Chapter 8: The Virtual Shaker Testing approach (VST)
8.1 E LECTRO - MECHANICAL LUMPED ANALYTICAL MODEL OF THE SHAKER
In order to develop an analytical and simplified model, with which to examine and compare more in details following results, Figure 44 shows an example of the desired shaker.
In fact, VST does not requires a priori a particular complex model as can be FE model or reduced FE.
Particularly, this approach is used in order to approaching to the MatLab/ Simulink code and the global procedure
Figure 44: Cross section of the shaker (left), example of electromechanical model (right) [Ref.5]
How is well described in “Chapter 7: ”, the main components of an electrodynamic shaker are:
• Rubber feet;
• Permanent magnet;
• Coil;
• Table;
• Rubber boot.
Particularly, a 4 DoFs has been considered. In fact, an appropriate modelling is represented by 3 mechanical vertical DoFs and 1 electrical DoF, at least.
pag. 77 In this way, is possible to appreciate the response of the 3 mechanical DoFs, in terms of displacements, speeds and accelerations, during the variation of the voltage.
Some equations, 3 Mechanical
Equations 𝑚𝑖𝑥𝑖̈ + 𝑑𝑖𝑥̇𝑖+ 𝑘𝑖𝑥𝑖 = 𝐹𝑖 (8.1)
1 Electrical
Equation 𝑒(𝑡) = 𝑅𝑖(𝑡) + 𝐿𝑑𝑖(𝑡)
𝑑𝑡 + 𝑉𝑏𝑒𝑚𝑓(𝑡) (8.2)
In which 𝐹 = 𝐵𝑙𝑛𝑖 = 𝜇𝐹𝑖 and 𝑉𝑏𝑒𝑚𝑓= 𝜇𝐹(𝑣𝑇− 𝑣𝐵) is the back electromotive force induced by the velocity of the coil in the magnetic field. Together are the expression of Lorentz’s law.
Practically, the electrical equation generates the source, in terms of applied force, that allows the coil motion.
Moreover, the relationship between the electrical and mechanical domains is not a “one-way street.” In fact, When the coil moves within the magnetic field, a voltage is generated across the coil in proportion to the velocity.
Figure 45: The implemented electro-mechanical model [Ref.33]
Observing the equation (8.1). the external force is composed by the unknown “i”, the current.
However, they will be pick up with the other unknowns.
Rewriting in matrix form, the mechanical equation of the analysed model, shown in Figure 45 appears as
[
𝑚𝑡+ 𝑚𝑖𝑢𝑡 0 0
0 𝑚𝑐 0
0 0 𝑚𝑏
] { 𝑥𝑡̈ 𝑥𝑐̈ 𝑥𝑏̈
} + [
𝑑𝑐+ 𝑑𝑠 −𝑑𝑐 −𝑑𝑠
−𝑑𝑐 𝑑𝑐 0
−𝑑𝑠 0 𝑑𝑏+ 𝑑𝑠 ] {
𝑥𝑡̇ 𝑥𝑐̇ 𝑥𝑏̇
} + [
𝑘𝑐+ 𝑘𝑠 −𝑘𝑐 −𝑘𝑠
−𝑘𝑐 𝑘𝑐 0
−𝑘𝑠 0 𝑘𝑏+ 𝑘𝑠 ] {
𝑥𝑡 𝑥𝑐 𝑥𝑏} = {
0 𝜇𝐹𝑖
−𝜇𝐹𝑖
} (8.3)
Introducing the electrical equation,
pag. 78 [
𝑚𝑡+ 𝑚𝑖𝑢𝑡 0 0 0
0 𝑚𝑐 0 0
0 0 𝑚𝑏 0
0 0 0 1
] { 𝑥𝑡̈ 𝑥𝑐̈ 𝑥𝑏̈ 0
} + [
𝑑𝑐+ 𝑑𝑠 −𝑑𝑐 −𝑑𝑠 0
−𝑑𝑐 𝑑𝑐 0 0
−𝑑𝑠 0 𝑑𝑏+ 𝑑𝑠 0 0 𝜇𝜈 −𝜇𝜈 𝐿]
{ 𝑥𝑡̇ 𝑥𝑐̇ 𝑥𝑏̇ 𝑑𝑖 𝑑𝑡}
+ [
𝑘𝑐+ 𝑘𝑠 −𝑘𝑐 −𝑘𝑠 0
−𝑘𝑐 𝑘𝑐 0 −𝜇𝐹
−𝑘𝑠 0 𝑘𝑏+ 𝑘𝑠 𝜇𝐹
0 0 0 𝑅 ]
{ 𝑥𝑡 𝑥𝑐
𝑥𝑏 𝑖
} = { 0 0 0 𝑒
} (8.4)
Or in more compact form
[[𝑴𝒎𝒆𝒄𝒉] 0
0 1] {{𝑥̈}𝑚𝑒𝑐ℎ 0 } + [
0
[𝑫𝒎𝒆𝒄𝒉] 0
0
0 𝜇𝜈 −𝜇𝜈 𝐿
] {{𝑥̇}𝑚𝑒𝑐ℎ
𝑑𝑖/𝑑𝑡 } + [
0
[𝑲𝒎𝒆𝒄𝒉] −𝜇𝐹
𝜇𝐹
0 0 0 𝑅
] {{𝑥}𝑚𝑒𝑐ℎ
𝑖 } = {
0 0 0 𝑒
} (8.6)
Then, calling {𝑞} = {{𝑥}𝑚𝑒𝑐ℎ
𝑖 }
[𝑴
𝑬𝑴]{𝑞̈} + [𝑫
𝑬𝑴]{𝑞̇} + [𝑲
𝑬𝑴]{𝑞} = 𝑭 ̅ (8.7)
In Table 20 the most relevant symbols, are recollected
Symbols Description
𝑚
𝑏,𝑑
𝑏,𝑘
𝑏 Mass, damping and stiffness of the body of the shaker𝑚
𝑡,
𝑚𝑖𝑢𝑡 Mass of the table and IUT𝑚
𝑐𝑑
𝑐,𝑘
𝑐 Mass, damping and stiffness of the coil𝑑
𝑠,𝑘
𝑠Damping and stiffness of mechanical connection between shaker body and table
(suspension)
R [
𝛺
], L [H] Resistance and inductance of the coil.𝜇𝑣 [V/(m/s)],𝜇𝐹 [N/A] Coupling constants (typically, with the same value) Table 20: Adopted symbols
Then, the State- Space system is obtained using the standard nomenclature
{𝑥̇} = [𝐴]{𝑥} + [𝐵]{𝑢}
(8.8)
{𝑦} = [𝐶]{𝑥} + [𝐷]{𝑢}
Where
pag. 79
[𝐴] = [ [
0 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 0
] [
1 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 1
] [𝑴
𝑬𝑴]
−1[𝑲
𝑬𝑴] [𝑴
𝑬𝑴]
−1[𝑫
𝑬𝑴]
] (8.9)
[𝐵] = [
[ 1 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 1
] [
0 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 0
]
[
0 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 0
] [𝑴
𝑬𝑴]
−1]
{ 0 0 0 0 0 0 0 1}
(8.10)
{𝑢} = 𝑒 (8.11)
The remaining matrices depends on which I want to observe.
In this case, I suppose to observe the acceleration, so
{𝑦} = { 𝑥
𝑡̈ 𝑥
𝑐̈ 𝑥
𝑏̈
}
(8.12)[𝐶] = [[𝑴
𝒎𝒆𝒄𝒉]
−𝟏[𝑲
𝒎𝒆𝒄𝒉] [𝑴
𝒎𝒆𝒄𝒉]
−𝟏[𝑫
𝒎𝒆𝒄𝒉]]
(8.13)[𝐷] = [
0 ⋯ 0
⋮ ⋱ ⋮ 0 ⋯ 0 ]
{
0 0 0 0
⋮ 0 0 1
}(8.14)
Obviously, in this case the damping values are supposed as known, but in general these quantities are difficult to impose. For this reason, is a common practice to propose the damping mechanical matrix as composed by “modal damping” in which the damping is assigned to the modes of the structure, or “proportional damping” in which the matrix is a linear combination of mass and stiffness whit appropriate coefficients.
[𝑫
𝒎𝒆𝒄𝒉] = [
2𝜁
1𝜔
10 … 0 0 2𝜁
2𝜔
2… 0
⋮ ⋮ ⋱ ⋮
0 0 … 2𝜁
𝑛𝜔
𝑛]
(modal damping) (8.15)[𝑫
𝒎𝒆𝒄𝒉] = 𝛼[𝑴
𝒎𝒆𝒄𝒉] + 𝛽[𝑲
𝒎𝒆𝒄𝒉]
(proportional damping) (8.16) In this framework, the pilot was the assembly composed by the IUT and the table into eq. (8.3) and the notcher is the coil.In these terms, during the virtual test when the frequency of the swept sine increases according to the sweep rule, if the acceleration level of the considered notcher exceeds its maximum
pag. 80 allowable, the control system reduces the magnitude of the input. Particularly, the input in this electro-mechanical model is a force able to impose an acceleration level to the pilot.
However, from a control point of view, it has to reach and maintain a predefined acceleration level.
The following pictures, related with the introduced mathematical, model are extracted using the control system provided by Siemens and described in “Chapter 9: Description of Siemens LMS Vibration Control Routines in Simulink environment”.
Quantity Value
𝑚𝑡 [kg] 4
𝑚𝑖𝑢𝑡 [kg] 1
𝑚𝑐 [kg] 3.5
𝑚𝑏 [kg] 150
𝑑𝑐 [N/s] 484
𝑑𝑠 [N/s] 643
𝑑𝑏 [N/s] 20
𝑘𝑐 [N/m] 546000
𝑘𝑠 [N/m] 145000
𝑘𝑏[N/m] 5000000
C 5
Ref.prof [g] 1
Notch level (global) [g] 0
S [oct/min] 2
Table 21: Implemented values
The simulation doesn’t consider the presence of notching. In this term, the output curves is the unnotched motion.
Figure 46: Pilot curves
pag. 81 Figure 46 shows the sinusoidal trend of the pilot curves. How it appears, all the curves are overlapped. This is exact, in fact, just one pilot is considered and the curve represents the same output. Actually, how will be examined in the next chapter, the Simulink model requires 4 signals coming from at least 1 input.
However, being the natural frequencies are about 20 Hz, 30 Hz, 90 Hz, pilots are able to maintain the required acceleration profile (1g) except around 30 Hz. Figure 47.
Figure 47: Acceleration profile reached up from the pilots (time and frequency) On the other hand, the time variation of the remaining DoFs is presented in Figure 48.
pag. 82 Figure 48: Coil and Body time variation
It shows the time response of the Coil and the shaker body. Particularly, how is possible to see, a great peek appears during the body response. Particularly, it appears at about 30 Hz in accordance with the second natural frequency. I fact, the first is not considered because the implemented model force the structure between the shaker body and the coil. This is equivalent as the constraint imposition between these two.
Lastly, Figure 49 shows the time and frequency variation of these unnotched DoFs. Again, the greatest peak appears if the input has the frequency of the second natural frequency of the global system
Figure 49: Coil and Body frequency variation
In fact, considering the previous mechanical system, the imposition of a force between the shaker body and the coil means to constraint a reduced system composed by only the coil and the sum of IUT and table. This concept is expressed by erasing the row and the relative column into the mass and stiffness matrix related to the shaker body. Then, performing a modal analysis, the natural frequency about 30 Hz is reached.
pag. 83