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When crossing a bridge, most people take for granted that the bridge will remain steady and support them, but history shows that this is not always the case. The first documented pedestrian bridge incident dates back to April 12, 1831 when one of Europe’s first suspension bridges, England’s Broughton Suspension Bridge, collapsed due to dynamical instability induced by marching troops. The prevailing wisdom since is that soldiers should avoid marching in step, in case their stepping frequency might resonate with a natural (vertical) vibration frequency of the bridge. It is now established practice that soldiers are given the command to “break step” upon crossing a bridge to avoid just such a phenomenon. Vertical vibrations of bridges due to random excitation from pedestrians are still of concern, but prior to the year 2000 lateral vibrations were given little attention. This was because, for normal walking, the lateral component of the ground reaction force is an order of magnitude smaller than the vertical component and in the absence of coherence between pedestrians the resulting bridge responses were assumed to be negligible.
The London Millennium Bridge was designed as a collaboration between engineers, architects, and artists, as a very low profile suspension bridge. Without visually intrusive vertical cables, the intention was that the structure would appear from the side to be like a mysterious long blade, spanning the river with little visible support. The unusual geometry of the slender span contributed to the bridge having greater flexibility than most bridges in the lateral direction, giving natural frequencies similar to typical pedestrian stride frequencies, while its relatively low mass also made it susceptible to significant vibrations. There is a widely available video that shows dozens of people rocking from side to side on the London Millennium Bridge’s opening day, seemingly in time with the bridge, which is often used as compelling evidence for pedestrian synchronisation in popular media19. However, we encourage the reader to look again. A distinction needs to be made between synchronisation of head and upper body movements (readily seen in videos) and synchronisation of footfalls on the deck. We are not aware of any video footage that establishes that footfall synchrony occurred. Indeed, there is possible evidence in that video of lack of footfall synchrony, because pedestrian forward velocities vary widely. Moreover, a walker providing an effective negative damping force to the bridge, necessarily at the bridge frequency, will exhibit a component of upper body motion at that frequency.
In fact this same phenomenon of a lateral instability of pedestrian bridges had been seen before, and there is evidence going back to 1972. The complete list of pedestrian bridges that are known to have developed lateral oscillation due to pedestrian motion runs to at least 30 separate examples; see Table 1 for a list of those for which there are detailed scientific reports and Table 2 for others for which quantitative evidence is not available. Note in the final column of these tables the scant evidence for pedestrian synchronisation being observed.
The geography of such crowd-induced instability events is truly worldwide. It includes the massive Bosphorus Bridge linking Asia and Europe35 and an icon of Lower Manhattan, the Brooklyn Bridge which started swaying as a crowd of pedestrians trudged across during the 2003 blackout. When packed shoulder to shoulder with pedestrians, the bridge started vibrating making pedestrians lose balance and feel seasick36. The Brooklyn Bridge repeatedly experienced crowd-induced instabilities during the 2011 protest and 2011 New Year’s celebration34 raising the concern that “Manhattans’s emergency exit”—as the bridge is sometimes called—is not built for crowds.
Coincidentally, one of the more recent examples of lateral pedestrian instabilities is Squibb Park Bridge, also in Brooklyn (it is a city of bridges, after all)37. Opened in 2013, this $3.9-million wooden park bridge was purposefully designed to bounce lightly but over time the increased bouncing and lateral swaying became a safety concern for pedestrians38. Three years after it was initially closed for $2.5-million repairs, the Squibb Park Bridge reopened in April 201739 but was later demolished in 2019 amid concerns of its structural integrity.
While the evidence of bridge instabilities is often anecdotal, some direct measurements of bridge response characteristics are available for recent crowd-induced instability events involving the Toda Park Bridge in Japan40, Solférino Bridge in Paris41, the London Millennium Bridge22, the Maple Valley Great Suspension Bridge in Japan25, Singapore Airport’s Changi Mezzanine Bridge21, the Clifton Suspension Bridge in Bristol, UK13, and the Pedro e Inês Footbridge in Portugal23.
A particularly notable observation was the instability due to crowds returning from an annual hot-air balloon festival across Bristol’s iconic Clifton Suspension Bridge13. Since vibrations of the bridge had been observed during previous crowd events, Macdonald was commissioned by the bridge’s operating trust to fit accelerometers to record the vibrations as the instability occurred. Observations showed that two lateral modes of vibration were excited simultaneously by the large pedestrian crowd, neither of which was tuned to the average walking frequency. Since then, the trust has stipulated that the bridge must remain closed to all pedestrians and other traffic at peak times during the balloon festival.
We have established a general expression for the average contribution to the bridge damping of the interaction force of a single pedestrian over one gait cycle. We have found that this increment σ can be written as the sum of three components (see Methods):
The terms σ2 and σ3 depend on the timing of stepping behaviour of pedestrians in response to the bridge motion. However, in all our simulations, we have found σ1 to be the most important effect in triggering large-amplitude vibrations (see the Supplementary Information). This effect is perhaps counter-intuitive, since it may be imagined that, in the absence of phase synchrony between the bridge and pedestrian, the lateral foot force on the bridge would average to zero. However, this is not the case; see Fig. 1 for a detailed explanation.
The figure contrasts the force transmitted to the bridge by two identical pedestrians who, when they simultaneously place their stance foot on the bridge (at the light blue and light red positions in an absolute co-ordinate frame), have equal and opposite gaits. As they place their feet, the lateral component of the foot force from each pedestrian is equal and opposite, so there is no net lateral force on the bridge. Suppose that during a time increment Δt the bridge moves to the left, so that the blue figure’s leg decreases its angle to the vertical within the frontal plane, whereas the red figure’s leg angle increases. Thus, during this bridge motion, the magnitude of the lateral component of the red figure’s lateral foot force increases whereas that of the blue figure decreases. Thus there is, on average, a change in resultant force in the direction of the bridge’s motion. Nevertheless, there can be large variations depending on a pedestrian’s foot placement strategy (see Figs. 5 and 6).
The figure contrasts the force transmitted to the bridge by two identical pedestrians who, when they simultaneously place their stance foot on the bridge (at the light blue and light red positions in an absolute co-ordinate frame), have equal and opposite gaits. As they place their feet, the lateral component of the foot force from each pedestrian is equal and opposite, so there is no net lateral force on the bridge. Suppose that during a time increment Δt the bridge moves to the left, so that the blue figure’s leg decreases its angle to the vertical within the frontal plane, whereas the red figure’s leg angle increases. Thus, during this bridge motion, the magnitude of the lateral component of the red figure’s lateral foot force increases whereas that of the blue figure decreases. Thus there is, on average, a change in resultant force in the direction of the bridge’s motion. Nevertheless, there can be large variations depending on a pedestrian’s foot placement strategy (see Figs. 5 and 6).
The expressions for σ1–σ3 should be evaluated individually for each pedestrian i and will depend on that pedestrian’s stride frequency ωi as well as the vibration frequency Ω of the bridge in the mode in question. Thus, we can write the total effective damping coefficient cT of the bridge with N pedestrians as
where c0 is the coefficient of natural (passive) damping of the bridge, \(\overline{\sigma }(\overline{\omega },{{\Omega }})\) is the average damping coefficient per pedestrian, and \(\overline{\omega }\) represents the mean pedestrian stride frequency.
We have found, over large ranges of pedestrian and bridge frequencies, that \(\overline{\sigma } \, < \, 0\) on average. Imagine a thought experiment in which pedestrians are added to a bridge deck one by one, then when we reach a critical number
of pedestrians, the overall modal damping cT of the bridge will become negative. Negative damping will cause the amplitude of the bridge vibration mode to grow exponentially.
To test this theory we have performed simulations on three different mathematical models describing a number of pedestrians coupled with a lateral bridge mode (see Methods for model descriptions). In each case we take a parsimonious assumption, justified in the relevant literature, that walking is fundamentally a process in which the stance leg acts as a rigid strut, causing the body centre of mass (CoM) to act like an inverted pendulum in the frontal plane18,31,42,43 during each footstep. Rather than fall over, the step ends when the other leg strikes the ground and, ignoring the brief double-stance phase seen in realistic gaits, the pedestrian switches to an inverted pendulum on that leg. We consider a single lateral vibration mode of the bridge, forced by the motion of N pedestrians walking in a direction perpendicular to this vibration. Any interaction between pedestrians other than indirectly through the bridge motion is ignored.
The modelling and simulation process are illustrated schematically in Fig. 2. We have simulated three different variants of the pedestrian model. Model 118,31 is the simplest, based on linearising the inverted pendulum in the frontal plane for small angles. It assumes the sagittal-plane dynamics is independent of the lateral foot position and that foot transitions occur at regularly spaced prescribed times. At each transition the new lateral foot position is governed by a biophysically inspired control law44 that enhances stability during horizontal ground motion. Model 2 is a new adaptation of Model 1, in which the timing of the foot placement alters as a kinematic consequence of the lateral bridge motion and foot placement. Finally, Model 320,45 assumes that the step timing is determined solely by the frontal-plane dynamics and that leg transition occurs each time the pedestrian CoM passes through a reference position defined as zero lateral displacement. A nonlinear feedback mechanism enables stable limit cycle motion in the absence of ground movement, and quasi-periodic motion on sinusoidally moving ground.
a Simulations are run for a coupled bridge-pedestrians system with pedestrians added sequentially at fixed time increments Tadd apart. The addition of the nth pedestrian (n = Ncrit) causes the overall damping coefficient to become negative hence the amplitude of motion to increase rather than diminish. b Inverted pendulum model of bridge mode and pedestrian lateral motion. Here, y is the lateral position of the pedestrian’s centre of mass (CoM), while p defines the lateral position of the centre of pressure (CoP) of the foot, both relative to the bridge. L is the equivalent inverted pendulum length and m is the pedestrian mass. The displacement x of the bridge in a lateral vibration mode is represented by an equivalent platform with mass M, spring constant K and damping coefficient C. \(\tilde{H}\) is the lateral component of the pedestrian’s foot force on the bridge deck. In return, the bridge motion causes an inertia force \(-m\ddot{x}\) on the pedestrian’s centre of mass. The pedestrians are depicted as “crash test” dummies with flexible hips; however, the actual inverted pendulum model is simpler, with pendulum-like legs connecting to the CoM.
a Simulations are run for a coupled bridge-pedestrians system with pedestrians added sequentially at fixed time increments Tadd apart. The addition of the nth pedestrian (n = Ncrit) causes the overall damping coefficient to become negative hence the amplitude of motion to increase rather than diminish. b Inverted pendulum model of bridge mode and pedestrian lateral motion. Here, y is the lateral position of the pedestrian’s centre of mass (CoM), while p defines the lateral position of the centre of pressure (CoP) of the foot, both relative to the bridge. L is the equivalent inverted pendulum length and m is the pedestrian mass. The displacement x of the bridge in a lateral vibration mode is represented by an equivalent platform with mass M, spring constant K and damping coefficient C. \(\tilde{H}\) is the lateral component of the pedestrian’s foot force on the bridge deck. In return, the bridge motion causes an inertia force \(-m\ddot{x}\) on the pedestrian’s centre of mass. The pedestrians are depicted as “crash test” dummies with flexible hips; however, the actual inverted pendulum model is simpler, with pendulum-like legs connecting to the CoM.
We choose parameters based on the set of controlled experiments on the London Millennium Bridge prior to reopening22. Up to N = 275 pedestrians were added individually at equally spaced time intervals Tadd. We performed our simulations for two different choices of pedestrian addition times Tadd = 20 s and Tadd = 10 s. These choices are consistent with the incremental pedestrian loading tests on the London Millennium Bridge22 and simulations conducted by Ingólfsson et al.46 in which pedestrians were added at average intervals of 7 and 12 s, respectively.
The pedestrian parameters are drawn from distributions (see Table 3) and multiple simulations are run for different bridge and mean pedestrian frequencies. The number of pedestrians at which the vibration amplitude begins to increase rapidly is noted for each simulation. Representative results for Tadd = 20 s are depicted in Fig. 3, with further results in the Supplementary Information (see Supplementary Fig. 1 for faster pedestrian addition time Tadd = 10 s and Supplementary Fig. 2 for the worst-case scenario of complete resonance).
See Methods for model details and parameter values. (Top row): Bridge vibration amplitude as a function of number of pedestrians N. The left-hand boundary of the pink shaded portion indicates the value Ncrit where cT crosses zero, and the blue shaded portion is where a degree of synchrony is observed. Insets show illustrative bridge x(t) (black) and a few representative pedestrian y(t)−p(t) (coloured) oscillations over three cycles. (Middle row): Computation of the total bridge damping cT given by Eq. (1) and the Kuramoto order parameter r given by Eq. (3) calculated for the phases of pedestrians’ CoP (Models 1 and 2) and CoM (Model 3). (Bottom row): instantaneous computed bridge and pedestrian foot placement frequencies. a Simulations of Model 1 which cannot synchronize. b Simulations of Model 2 which permits weak synchronization. c Simulations of Model 3 with strong propensity for synchronization.
See Methods for model details and parameter values. (Top row): Bridge vibration amplitude as a function of number of pedestrians N. The left-hand boundary of the pink shaded portion indicates the value Ncrit where cT crosses zero, and the blue shaded portion is where a degree of synchrony is observed. Insets show illustrative bridge x(t) (black) and a few representative pedestrian y(t)−p(t) (coloured) oscillations over three cycles. (Middle row): Computation of the total bridge damping cT given by Eq. (1) and the Kuramoto order parameter r given by Eq. (3) calculated for the phases of pedestrians’ CoP (Models 1 and 2) and CoM (Model 3). (Bottom row): instantaneous computed bridge and pedestrian foot placement frequencies. a Simulations of Model 1 which cannot synchronize. b Simulations of Model 2 which permits weak synchronization. c Simulations of Model 3 with strong propensity for synchronization.
For each simulation, we numerically validate our general expression (1) for the total effective damping cT by calculating \({\sigma }_{1}^{(i)},\) \({\sigma }_{2}^{(i)},\) and \({\sigma }_{3}^{(i)}\) for each pedestrian i via (9)–(11). We also compute the Kuramoto order parameter10 r, defined using
where φi is the numerically calculated phase of the ith pedestrian’s CoM or CoP (the distinction is made in Fig. 3), ψ is the average phase, and 〈 ⋅ 〉 denotes time average. Note that r = 1 implies complete synchrony, and r = 0 implies uncorrelated motion.
The simulations in Fig. 3 show how the onset of large amplitude bridge motion coincides with when the computed cT becomes negative, at N = Ncrit. For Model 1, in which there is no adjustment to the gait frequency, the bridge’s vibration amplitude grows unrealistically without bounds. In contrast, for Model 3, the onset of moderate amplitude motion starts a process of increased coherence (or phase pulling15) between the pedestrians’ and bridge motion. The order parameter and inset sample solution traces indicate that increased synchrony then occurs between each pedestrian and the bridge. The amplitude of bridge vibrations then saturates. Model 2, which is a more realistic version of Model 1 for higher than moderate amplitude of bridge motion, shows similar amplitude saturation and coherence after instability occurs. Further simulations of Models 2 and 3 for different frequency parameters show that instability is at approximately N = Ncrit defined by (2), leading to a varying amount of synchrony as the amplitude grows. Thus, the negative-damping criterion can be understood as the cause of instability in all cases. Also, the varying degrees of synchrony are a consequence, not the cause of the instability.
Note that the previous analysis20 of the London Millennium Bridge instability based on Model 3 predicted the critical crowd size but, with some caveats, supported the synchronisation hypothesis. However, this analysis was performed for fixed crowd sizes such that a fixed number of pedestrians were placed on the bridge and the system was integrated for a sufficiently long time. Then, the crowd size was increased, and the simulations were repeated again. The key difference between these previous results20 and our paper is that despite the strong propensity of Model 3 for synchronisation, our results demonstrate that bridge instability occurs prior to the onset of crowd synchrony when pedestrians are added sequentially and the crowd size gradually increases in time, as in the controlled experiment on the London Millennium Bridge22. The previous work17 also studied the London Millennium Bridge instability for fixed crowd sizes. In particular, this work used an energy-optimised pedestrian model with a linear feedback controller to demonstrate that heterogeneous pedestrians incapable of synchronising even at large crowd sizes can shake the bridge without synchronisation17. This effect was also reported in an earlier paper by Baker28 and described for Model 1 in Macdonald18. Remarkably, our results indicate that pedestrians with a weak (Model 2) or strong (Model 3) propensity for synchronisation can first initiate the bridge vibrations at a critical crowd size and then become synchronised at larger crowd sizes when added sequentially (also see the extreme case of identical pedestrians in Supplementary Fig. 2 in the Supplementary Information).
A natural question is to seek to understand how the negative-damping coefficient depends on bridge and mean pedestrian stride frequencies Ω and \(\overline{\omega }\), and whether it can be enhanced or suppressed by resonance effects. Figure 4 shows the results of many ensemble runs. For each model we show in an upper plot the computed value of \(\overline{\sigma }\) as a function of the ratio \({{\Omega }}/\overline{\omega }\) of bridge to average pedestrian frequency.
Simulations of Models 1 and 2 (a) and 3 (b) indicate the range of frequency ratio \([{{\Omega }}/\overline{\omega }]\) in which \(\overline{\sigma }\) is negative so that a single pedestrian, on average, contributes to bridge instability. Each ratio of \([{{\Omega }}/\overline{\omega }]\) corresponds to different combinations of Ω and \(\overline{\omega }\) (blue dots). Black dotted lines indicate the average of \(\overline{\sigma }\) and Ncrit for a given ratio. The red curve indicates the 5th percentile of the \(\overline{\sigma }\) distribution. The green curve is the analytical expression (36) for \(\overline{\sigma }\) (top plot) and analytical estimate (37) for Ncrit (bottom plot), given in the Supplementary Information and calculated for Model 1 with identical pedestrians with fixed ω = 5.655 rad/s and S.D. = 0. The magenta dot corresponds to the initial ratio \([{{\Omega }}/\overline{\omega }]\) used in Fig. 3, the yellow dot corresponds to \({{\Omega }}/\overline{\omega }=1\). See the Supplementary Information for the details of the calculations.
Simulations of Models 1 and 2 (a) and 3 (b) indicate the range of frequency ratio \([{{\Omega }}/\overline{\omega }]\) in which \(\overline{\sigma }\) is negative so that a single pedestrian, on average, contributes to bridge instability. Each ratio of \([{{\Omega }}/\overline{\omega }]\) corresponds to different combinations of Ω and \(\overline{\omega }\) (blue dots). Black dotted lines indicate the average of \(\overline{\sigma }\) and Ncrit for a given ratio. The red curve indicates the 5th percentile of the \(\overline{\sigma }\) distribution. The green curve is the analytical expression (36) for \(\overline{\sigma }\) (top plot) and analytical estimate (37) for Ncrit (bottom plot), given in the Supplementary Information and calculated for Model 1 with identical pedestrians with fixed ω = 5.655 rad/s and S.D. = 0. The magenta dot corresponds to the initial ratio \([{{\Omega }}/\overline{\omega }]\) used in Fig. 3, the yellow dot corresponds to \({{\Omega }}/\overline{\omega }=1\). See the Supplementary Information for the details of the calculations.
Note that Models 1 and 2 are effectively identical for small amplitude bridge motion. For Model 1, McRobie47 derived an exact analytic expression for \(\overline{\sigma }\) (shown as the green curve in the top panel of Fig. 4a). At the resonance condition where \(\overline{\omega }={{\Omega }}\) (represented by the yellow dot), the theory47 predicts a large range of \(\overline{\sigma }\)-values, depending on the relative phase between the bridge and pedestrian. The hypotheses behind our general calculation of \(\overline{\sigma }\) fail precisely at this resonance (see the Supplementary Information). The simulation results for Model 2 (represented by the blue dots), show features of large negative values of \(\overline{\sigma }\) just below \({{\Omega }}/\overline{\omega }=1\) and large positive values slightly above. These are believed to be due to the adaptation of the step timing, in response to perturbations from bridge motion, giving similar effects as previously found numerically for an inverted pendulum walking on a vertically oscillating structure48 and experimentally for subjects walking on a laterally oscillating treadmill15.
Also observe the paucity of data in certain regions of the lower panel of Fig. 4b and the apparent bi-modality of the data. This is because, for Model 3, limit cycle pedestrian motion is an emergent property of the simulations, rather than essentially an input parameter as it is for Models 1 and 2. Also note this model is liable to hysteresis between limit cycles of different period45.
For all three models, we find the average value of \(\overline{\sigma }\) to be mostly a function of the frequency ratio, being only a weak function of the pedestrian or bridge frequencies independently. Using this value in Eq. (2) gives the predicted critical number Ncrit of pedestrians required to trigger an instability. The lower plots indicate the success of this prediction, by comparing it with the value of N at which the vibration amplitude begins to increase rapidly in the simulations.
Also note the large spread of the model outputs for both \(\overline{\sigma }\) and Ncrit, especially for Model 2. Our theoretical calculations only consider the long term averages of the effective damping coefficient cT. This is only part of the story, because true walking behaviour is transient and involves changes to the trajectory of the walker’s CoM and the foot placement strategy. On stationary ground, a walker’s CoM will oscillate laterally with a dominant component at half the footfall frequency. Without changing the footfall frequency, the platform motion introduces a second frequency inducing the walker to adopt a two-frequency quasiperiodic pattern of footfall placement (Fig. 5). Depending on the phase of this quasiperiodic pattern, we have found that pedestrians can show large deviations from the long term average (see next section).
Panel a is for a stationary platform, while panels b and c are for a bridge oscillating at 6 mm amplitude at 0.4 Hz, with walkers adopting Hof et al.’s44 balance laws based on relative and absolute velocity, respectively. The bridge motions induce quasiperiodic placement patterns. The walker’s centre of mass and the bridge displacements are shown in red and green, respectively. The lower panels show the corresponding forces applied to the bridge. Walker parameters: m = 74.4 kg, fwalk = 0.86 Hz, L = 1.2 m, b = 15.7 mm.
Panel a is for a stationary platform, while panels b and c are for a bridge oscillating at 6 mm amplitude at 0.4 Hz, with walkers adopting Hof et al.’s44 balance laws based on relative and absolute velocity, respectively. The bridge motions induce quasiperiodic placement patterns. The walker’s centre of mass and the bridge displacements are shown in red and green, respectively. The lower panels show the corresponding forces applied to the bridge. Walker parameters: m = 74.4 kg, fwalk = 0.86 Hz, L = 1.2 m, b = 15.7 mm.
Nevertheless, for all three models, note that Ncrit is minimised not when there is a frequency match between the pedestrian and bridge frequencies, \({{\Omega }}/\overline{\omega }=1\), but when the pedestrian frequency is less than the bridge frequency, \({{\Omega }}/\overline{\omega }\approx 1.3\) for Models 1 and 2 and \({{\Omega }}/\overline{\omega }\approx 1.1\) for Model 3. Notice the red 5th percentile curves in Fig. 4 (top row) that indicate that negative damping can be observed at any frequency in the considered range of frequency ratios. Note too that there are some frequency ratios for which \(\overline{\sigma }\) is positive. If pedestrians walked at those frequencies, then their motion would enhance that bridge mode’s stability rather than reduce it.
An explanation of this frequency dependence can be summarised as being a question of timing. The argument in the caption of Fig. 1 implicitly assumes that the bridge is moving in a single direction during each step and that the bridge and pedestrian stride frequencies are similar. Particular tunings of this frequency ratio can in fact lead to a reversal of the effect in Fig. 1. Nevertheless, over the frequency range considered, both the size of the regions of pedestrian-induced negative damping and its average value greatly outweigh that of positive damping.
Figure 1 explains how bridge motion breaks the symmetry of the loading applied by mirror-imaged walkers such that long-term averages need not equal zero. That is only part of the explanation, as it does not consider the motion of the walkers’ centres of mass nor the various foot placement strategies that may be adopted to maintain balance. In principle, the foot placement as defined by Hof et al.44 is dependent on the lateral velocity of the pedestrian’s centre of mass. However, uncertainty remains as to whether the velocity should be defined in reference to the oscillating bridge (relative velocity) or a stationary point against which the bridge is moving (absolute velocity). Therefore, Fig. 5 shows results from Model 1 for both of these conditions.
The corresponding forces applied to the bridge in these three cases are also shown in Fig. 5. Since bridge motions are small, the forces are similar in all three cases. By taking the difference in forces, Fig. 6 highlights the small change in the applied forces that are the result of the bridge motion, and correlates these with bridge velocity. The walker adopting the relative velocity control law creates forces which are negatively correlated with bridge velocity, leading to a positive damping effect. By contrast, the additional forces generated by the walker adopting the absolute velocity balance law are positively correlated with the bridge velocity, leading to the negative damping effect which feeds energy into the bridge.
The bridge velocity is shown in red. Lower panels: the correlation between the bridge velocity and the induced forces. The red lines indicate the average effective damping coefficient \(\overline{\sigma }\). Panel a corresponds to panel b in Fig. 5; panel b corresponds to panel c in Fig. 5.
In summary, the bridge motions cause the walkers to adjust their foot placements which induces small quasiperiodic forces which have a component at the bridge frequency. Depending on the balance law adopted (and the frequency of bridge motion and other parameters), the phases of these additional forces can either add or extract energy to/from the bridge.
Experimental evidence is limited as to which balance law is more realistic for a walker on a moving platform, but the laboratory experiments augmented with Virtual Reality by Bocian et al.15 provide some evidence for the absolute velocity control law. Walkers following either law could be present on the bridge. Also, the energy flows vary within different regimes of the quasiperiodic motions, such that the short-term effective damping may vary markedly from its theoretical long-term average value. Bridge designers should thus be aware that there could be dangerous instances of the negative damping effect at any bridge frequency.
This is the underlying cause of the instability of footbridges and does not entail walkers making any change to the frequency of their footsteps. Instead, gait widths are amplitude modulated, introducing complicated phase relationships between foot placements and bridge motions, many of which have the effect of negative damping and feed energy into the bridge.
As bridge amplitudes grow, adjustment of footfall timing is an additional possibility and this is included in Models 2 and 3. Potential outcomes include the now-classical Kuramoto transition to synchronisation, as well as phase pulling phenomena where footfalls do not fully synchronise to the bridge motions, but spend proportionally longer at some relative phase offsets15,34. Walkers who synchronise or exhibit phase pulling can add differing amounts of energy to the bridge, depending how their footfall phases relate to that of the bridge velocity. Phase synchronisation can be triggered by bridge motions excited by the more fundamental mechanism of amplitude-modulated gait width, and this can lead to dangerous amplification of the bridge motions. It may also be noted that there exist parameter regimes where walkers synchronise at phases that lead to energy absorption or where they synchronise with a certain phase but the amplitude of the forcing does not grow indefinitely with the bridge amplitude, thereby limiting the bridge response. However, there is insufficient evidence for this to be relied upon in bridge design.
I just read the news – over 500,000 people are now homeless in Europe, thanks to current events.
In times like these, I always turn to the music of Tom Pacheco. His songs are prayers, heartfelt prayers of understanding and appeal.
When I went to www.tompacheco.com, I heard a perfect song for our situation! Thank you Tom Pacheco!
He wrote the song below for sisters and brothers everywhere. We all need a little help now and then, especially those of us in war-torn Ukraine.
SHADOW OF A SEAGULL
Heavenly father, spirit of all I can see
Watch over my sister the way you have watched over me.
Give her protection, through any danger she meets.
Though she may stumble, let her always land on her feet.
She’s been unlucky. She’s been betrayed.
This time, please give her a good hand to play.
Make every cloud she cannot outrun
be just a shadow of a seagull in the sun.
I have been worried, knowing she’s out there alone
searching for something inside that is deeper than bone.
How long can somebody suffer so much for so long,
before they believe there’s no reason at all to go on.
Show her, her value, to her own eyes.
Give her the wings that will help her to fly.
Make every cloud she cannot outrun
Be just a shadow of a seagull in the sun.
She has taken far too many falls.
Worked so long and hard just to lose it all.
Every crop she planted did not yield.
This time, let a treasure fill her fields.
Guide her through valleys, clear a few trees from her path.
Spare her the merciless winds and the cold winters wrath.
Lead her to someplace of beauty where healing can start.
Let the moon shine off the rivers and into her heart.
Let her find purpose and let her find peace.
From every prison, may she be released.
Make every cloud she cannot outrun
Be just a shadow of a seagull in the sun.
These lyrics and music are copyrighted by Tom Pacheco.
I urge you to listen to some of Tom’s songs. He is good. His heart goes out to the men and women and children suffering in Europe now.
Tom’s energy and work is just the person to motivate us to bring peace in our hearts for Ukraine.
I know this, first hand. Tom stepped up to the plate in Woodstock to help those in need more than once. Tom only knows to give for the benefit of others.
I wrote a memoir about hunger and Tom is in it twice, I think. He is going to be in my upcoming book “Ketchup Sandwich Chronicles”.
Do you have any of his CD’s? If so, play the music to receive peace.
This book sells on my website, www.thurmangreco.com. When you purchase a copy, I’ll send it to the address you give. And, I’ll forward the proceeds to Tom as a small tribute to the goodness he brings to the planet.
Tom’s energy ripples out everywhere and this is something we can all use in these times.
My T-shirts are also available now. We need more of this energy and the proceeds can ripple out as well.
It’s possible that some of you will not understand this appeal. But, some of you will.
Keep on!
THANK YOU FOR ALL YOU ARE DOING TO PROMOTE PEACE ON OUR PLANET. EVERY SMALL VIBE COUNTS AND MULTIPLIES THE STRENGTH.
Thurman Greco
Please share this article with your preferred social media network. Please forward it to your friends and family.
Find out more about Thurman at www.thurmangreco.com.
“Shadow of a Seagull” by Tom Pacheco
The post “Shadow of a Seagull” – It’s Time for Tom Pacheco! appeared first on Hunger Is Not a Disease.
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During NYFW, Dominican beauty Lineisy Montero takes Vogue along for her pre-catwalk routine.
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A Merced County family whose 18-year-old son died in a 2016 skydiving accident in Lodi was awarded $40 million by a San Joaquin County judge Wednesday.
Paul Van Der Walde, the Campbell-based attorney who represented the family of Tyler Turner in the lawsuit, told the Fresno Bee on Wednesday that the judgment was significant not only because of the amount that Lodi Parachute Center Bill Dause will be required to pay, but because it specifically targets him.
“We are hoping this will allow us to get this place closed or be sold to a responsible owner who can operate it safely,” Van Der Walde said.
The Turner family sued the parachute center, Dause and Skydivers Guild, Inc. on the grounds of negligence in Tyler Turner’s death in 2016.
A phone call to Dause and the parachute center was not returned to the News-Sentinel. Dause told the Bee on Wednesday he had no comment.
On Aug. 6, 2016, Los Banos resident Tyler Turner went to the parachute center, located at 23597 Highway 99 in Acampo, with his mother and three high school friends to celebrate his 18th birthday.
It was the first time all four of the teens had been skydiving, according to the lawsuit, and they all planned to perform tandem jumps with an instructor.
The suit alleges that once Francine Turner paid the fees for the jumps. The teens were taken to a room to watch a safety video. During the video, they were given a document to sign that contained a release of liability.
As soon as the last of the group signed the document — which was allegedly before the video ended, according to the suit — the teens were ushered into another room to be fitted for gear.
The airplane the teens boarded climbed to an altitude of 13,000 feet, and the suit states that there were no complications with their exit.
Tyler Turner’s tandem partner, Yong Kwon, pulled the ripcord for a drogue attached to the primary chute. The drogue is designed to inflate and help reduce velocity as jumpers descend, as it will inflate with air rushing into it.
However, the drogue did not inflate, and instead twisted violently in the wind, the lawsuit states. Kwon was then supposed to pull the release handles for the chute, but never did, the suit claims.
When the pair had descended to 3,000 feet, Kwon pulled the main chute’s “cutaway” and the release for the reserve drogue. Because the main chute was never deployed, it could not be “cutaway” to clear for the release of the reserve parachute.
As a result, the suit claims, the reserve drogue became entangled with the main chute drogue.
An investigation into the accident found that Kwon did not have proper training or certification to make the tandem jump. In addition, some of Tyler Turner’s friends were younger than 18 at the time, and signed their own liability waivers, rather than their parents the day of the accident, the investigation found.
“He (the instructor) was still under a probationary period when they did the jump,” Van Der Walde told the Bee. “And he did not have the appropriate emergency training.”
Francine Salazar Turner told the Bee she misses her son terribly and wants to use the judgment money to start a scholarship fund in his name.
He was a straight-A student at Pacheco High School with a 4.3 grade point average, she said, and he earned a full scholarship to UC Merced, where he planned to focus on biomedical engineering.
“He was an amazing person,” she said. He had an adventurous spirit and wanted to do so much in this world.”
Francine Turner said she also hopes the skydiving center will be operated by someone else other than Dause.
“He showed no compassion whatsoever,” she said. “It was just horrible.”
According to records, there have been 20 deaths at the skydiving center since it opened in 1981, and 15 since 1999.
The most recent accident happened in 2019, when a 28-year-old woman was struck by a semi-truck on Highway 99 as she was attempting to land.
It was unclear how the woman drifted from the skydiving center and onto the highway, but an unofficial report from a wind sensor at Stockton Airport reported that winds were as high as 15 miles an hour that day.
Assembly Bill 295, also known as “Tyler’s Law,” was signed by Gov. Jerry Brown in September 2017. The bill, authored by Assemblywoman Susan Talamantes Eggman, D-Stockton, will hold parachute centers accountable in state court if they fail to abide by federal safety regulations.
Eggman proposed the law after it was discovered that Kwon was not properly certified as a tandem jump instructor.
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Costa Ricans head into national elections Sunday facing a dizzying array of 25 presidential candidates, no dominant issue to drive turnout and nearly one-third of those intending to vote saying they are still undecided.
Those factors make it highly likely that Costa Rica will hold a second round of voting April 3 between the top two vote getters to determine its next president.
The last poll published Tuesday by the University of Costa Rica’s Center of Political Research and Studies showed that 31% people who planned to vote Sunday remained undecided and no one candidate polled above 17%. The poll surveyed 1,006 people by telephone between Jan. 27 and Feb. 1, and has a margin of error of plus-or-minus 3.1 percentage points.
“Many options are being left open, there are many possible scenarios,” said Ronald Alfaro, one of the center’s political scientists. “This is very much in the style of the elections we’ve had recently.”
At least in 2018, the diametrically opposed positions of now-President Carlos Alvarado and religious conservative Fabricio Alvarado propelled Costa Ricans to the polls. This time there is no dominant issue.
Costa Ricans are frustrated by high unemployment, recent public corruption scandals and another surge of COVID-19 infections, but no candidate has captured the public’s imagination.
Fabricio Alvarado, who lost to Carlos Alvarado in a second round of voting four years ago, is running this time for his New Republic party. But he only garnered about 10% support in the poll.
Slightly above him is José María Figueres, candidate for the National Liberation Party founded by his father José Figueres Ferrer, who himself served as the country’s president on three occasions in the 1940s, 50s and 70s.
The younger Figueres was Costa Rica’s president from 1994 to 1998, but has been questioned over a $900,000 consulting fee he received after his presidency from the telecommunication company Alcatel while it competed for a contract with the national electricity company. He was never charged with any crime and denied any wrongdoing.
Also among the contenders is another experienced politician, Lineth Saborío for Christian Social Unity. Saborío served as Costa Rica’s vice president during the administration of Abel Pacheco from 2002 to 2006.
Previously, Saborío led Costa Rica’s Judicial Investigation Department, which oversees criminal investigations. With that background she has tried to project herself as a strong figure, but her performance in debates has appeared to weaken her appeal.
The candidate for President Alvarado’s incumbent Citizen Action party is barely registering at 1%.
However, with at least 40% of votes required by a candidate to avoid a runoff vote the election remains very much up for grabs.
Costa Ricans will also select the 57 members of the country’s Legislative Assembly.
Turnout remains uncertain with new COVID-19 infections running around 6,000 a day. An election official had encouraged those infected to abstain from voting, but others have acknowledged there’s no way to keep people from exercising their constitutional right.
The government announced Wednesday that it would lift some pandemic-related mobility restrictions that kept a portion of vehicles off the road each day between midnight and 5 a.m.
