Structure Flexibility Impacts on Robust Active Vibration Isolation Using Mixed Sensitivity Optimization

[+] Author and Article Information
Claes Olsson

 Volvo Car Corporation Chassis & Vehicle Dynamics, Dept. 96260, PV4B7, 405 31 Gothenburg, Swedencolsson5@volvocars.com

J. Vib. Acoust 129(2), 179-192 (Sep 08, 2006) (14 pages) doi:10.1115/1.2424970 History: Received February 04, 2005; Revised September 08, 2006

Active vibration isolation from an arbitrarily, structurally complex receiver is considered with respect to the impacts of structure flexibility on the open- and closed-loop system characteristics. Specifically, the generally weak influence of flexibility on the open-loop transfer function in the case of total force feedback, in contrast to acceleration feedback, is investigated. The open-loop system characteristics are analyzed based on open-loop transfer function expressions obtained using modal expansion and on modal model order reduction techniques. To closely demonstrate and illustrate the impacts of flexibility on the closed-loop system performance and stability, a problem of automotive engine vibration isolation from a flexible subframe is presented where the neglected dynamics are represented as an output multiplicative model perturbation. A physical explanation as to why the contribution of flexibility to the open-loop transfer function could be neglected in the case of total force feedback in contrast to acceleration feedback is given. Factors for an individual eigenmode to not significantly contribute to the total force output are presented where the deviation of the mode direction relative to the actuator force direction is pointed out as a key one in addition to modal mass and damping coefficient. In this context, the inherent differences between model order reduction by modal and by balanced truncation are being stressed. For the specific automotive vibration isolation application considered, the degradation of robust performance and stability is shown to be insignificant when obtaining a low-order controller by using total force feedback and neglecting flexibility in the design phase.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

A model of a car engine and subframe suspension system including rigid body engine and gear box, flexible subframe, rigid body torque rod, six rubber bushings, two engine mounts, and force actuator output Fs acting in the longitudinal direction of the torque rod where the doubled arrowed line indicates force action and reaction

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Figure 2

RHS mount static translational stiffness characteristics, in x, y, and z directions

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Figure 3

Transfer functions from Fs (see Fig. 1) to total force transmitted to subframe in the longitudinal direction of the torque rod (solid), and to subframe acceleration of the actuator attachment point in the direction of the actuator (dashed). The transfer functions are generated using the 42-DOF model of the car engine and subframe suspension system.

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Figure 4

The effects of subframe flexibility on the open loop transfer functions shown in Fig. 3. Solid lines show the transfer function differences in gain and phase due to subframe flexibility while comparing the transfer function obtained using the 42-DOF model (solid line in Fig. 3) with one obtained using a rigid subframe representation and the same input and output (i.e., input equal to Fs and the output given by the total force transmitted to subframe). The dash-dotted lines show the corresponding differences for the transfer function from Fs to subframe acceleration.

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Figure 5

Gains of the true system given by Eq. 28 (solid), of the system reduced using modal truncation (dashed), and of the system reduced by balanced truncation of the states corresponding to the two smallest HSVs γ3 and γ4 in Eq. 32 (dash-dotted)

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Figure 6

Gains of the true system given by Eq. 28 (solid), of the system reduced using modal truncation (dashed), and of the system reduced by balanced truncation of the states corresponding to the HSVs γ2 and γ3 in Eq. 32 (dash-dotted)

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Figure 7

Magnitudes of three transfer functions; ∣H1Dss(iω)∣ (solid), ∣1+HΣ1(iω)∣ as defined by Eq. 39 (dashed), and ∣HΣ2(iω)∣=∣H1Dss(iω)−HΣ1(iω)∣ (dash dotted)

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Figure 8

Magnitudes of three transfer functions; ∣H1D(iω)∣ (solid), ∣HΣ1(iω)∣ defined by Eq. 39 (dashed), and ∣GD(iω)∣ (dash dotted)

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Figure 9

Magnitudes corresponding to ∣h10ss∣ (solid), ∣h11ss∣ (dashed), and ∣h12ss∣ (dash dotted)

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Figure 10

Some high-frequency poles and zeros of the system shown in Fig. 1 corresponding to the two transfer functions presented in Fig. 3, i.e., from actuator force to total transmitted force and receiver acceleration, respectively

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Figure 11

Magnitudes of the weighting functions W1−1(s)V−1(s) (solid) and W2−1(s)V−1(s) (dashed)

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Figure 12

The sensitivity function S(s) evaluated with the 8th-order design model G(s) (dashed) and with the complete 84th-order model (solid) using the same 12th-order controller. The dominant spectral contents of the engine excitation are assumed to be contained in the frequency region between the two vertical lines.

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Figure 13

Magnitude of the transfer function from disturbance to plant input evaluated using the 12th-order controller and the 8th-order design plant model G(s) (dashed) and the complete 84th-order model (solid)

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Figure 14

The upper bound on relative model perturbations for closed-loop stability evaluated using the 8th-order design model G(s) (dash dotted), and on model perturbations taking the errors between the 8th-order model and the flexible full 84th-order model into account (dashed). Also shown is the relative error between the 8th- and 84th-order models (solid).



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