Non-Linear Dynamic Analysis of Timber Frame Structure with Bolted-Fastener Connections

Abstract

Understanding the dynamics of timber structures is essential for the timber structural engineering field, where it is necessary to build predictive numerical models and digital twins. Three similar-sized representative post-beam bracing frames with wood–metal assemblies were tested. Experimental modal analysis gave some indication of the non-linear behaviour of the structure. Then, the frame was submitted to a logarithmic sine sweep, which highlighted some specificities of the non-linear modes: dependence on the sweep direction and amplitude, jump, etc. These phenomena can be explained by friction and shocks in the assemblies. An accurate model of these non-linearities could lead to resilient and more earthquake-resistant timber structures, as the equivalent damping of a non-linear structure is way lower than for a linear one.

1. Introduction

1.1. Industrial Context

Timber is one of the oldest construction materials and is widely used in some countries for traditional buildings. However, designing timber structures presents more challenges for engineers compared with concrete and metal, especially for larger structures, due to the variability in its mechanical properties and its sensitivity to moisture. Technical progress and political commitments to using environmentally friendly materials have led to the development of numerous medium- and high-rise timber building projects. In France, the construction sector is the third-largest contributor to pollution, after agriculture and transport. Building materials account for a significant proportion of the sector’s carbon emissions. Integrating bio-based materials into construction could potentially reduce the emissions from this sector, reducing France’s overall greenhouse gas emissions.

Timber structures are lighter than concrete or metallic ones, offering several environmental and logistical benefits: reduced transportation emissions, quicker on-site assembly, and the possibility of prefabrication. However, medium- and high-rise wooden buildings can experience significant vibrations and accelerations due to human-induced loads, wind forces, or earthquakes. The dynamics of such wooden structures often involve specific non-linearities and relatively large vibration amplitudes. Despite this, modelling and understanding of these behaviours remain limited, particularly in estimating modal damping.

These non-linearities are often considered a drawback, as they make the dynamics of timber structures harder to predict. Nevertheless, they represent an opportunity to reduce stresses and vibrations caused by dynamic loads [1,2,3].

1.2. Timber Structure Dynamics and Non-Linear Modes

The commonly accepted experimental method for measuring the natural frequencies, mode shapes, and modal damping ratios of a building involves using accelerometers and performing linear experimental or operational modal analysis [4]. Experimental modal analysis conducted on medium- and high-rise structures reveals a wide range of modal damping coefficients and frequencies [5]. These variations depend on factors such as the building’s height, the bracing frame technology employed, and the magnitude of the applied load. Wind-induced vibrations can cause discomfort for occupants, particularly in high-rise timber structures exceeding 50 m in height [6].

A large number of specific assemblies and devices are known to improve the dynamics of tall buildings, especially timber structures, in order to minimise damage in the event of an earthquake. For example, dissipative bracing uses the plasticity of metal as a resilient method to dampen the seismic load [7,8]. Especially in earthquake-prone countries, special assemblies are designed to provide elastic dissipation by allowing damage-free displacement [9]. External devices can then improve the damping of tall buildings. The best known is the TMD (Tuned Mass Damper), but a large number of systems exist or are under development [10]. However, the implementation of this type of device is expensive. In this paper, a structure with traditional bracing and assemblies is presented. Its non-linear behaviour seems to give it damping potential.

The concept of non-linear modes has existed for a decade [11,12], but it is currently experiencing renewed interest due to its potential applications across various industrial fields [13,14,15,16], especially in aeronautics. The primary distinction between linear and non-linear modes lies in the dependence of non-linear mode parameters—such as frequency, equivalent damping, and amplitude—on the magnitude of the input signal and the loading history. Understanding the mechanisms behind the dissipative properties of timber structures, particularly within their connections, is crucial for promoting the use of wood in high-rise construction projects. Non-linear modes offer a promising tool for analysing and predicting the dissipative behaviour of timber structures, facilitating the design of more resilient and dissipative buildings.

The dynamics of wood structures has been extensively studied [17,18], often through experimental approaches. Timber structures are frequently modelled as linear systems, despite the well-known fact that damping in these structures strongly correlates with the amplitude of the input signal, as observed in the DynaTTB project [19]. This project aimed to model complex high-rise timber buildings through experimental measurements. However, to the best of the authors’ knowledge, non-linear modes have not yet been clearly identified in this type of structure.

This study investigated the dynamics of bracing frames with bolted connections, referred to as frames. Three identical frames were tested on a shaking table, primarily using sine-swept excitation signals. Non-linear modes were identified, and the origins of these phenomena are discussed.

2. Experimental Tests

The frames were two-dimensional braced frames representing timber structures, with an added mass of 1.5 tonnes equally distributed along the upper horizontal beam (Figure 1a and Figure 2). The main characteristics of the structure are provided in Appendix A. The connections of the structure were made with 12 mm bolted connections and steel plates (Figure 1b). The bolts were force-fitted into the wood but had clearances in the steel panel holes. The structure was designed to ensure that damage occurred only in the diagonal joints.

The tests were conducted on the unidirectional shaking table at FCBA. The braced test frame was rigidly fixed to the table (Figure 2). A horizontal metal plate was welded to the base joint and clamped to the shaking table with four bolts, each 16 mm in diameter. A steel support structure was positioned on each side of the braced frame to guide its movement and prevent lateral collapse. Several boxes filled with lead bags were attached to the top bar to simulate the load of upper stories (one on the top and two on the sides of the structure). For safety, the top box was secured to the bridge crane with a steel chain.

The choice of a two-dimensional structure was made for economic reasons. Two pieces of Teflon were placed at the top of the support structure, a few centimetres apart, to prevent the major part of the transversal vibration (perpendicular to the load direction) and to minimise friction. However, the residual friction could affect the overall damping of the structure. Then, the structure was much more flexible in the transverse direction than in the load direction, so we assumed that the transverse and longitudinal bending modes were uncorrelated.

Accelerometers were installed at various points on the structure, particularly at the edges of the specific joints (Figure 3). The displacements of the upper point of the structure were measured using a wire sensor, while the displacement, acceleration, and force of the shaking table actuator were monitored. All the sensors measured values in the longitudinal direction, except 1 and 3, which were three-dimensional. The transverse acceleration was one order of magnitude lower than the longitudinal acceleration.

3. Linear Modal Analysis

The dynamic properties of the bracing frame were identified using two modal analysis methods. The primary objective of this study was to determine the structure’s modal frequency and damping. First, the frame was subjected to white noise excitation. Then, an experimental modal analysis was performed using an impact hammer. These two methods are compared in this section.

A linear finite element model was developed to compare with the experimental modal analysis results. The bracing frames were assumed to be clamped at the base, and the joints were modelled as linear springs with an experimentally measured stiffness (longitudinal stiffness $k_s=80$ kN/mm). The other assemblies were considered as toggle joints.

By computing a modal analysis with the finite element software Cast3M-2021 [20], a frequency of 8 Hz was found for the first bending mode and about 16 Hz for the second bending mode. A comparison between the experimental and numerical mode shapes is shown in Figure 4.

3.1. White Noise

The white noise signal excited the entire frequency range from 0.5 Hz to 20 Hz, with a consistent shaking table acceleration magnitude. The frequency response function (FRF) between the shaking table and the top of the structure (accelerometers 1, 2, and 3 in Figure 3) is plotted in Figure 5. In the case of a linear structural response, several well-defined peaks would be expected. Instead, a very damped and noisy frequency around 5 Hz was observed.

The first bending mode expected at about 8 Hz was not detectable at all on the transfer function. The second flexural mode, expected near 16 Hz, corresponded to a peak in Figure 5 but had a very small amplitude. The structure dissipated most of the input signal’s energy, and the damping phenomenon primarily affected the lower frequencies. To gain further insight, an additional experimental modal analysis was conducted to evaluate the linear modes of the structure.

3.2. Experimental Modal Analysis

Experimental modal analysis (EMA) was performed using an impact hammer and PULSE-23 [21] software. Accelerometers were positioned along the frame, and several impacts were applied using an instrumented impact hammer (Figure 3). For each position, five hammer strikes were executed perpendicularly to the beam, and the mean transfer function was recorded. Each impact was required to have a roughly equivalent magnitude (700 N). Mode identification by the software was performed using the Rational Fraction Polynomial Method [22] with the help of a stability diagram (Figure 6). The stability diagram method is an efficient approach for identifying multiple modes. It is constructed by progressively increasing the number of assumed modes and applying curve-fitting methods to identify modal frequencies from a set of frequency response functions (FRFs). Modes whose characteristics remain consistent across iterations are considered “real” modes. Additionally, the CMIF (Complex Mode Indication Function) was used to facilitate the identification of the system’s experimental eigenvalues [23]. This study focused on the first four modes, as they were the only ones clearly identified through the experimental modal analysis.

The EMA using the stability diagram analysis identified two main resonance frequencies: $f\approx 8$ Hz and $f\approx 16$ Hz. These were also clearly identifiable on the impact response transfer function (Figure 7). At higher frequencies, the modal identification became noisy. Impact 1 occurred at the top of the left post and impact 3 at the centre of the horizontal beam, which explained the difference in magnitude. However, it did not change the modal identification.

These results contradict the white noise tests (Figure 5), but were globally consistent with the numerical linear finite element model. For a linear structure, the white noise test and experimental modal analysis were expected to yield the same resonance frequencies. The experimental mode shapes were compared with the simplified linear model of the braced frame, showing strong similarity. The MAC is a widely used indicator to compare numerical and experimental mode shapes with values ranging from 0 (fully uncorrelated) to 1 (fully correlated) [24]. The modal assurance criterion (MAC) was approximately 1 for mode 1 (

Table 1. Frequency and damping of experimentally identified modes using experimental modal analysis compared with the numerical frequency.

Mode	Exp. Frequencies (Hz)	Exp. Damping	Num. Frequency	MAC
1	7.3	1.8	8	0.99
1	7.6	7.3	∅	∅
2	15.2	0.9	∅	∅
3	16.3	2.1	16	0.15

). However, the MAC was very low for the other modes, likely due to the non-linear behaviour of the structure and the limited number of measurement points.

A particular finding of the experimental modal analysis was the presence of two very close, nearly identical frequencies for the first flexural mode (Table 1, 7.3 Hz and 7.6 Hz), but with significantly different damping ratios (1.6 and 7.3%). This characteristic was observed previously in non-linear modes [16], suggesting a non-linear behaviour in the structure. Further analyses were conducted to gain a deeper understanding of this phenomenon.

4. Sine-Swept Analysis

4.1. Frequency Response Function

Non-linear dynamics in structures can be accurately identified by applying a logarithmic sine sweep at the base [16,25], provided that the sweep rate is low [26]. As the first resonance frequency was not identified with the white noise test, it was decided to use a displacement constant sinusoidal sweep that produced an acceleration that grew with the frequency of the signal (see “shaking table” signal in Figure 8 and Equation (1)). The sweep rate was fixed to 1 octave/minute from the frequency of 1 Hz.

$$\begin{array}{c}s=Asin\left(2\pi f_1\tau (e^{t/\tau }-1)\right)\\ \tau ={\displaystyle \frac{T}{log\left(\frac{f_1}{f_2}\right)}}\end{array}$$

(1)

where A is the amplitude of the signal as a displacement, $f_1$ and $f_2$ are the initial and final frequencies, and T is the duration of the signal.

Three frames with identical geometry, as specified in Appendix A, were tested. These frames differed due to the variability in the wood’s mechanical properties and differences in clearances and tolerances. During testing, a resonance was clearly identified at approximately 3 Hz (Figure 8), with amplified vibrations at the top of the structure and increasing noise. For frequencies above 5 Hz, the structure seemed to “filter out” the vibrations generated by the shaking table. The upper part of the structure, including the added mass, had almost zero displacement. The movement remained unidirectional, with negligible acceleration perpendicular to the plane of the frame. Figure 8 shows the acceleration of the frame’s upper part in relation to the frequency of the shaking table’s input signal.

The sweep direction influenced the structure’s dynamics. For a table displacement amplitude of $\pm 0.5$ cm, the signal amplification was maximal around 4 Hz during the upward sweep, whereas it peaked around 3 Hz during the downward sweep. During the downward sweep, a sudden jump in the signal deviated from the symmetrical response typical of linear oscillators. Additionally, the shape of the output signal and the resonance frequency were highly dependent on the input signal’s amplitude. These characteristics are indicative of a non-linear mode, as demonstrated by Noël and Kerschen in aeronautical applications [16] and predicted by theory [11,27]. The frequency response function (FRF) of three frames for the three input signal amplitude is shown in Figure 9. The output signal shape varied with magnitude: at low amplitudes, the structure exhibited a softening behaviour, with the backbone curve connecting the amplitude peaks sloping towards lower frequencies. At higher amplitudes, the magnitude peak flattened. This change in behaviour was observed by Renson [15] in a two-degrees-of-freedom non-linear system with clearance, further confirming the non-linear behaviour of the structure.

The non-linearity was more pronounced in the first frame than in the others. Several tests (white noise, harmonic motion, experimental analysis) were conducted on this first frame before the sine sweep test, likely causing slight pre-existing damage, particularly in the clearances within the joint.

Two joints of the structure were instrumented. Accelerometers were positioned on both edges of the two connectors of the bottom part of the left pole (accelerometers 8, 9, 10, and 11 in Figure 3). The purpose of these measurements was to detect a potential lack of transmissibility in the joint. Transmissibility is defined as the ratio between the acceleration signal on each side of a joint. The transmissibility remained globally equal to one in the frequency interval of the input signal, despite a very small decrease around the frequency of the non-linear mod (Figure 9).

For each point of the structure, the maximum amplifications of the excitation signal were very limited (between magnitudes 3 and 7), corresponding to equivalent linear damping between 7 and 15%, which was much more significant than the damping measured on a mid-rise wooden building (1.5 to 3%) [5]. The equivalent linear damping was calculated using the 3 dB method. Thus, this structure was less sensitive to vibration than its idealised linear equivalent in the frequency range of the input signal. However, as the transmissibility of the assemblies was close to 1 at low frequency, other phenomena were involved in the non-linearity of the system. Time–frequency analysis was carried out to understand them.

4.2. Time–Frequency Analysis

Time–frequency analysis was performed using the EasyMod toolbox [28]. A Fast Fourier Transform (FFT) of the signal was computed over small periods of time and the acceleration magnitudes of the response were displayed with different colour intensities in dB relative to the maximum acceleration. A rectangular window was used to avoid attenuating meaningful data. Figure 10 shows the time–frequency plots of frame 3 for a magnitude of the input signal of 2.5 cm, Figure 11 shows the time–frequency diagrams of frame 3 for a magnitude of the input signal of 2.5 cm, and Figure 10 presents the time–frequency graphs for the three frames for the three levels of amplitude. As mentioned before, frame 1 had been subjected to different tests before the sine-swept experiments, and its dynamical properties might have been altered (damages on joints). Furthermore, the sample rate was four times lower for frame 1 (256 Hz) than for frames 2 and 3 (1024 Hz).

The input signal was a logarithmic sine sweep that rose and fell between 1 and 10 Hz according to Equation (1). This explains the arrow shape on each graph for low frequencies. Several harmonics at higher frequencies were also visible. The time–frequency diagrams display three domains: a linear zone, a transition zone, and a non-linear zone (Figure 10). When the frequency of the input was lower than the frequency of the non-linear mode (for a time less than 100 s or greater than 320 s in Figure 10), the behaviour of the structure appeared linear. In these zones, the energy remained mainly at the frequencies of the input signal. The input signal also slightly activated the first linear mode of the structure at 8 Hz and 16 Hz (see Figure 10).

When the input signal frequency exceeded the frequency of the non-linear mode, the higher frequencies of the structure were strongly excited, reaching several hundred hertz, particularly the harmonics of the input signal. This effect became more pronounced as the input signal amplitude increased. Experimentally, when the structure vibrated in this frequency range, it emitted a loud noise that resembled numerous impacts. Consequently, the linear resonances at 8 Hz and 16 Hz were indistinguishable in this zone.

Between the linear and non-linear zones, a transition zone could be observed in some cases, particularly when the input signal had a high amplitude (Figure 10). This zone corresponded to the peak amplitude during the non-linear mode resonance. The structure’s acceleration appeared particularly chaotic in this zone. The maximum amplitude, which corresponded to frequencies between 30 Hz and 100 Hz, was ten to thirty times greater than the input signal amplitude. While all the frames exhibited similar dynamics, frame 2 displayed sub-harmonic resonance, which is discussed in the next subsection.

4.3. Sub- and Super-Harmonic Resonance

An interesting phenomenon occurred with frame 2 during the sine sweep tests at low amplitudes. For input frequencies below 5 Hz, the frame behaved similarly to the others, with a peak in the non-linear mode around 3 Hz. However, when the input signal reached 6 Hz, the structure began oscillating at half the input signal frequency, with a significant increase in amplitude. This phenomenon is visible in the temporal evolution of the displacement in Figure 12. The sub-harmonic resonance remained prominent in the frequency range between 6 and 8 Hz when the amplitude of the input signal was equal to $\pm 0.5$ cm or $\pm 1$ cm.

In the first part of Figure 12, the structure displacement followed the frequency of the input signal (5.9 Hz to 6 Hz). A brief transition phase (6 Hz to 6.1 Hz) was then observed, during which an asymmetry emerged in the structure’s displacement. For frequencies above 6.1 Hz and up to approximately 8 Hz, the structure’s displacement frequency stabilised at half the input signal frequency, and a resonance occurred: the structure’s displacement amplitude surpassed the input signal amplitude. The time–frequency diagrams in Figure 11 also clearly illustrate this phenomenon: for tests on frame 2, distinct lines appeared starting at 150 s, corresponding to a frequency equal to half the input signal frequency.

This phenomenon also occurs in rotor systems in the presence of clearances and/or defects [29,30]. Non-linear terms in the dynamics equation cause these resonances and could be due to Hertzian contact [29] or stiffening, biasing, and smoothing effects [31]. Due to the wide tolerance (1 mm) on the gaps in the frames, it could be understood that this phenomenon was only observed for frame 2.

5. Discussion

The time–frequency graphs (Figure 11) revealed a threshold phenomenon, suggesting that friction might be involved. Additionally, the high-frequency content observed in the non-linear zone, along with the noises heard during the tests, indicates significant contacts and impacts within the assemblies. Since the bolts were free to move within the clearances of the steel plates, there may have been adherence between the bolts and the steel plates at low accelerations or velocities. Under these conditions, the assemblies behaved like stiff linear springs, which resulted in the overall linear behaviour of the structure. During the experimental modal analysis, it could be assumed that adherence occurred in the assemblies due to the low and rapid loads applied. Consequently, the modes identified during these tests aligned with the theoretical modes predicted by a linear model.

The main hypothesis used to explain the non-linear zone is that beyond a certain level of acceleration, the bolts began to slide within the steel plate holes. In this state, the assemblies behaved like non-linear springs influenced by friction and clearances, which significantly altered the overall dynamics of the structure. The transition zone observed in the time–frequency graphs corresponded to the shift from adherence to sliding, which generated chaotic motion—a stick–slip phenomenon [32]. After this transition, the motion of the structures was mainly due to the harmonics of the input signal, suggesting that the whole dynamics of the structure could be modelled. Further studies should be carried out to understand the behaviour of the structures in the amplitude and frequency domains.

The bolts can be represented as cylinders in a bearing with clearance and friction. This type of system is common in machine theory [33]. Friction mechanisms can produce sub-harmonic resonances. As demonstrated by Sakai and Kim [34,35], the difference between static and dynamic friction coefficients can trigger stick–slip behaviour, causing the system to transition between states. This generates localised negative damping, which leads to resonance. This phenomenon could explain the sub-harmonic resonance observed in frame 2.

In an experiment conducted at the University of Turin by Anastasio and et al. [36,37], a non-linear system with friction and cubic or quadratic stiffness was developed. The system was analysed using a logarithmic frequency sweep (both sweep up and sweep down) with a constant displacement amplitude. The resulting spectrogram exhibited features similar to those observed in our wooden frame tests, including a chaotic limit at specific frequencies and evidence of sub-harmonic resonance. The frequency response of this experiment also closely mirrored that of the wooden frame, with both displaying a softening effect and a noticeable difference between the sweep-down and sweep-up responses.

Timber frame structures are complex due to the heterogeneity of the wood, the assembly of multiple mixed materials, gaps, friction, and local forces. Nevertheless, their behaviour is similar to a simpler structure for which a theoretical study exists. An analytical and/or numerical model seems feasible to predict and model the non-linear modes of timber frames.

Due to the complexity of the structure studied, sub-harmonic resonances were only visible in frame 2. It can be assumed that for the other two frames, the variability in the properties of the assemblies hid this phenomenon. Furthermore, for an amplitude of the signal of $\pm 2.5$ cm, it can be assumed that the internal shock in the assemblies had a dominant effect compared with the friction.

6. Conclusions and Future Work

This study highlighted the complex dynamics of a wooden structure with bolted assemblies. Experimental modal analysis suggested that the structure behaved linearly during free oscillations. However, normal modes were not observed when the structure was subjected to forced oscillations.

Non-linear modes were experimentally identified using sine-sweep tests on a shaking table. These tests revealed several characteristics of non-linear modes, including jumps, sensitivity to amplitude, and sensitivity to sweep direction. The primary indicator for the non-linear behaviour of these structures was the presence of clearances and friction within the bolted assemblies. Similar behaviour was observed in experiments in the literature that involved friction and gaps, but to the author’s knowledge, this is the first time it has been identified in a timber structure. It is clear that the number of samples was insufficient to assess the effect of the variability. A larger number of tests should be carried out on smaller specimens (e.g., on a single joint) to determine these effects.

These structures have promising applications in civil engineering. Non-linear modes “filter out” normal modes and result in a weak amplification of the input signal. By leveraging the non-linear characteristics of timber assemblies, it may be possible to significantly enhance the energy dissipation capacity of timber structures, thereby improving their ability to withstand earthquake- or wind-induced loads. Moreover, for frequencies above the non-linear mode frequency, the energy from earthquakes or wind is transferred to high frequencies, resulting in minimal displacements, stresses, and disturbances (apart from potential acoustic nuisances).

The next phase will involve developing theoretical and numerical models to capture the dynamic non-linearity of the assemblies. This will provide a deeper understanding of the phenomena involved and enable the design of assemblies with optimised dynamic properties.