Chapter
1
Description of problem
Contents
1.1 Background . . . 1
1.2 Approach . . . 3
1.2.1 Simulated ideal state . . . 3
1.2.2 Simulated tested state . . . 5
1.2.3 Analysis . . . 5
1.2.4 Damping . . . 5
1.3 Interpretative analysis . . . 7
1.1
Background
The substructures, which are vital parts of assembly, must be designed carefully before starting to operate in order to ensure a successful live cycle. For this kind of substructure the FE models have to be checked by test in the laboratory to avoid any errors during the designing. The testing is also applied for substructure subjected to fatigue or high amplitude failure. Ideally the latter tests are carried out on the complete assembly where the real vibration environment is generate. Unfortunately, this is not possible for many reasons mainly concern the cost and the time. Also for intellectual property, most of the substructure cannot be tested using the complete vibration environment but only its envelope. The main ways to carry out the test according to the preliminary remarks are following discussed.
When the substructure, also called accessory or box, is supplied by external company, for intellectual property, the external company can only obtain the envelope of the mounting point responses. The mounting points are the points where the
accessory is attached to the assembly, also called system or engine. The mounting point responses are used to test the accessory on the shaker, see Figure 1.1. An example of this, it is represented from the company launching rockets, which supplies of the envelope of the vibration environment at the mounting point to the company manufacturing the payload.
Figure 1.1: Envelope of vibration environment available. B, E and S are box, engine and shaker respectively
When the accessory is tested to high amplitude issue inside the same company which produces the assembly, all vibration environment can now be known. In most of the cases the vibration environment is represented by Frequency Response Function of the mounting points obtained running the full system with or without the accessory mounted on it. As discussed in the preliminary introduction, it is cheaper and quicker test the accessory on the shaker subjected to the mounting point responses from the system. However the shaker can not represent the complete response of the system. For instance in reality one mounting point has 6DOF s, three of translational and three of rotation, but the shaker can apply only one of these to the accessory, generally one translational displacement orthogonal to the plane where the shaker is fixed. Also the vibration environment applied to the accessory by the system is different for each mounting point, but the shaker can only apply the same vibration for each mounting point, see Figure 1.2. An example of this is represented from the company produced car engine, when they want to test the components like pomp, probe, electronic control unit, etc.
The previous ways to test are still valid for the fatigue issue, but in this case a different method can be used, the displacement control method. Supposing to have the assembly, it is possible to measure the displacement in one direction of one chosen point of the accessory, then find the same displacement for the same point for the accessory on the shaker leading to better simulation of the ideal state at least in the direction of interest, see Figure 1.3. An example of this is represented
1.2. APPROACH
Figure 1.2: Vibration environment available
from the company produced aircraft engine like Rolls Royce PLC, which tests some components by itself.
For each of the previous test methods the substructure has a different dynamic behaviour when it is installed on the assembly than on the shaker. This because the shaker is rigid body without dynamic while the assembly has a its dynamic. The aim of this study is to carry out an harmonic analysis on both configurations to quantify the different dynamic behaviour of the accessory, called gap.
1.2
Approach
To allow the comparison and to quantify the gap of the substructure two different configurations were constructed: simulated ideal state and simulated tested state.
1.2.1 Simulated ideal state
The simulated ideal state concern the model composed of the accessory on the system. This is also called real configuration because represents the real system which will operate, see Figure 1.4. This model is used to get the mounting point
Figure 1.3: Displacement control method
responses which will be used as input for the shaker. In some cases, most of the times, the mounting point responses is measured from the system without the accessory mounted on it as shown in Figure 1.1. Also, the simulated ideal state is used to get the monitoring point response that is the point of interest on the box
1.2. APPROACH
1.2.2 Simulated tested state
The simulated ideal state concern the model composed of the box on the shaker. This is also called test configuration. As discussed before, there are different ways to test the accessories depending whether from the issue investigated, as high amplitude excitation or fatigue, and from the data available, as the FRF or its envelope. Some example are shown in Figure 1.1, Figure 1.2 and Figure 1.3, which represented respectively: FRF envelope available, FRF available and displacement control.
1.2.3 Analysis
Two kinds of analysis are used to study the different configuration. First of all a modal analysis used to find the natural frequencies with relatives modes shape, eigenvalue and eigenvector respectively, secondly a harmonic analysis used to find the FRF or the displacement of the point of interest. The theory about modal and harmonic is omitted, see [1], [2], [3], [4], [5] and [6].
1.2.4 Damping
In contrast with the theory about modal and harmonic analysis, a brief discussion about the damping used for the harmonic analysis is shown in the current section.
The damping matrix [C] in its most general form is:
[C] = α[M ] + (β + βc)[K] + Nm X j=1 βjm+ 2 Ωβ ξ j [Kj] + Ne X k=1 [Ck] + [Cξ] (1.1) where:
[C] = structure damping matrix α= mass matrix multiplier [M ] = structure mass matrix β = stiffness matrix multiplier
βc = variable stiffness matrix multiplier
[K] = structure stiffness matrix Nm = number of materials
βjm = stiffness matrix multiplier for material j
βjξ= constant (frequency-independent) stiffness matrix coefficient for material j Ω = circular excitation frequency
Kj = portion of structure stiffness matrix based on material j
Ck= element damping matrix
Cξ= frequency-dependent damping matrix
βc, the variable stiffness matrix multiplier, is used to give a constant damping
ratio, regardless of frequency. The damping ratio is the ratio between actual damping and critical damping. The stiffness matrix multiplier is related to the damping ratio by: βc = ξ πf = 2 Ωξ (1.2) where:
ξ= constant damping ratio
Constant structural damping in the all system
In this section structural damping is considered as only cause of damping and also the same value of damping is assumed for each component of the system. The structural damping, [1], [4], [5], is attributed to internal friction in structural material (such as steel and aluminium alloys) that are not perfectly elastic. As experiments suggest, it can be represented as a independent. Following frequency-independent and considering the same damping for each component of the structure, the Equation 1.1 can be written as:
[C] = βc[K] (1.3)
where:
βc = variable stiffness matrix multiplier, see the Equation 1.2
[K] = structure stiffness matrix
It should be noted that the damping matrix [C] is multiplied for Ω on the equation of motion that’s why the structural damping becomes frequency-independent. Different structural damping for each component of the structure
In current section different structural damping is considered for each component of the system. Following this hypothesis and considering the frequency-independent of the structural damping, the Equation 1.1 can be written as:
[C] = Nm X j=1 2 Ωβ ξ j [Kj] (1.4)
1.3. INTERPRETATIVE ANALYSIS
where:
Nm = number of materials
βjξ= constant (frequency-independent) stiffness matrix coefficient for material j Ω = circular excitation frequency
Kj = portion of structure stiffness matrix based on material j
1.3
Interpretative analysis
There is an overriding complication to plotting FRF data which derives from the fact that they are complex and thus there are three quantities: frequency, real and imaginary part. These can not be fully displayed on a standard x − y graph. Because of this, any such simple plot can only show two of the three quantities and so there are different possibilities available, several of which are used from time to time [3]. The form of presentation used in this study is: Modulus [m] vs. Frequency [Hz]. To quantify the amount of different behaviour between those two configuration, called gap, the following relation is used:
gap= 1 − |xB− xEB| max|xB− xEB|
(1.5) where:
xB = modulus of displacement for the test configuration, or only box
