The determination of an external force is a very important task for the purpose of control, monitoring, and analysis of damages on structural system. This paper studies a stochastic inverse method that can be used for determining external forces acting on a nonlinear vibrating system. For the purpose of estimation, a stochastic inverse function is formulated to link an unknown external force to an observable quantity. The external force is then estimated from measurements of dynamic responses through the formulated stochastic inverse model. The applicability of the proposed method was verified with numerical examples and laboratory tests concerning the wave-structure interaction problem. The results showed that the proposed method is reliable to estimate the external force acting on a nonlinear system.

In the field of engineering, it is often very important to estimate external loads acting on dynamic structural systems for a design and analysis of the structural systems. The proper determination of external loads may contribute to extensive practical applications in control, monitoring, and analyzing engineering systems such as bridges under static or dynamic loading, floating structures on waves, and supply systems such as pipes.

However, for some structural systems, it is sometimes difficult to directly measure an external force for some reasons such as installation difficulty of the measurement devices and large magnitude forces. These difficulties led to a necessity of alternative methods to estimate external forces. Therefore, many methods have been developed by means of the inverse problem formalism, inferring an unobservable value from an observable value, except very few cases where the force can be measured directly.

Stevens [

The aforementioned studies assume that structural systems of interests are linear. However, in practice, nonlinearities are always present in various forms for mechanical or structural systems. Therefore, it is necessary to consider these nonlinearities for reliable system design and analysis. To cite a few examples of the force identification for nonlinear systems, Ma and Ho [

It is worth here noting that the aim of the above methods, regardless of considering a linear or nonlinear system, is to make the solution stable by modifying the original formulation. These methods yield only a single estimate of unknowns without quantifying associated uncertainties in the solution. Thus, these kinds of methods are often referred to as the deterministic inverse method.

Due to increasing demands on robustness and reliability as well as due to the rapid growth of computational capacity, it has been more and more important to solve problems under conditions of uncertainty. As a result, recently, problems considering uncertainties have thus attracted much research attention in diverse fields such as electric impedance tomography [

In this paper, an original method based on a stochastic inversion is developed to determine external forces acting on a nonlinear system from measurements of a system response. The method studied here comprises the following four parts: (i) construction of a mathematical inverse model linking an external force to a motion response, (ii) formulation of a stochastic inverse function that allows to determine an external force, (iii) design of probabilistic model to quantify various uncertainties arising from an insufficiency of information on parameters of interests and measurement errors, and (iv) computation of inverse solutions by designing computational tools with full probabilistic description.

In the first part of this paper, the mathematical formulation on the proposed method is presented. Complete description of the proposed method is illustrated through numerical examples. In addition, mathematical characteristics of the constructed model are also discussed. In the second part of this paper, the proposed method is validated through a practical application to a real problem regarding the wave-structure interaction. Experiments for a ship subjected to wave excitation are performed in the ocean engineering basin. Results of the analysis of the experimental data show that wave excitations can be successfully estimated from measurements of the response alone, using the proposed method.

The following nonlinear physical model is first considered:

If

In (

If a motion response

Equation (

In this study, we formulate a stochastic inverse function to achieve a stable solution procedure. By considering all variables of interests to be random variables, (

For an arbitrary nonempty sample space

Equation (

A single realization

The probabilistic expression (

For the purpose of modeling, components in the right-hand side of (

If a measurement error

Finally, it needs to consider the prior distribution

Using the models for the likelihood (

The choice of hyperparameters

Equations (

The MCMC methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain, which is characterized by the rule that the next state depends only on the current state. The generated samples from the algorithm are used to approximate the target distribution. Metropolis-Hastings and Gibbs algorithms [

Initialize

For

Sample

Sample

Sample

Sample

Sample

if

else

Sample

Sample

if

else

In this section, we want to numerically investigate the proposed method to determine external forces acting on nonlinear system through a particular numerical example.

As a numerical example, the following physical model is considered:

At first, the motion responses were simulated with zero initial conditions

External force and corresponding responses.

The aim is now to estimate the external force from the measurement of system responses using the proposed stochastic estimation method in Section

Exact and two noisy data for

In this subsection, the characteristic of the deterministic inverse model (

Equation (

Numerical solutions via the pseudo inverse.

The cause and the degree of instability in the solution can be understood by analyzing the behavior of singular systems, more specifically, each component in the right-hand side of (

Picard plots for the singular system with two noisy data in Figure

In summary, for a stable solution, the approximated system should satisfy the condition that coefficients

As pointed out, the inverse model (

Accordingly, the inverse solution, unknown external force

Numerical results are shown in Figure

Numerical solutions via the proposed method.

Figure

Examples of trace plots for the MCMC simulation and their posterior distributions.

The results in the previous section were fairly good. However, it has a limitation that numerical simulations were conducted based on the synthesized data. In this section, to ensure the practicability, the proposed method is also applied to the real practical problem regarding the problem of estimating wave exciting forces on a ship in roll motion. The estimation of wave exciting forces is one of the most important issues in offshore structure design since waves are the most important environmental phenomena that affect the system stability of offshore structures.

The mathematical model of ship rolling in waves (Figure

Roll motion of a ship in waves.

Equation (

Analogy of (a) the general mechanical system and (b) the ship motion in water.

More specifically, forces acting on ships can be decomposed into hydrostatic and hydrodynamic forces. Hydrostatic force is the force induced by hydrostatic pressure (the static buoyancy force) exerted by a fluid at equilibrium due to the force of gravity. This hydrostatic force is related to the restoring mechanism

Hydrodynamic forces can be considered as the forces resulted from the pressure field disturbed by a moving body. Hydrodynamic forces are divided into three main components. The first is the Froude-Krylov force due to the pressure field in the incident wave. The second is the diffraction force due to the scattering of the incident wave field. The last is the radiation force due to the waves generated by the oscillatory motions of the body. Among them the radiation force is related to the added mass

In this section, the attention is focused on the estimation of the wave excitation

For the purpose of practical application, laboratory tests regarding ship rolling motions in waves were conducted in the Ocean Engineering Basin at The University of Tokyo. The basin, which is often called the towing tank, is designed to perform tests related to various kinds of floating structures. Figure

Particulars of the model.

Length |
2.500 m |

Breadth |
0.387 m |

Draft |
0.132 m |

Displacement volume |
0.110 m^{3} |

GM | 0.074 m |

Resonance frequency |
6.905 rad/s |

Illustration of (a) the test model for experimental study and (b) its body plan.

Figure

Overview of experimental setup.

Figure

Typical example of time history of data for a test model: wave elevation measured at far wave-probe (a) and at near wave-probe (b) and the corresponding responses (c).

As a first attempt, the proposed method was applied to the roll response data when the frequency

The measured roll response data with

Observable quantity

In the same way used in the previous section, the information on unknown wave exciting moment

Figures

Wave exciting moment obtained by pseudo inverse for the response data with

Wave exciting moment obtained by stochastic inversion for the response data with

For the purpose of validation, roll response was resimulated using the posterior mean of the identified wave exciting moment since the exact solution for

Comparison of the re-simulated response with the measured one for the response data with

The same procedure was also applied to the other response data when

Numerical results for the response data when

Numerical results for the response data when

When calculating the wave exciting force acting on a ship, the strip theory [

To ensure the validity of the present method, a qualitative comparison is also made in Figure

Qualitative comparison of wave exciting moment.

It is worth emphasizing that the proposed method is not limited to the case of the stable sinusoidal responses. It is also applicable to the estimation for any form of wave excitation, provided that the motion response is properly recorded. To test this, we also conducted another experiment regarding the ship motion subjected to the wave excitation induced by irregular waves.

Figure

Time history of measurements for the case of irregular wave.

Estimated wave exciting moment through the proposed method.

Comparison of the re-simulated response with the measured one.

A new method has been proposed to estimate the external forces acting on a nonlinear vibrating system, based on formulating the stochastic inverse model from the measured roll response data. The method has been evaluated through an analysis of some digitally simulated data for a numerical example. It has been shown that the proposed method can be used to not only identify the external force but also detect uncertainties in solution arising from measurement error or insufficient information on the system.

To ensure applicability, the wave-structure interaction problem regarding the roll motion of a ship has also been treated as a practical application. To this end, a series of experiments were conducted in the ocean engineering basin. The analysis results demonstrate that the proposed method has been successfully applied to estimate the various types of wave excitation. The estimated results were qualitatively and quantitatively good in all cases.

The authors would like to thank the Editor and anonymous reviewers for their valuable comments, suggestions, and constructive criticism, which were very helpful in improving the paper substantially.