1 Introduction

With the increase in loan of artworks between museums, art conservators are increasingly concerned about the behaviour of paintings in their care in response to shock and vibration during transport [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. By monitoring the motion of the painting during transport, they try to define safety limits of shock and vibration for the painting [1,2,3,4], predict the damage of the painting [13, 14, 17], or evaluate the vibration isolation performance of the packaging system [18,19,20]. Combined with the dynamics characteristics of the painting [11, 21,22,23], art conservators can also improve the packaging systems to prevent the eigenmodes of the painting from being excited, and avoid irreversible damage [5,6,7, 18]. Therefore, how to effectively monitor the response of a painting to shock and vibration excitation during transport is of great significance. Constrained, among others, by the cramped design of the packing case, it is difficult to directly monitor them during transport, as the equipment required is generally too bulky to be included alongside the painting in a standard packing case. Attaching sensors to the canvas is in any case unacceptable and dangerous. Furthermore, the lack of an inertial reference makes contactless measurements very difficult. In [4], an advanced small laser displacement sensor was used, but it was found that the fixture of the sensor may possibly change the dynamic characteristics of the crate and severely limits the value of the obtained information. A common practice is, to use accelerometers attached to the strainer [5, 9,10,11,12]. These accelerometers capture the response of the strainer to shock and vibration excitation during transport, which assists art conservators to make predictions, evaluations, and improvements. However, during this process, the response of the canvas itself is always unknown and practically impossible to monitor. If more accurate information about the vibration response or vibration level of the canvas during transport would be known, it would allow to optimize the transport conditions and further protect the painting. Based on this motivation, in this contribution, a method is proposed to reproduce in the laboratory the main vibration of the strainer during transport, so that the vibration of the canvas can be directly observed in the laboratory. This serves as a reference for the actual vibration of the canvas during transport.

Given the fragility of the paint layer and considering that the application of sensors on the paint layers or canvases of original valuable paintings is prohibited from a conservatory perspective, few works directly monitored the response of the canvas to shock and vibration excitation during transport. Only little work has been attempted to reproduce in the laboratory the excitation to the painting during transport, so as to observe the response of the canvas. In [5], a uniaxial electrohydraulic shaker applying random vibration to simulate road excitation had been developed to test a painting in the laboratory. However, the random vibration can hardly represent the characteristics of shock and vibration excitation under different road conditions. There is also a big difference between the uniaxial random vibration of the painting produced by the shaker and its spatial vibration during transport. A transport simulator with linear movement along a single axis was designed in [9], which allows performing the simulation sequentially along each axis of the painting to reproduce the vibrations of each degree of freedom acquired by the acceleration sensor during transport. Combined with some advanced measurement techniques, this simulator has made contributions to the evaluation on the painting transport effects [13, 14, 17, 19]. However, this simulator is still only capable of single axis movement. It is also stated in [9] that the vibration reproduction accuracy of the simulator is not ideal. Therefore, in this contribution, multi-dimensional vibrations directly related to canvas damage will be reproduced with sufficient accuracy based on the data from multiple accelerometers on the strainer measured during transport.

In this contribution, the investigated painting was first packed into a commercial packing case for a real transport experiment. Four accelerometers were attached to different positions of the painting hanging system. The strainer is considered as a rigid body, thus the data from the four accelerometers are sufficient to record the necessary vibration during transport. After the transport experiment, the investigated painting was taken out of the packing case without removing the hanging system and then installed in the laboratory. Four electrodynamical shakers were connected to the hanging system as actuators. Keeping the positions of the four accelerometers unchanged, if the measured accelerations for the actuation with the shakers are the same as the accelerations measured during transport, the vibration of the painting during transport has been reproduced in the laboratory. This allows to measure the paint layer vibration with contactless methods, e.g. a laser Doppler vibrometer (LDV), since an inertial reference is available in the laboratory. Therefore, the analysis of transport-induced vibration is possible. In order to realize the simultaneous tracking control of all four actuators, a feedforward controller was designed based on the multi-channel Filtered-x Least Mean Square algorithm (FxLMS) in MATLAB/Simulink. Loading this controller into the dSpace real-time system enables hardware-in-the-loop simulations, and reproduces the necessary vibration of the investigated painting in the laboratory as if it would have been transported on the road.

The novelty of this contribution lies in the use of a multi-channel feedforward controller to reproduce the canvas vibration yielding the canvas damage during transport with sufficient accuracy in the laboratory. The reproduction results provide a reference for the vibration of the canvas, which is practically impossible to monitor directly during transport. This approach will allow for a more direct and efficient assessment of transport effects and possible damage of paintings.

This contribution is organized as follows. Firstly, the procedures and equipments used for the real transport experiment and the vibration reproduction experiment are described in Sect. 2. In Sect. 3, the multi-channel tracking control algorithm for realizing the vibration reproduction is presented. Based on these, in Sect. 4, the vibration reproduction results for the investigated painting are discussed. Finally, a brief summary is given in Sect. 5.

2 Experimental Setup

This contribution mainly focuses on reproducing the vibration of the painting in the laboratory. Therefore, the setup and ideas used in this experiment will be introduced in detail. In contrast, the setup and procedures for the transport experiment, which provides the real input data for the laboratory experiment, will only be briefly described.

2.1 Transport experiment

The investigated painting on canvas in this contribution shows a landscape and has outer dimensions 49 cm \(\times \) 65 cm \(\times \)1.8 cm. The artist who created this painting is unknown, but art conservators estimate that it was painted in the early 20th century. A knowledge of the material composition of paint layers and the canvas is not relevant to this work, so it will not repeated here. This real painting though of negligible artistic value, has become fragile and massively damaged in the course of natural ageing. It would not have been transported in this condition as part of the museum loan system without consolidation measures. The real transport experiment was carried out in cooperation with hasenkamp Holding GmbH. They provided a proper packing case called Vario that has a flexible, shock-mounted and size-adjustable aluminium inner frame. The investigated painting was fixed in this packing case without a protective backing using a patented hanging system, as shown in Fig. 1. The overall portfolio ranges from simple protective packaging to shock-absorbing and fire-retardant climate boxes that guarantee constant temperature and humidity conditions inside the packing case. In addition, both strict conservational and ecological requirements are taken into account.

Four three-dimensional accelerometers of model PCB-356A03/NC were attached to the hanging systems. The connection between the hanging system and the strainer of the investigated painting is considered as rigid, and the strainer is thus assumed to be a rigid body in this contribution. Therefore, the data collected by the four three-dimensional accelerometers is sufficient to represent the spatial vibration of the investigated painting during transport. Additionally, the temperature and humidity inside the packing case were recorded using MSR165 data loggers. Then, the packed transport case was secured in the centre of the transport car with belts. The supporting data acquisition equipment and power supply facilities were also placed in the car. All packaging, handling as well as transport was carried out by professionally trained staff of hasenkamp, just as for any professional art transport. The transport experiment was carried out within the city of Cologne and lasted for one hour and the data was recorded with a time resolution of \(\Delta t=0.2\) ms. During the transport experiment, various street conditions such as highway, bad roads, sudden braking manoeuvres, and waiting for traffic lights were experienced. The collected vibration data and climate data were used for the laboratory reproduction in the next section.

Fig. 1
figure 1

The experimental setup used for a real transport experiment of the investigated painting

Full size image

2.2 Laboratory experiment

2.2.1 Simplified vibration of the painting during transport

The general rigid body motion of the strainer consists of three independent translations and three independent rotations. This motion is transferred to the canvas via the strainer, which leads to the canvas motion in the painting plane as well as perpendicular to this plane. The damage to the paint layer is mainly caused by the canvas movement and the associated deformation of the paint layer. Due to a supporting effect provided by the tensile strength of the canvas, the movement in the painting plane leads to only very small stretches in the paint layer. In contrast, when the canvas moves perpendicularly to the painting plane, the canvas is flexible, so that there is only little supporting effect and a greater deformation, hence a damaging effect is to be expected [24]. However, this hypothesis needs to be clarified by additional research in the future with the fully operational laboratory test bed. Therefore, in this contribution, we limit ourselves to the reproduction of the acceleration components perpendicular to the strainer, because only those excite the canvas motion, and consequently the paint layer motion, perpendicular to the painting plane. The strainer motion can be divided into three independent movements: a translational movement in the transverse direction (z-axis), a rotational movement around the longitudinal axis (x-axis), and a rotational movement around the vertical axis (y-axis), as shown in Fig. 2. The translation in y- and x-direction and the rotation around the z-axis are not reproduced in the laboratory setup.

Fig. 2
figure 2

Rigid body movements of the strainer related to damage

Full size image

Analysing the data collected during transport, it is found that the root mean square value of the vibration displacement at each accelerometer does not exceed 1.5 mm, which is relatively small compared with the size of the investigated painting. Therefore, in Fig. 2, all kinematics are completely linearized. Obviously, the acceleration of the transverse translation of the strainer (\(\varvec{a}_z\)) is fully captured by the components of the four accelerometers perpendicular to the painting plane (\(\varvec{a}_{\textrm{T}zi}, i=1,2,3,4\)). The angular accelerations \(\varvec{\alpha }_x\), \(\varvec{\alpha }_y\) of the strainer rotating around the x-axis and the y-axis satisfy

$$\begin{aligned} \varvec{a}_{\textrm{R}xi}=\varvec{\alpha }_x \times \varvec{r}_{i},\quad \varvec{a}_{\textrm{R}yi}=\varvec{\alpha }_y \times \varvec{r}_{i}, \quad i=1,2,3,4, \end{aligned}$$
(1)

where \(\varvec{r}_{i}\) is the position vector for each accelerometer, \(\varvec{a}_{\textrm{R}xi}\) and \(\varvec{a}_{\textrm{R}yi}\) are the components of the four accelerometers perpendicular to the painting plane and \(\varvec{\alpha }_x\), \(\varvec{\alpha }_y\) are angular accelerations. Because the positions of four accelerometers relative to the strainer are fixed, the angular accelerations \(\varvec{\alpha }_x\), \(\varvec{\alpha }_y\) of the strainer around the x-axis and y-axis according to Eq. (1) are also fully captured by the components of the four accelerometers perpendicular to the painting plane. Furthermore, these three rigid body movements of the strainer are independent of each other, so during the practical measurement, the component of each accelerometer perpendicular to the painting plane \(\varvec{a}_{zi}\) satisfies

$$\begin{aligned} \varvec{a}_{zi}=\varvec{a}_{\textrm{T}zi}+\varvec{a}_{\textrm{R}xi}+\varvec{a}_{\textrm{R}yi},\quad i=1,2,3,4. \end{aligned}$$
(2)

Thus, in summary, tracking only the component of each of the four accelerometers perpendicular to the painting plane enables the reproduction of the three painting movements that are particularly relevant to damage.

2.2.2 Experimental setup for vibration reproduction

After the real transport experiment, the investigated painting was taken out of the transport crate without removing the hanging systems and transferred to the laboratory. All the experimental setup used for vibration reproduction is shown in Fig. 3. The ensemble consisting of the investigated painting and the hanging systems was suspended with rubber bands on an aluminium support, forming a very soft suspension which can be approximately considered as a free suspension. Since only the component of each three-dimensional accelerometer perpendicular to the painting plane is tracked, uniaxial accelerometers of model PCB-333B30 were used here instead. The position of each accelerometer remained the same as in the transport experiment to ensure the correct comparison of the reproduction results. Four electrodynamical shakers used as actuators were connected to the hanging systems and were as close as possible to the corresponding accelerometer to excite the vibration there. Each electrodynamical shaker and the hanging system were separated by an elastic O-ring to avoid the armature bending due to an over-tight connection. The electrodynamical shakers are of model PCB TMS-K2007E01. Their output frequency range is DC-9 kHz, and the maximum output acceleration under 0.907 kg load is 3.3 g peak to peak, which fully meets the experimental requirements.

Fig. 3
figure 3

The experimental setup used for vibration reproduction of the investigated painting in the laboratory

Full size image

During the experiment, the signal conditioner of model PCB-482C24 powers the accelerometers, amplifies their output voltages, and transmits them to dSpace. The dSpace device used for hardware-in-the-loop simulation consists of a DS1006 Processor Board, a DS2102 DAC Board, and a DS2004 A/D Board. The accompanying dSpace Controldesk software provides access to the simulation platform as well as to the connected control system. The simulation platform restores the voltage signals collected by dSpace to the actual acceleration signals. The multi-channel vibration tracking controller in the simulation platform updates the control voltage of each actuator according to the actual acceleration signals and the desired acceleration signals collected in the transport experiment. The simulation platform was built in MATLAB/Simulink and loaded into dSpace.

During stably reproducing the vibration of the strainer, the vibration responses of arbitrary points on the canvas were measured by a Polytec Vibroflex QTec laser Doppler vibrometer (LDV). The innovative Polytec Vibroflex QTec LDV has the advantage of using a helium-neon laser source and a multiple channel interferometer which virtually avoids laser drop outs even when measuring uncooperative surfaces. The yellow point on the canvas of the investigated painting in Fig. 3 was selected to show the measurement results. This point is selected arbitrarily, but care was taken to avoid selecting a point on a vibration node of a lower eigenmode of the canvas.

Studies have shown that the dynamics characteristics of the painting are affected by climate [1]. The data recorded in the transport experiment shows that the relative humidity in the packing case was stable at 52 \(\pm \,\)0.7% and the temperature was stable at 19.9 ±  0.5 \(^{\circ }\)C. In order to avoid the influence of changing climate on the experimental results as much as possible, all the vibration reproduction experiments were carried out in a climate chamber. The temperature and humidity in the climate chamber were adjusted to be the same as those in the transport crate during the transport experiment.

3 Multi-channel vibration tracking controller

It is the purpose of the controller discussed in the following to move the painting using the four shakers in the same way as it was measured in the crate during transport. The vibration reproduction controller in the simulation platform has to simultaneously control four actuators and reproduce the acceleration signals obtained from the transport experiment with high accuracy. There are many algorithms that can be used to achieve this function. In the early stage of this research, a feedback controller based on an adaptive PID algorithm was tried, but high-precision tracking could not be achieved. Besides a problem with integrator windup, long-term experiments revealed error propagation of the adaptive algorithm, which prohibited the use of an adaptive feedbakc control.

Instead, a feedforward controller was used. One commonly used algorithm in active noise control is the Filtered-x Least Mean Square (FxLMS) algorithm. It is derived from the Least Mean Square (LMS) algorithm, which is simple to implement, has a small amount of calculation, and has good robustness [25]. The FxLMS algorithm achieved extremely good results in single-channel vibration reproduction. Therefore, an extension of this algorithm to multi-channel control is applied in order to realize the vibration reproduction of the investigated paintings. The control logic of this controller is introduced in this section.

3.1 Multiple-input multiple-output FxLMS algorithm

For the vibration reproduction system designed in this contribution, there are four actuators and four accelerometers. Preliminary experimental results revealed that the use of multiple single-channel controllers, each linked to a single accelerometer and an actuator, did not work effectively. Instead, the single-channel algorithm had been extended in order to enable it to be used in a multiple-input multiple-output (MIMO) control scheme. Here, a feedforward controller based on the multi-channel FxLMS algorithm is adopted and its arrangement is illustrated in Fig. 4 [25]. The control signal supplied to each actuator is generated by a separate finite impulse response (FIR) filter, and it is assumed, without loss of generality, that each actuator filter has M stages [26]. The desired signal for each accelerometer is derived from the acceleration collected in the transport experiment. Therefore, the error signal \(e_i(k)\) of the ith accelerometer at time k is the difference between the desired signal \(d_i(k)\) and the sum of contributions from each of the N actuators \(s_{ij}(k)\) with \(N=4\), in this contribution

$$\begin{aligned} e_i(k)=d_i(k)-\sum _{j=1}^{N} s_{ij}(k)\quad i,j=1,2,3,4. \end{aligned}$$
(3)

The jth actuator component of the ith error signal is not, in general, equal to the output of the jth actuator, but is rather equal to a version of the control signal which has been modified by the pathway transfer function between the output of the jth actuator filter and the collected ith acceleration signal (dSpace \(\rightarrow \) actuator \(\rightarrow \) accelerometer \(\rightarrow \) signal conditioner \(\rightarrow \) dSpace). Modelling this transfer function as an m-stage finite impulse response function \(\varvec{c}_{ij}\) and assuming that the system is time invariant yields

$$\begin{aligned} s_{ij}(k)=\varvec{y}_j^\textrm{T}(k)\varvec{c}_{ij}, \end{aligned}$$
(4)

where \(\varvec{y}_j(k)\) is a vector of the most recent outputs from the jth actuator filter

$$\begin{aligned} \varvec{y}_j(k)=[y_j(k) \quad y_j(k-1) \quad \cdots \quad y_j(k-m+1)]^\textrm{T}. \end{aligned}$$
(5)
Fig. 4
figure 4

Block diagram of an adaptive MIMO feedforward control system based on the FxLMS algorithm

Full size image

The output of the jth actuator filter can also be expressed as

$$\begin{aligned} y_j(k)=\varvec{x}_j^\textrm{T}(k)\varvec{w}_j(k), \end{aligned}$$
(6)

where \(\varvec{w}_j(k)\) is the vector of the jth actuator filter weight coefficients at time k. Then, \(s_{ij}\) can be expressed in the ‘filtered reference signal’ format

$$\begin{aligned} s_{ij}(k)=[\varvec{X}_j^\textrm{T}(k)\varvec{w}_j(k)]^\textrm{T}\varvec{c}_{ij} =\varvec{w}_j^\textrm{T}(k)[\varvec{X}_j(k)\varvec{c}_{ij}]=\varvec{w}_j^\textrm{T}(k)\varvec{f}_{ij}(k), \end{aligned}$$
(7)

where \(\varvec{X}_j(k)\) is an \((M \times m)\) matrix of the m most recent reference signal vectors

$$\begin{aligned} \begin{aligned} \varvec{X}_j(k)&=[\varvec{x}_j(k) \quad \varvec{x}_j(k-1) \quad \cdots \quad \varvec{x}_j(k-m+1)] \\&= \begin{bmatrix} x_j(k) &{} x_j(k-1) &{} \cdots &{} x_j(k-m+1) \\ x_j(k-1) &{} x_j(k-2) &{} \cdots &{} x_j(k-m) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ x_j(k-N+1) &{} x_j(k-N+2) &{} \cdots &{} x_j(k-m-N+2) \\ \end{bmatrix} \end{aligned} \end{aligned}$$
(8)

and \(\varvec{f}_{ij}\) is the ijth filtered reference signal vector, ‘filtered’ by the pathway transfer function between the jth actuator filter output and ith acceleration sensor output. In practical application, the pathway transfer functions \(\varvec{c}_{ij}\) are usually unknown, and their estimates \(\hat{\varvec{c}}_{ij}\) must be used in calculating the filtered reference signal [27]. Thus, each element \(f_{ij}\) in this vector is

$$\begin{aligned} f_{ij}(k)=\varvec{x}^\textrm{T}(k)\varvec{c}_{ij} \approx \varvec{x}^\textrm{T}(k)\hat{\varvec{c}}_{ij}. \end{aligned}$$
(9)

The key to the practical application of this adaptive filter is how to construct the desired signal and reference signal. For the control system in this contribution, the goal of each actuator is only to make its corresponding accelerometer achieve the expected tracking effect. Thus, the desired signals at each accelerometer are derived from the real transport experiment. As a feedforward controller, it is desirable that its output converges to its input as precisely as possible. So the reference signal of each actuator filter is the same as the desired signal of its corresponding accelerometer, namely,

$$\begin{aligned} d_i(k)=x_j(k), \quad i=j. \end{aligned}$$
(10)

The cost function used by the LMS adaptive filter is the mean square error. For the MIMO control arrangement, its aim is to realize the error criterion, which is to minimize the sum of the mean square errors of the signal from each accelerometer, \(\Xi \), that is

$$\begin{aligned} \Xi =\sum _{i=1}^{N} \textrm{E} \{e^2_{i}(k) \}. \end{aligned}$$
(11)

Differentiating this with respect to the jth weight coefficient vector produces the gradient estimate, and using it in the standard gradient descent format produces the MIMO filtered-x LMS algorithm. For the jth actuator filter, this is expressed as described in [28, 29] by

$$\begin{aligned} \varvec{w}_j(k+1)=\varvec{w}_j(k)+2\mu \sum _{i=1}^{N} \varvec{f}_{ij}(k) e_{i}(k), \quad N=4, \end{aligned}$$
(12)

where \(\mu \) is the convergence coefficient.

3.2 Offline multi-channel pathway modelling

The remaining problem in the above MIMO FxLMS algorithm is how to obtain the estimated pathway transfer functions \(\hat{\varvec{c}}_{ij}\). Although the FxLMS algorithm does not need the pathway model to be very accurate to maintain the stability of the system, a fast and reasonably accurate estimation of the pathway transfer function is important to ensure adequate performance, stability, and convergence speed of the system [25]. For the multiple channel control system adopted in this contribution with 4 actuators and 4 accelerometers, there are 16 pathways that need to be modelled. Considering the inter-channel coupling effect when the control signals are fed to several actuators, here, all pathways are decided to be modelled simultaneously. Moreover, the variations of all existing pathway transfer functions are considered to be within a small range. Therefore, it is not necessary to do pathway modelling online or by training/learning algorithms. The model obtained offline can be used permanently in the actuator filter weight-update algorithm. The offline pathway modelling method is based on the Least Mean Square (LMS) adaptive algorithm as shown in Fig. 5 and described in [25, 26, 30].

Four narrowband noise sources are fed to the four actuators. The bandwidth of the narrowband noise includes at least the working bandwidth of the MIMO control system. The residual error signal for the pathway modelling at the ith accelerometer can be written as

$$\begin{aligned} e_{Ci}(k)=\sum _{j=1}^{N} \varvec{u}_j^\textrm{T}(k)[\varvec{c}_{ij}(k)-\hat{\varvec{c}}_{ij}(k)], \quad i,j=1,2,\cdots ,N,\quad N=4. \end{aligned}$$
(13)

Each pathway uses the LMS algorithm to estimate its model, see [25], namely,

$$\begin{aligned} \hat{\varvec{c}}_{ij}(k+1)=\hat{\varvec{c}}_{ij}(k)+2 \mu \varvec{u}_j(k) e_{Ci}(k). \end{aligned}$$
(14)
Fig. 5
figure 5

Block diagram of modelling all pathways for the MIMO FxLMS control system

Full size image

To carry out pathway modelling for four actuator outputs simultaneously, four uncorrelated narrowband noise signals must be used to remove the inter-channel coupling effect [25]. Besides, uncorrelated narrowband noise signals also contribute to realize the error criterion at each accelerometer when modelling the pathways. After the residual error signal in Eq. (13) is minimized by the adapting algorithm in Eq. (14), the following is obtained

$$\begin{aligned} \varvec{c}_{ij} \approx \hat{\varvec{c}}_{ij}. \end{aligned}$$
(15)

Thus, in this case, correct models are obtained for all the pathways between the actuators and the accelerometers, which can be used in the MIMO control system described above.

4 Vibration reproduction results

Based on the experimental setup built above and the designed multi-channel vibration tracking controller, some vibration reproduction experiments were carried out. According to the experimental results, the performance and effect of this controlled system will be discussed in this section.

4.1 Preprocessing of the desired signal

The desired signals for each accelerometer were obtained from the transport experiment and were acquired at a sampling rate of 5 kHz corresponding to \(\Delta t=0.2\) ms. However, such a high sampling rate need not and cannot be adopted during vibration reproduction due to hardware limitations of the real-time environment. In the multi-channel vibration tracking controller, all the actuator filters and the pathway filters are set to 100 stages. Therefore, a total of 100-order matrix operations of 16 pathways are involved in the update process of the control algorithm. Coupled with other operations such as data acquisition, dSpace cannot complete all operations within a sampling period of 0.2 ms and real-time simulation cannot be performed. Clearly, reducing the stage of the FIR filter increases the usable sampling frequency, but also degrades the performance of the controller. In this contradiction, the sampling frequency during the vibration reproduction experiment was set as 1 kHz, i.e. \(\Delta t=1\) ms, to maintain the unchanged stage of the FIR filter.

Limited by the output performance of the electrodynamical shakers used in this contribution, their acceleration output waveform has a large distortion below 5 Hz. Besides, considering that the first eigenfrequency of the investigated painting is about 7.5 Hz, the minimum operating frequency of the multi-channel vibration tracking controller was fixed at 5 Hz. According to the Nyquist–Shannon sampling theorem, when the sampling frequency is 1 kHz, there is no aliasing effect for signals below 500 Hz. The analysis of acceleration signals in the transport experiment shows that their components of high frequency are usually small. Furthermore, high frequency excitation is assumed to have neglectable effect on the damage of the canvas, because the vibration displacement of the canvas under high frequency excitation is typically weak due to the paintings inertia. Therefore, the highest frequency of vibration reproduction is set as 100 Hz in this contribution. A large number of canvas eigenmodes are already contained in this frequency range [23]. This does not mean that the designed multi-channel vibration tracking controller can only work within this range. In fact, the experimental results have shown that it can work stably between 5 and 500 Hz on our hardware. However, affected by noise, small vibration signals of high frequency are difficult for the controller to reproduce and are not relevant for the painting.

Thus, before performing the vibration reproduction experiment, the original acceleration signals in the transport experiment were preprocessed. The components between 5 and 100 Hz are first extracted using a bandpass filter, and then resampled using a sampling frequency of 1 kHz. The comparison before and after preprocessing of a short but representative section of an acceleration signal is shown in Fig. 6. Except for the loss of very high frequency components above 100 Hz, the dominant characteristics of the acceleration signal do not change much.

Fig. 6
figure 6

Comparison of a short section of the acceleration signal from the transport experiment before and after processing

Full size image
Fig. 7
figure 7

Reproduction results for a short section of the normal vibration response

Full size image
Fig. 8
figure 8

Amplitude frequency response curves of the signals in Fig. 7

Full size image
Fig. 9
figure 9

Reproduction results for a short section of the lower vibration response

Full size image
Fig. 10
figure 10

Reproduction results for a short section of the shock response

Full size image

4.2 Reproduction for vibration response

The excitations of the painting during transport can be roughly divided into shock and vibration. The characteristics of the painting response under vibration excitation is that its amplitude and frequency change much slower than for the shock response and repeat to a higher degree than for single shock response. For example, when a transport vehicle is driving at a relatively steady speed on a highway, if the road condition is good, the measured response corresponds to the vibration response of the painting. The acceleration signals with a duration of 840 s in the transport experiment were reproduced, and the reproduction results of a small section of the vibration response are shown in Fig. 7.

The ‘Upper Left’ in Fig. 7a refers to the data of the accelerometer in the upper left corner when facing the investigated painting, and analogously for the other plots in Fig. 7. The FxLMS algorithm achieves good to satisfactory agreement for all four accelerometers, although the agreement for the two lower accelerometers is somewhat degraded due to the larger extend of high frequency components. The amplitude frequency response curves of all signals within the time range shown in Fig. 7 are plotted in Fig. 8. It further shows that the FxLMS algorithm achieves good reproduction effect on the dominant components in the signal, but the reproduction results on the components with smaller amplitudes, especially high frequency components, are poor.

From the transport experiment, it was observed that the vibration response amplitude of the painting rarely exceeds 0.25 g. Thus, the results presented in Fig. 7 show the tracking quality for the higher levels of vibration amplitude. For smaller amplitude around 0.01 g, as presented in Fig. 9, the tracking effect still remains good. For the weaker vibrations, the effect of vibration reproduction will deteriorate due to the influence of noise in the sampling channel. However, this is considered as uncritical since these weak vibrations do not contribute to the damage of the painting.

In general, the reproduction effect for the vibration response of the investigated painting by the multi-channel vibration tracking controller has reached the expectation.

4.3 Reproduction for shock response

The response of the painting when being subjected to shock excitation during transport is a highly transient behaviour. The response is characterized by rapid rise and decay of the amplitude in a short time, and the energy is distributed over a wide frequency band. For example, when the transport vehicle brakes suddenly or passes a pothole in the street, the response of the painting is dominated by the shock response. According to this characteristic, strong shock responses can easily be distinguished from normal vibration responses. However, for events of smaller shock amplitude this distinction becomes more and more difficult. The maximum amplitude of the shock response observed in this transport experiment reached 0.5 g. The rapid changes on amplitude and frequency pose great challenges to the multi-channel vibration tracking controller. The reproduction results of a higher shock response during the transport experiment are shown in Fig. 10.

Obviously the overall behaviour is again represented well. Particularly for the two upper accelerometers, the agreement is very good. The two lower accelerometers, however, show noticeable overshoot. Actually it is also observed in Figs. 7 and 9 that the vibration reproduction at the upper two accelerometers is always better than that at the lower two accelerometers. This must be related to the characteristics of the slower convergence speed of the FxLMS algorithm, but it should also be pointed out that for the multi-channel FxLMS algorithm the cost function in Eq. (11) aims to minimize the sum of the mean square errors of the signal from each accelerometer, and cannot completely eliminate the inter-channel coupling effect. Especially during transport, the amplitude of the upper two accelerometers is usually about twice that of the lower two accelerometers, which makes vibration reproduction at the lower two accelerometers more affected by inter-channel coupling effects. Moreover, in terms of hardware, the hanging system of the investigated painting during the transport experiment was rigidly connected to the crate. Although in the laboratory the hanging systems and the shakers were also rigidly connected, the connection between the armature of the shaker and its support flexure was softer than the connection in transport. This elastic connection may have increased the difficulty of reproducing the shock response.

In general, the reproduction results still reflect most of the information about the shock response of the investigated painting. The reproduction can be considered as being fully acceptable for the intended use.

4.4 Reproducibility test

The purpose of the vibration reproduction system designed in this contribution is to reproduce the vibration of the strainer during transport as precise as possible, so as to observe the response of the canvas in the laboratory and provide a reference for the unknown response of the canvas during transport, eventually using it to explore damage mechanisms of paintings during transport. Therefore, it is important to investigate whether the vibration reproduction system itself has a stable reproducibility, i.e. whether repeated experiments really yield identical behaviour. In theory, although the convergence speed of the multi-channel FxLMS algorithm is slow, as long as the iteration step size is set reasonably, the parameters of the controller will continuously be updated to offset external disturbances and there will be no error propagation. In order to verify its reproducibility, a repeated vibration reproduction experiment was carried out. The acceleration signals with a duration of 730 s were reproduced repeatedly several times and the velocity response of each cycle of the selected measurement points on the canvas was observed by the laser Doppler vibrometer. The comparison of the velocity response over a small period of time for the 3rd, 5th and 7th cycles is shown in Fig. 10.

Fig. 11
figure 11

Comparison of the velocity response of different cycles on the selected measurement points

Full size image

Evidently, there are only very minor differences between the different vibration reproduction cycles. The source of the difference may be the interference of external environmental noise and the influence of climate fluctuations. Anyway, this proves that the designed vibration reproduction system has a stable reproducibility.

5 Conclusion

Aiming at the problem that it is difficult to monitor the vibration response of the canvas during transport, this contribution proposes to reproduce the measured vibration of the strainer by driven shakers in the laboratory, so as to conveniently observe the response of the canvas in the laboratory. The real-time simulation platform based on dSpace uses a multi-channel FxLMS algorithm to simultaneously control four actuators, thereby reproducing the acceleration data collected on the strainer during transport. Limited by the complexity of the control algorithm and the performance of dSpace, the sampling rate of the simulation platform is set to 1 kHz, and the effective working frequency band is 5–100 Hz, which fully meets the requirements. The experimental results show that the control system has sufficient precision in reproducing the vibration response, even for smaller vibrations around 0.01 g. There is some overshoot in the reproduction of the shock response, but the experiment still reflects the overall characteristics of the shock response very well, so it can be considered as fully acceptable. Furthermore, the long-term cyclic reproduction experiment demonstrates the high level of reproducibility of the control system.

The multi-channel vibration reproduction system designed in this contribution provides a reference for the unknown response of the canvas during transport. This will help art conservators to study the dynamic characteristics of paintings, evaluate the transport effect and the vibration isolation ability of the transport crate. It will even make it possible to study the damage mechanisms of the painting during transport through long-term experiments in the laboratory. Therefore, future research will focus on the damage mechanisms in the paint layer made observable in the laboratory by the developed method and test bed.