Integration of different approaches to simulate active structures for automotive applications Sven Herold, Heiko Atzrodt, Dirk Mayer, Martin Thomaier Fraunhofer-Institute for Structural Durability and System Reliability LBF, Mechatronics/Adaptronics, Bartningstraße 47, D-64289 Darmstadt, Germany, e-mail:
[email protected],
To solve a wide range of vibration problems with the active structures technology, different simulation approaches for several models (depending on frequency range, modal density and control target) are needed. These simulation approaches can be different for each individual part of the active system: the mechanical structure, the active components (sensors, actuators), and for the controller. To ensure a highly efficient and accurate simulation of the active system, the sub models must harmonize. For this purpose, structural models considered in this article are modal state space formulations and transfer based or impedance based models. Consequently, the impedance based models are derived directly from the measured transfer functions. The modal state space formulations are derived from finite element models and/or experimental modal analysis. To couple mechanical and electrical systems with active materials, the concept of impedance models was successfully tested. The impedance models are enhanced by adapting them to the measured electrical impedance. The controller design depends strongly on the frequency range and the number of modes to be controlled. To control systems with a small number of modes, techniques such as active damping or integral force feedback may be used, whereas in the case of systems with a large number of modes or with modes that are not well separated, other control concepts (e.g. adaptive controllers) are more convenient. The approaches introduced here will be verified by simulating academic test examples.
1
Introduction
based models derived from measured frequency response functions. Impedance based models are used to represent the actuators, which are responsible for the coupling of mechanical and electrical systems by means of active materials [1]. With the model of the mechanical system including actuators and sensors the development of the controllers by means of simulation is possible. The controller design depends strongly on the frequency range and the number of modes to be controlled. Thus, controller concepts like active damping, integral force feedback and adaptive control can be developed and tested [12, 9]. The modelling approach and the simulations will be verified by experimental results.
For most of industrial applications the product cycle became shorter in the last decades. This leads also to shorter development times while the complexity of the products is enhancing. An increasing relevance of dynamical and acoustical properties of products related to comfort aspects can also be observed. The active structures technology can help to solve problems discussed above. To adopt this technology a huge amount of special knowledge is required. For instance mechanical structures, sensors, actuators and controllers have to be developed. For an appropriate handling of the active structures technology an approach for the whole system must be done. The conflict between shorter development times, higher complexity and the importance of dynamical and acoustical properties of the products can be solved by introducing new fast development strategies including new modelling and experimental methods. A general modular method for the design of active structures will be discussed here. The possible integration of different approaches to model the mechanical part, actuators and controllers is an advantage of the method. An easy exchange of any part of the active system will be possible, e.g. for model updating. An other advantage is the simulation in time domain, which allows the implementation of time variant systems. To establish a well running simulation the sub models must match. Models for the mechanical structure can be implemented dependent from the modal density as state space models derived from finite element (FE) models or as transfer function
2
Active systems
Active systems can be described in general with coupled physical properties in more than one direction (sensing and actuating). In most of the cases in the field of vibration analysis and control mechanical and electrical quantities are coupled. The active systems consists of mechanical structures, sensors, actuators, signal processing units and controllers. Figure 1 shows a scheme of a general active system. For active structure development (structural optimization, controller development) active system simulations in parallel to experimental dynamical analysis are necessary. To establish active system simulations each of the parts has to be modelled. An advantage of the approach pre-
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FF2 2
xx2 2 ZZI I
FF1 1 FFE + E +
+
xxn n
Main geometrical parameters and distances of the rubber mounts from each other are explained in Table 1. The minor distance of the driving point to the edges of the plate in x-direction is lxd = 0.07 m and in y-direction it is lyd = 0.06 m. A stainless steel X6CrN iM oT i 17−12−2 (1.4571) material is chosen for manufacturing the plate. The rubber mounts supporting the plate are commercial available products with a diameter of dr = 0.02 m, a length of lr = 0.03 m and a hardness of h = 55 Sh.
controller controller
structure Structure222 structure
Herold, Atzrodt, Mayer, Thomaier
UUA A xx1 1
+
structure Structure111 structure
3
Figure 1: Active system description
Modelling of components
In this section the modelling of mechanical structures, actuators and controllers with the aim of a active system simulation and design will be discussed. More advanced simulations also include the behavior of sensors, signal condition hardware and amplifiers [14, 4].
sented in Figure 1 is the possibility of modular modelling of each component. Thus, it becomes easy to modify the components and their modelling strategy, which is often necessary in any structure development process.
exciter
3.1
mass force sensor hybrid interface
Mechanical structure modelling
Depending on the number of modes in the frequency range of interest, the modelling of mechanical systems can be different. For a low number of modes the evaluation of reduced order simulation models can be proceeded by using analytical models or FE models [3]. This method provides a complete model of the whole mechanical structure but it is complicated to update the model for a large number of modes.
plate
rubber mounts
driving point
Figure 2: Experimental setup A rectangular plate supported by four passive rubber mounts and a hybrid active interface is chosen as a test example (see Figure 2) to demonstrate modelling strategies. This test example is used to show some effects in modelling the active components and bring them together to perform an active system simulation. Such simulations are necessary for instance to compute dynamic load cases or for controller development.
mounting positions
Figure 3: Finite element model Geometrical and material parameters of the mechanical structure described in Section 2 are used for setting up a FE model (Figure 3). The properties of the rubber mounts are identified by experimental analysis. A numerical modal analysis of the mechanical structure supplies eigenvectors and eigenfrequencies, while modal damping coefficients can be taken from experimental modal analysis. ˙ q˙ ζ(t) = A ζ(t) + B u(t) ζ(t) = q y(t) = C ζ(t) + D u(t) (1) −diag (2 ϑω0 ) −diag ω 20 with A = I 0
Table 1: Parameters of the test example Quantity geometric length width thickness distance of mounts length width
symbol
unit
value
lx ly lz
m m m
0.295 0.250 0.003
lxm lym
m m
0.250 0.225
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notes the description of transfer functions in terms of the SOS formulation:
The modal transformation allows an easy reduction of the modal matrix Φ, because the problem is decoupled. The state space formulation is used to implement the modal decomposed model in the simulation environment. Equation 1 shows the system matrices in modal state space formulation. In consequence only the matrix of eigenvectors Φ, the vector of eigenfrequencies ω 0 and the vector of modal damping coefficients ϑ are transferred to the simulation environment. All modal parameters are introduced into the system matrices of state space [12, 4]. The matrix B includes the transposed modal matrix ΦT , while the matrix C is identical to the modal matrix Φ. The vector y is T the vector of physical deformation and equal to [x˙ x] . Figure 4 shows the comparison between simulated (modes obtained from the FE model) and the measured driving point frequency response function (FRF) of the plate. The model fits the measured FRF in the frequency range from 20 to 450 Hz very well. At higher frequencies the model errors increases as discussed above.
H=
K1n
The model is now identified and transferred to the s-domain. For a SOS model an easy implementation in terms of the state space formulation is possible. The state space matrices (A, B, C, D) for implementing the driving point FRF model are pointed out in Equation 5. −diag (2 ϑω 0 ) −diag ω20 A= I 0 h i B = 1 . . . 1 ... 0 . . . 0 (5) h iT C = K11 · · · K1n ... K01 · · · K0n
measured reduced FE−model
| [dB]
−60
adm
|H
−40 −50
−70 −80 −90
200
400
600
800 f [Hz]
1000
1200
(3)
A least squares solution is implemented to compute the coefficients K1n and K0n of the numerator functions Nn of the SOS model. To obtain real coefficients Equation 3 has to be rewritten in terms of real and imaginary parts of Dn and H. To solve the system also the evaluation of the variable s at discrete points sk is necessary: K01 .. . .. sk 1 < . Dn (sk ) Dn (sk )
