# A frequency domain approach for parameter identification in multibody dynamics

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## Abstract

The adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, as, e.g., parameter identification. In case of the identification of parameters in oscillating multibody systems, a combination of Fourier analysis and the adjoint method is an obvious and promising approach. The present paper shows the adjoint method including adjoint Fourier coefficients for the parameter identification of the amplitude response of oscillations. Two examples show the potential and efficiency of the proposed method in multibody dynamics.

## Keywords

Parameter identification Frequency domain Multibody dynamics Adjoint system Optimization Fourieranalysis Window functions Engine orders Order analysis## 1 Introduction

Applications of the adjoint method to solve a variety of optimization problems in engineering sciences are widespread. Much attention to this approach has been paid recently in the context of multibody systems (see, e.g., [1, 2, 3, 4, 5, 6, 7]) in the field of optimal control, sensitivity analysis, and parameter identification. In [8], the adjoint method is seen as a special case of linear duality, which dramatically improves the efficiency of the computation only solving the dual problem. The basic idea of the adjoint method, e.g., as presented in Nachbagauer et al. [7], is the enhancement of the cost function by the system equations of motion including specific system parameters or controls to identify. By including the system equations of motion in the cost function, adjoint variables have to be introduced, leading to the dual problem when solving for these adjoint variables. Minimizing the cost function leads to a classical optimization problem identifying unknown parameters of the system, as, e.g., the mass or inertia parameters of a body or stiffness and damping parameters of a spring-damper force involved. These prescribed examples lead to an identification of parameters in time domain. Various applications of the adjoint method in multibody dynamics for optimal control problems and parameter identifications in time domain can be found in recent works, e.g., in [7]. Adjoint sensitivities have been used in a penalty formulation in time domain for a full vehicle model in [9].

Moreover, from the experiment point of view, the frequency domain plays a key role when analyzing complex multibody systems. Very often only frequency ranges can be investigated in detail. Too low frequencies may not be measured, e.g., with acceleration sensors, and too high frequencies are mainly caused by measuring noise. Identification in the time domain would lead to some kind of best-fit solution. Hence, the goal of the identification is in general to fit a special frequency range. In [10], a system identification for vehicle dynamic applications has been presented based on impulse–momentum equations using a transfer function written as a frequency response function in order to take into account low and high frequency ranges. Spectral element techniques for parameter identification can also be found in the field of layered media in structural dynamics [11]. Therein, the characteristic function of the system, combining the response and impulse force function of the system, is represented in the frequency domain. The transfer function which characterizes the system in the frequency domain is then given as a Fourier transformation. The wavelet transform is used in [12] as a time–frequency representation for the determination of modal parameters of a vibrating system. Therein, natural frequencies, damping ratios, and mode shapes are estimated in the time domain from output data only. A wavelet-based approach for parameter identification is as well presented in [13]. Systems with cubic nonlinearities and systems undergoing both continuous and stick/slip motion have been addressed therein.

The latter mentioned works emphasize the importance of the spectral analysis of the system in order to understand the behavior of the system and consequently be capable of efficient parameter identification. The present paper shows a method for parameter identification in complex multibody systems in the frequency domain. A combination of the adjoint method and classical Fourier analysis for the identification of the amplitude response is presented herein as a novel approach and is applied to engineering problems.

## 2 Problem definition: cost function in terms of Fourier coefficients

## 3 The adjoint gradient computation

## 4 Application to multibody systems

## 5 Numerical examples

### 5.1 Cart pendulum system

### 5.2 Identification of torsional vibration damper (TVD) parameters

#### 5.2.1 Model structure

### Crankshaft

### Conrod

### Piston

In the used model, each piston features only one translational degree of freedom. The mass of each piston is denoted as \(m_{\text{p}}\) and the piston’s effective area as \(A_{P}\).

### Dual mass flywheel

The primary side of the dual mass flywheel (DMF) is mounted on the right end of the crankshaft (see Fig. 7). Hence, its moment of inertia is assigned to \(q_{1}\). Instead of introducing a degree of freedom for the secondary side of the DMF, the prescribed angle \(q_{\text{runup}}(t)\) is used. A nonlinear torsional spring and a linear damping element is used for connecting the primary with the secondary side.

### Torsional vibration damper

### Pulley wheel

The pulley wheel used for driving additional aggregates introduces another degree of freedom (\(q_{26}\)), which is connected to the TVD hub using a linear spring/damper with parameters \(c_{PW}\) and \(d_{PW}\).

### Cylinder pressures

The cylinder pressure is given by a two-dimensional map depending on the rotational speed and the crankshaft angle. The pressure is applied on each piston in accordance with the firing order.

### Run-up of the engine

#### 5.2.2 Results of the parameter identification

## 6 Conclusions

The present paper shows a new method for parameter identification using the amplitude response of oscillations in multibody system dynamics. The proposed method combines the classical Fourier analysis with the adjoint sensitivity analysis for the gradient computation in an optimization problem. Using the spectrum of a system output helps in understanding the behavior of a system whereas the interpretation of the time domain data is not always promising because of excessive time ranges, noisy signals, or systematic errors.

The method is applied to a nonlinear cart–pendulum system. In order to demonstrate that the method copes with system uncertainties in the example, one parameter, which is not part of the identification, is set to a value different from the one used for generating the measurement. Therefore, the desired spectrum cannot be obtained to its full extent, but only in the given frequency range. In the time domain this would lead to an optimization problem that is not well-posed.

Further, an identification is performed using the model of a four-cylinder engine. Here, the robustness of the method and also the capability to deal with larger systems are pointed out.

## Notes

### Acknowledgements

Open access funding provided by University of Applied Sciences Upper Austria. K. Nachbagauer acknowledges support from the Austrian Science Fund (FWF): T733-N30.

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