Experimental research on the influence of modal nonlinearities of paintings under mechanical loads

Paintings are inevitably subjected to shock and vibration during transportation. Excessive shock or vibration may cause irreversible changes in the complex materials of canvas and paint and accelerates their natural aging [1‐7]. A transportation experiment has shown that once the eigenmodes of the painting are excited, the vibration intensity of the canvas can be much higher than the vibration intensity transferred to the frame through the crate [8]. Therefore, it is necessary to study the modal characteristics of paintings and then improve the packaging system accordingly, so as to prevent the eigenmodes of the painting from being excited during transportation. For many works on the transportation of paintings, there is the hypothesis that since the vibration displacement is small, paintings can be treated as a linear system, i.e. their modal parameters can be always considered constant [9‐14]. However, other research has shown that the modal characteristics of paintings are not only affected by climate [15‐17], but also vary with different vibration intensities [6, 18]. This indicates that there are nonlinear aspects in the paintings modal characteristics and the linearity hypothesis remains to be verified. Therefore, in order to improve the packaging system of paintings, it is of great significance to study whether the modal nonlinearities of paintings will show during transportation and whether these modal nonlinearities need to be considered in the design of the packaging system. In this contribution, the modal nonlinearities of paintings under different shock and vibration intensities during transportation will be carefully studied.

There have been few researches on the modal nonlinearities of paintings under different vibration intensities in the last decades. In [6], research about the modal nonlinearities of paintings is carried out, and in [18] modal nonlinearities were discovered as well. In [6], through sweep experiments to simulate the vibration of the painting, the canvas showed to have strong progressive stiffness characteristics and degressive damping characteristics. Namely, with the increase in external excitation, the resonance frequency of the painting increases, and the amplitude of the resonance peak decreases. It is also mentioned in [6] that the resonance peaks are asymmetric, and the resonance peaks obtained by up-sweep and down-sweep, i.e. sweep with increasing and decreasing frequency, respectively, are different. These nonlinear phenomena are discussed by the Duffing oscillator model in [6]. However, whether these nonlinear phenomena will occur during transportation and whether the design of painting packaging systems needs to consider these nonlinear factors still need to be studied. Moreover, paintings are not only subjected to vibration but also to shock during transportation, which means that the general conclusion about the modal nonlinearities of paintings under different types of excitations deserves further study.

Based on the above motivation, in order to simulate the state of a painting during transportation, hammer experiments and sweep experiments are conducted in this contribution. The hammer experiment on the frame is realised by an automatic impulse hammer which is used to simulate a shock on the painting during transportation. The sweep experiment is realised by an electrodynamical shaker, as well as a loudspeaker and is used to simulate the vibration on the painting during transportation. Both the hammer and sweep excitation differ from the undeterministic excitation occurring during real transportation. However, they represent limit cases that are uniquely defined and can easily be compared for different measurements. Broadband excitation, e.g. with noise, would have been possible but was not conducted in the measurements presented here. Since the aim is the evaluation of nonlinear behaviour, a simple axial excitation is chosen instead of a multiaxial excitation. This allows to directly relate excitation amplitude with (possibly nonlinear) response. The excitation is chosen perpendicular to the painting for nonlinear effects are more likely to occur in this direction. In order to avoid excessive excitation amplitudes, these are chosen in the range which was recorded during a real transport, [8].

The sweep experiment with shaker is a common experiment which focuses on the modal nonlinearities of a painting under different vibration intensities during transportation. The sweep experiment with loudspeaker is a contactless measurement method, more suitable for real precious paintings that do not allow use the above mentioned contact measurement methods. In this contribution, the feasibility of this contactless method to measure the modal nonlinearities of paintings under different vibration intensities will be verified.

In order to draw a general conclusion about the modal nonlinearities of paintings, two real paintings with natural aging and a recently manufactured dummy painting are tested in this contribution. The two real paintings are of negligible artistic value, and thus, they can be tested under lab conditions. The dummy painting was made by an experienced art conservator, with uniform tension and uniform and homogeneous distribution of primer and paint. It does not yet show any deterioration in terms of aging. In all experiments, the vibration acceleration on the three tested painting frames as well as the vibration velocity and vibration displacement at selected measurement points on the canvases is collected. The vibration velocity of the canvas is combined with the corresponding excitation force to calculate the frequency response function (FRF) and analyse its modal nonlinearities. The vibration displacement of the canvas can intuitively express the vibration intensity of the tested paintings. The vibration acceleration of the frame is compared with those recorded during a real transportation in [8], which is then used to assess whether the modal nonlinearities will occur during the transportation. Based on all the experimental results for the three tested paintings, the general behaviour of the modal nonlinearities of paintings under different shock and vibration intensities are summarised. Then, a conclusion of whether the modal nonlinearities of paintings should be considered during the transportation is derived.

This contribution focuses on experimental research on the modal nonlinearities of paintings. The novelty of this contribution is the investigation to what extent these nonlinearities occur for excitation amplitudes of typical art transports, and consequently, a discussion about whether the modal nonlinearities of paintings should or must be considered during the transportation as well as the related consequences of the modal nonlinearities for painting transportation. Moreover, the most suitable experimental methods for measuring the modal nonlinearities of paintings are also summarised.

This article is divided as follows. Section 2 contains the description of the three tested paintings and the experimental setup. Section 3 contains a discussion whether paintings can be treated as a linear system during transportation through a short introduction to the results of a hammer experiment and a sweep experiment. In Sect. 4, the general behaviour of the paintings modal nonlinearities is summarised and their consequences for painting transportation are discussed. Section 5 gives a brief summary.

In this contribution, three paintings are investigated as shown in Fig. 1. The first painting shows a seascape with the dimensions 50\(\times \)60.7\(\times \)2 cm. The second painting is a still life painting with dimensions 57\(\times \)46\(\times \)2 cm. Both of them are real paintings, though of negligible artistic value, and have become fragile and massively damaged after natural aging. They would not have been transported in this condition as part of the museum loan system without consolidation measures. The purpose of studying two real paintings simultaneously is to discover the similarities and the differences in modal nonlinearities between real paintings. The last painting is a dummy painting with the logo of the authors’ institute, which measures 60\(\times \)70\(\times \)4 cm.

In contrast with the previously described paintings which were built, primed, painted etc. by the artist and were also left in the pre-tension state as of purchase/loan, the purpose of the dummy painting is to set up an intensionally simplified dummy model in order to assess whether some of the observed properties change with those simplifications. The production process of the dummy painting is similar to that in [18]. The canvas of the dummy painting is decatised and then uniformly tensioned on a biaxial stretcher. Next, the stretched canvas is treated with a so called pre-gluing to prevent the primer from permeating into the canvas in the next step. The primer is heated and then applied on the canvas in three layers with a brush from different directions. The blue oil paint layer and the white paint layer are also applied sequentially on the primer layer with a brush. Lastly, the canvas is sandwiched between two wooden frames to form the final dummy painting. Because the dummy painting has only been dry for less than a month and there is almost no deterioration in terms of aging, through in the research related to this dummy painting, the similarities and the differences in modal nonlinearities between the real paintings after natural aging and the young painting can be discovered. Each painting is equipped with 4 perforated metal plates which are used to simulate the state of the painting being fixed on a transportation plate through the perforated metal plate, and it is also convenient to hang the painting during the experiment.

Figure 2 depicts the experimental setup used to study the modal nonlinearities of paintings. It consists of three different paintings, two different excitation mechanisms, a Polytec VibroFlex laser Doppler vibrometer (LDV), three triaxial accelerometers and a climate box. The investigated painting is mounted with rubber bands on an aluminium support, forming a very soft suspension which can be approximately considered as a free suspension. Although the described suspension differs from typical boundary conditions during transportation, the vibration of the painting (and the paint layer) could still be generated to be essentially the same as for a transportation, if multidimensional excitation forces were produced such that the multidimensional accelerations measured in the laboratory were the same as measured during the actual transport. As long as the desired motion of the painting is generated, the difference in boundary conditions has no negative impact on the measurements. Three triaxial accelerometers are attached on perforated metal plates in order to monitor the acceleration of the tested painting and compare it with the vibration acceleration during the transportation experiment in [8]. The model of the triaxial ICP accelerometers is PCB-356A03/NC.

The vibration velocity of the canvas and the excitation force are collected to study the modal nonlinearities of tested paintings, i.e. so-called mobilities are measured. To rule out misinterpretations by leakage effects due to finite measurement time, different window functions were applied to the time domain signal. Among others, a Bartlett and a Von Hann window was used for the sweep experiments and an exponential window was used for the impact experiments. In the frequency range of interest, apart from the most subtle differences, the mobilities were identical. Also, coherence was checked to evaluate the quality of the measurement. No detrimental effects were found.

In principal, a mobility measurement would have to be performed for many points to fully capture the spacially distributed motion of the paint layer and to also fully evaluate the spatial distribution of nonlinearities. However, measurements conducted at different points on the canvas revealed similar results as to the extent of nonlinear behaviour. Therefore, for simplicity, the green point labelled MP on the tested painting shown in Fig. 2 is selected as a representative point to present the experimental results in this contribution. In order to avoid the loss of modal information during the experiment due to the measurement point being located on one of the nodal lines of the first eight eigenmodes, this measurement point is selected at one fourth of the width and height. The seascape painting is less damaged at this point, while the still life painting has obvious cracks at this point. The vibration velocity signals at the measurement point are measured with the Polytec VibroFlex LDV. The VibroFlex LDV has the advantage of using a \(\hbox {CO}_2\)-based Laser source and a multiple channel interferometer which virtually avoids laser drop outs even when measuring uncooperative surfaces. Rigorous inspection of the measurement signals in time domain revealed no laser drop outs. Velocity and displacement were measured simultaneously.

In this contribution, two different excitation mechanisms are adopted for experimental research. The first type of excitation is an impulse excitation generated by an automatic impulse hammer as shown in Fig. 3a. The automatic impulse hammer is composed of an electrodynamical shaker and a modified tip mechanism that generates a very short hit and avoids double hits. The impulse excitation directly acts on a metal plate on the tested painting from the back, which is used to simulate the shock of the painting during transportation. The excitation point is shown as the yellow point labelled EP1 in Fig. 2. The contact force between the excitation point of the tested painting and the tip of the automatic impulse hammer is measured by an ICP force sensor PCB 086E80. The excitation is chosen such that measured acceleration does not exceed the acceleration measured for shock events in [8].

The second type of excitation is a sweep excitation generated by an electrodynamical shaker or loudspeaker. The sweep excitation is used to simulate the vibration of the painting during transportation. When the excitation is generated by the electrodynamical shaker, the excitation position is the same as for the hammer experiment on the frame, but the brass stinger of the shaker is firmly connected to the perforated metal plate with two nuts as shown in Fig. 3b. The input force of the shaker is measured by an ICP force sensor PCB 208C01. In order to obtain a realistic excitation intensity, root mean square values of the painting frame response as measured in [8] are used. When the excitation is generated by the loudspeaker, the loudspeaker simply faces to the painting in a distance of round about 1 cm, as shown in Fig. 3c. This contactless experimental method is particularly suitable for measuring the modal nonlinearities of real precious paintings. However, because the input pressure generated by acoustic excitation is difficult to measure, the modal nonlinearities can only be analysed and evaluated based on the velocity response of the canvas instead of the mobility which would require a force or pressure measurement. The impulse signals or sweep signals of the above three excitation mechanisms are provided by a function generator, and the intensity of shock and vibration are adjusted by the input voltage. In order to avoid the influence of changing climate on the modal characteristics of the tested paintings as much as possible, all experiments are carried out in a climate box. The potential relative humidity adjustment range of the climate box is 35–80%, while 52.5%±0.4% is used for the experiments if not separately mentioned, and the temperature is kept constant at 22 \(^{\circ }\)C. In this range, the changes in modal characteristics caused by changing climate have been checked to be negligible.

In the following, by means of the hammer experiment and the sweep experiment, the dynamical behaviour of the three considered paintings under different shock or vibration intensities are identified. By comparing the shock and vibration intensity of the seascape painting in the experiment with the transportation in [8], it is clarified that the dynamic behaviour of the three tested paintings observed in the experiments covers the dynamic range that occurs during transportation. This way, the conclusion of whether the painting can be treated as a linear system during transportation can later be derived.

After the observation that nonlinearities occur for sufficiently large excitation energies in the previous section, here, the sweep experiment is further analysed for the three tested paintings. The general behaviour of the paintings modal nonlinearities is discussed and some consequences of the modal nonlinearities for transportation are deduced.

In this contribution, through the hammer experiment and the sweep experiment, the modal nonlinearities of two real paintings and a dummy painting under different shock and vibration intensities have been studied experimentally. The experimental results show that the painting exhibits obvious modal nonlinearities only when being subjected to vibrations of different intensities. The comparison of the vibration acceleration of the seascape painting in the sweep experiment and the transportation experiment demonstrates that a painting can be treated as a linear system when being subjected to shock, but its modal nonlinearities are not negligible when being subjected to vibration. First of all, the most important problem caused by the modal nonlinearities is the significant resonance frequency offset, which is likely to directly lead to the failure of the original vibration isolation measures for the eigenmodes of the painting. Then, as the vibration intensity increases, the amplitude of the resonance peak decreases and the resonance peak becomes more flat, which increases the probability that the eigenmode is excited during transportation. Moreover, it was observed in the painting that the superharmonics and subharmonics may be of the same order of magnitude as the fundamental vibration or even higher. This indicates that it is also necessary to attenuate the harmonic vibration during the painting transportation. Lastly, it is worth noting that under the same vibration intensity, different modal characteristics exist for the paintings when sweeping with increasing and decreasing frequencies. This makes the analysis of different excitation schemes complicated. Therefore, the identification of the modal nonlinearities of the painting is absolutely necessary and important. The best identification method is the sweep experiment with shaker or loudspeaker. The modal nonlinearity observed in the sweep experiment with shaker is most comprehensive, while the sweep experiment with loudspeaker is particularly suitable for real precious paintings.

So far, only the occurrence of nonlinear effects have been shown. It also shown that those nonlinearities are non-negligible for real paintings and excitation intensities observed for real transports. As to the physical origins of these non-linearities no conclusion can be drawn. In [6] some of the nonlinear phenomena were compared with the behaviour of a Duffing oscillator which is motivated by the cubic stiffness term that appears in the nonlinear description of membranes. This gives a hint about the origin of the general behaviour found in this contribution, e.g. the increase in the resonance frequency with excitation intensity. Other observed phenomena like the inverse relationship for small excitation intensities, different behaviour for different resonance frequencies an alike, are more involved and may be a nonlinearity of a different kind for which a manifold origins are conceivable. They remain sources for further research.