue.

2.8.1 Box attached at one point to the engine

This section compares the behaviour of the box mounted on the engine, the real configuration, with the box mounted on the shaker subjected to the vibration environment of the box connection point of the engine without the box, the test configuration. The box was fixed on a stiff point and on a flexible point of the engine. For each mount different types of brackets were applied: soft anti-vibration mounts and hard undamped systems. The resulting behaviour for the box was different between test and real configurations. In some frequency ranges the displacement of the box was higher on the test configuration than on the real configuration leading to over testing, while in other frequency ranges under testing occurred. Over and under testing happened mainly close to the neo natural frequencies from the connected engine and box system. The amount of over or under testing depended on the mounting point of the engine and on the type of bracket used. The largest differences were found for the box fixed on a flexible part of the engine by a bracket without an anti-vibration system. Test results closer to the real configuration could be obtained with a bracket with an anti-vibration system. For the stiff part of the engine there was no over or under testing. The next step was to compare the displacements of the connecting point for the engine with a mounted box to the engine without the box. Through this comparison, it was clear that the resulting over or under testing was due to the different displacement of the connection point on the engine, which in turn was influenced by the presence of the box. This behaviour was confirmed

Figure 2.32: Monitoring response

when it was used as input for the shaker the connection response obtained from the whole system, engine plus box, as the box showed the same behaviour as when on the engine.

2.8.2 Box attached at two points to the engine

For the box attached at two points to the engine, the over or under testing originated one more from the different input spectrum for the shaker, which was caused by the presence of the box on the mounting points. A further issue investigated was problem that the box was attached at two points to the engine but one displacement can be produced by the shaker for all mounting points. The box was subjected to the vibration of both connection points of the engine with the box mounted, one after another. The resulting over or under testing was mainly due to the difference between the displacement of the two mounting points. If the connection response of two points had been the same, the box would not have had a different behaviour in real and test configurations.

Figure 2.33: Comparison output from the box attached at four points, output from the whole system as input for the shaker

2.8.3 Box attached at four points to the engine with displacement control

Only one displacement control was possible on the shaker. In reality the box was attached at more than one point to the engine. The box was attached in the real configuration at four points to the engine and in the test configuration at four points to the shaker. Once more the shaker only generated one displacement, equal for each mounting points, which was modulated to obtain the same response of the target point on the box as in the real configuration. It could be seem that the input of the shaker required for a displacement control of the target point was larger from the displacement of the box on the engine, which meant more fatigue for the mounting point.

Table 2.17: Connective matrix data for the whole system attached at four points 1st DOF 2nd DOF N◦ _{of the element}

1 2 I 2 3 II 2 4 III 3 5 IV 3 6 V 2 6 VI 4 6 VII 4 7 VIII 5 8 IX 6 9 X 7 10 XI 8 11 XII 9 12 XIII 10 13 XIV 11 14 XV 12 14 XVI 12 15 XVII 11 16 XIX 14 17 XX 15 18 XXI 13 19 XXII 16 20 XXIII 17 20 XXVI 18 21 XXV 19 21 XXVI 20 22 XXVII 20 23 XXVIII 21 23 XXIX 23 24 XXX

Table 2.18: Natural frequencies of the system attached at four points Engine plus box [Hz] Box [Hz]

27.8 113.2 39.3 128.3 57.1 183.8 98 563.5 99.5 601.3 112.3 746 117.9 798.5 121.8 935.6 130.9 955.5 133.4 184.5 205.6 261.3 353.1 585.3 608.4 673.3 694.8 761.5 818.8 924.5 943 1016

Figure 2.36: Check with Ansys, output from the engine

Figure 2.37: Check with Ansys, output from the whole system connected by one point

Figure 2.38: Check with Ansys, output from the box on the shaker connected by one point

Figure 2.39: Check with Ansys, target point, box on the shaker connected by four points