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Review
Structural Vibration Comfort: A Review of Recent Developments
Weiping Xie and Yumeng Hua *
School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China;
xiewp@whut.edu.cn
* Correspondence: huaym@whut.edu.cn
Abstract: With continuous improvements in the social economy and living standards of individuals,
the vibration comfort of building structures has gradually been emphasized by academic and engi-
neering communities, such as vehicle-induced vibrations in buildings near traffic, human-induced
vibrations in large-span structures, wind-induced vibrations in super-high-rise buildings, and ma-
chinery-induced structural vibrations. Comfort-based structural analysis is distinct from traditional
safety-based structural analysis, and its theoretical systems and unified guidelines have not yet been
established. This paper reviews recent research on structural vibration comfort, including major
load categories and their impacts, comfort-based structural analysis, evaluation methods, and vi-
bration-mitigation measures. By presenting the shortcomings of the existing research, potential top-
ics for future study are suggested.
Keywords: vibration comfort; serviceability; environmental vibration; vehicle-induced vibration;
human-induced vibration; wind-induced vibration; machinery-induced vibration; structural
analysis; comfort evaluation; vibration control
1. Introduction
With the upgrading of construction technology, application of lightweight and high-
strength materials, realization of new structural systems, and pursuit of architectural aes-
thetics, modern engineering structures tend to develop in light, flexible, large-span, and
towering designs. This change has resulted in an apparent decrease in the structural self-
vibration frequency, which is close to the frequency of traffic vehicles, wind, dynamic
machinery, and people’s daily activities; thus, it is prone to resonance and generates a
significant dynamic response. Such structural vibration is generally not sufficient to cause
safety problems, but it can cause uncomfortable feelings for the people in the building,
which is a comfort problem. People in such vibrating environments can appear nervous,
annoyed, or panicky, which may result in mental illness over time. Therefore, in the de-
sign and construction of “sensitive buildings” (e.g., buildings near traffic, large-span
structures, super-high-rise structures, and buildings with dynamic machinery inside), the
structural dynamic response is predicted or measured, and vibration control measures
are applied if the response levels exceed comfort standards. Such engineering cases have
increased in number in recent years.
Research on vibration comfort began in 1931, when Reiher and Meier conducted pi-
oneering experiments, developed preliminary conclusions on how vibration frequency
affects human comfort, and provided a paradigm for comfort studies [1]. Subsequently,
scholars in the United States, Europe, and Japan have conducted numerous studies and
issued a series of specifications and guidelines [2–7]. Griffin’s team conducted a repre-
sentative study, investigating the effects of frequency, direction, and duration of vibra-
tion, human posture, and other factors of the human vibration perception threshold
through a number of experiments [5]. The Millennium Bridge vibration event in London,
UK, in 2000 brought widespread aention to the comfort of pedestrian bridge structures
Citation: Xie, W.; Hua, Y. Structural
Vibration Comfort: A Review of
Recent Developments.
Buildings 2024, 14, 1592. hps://
doi.org/10.3390/buildings14061592
Academic Editors: Fabrizio Gara,
Haoran Zuo, Kunjie Rong, Ruisheng
Ma and Siyuan Wu
Received: 29 March 2024
Revised: 18 May 2024
Accepted: 23 May 2024
Published: 31 May 2024
Copyright: © 2024 by the authors.
Submied for possible open access
publication under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (hps://cre-
ativecommons.org/licenses/by/4.0/).
Buildings 2024, 14, 1592 2 of 28
by scholars and engineers [8,9], subsequently drove the study of structural comfort caused
by human-induced loads, and provided a strong impetus to research in this field. Re-
searchers have accumulated numerous results on structural vibration comfort.
High-speed railways and urban rail transits that have developed rapidly in China
since the 1980s facilitate resident travel, but also induce environmental vibrations that
have caused complaints from nearby residents [10,11]. With improvements in quality of
life, people’s demand for living environments has also increased. Environmental vibration
is defined as an environmental ground motion of small amplitude resulting from natural
and/or anthropogenic causes (traffic, human-induced excitation, wind, and machinery ex-
citation) [12]. To address this problem, Chinese scholars have focused aention on re-
search into vibration comfort.
In addition to comfort problems, environmental vibrations cause negative impacts,
including secondary noise, failure of precision instruments, damage to non-structural
components, safety of ancient buildings, and wildlife preservation. Most of these issues
do not involve structural safety and primarily affect architectural functions or structural
serviceability. In engineering, these non-safety issues are sometimes all included in com-
fort analyses, but some of the above analyses do not conform to the definition of comfort.
Environmental vibrations are the cause of comfort problems; however, comfort problems
are not the only hazards caused by environmental vibrations. In the literature regarding
structural vibration comfort, the term “serviceability” is used, but it differs from the con-
notation of comfort. In the International Organisation for Standardisation (ISO) publica-
tion ISO 5805-1997 Mechanical vibration and shock—Human exposure—Vocabulary [13],
“comfort” is defined as the subjective state of well-being or absence of mechanical disturb-
ance in relation to the induced environment (mechanical vibration or repetitive shock).
This indicates that “comfort” is the subjective feeling of vibration by the user in the struc-
ture. “Serviceability” has a broad scope. The ISO publication ISO 10137: 2007 Bases for
design of structures—Serviceability of buildings and walkways against vibrations [14] de-
scribes the receivers in vibration serviceability problems, which can encompass the build-
ing structure (or components such as beams, slabs, walls, and windows, among others),
contents of the building (instrument and machinery among others), or human occupants
of the building. It indicates that “serviceability” includes not only human comfort but also
vibration-induced damage to structures or components (e.g., safety of ancient buildings,
non-structural component cracking) and the regular operation of precision instruments.
This paper reviews the progress of research on structural vibration comfort. Section
2 systematically illustrates the comfort problems induced by loads of traffic vehicles, hu-
mans, wind, and dynamic machinery. Section 3 discusses comfort-based structural anal-
ysis methods and describes the key modeling and calculation details. Section 4 summa-
rizes the comfort evaluation methods. Section 5 illustrates the existing vibration mitiga-
tion measures employed in vibration comfort control. Section 6 suggests potential topics
and directions for study in light of the shortcomings of current research.
2. Load Categories and Impacts on Vibration Comfort
Statistics in China in 2020 indicate that the sources that induce vibration comfort
problems include traffic (26.4%), humans (34.5%), wind (19.1%), dynamic machinery
(12.7% commonly and 5.8% for construction), and others (1.5%) [15]. Load categories dif-
fer in terms of source characteristics, action position, corresponding sensitive structure
type, and dynamic response law. This section surveys the vibration comfort problems in-
duced by each of the four load categories.
2.1. Vehicle-Induced Vibration
Vehicle-induced vibration is the dynamic response of structures caused by the move-
ment of trains, such as high-speed, subway, and maglev trains, or motor vehicles, such as
cars, buses, and heavy-duty trucks. In this subsection, research on the vehicle load model,
Buildings 2024, 14, 1592 3 of 28
vibration propagation in soil and soil–structure interaction (SSI), and a simplified calcula-
tion approach are presented.
2.1.1. Vehicle Load Model
Research on vehicle-load models began with theoretical and experimental studies of
vehicles crossing bridges by Willis and Stokes in 1849 [16,17]. The initial classical theoret-
ical methods simplified the vehicle as a moving constant force, moving harmonic force,
moving mass, or moving sprung mass and used analytical methods to obtain dynamic
responses, as shown in Figure 1a–d. In the moving mass model illustrated in Figure 1c,
the force applied on the ground is represented by Equation (1), considering the lateral
inertial of the moving object. By disregarding the lateral inertia, Equation (1) simplifies to
Equation (2), transforming the model into the moving load shown in Figure 1a [18,19].
𝐹𝑥,𝑡𝑀𝑔−d𝑤
d𝑡𝛿𝑥−𝑥 (1)
𝐹𝑥,𝑡𝑀𝑔𝛿𝑥−𝑥 (2)
where 𝑔 is the gravitational acceleration; 𝑤𝑤𝑥,𝑡 denotes the lateral deformation of
the underlying structure; 𝛿 denotes the Dirac delta function; 𝑥 represents the longitu-
dinal coordinate of the moving mass on the ground.
(a) (b)
(c) (d)
(e)
Figure 1. Vehicle load models: (a) moving constant force; (b) moving harmonic force; (c) moving
mass; (d) moving sprung mass; (e) 10-DOF multi-rigid-body model (DOFs are marked in red; mass,
stiffness, and damping coefficients are marked in blue).
Since the 1970s, with the development of computer and numerical techniques, vehicle
models based on multi-rigid-body dynamics have gradually become the mainstream
methods. Chu et al. [20] initially simplified the railway vehicle as a three-degree-of-free-
dom model system consisting of the car body and wheel–axle sets, after which the degree
Buildings 2024, 14, 1592 4 of 28
of freedom (DOF) of developed vehicle models was continuously increased. Figure 1e
shows a plane 10-DOF train model popularly used in the prediction of vehicle-induced
environmental vibration [10]. This method considers the train–track (or car–road) cou-
pling effect, sets the wheel–rail (or road) contact relationship, uses track irregularity (or
road surface roughness) as the main source of excitation, and calculates the equations of
motion using iterative or energy methods. Over the years, this vehicle modeling method
and its computational theories have tended to be complete and are now applicable to var-
ious vehicle types, such as trains and cars, and many related monographs have been pub-
lished [10,21,22].
Research in this field has mainly been conducted by researchers in the disciplines of
vehicle, railway, and bridge engineering. Their primary objective in developing vehicle
load models and dynamic analyses was to determine passenger comfort or the safety and
durability of train running, railways, or bridges. Table 1 lists the typical train load models
in recent years. Therefore, their vehicle load models are sufficiently detailed, with the lat-
est model having 42 DOFs [23]. However, when the purpose of dynamic analysis is turned
to the comfort of building structures, theoretically, a vehicle model with fewer DOFs and
simplified track and wheel–rail contact relationships can satisfy the accuracy requirement
for comfort analyses because of the filtering properties of soil for high-frequency vibra-
tions and the low-frequency characteristics of structures. In the existing literature on com-
fort analysis, the train models used for calculations include several types of multi-rigid-
body models, such as plane models (e.g., 10 DOFs [10,24]) and space models (e.g., 35 DOFs
[25,26]). Furthermore, the track irregularity spectrum corresponds to the vertical irregu-
larity of a single track or the vertical, horizontal, alignment, and rail gauge irregularities
of double tracks, according to the selected plane or space vehicle model. In the above lit-
erature, accurate structural responses have been obtained using different vehicle load
models. However, an overly refined vehicle and track model can result in low calculation
efficiency. For comfort-based analysis, the selection of suitable vehicle load models to bal-
ance the calculation accuracy and efficiency remains to be investigated.
Table 1. Typical train load models and development objectives.
Development Objective References
running safety Ma et al., 2022 [27]; Zeng et al., 2022 [28]; Ju et al., 2023 [29]; Zhao et al., 2023 [30];
Tang et al., 2023 [31]; Li et al., 2024 [32]; Jiang et al., 2024 [33]
safety and durability of railways
or bridges Gou et al., 2023 [34]; Chen et al., 2024 [35]; Zhang et al., 2024 [36]
passenger comfort Gou et al., 2023 [34]; Xin et al., 2023 [37]; Li et al., 2024 [32]
environmental vibration Qu et al., 2022 [38]; Ren et al., 2023 [26]; Xu, 2023 [39]; Hu et al., 2024 [40]; Malmborg
et al., 2024 [41]
2.1.2. Vibration Propagation in Soil and SSI
Buildings sensitive to vehicle-induced vibration can be classified as “vehicle-struc-
ture” and “vehicle-soil-structure” types based on the difference in the vibration propaga-
tion paths from the source to the structure. The difference between the two is whether the
vibrations propagate primarily in the soil. The representative buildings of the first type
include integrated station–bridge structures [42–45], buildings with carriageways inside
[46,47], and railway overtrack buildings [48–50], among others. Their common character-
istic in structural construction is that the track or carriageway is rigidly connected to the
structure, and the vibration is primarily propagated within the structure. For instance,
railway trains can cause comfort problems in waiting rooms and commercial areas within
stations. Arriving and departing trains in stations have low speeds, and thus, integrated
station–bridge structures will be more significantly impacted than the stations separated
from rail bridges. The second type of building is characterized by vehicle-induced vibra-
tions that must propagate in the soil and then into the structure. Buildings near railways
Buildings 2024, 14, 1592 5 of 28
or roads mostly belong to this type [51,52]. This type of building is more common and
numerous, and there are still frequent complaints and exceeded standards owing to cur-
rent imperfections in environmental vibration assessment and structural design methods.
For vehicle-induced vibrations in the second type of building, the vibration propagation
behavior in the soil and SSI is critical.
The deformation of soil caused by vehicle-induced environmental vibrations is gen-
erally in the elastic phase. Waves in the soil are elastic waves that propagate as body and
surface waves. This issue began with Lamb’s work in 1904, that is, the wave solution for
an elastic semi-infinite spatial body acting on its surface or interior using a point or line
load [53]. Subsequent researchers considered more complex soil and vibration source
properties to extend and deepen Lamb’s problem and initially obtained a vehicle-induced
vibration propagation aenuation law in soil layers [54–57]. However, the complex dy-
namic properties of soil, difficulty in detecting inhomogeneous soil layers, and effects of
surrounding structures result in difficulties in calculations and predictions. Prediction
methods for environmental vibration include empirical formulas [58,59], analytical or
semi-analytical methods [60,61], and numerical simulations [62–65], among which the fi-
nite element model (FEM) is the more accurate and commonly used prediction method
[63–65]. With further research, the modeling of the dynamic properties of natural soil has
progressed from elastic bodies [53] to viscoelastic bodies [66], and then to saturated or
unsaturated porous media [67]. Although these existing studies have effectively improved
accuracy, soil models still have many assumptions compared to natural soils. At present,
field measurements remain an effective approach for obtaining vibration aenuation laws
for the geological conditions of a region. Therefore, the current environmental impact as-
sessment (EIA) standard in China states that tests are preferentially used to obtain vibra-
tion aenuation laws over distance [68].
Vehicle-induced environmental vibrations were input from the structural founda-
tion. Therefore, the SSI becomes a non-negligible issue in the calculations. The SSI issue
initially received aention in seismic response analyses [69]. Compared to earthquakes,
vehicle-induced environmental vibrations are mainly focused in the vertical direction and
have smaller amplitudes and higher frequencies. Under these two excitations, the soil
properties and contact relationships with the structural foundation are different. Cur-
rently, the impact of SSI in the context of vehicle-induced vibrations has not been ade-
quately addressed. Most existing models assume that the SSI is negligible, with the soil
and structure subsystems being uncoupled during calculations. This uncoupled approach
is more suited to far-fault ground motions, where the receivers are distant from the source
[70]. However, in the case of vehicle-induced vibration, shorter source–receiver distances
and higher frequencies result in greater interaction [71,72]. The effects of SSI are mainly
reflected in three aspects: changing the original vibration field, elastic boundaries on the
structure, and coupled damping [63–75]. Yao et al. [76] calculated that the vibration of
building floors could decrease by 10–12 dB when the nonlinear contact interaction be-
tween soil and building foundation is considered. The interaction is influenced by dy-
namic soil characteristics, structural foundation types, structural mass, source–receiver
distances, among others [73,77,78]. Coulier et al. [77] investigated that the interaction can
be significant when the source–receiver distance is smaller than the dilatational wave-
length in the soil for underground railway tunnels, or smaller than six Rayleigh wave-
length for railway track at grade, through a case study of a building on a shallow founda-
tion. Numeric contact elements or simplified spring and damper components can be uti-
lized to model the SSI. Aubry and Clouteau [79] proposed a sub-structuring approach to
simulate the SSI, which has been applied to structural vibrations under traffic excitation.
Pyl et al. [80] first studied the SSI using a 3D model under road traffic excitation. Colaço
et al. [72] used the equivalent stiffness and damping method to consider SSI in calculating
railway-induced vibrations.
Vibration propagation in soil and the SSI cause non-uniform excitation of vehicle vi-
bration loads on the structure. Previously, uniform excitation methods have been used to
Buildings 2024, 14, 1592 6 of 28
apply vibrations at the ground or structural boom to a structure calculation model
[81,82]. However, Hua et al. [83] indicated that the impact of non-uniform excitation
should be considered when calculating the structural dynamic response induced by sub-
ways. The non-uniform excitation effect can result in differences in the distribution of struc-
tural responses, and its load model and calculation methods need to be studied further.
2.1.3. Simplified Calculation Approach
For comfort-based structural analyses, the primary shortcoming of current vehicle-
induced vibration load models is the calculation efficiency rather than accuracy. The ac-
curacy of the load models developed for vehicle comfort and track safety satisfied the
requirements of structural comfort analyses. However, the large number of DOFs in soil
FEM, the iterative method, and the short calculation time steps result in a large amount of
computation and increase the difficulty of modeling, debugging, and calculation of the
model, which makes it difficult to extend this calculation method to structural design.
To improve calculation efficiency, researchers have proposed solutions from several
perspectives. Zeng et al. [84] proposed a time-integration analysis scheme for the vehicle–
bridge-interaction problem to prove the computational efficiency of existing algorithms;
Wang et al. [85] developed a multi-point synchronous algorithm to solute the large sparse
linear equations of the train–track–soil coupled system, resulting in a five- to tenfold in-
crease in computation efficiency; Touhei [86] proposed a 2.5-dimension (2.5D) FEM to use
wave number transformations to track and soil along the line direction. However, this
method is challenging to apply to building structures [87]. Currently, simplified load
models and efficient calculation methods are pressing requirements.
2.2. Human-Induced Vibration
Human-induced vibrations are a dynamic response of structures to human activity.
In this subsection, the human-induced load model, human–structure interaction (HSI),
and crowd load model are presented.
2.2.1. Human-Induced Load Model and HSI
Several landmark events, such as the Millennium Bridge vibration event (London,
UK, 2000) [8] and the Techno Mart building vibration event (Seoul, Republic of Korea,
2011) [88], have aracted extensive aention regarding the comfort problem induced by
humans. Human-induced loads generally have a low intensity and thus cause significant
dynamic responses by exciting structural resonance. Because the main frequency of hu-
man-induced loads is 1–3 Hz, light and flexible engineering structures, such as pedestrian
bridges, connective corridors, large-span buildings, and cantilevered buildings, are the
primary study subjects [89,90]. In the structural design of such buildings, it is recom-
mended that the fundamental frequency of the structure should exceed 3 Hz, thus avoid-
ing human-induced comfort problems [14,91]. Some studies have proposed that high-fre-
quency floor slabs may also experience human-induced vibration comfort problems [92];
however, this phenomenon requires more engineering cases to be verified.
The human body is an adaptive system, and human behavior is intensely subjective
and random, resulting in complex human-induced loads. The establishment of human-
induced loads is based on experiments and statistics on human body dynamics parameter
calibration and plantar force measurement with different body postures. Furthermore,
HSI is a central issue in the study of human-induced vibrations [93]. In particular, when
light and strong building materials are applied, the structural frequency decreases, and
the ratio of human–structure mass increases, resulting in a more significant effect of HSI.
The relevant tests indicate that humans can alter the mass distribution and damping of
the structural dynamic system [94–97]. The coupling mechanism and applicable conditions
remain to be addressed to develop simplified calculation methods for structural design.
Buildings 2024, 14, 1592 7 of 28
By simplifying the dynamic characteristics of the human body and the HSI, research-
ers have developed human-induced load models that correspond to various human activ-
ities, including walking, running, jumping, and bouncing, among others [89,98,99]. Table
2 lists part of the typical human-induced load models. In particular, the vertical walking
load is the most common type of human-induced load and has received the most aention
from researchers [89]. The time history of walking force during single-foot landing is dou-
ble-peaked “M” shapes, consistent with the physical process from heel landing to toe off
the ground when walking. In the frequency domain, the walking force has main harmonic
and multi-order subharmonic peaks. Types of load models include the Fourier series
model, spring-mass-damping model, and bipedal walking model, among others, as listed
in Table 2.
Table 2. Typical human-induced load models.
Human Ac-
tivities Model Type Originator, Year
walking
(vertical load)
Fourier series model Zivanovic et al., 2007 [100]; Chen et al., 2014 [101]
spring-mass-damping model Silva et al., 2013 [102]; Shahabpoor et al., 2023 [103]
bipedal walking model Whiington et al., 2009 [104]; Kim et al., 2011 [105]; Qin et al. 2013 [106]
multibody model Máca et al., 2011 [107]
power spectrum model Brownjohn et al., 2004 [108]; Piccardo et al., 2012 [109]; Ferraroi et al.,
2016 [110]; Chen et al., 2019 [111]; Wang et al., 2020 [112]
walking
(lateral load)
synchronization-based model
Fujino et al., 1993 [113]; Dallard et al., 2001 [114]; Nakamura et al., 2004
[115]; Stroga et al., 2005 [116]; Blekherman, 2005 [9]; Piccardo et al.,
2008 [117]
self-excitation model Pizzimenti et al., 2005 [118]; Ingólfsson et al., 2011 [119];
inverted pendulum model Macdonald, 2008 [120]; Bocian et al. [121]; Carroll et al., 2011 [122]
running Fourier series model Racic et al., 2014 [123]
jumping Fourier series model Ellis et al., 2004 [124]
power spectrum model Xiong et al., 2018 and 2021 [125,126];
bouncing Fourier series model Ji et al., 1994 [127]; Parkhouse et al. 2006 [128]; Duarte et al., 2009 [129];
SMD model Dougill et al., 2006 [130]; Wang et al., 2019 [131]
The modeling of human-induced loads is generally based on tests performed on the
rigid ground. However, when a human is in a vibrational environment or on a flexible
structure, the plantar force can differ from the test results on rigid ground because of the
HSI and the spontaneous regulation of the human body system [132]. This influencing
factor should be studied to refine the existing human-induced load models.
Human-induced load models applied to residential buildings with common func-
tions have been extensively studied. In recent years, people have expected buildings to
have more functionality; therefore, the number of new-functional and comprehensive
buildings has increased. A human-induced load model must correspond to a building
function. For instance, some special-shaped landscape bridges require staircase walking
loads [133]; comprehensive buildings with indoor gymnasiums require sports loads (e.g.,
basketball, badminton, and volleyball); and comprehensive buildings with performance
halls inside require human wave and swing loads. With the promotion and construction
of these new buildings, novel human-induced load models required for structural design
have become a future research trend.
Buildings 2024, 14, 1592 8 of 28
2.2.2. Crowd Load Model
Because of simplifications in modeling, existing single-human-induced load models
cannot completely simulate the dynamic characteristics of the human body yet. However,
the mass ratio of one single person to a structure is too small to induce comfort problems.
In engineering, the comfort problems of buildings are generally caused by crowd load
excitation. Therefore, to improve the accuracy of human-induced vibration calculations,
the primary objective is to consider crowd behavior in load modeling instead of further
refining the single-human model.
Human dynamic parameters are related to factors such as gender, age, height, and
weight, and their distribution in a crowd is strongly random. Moreover, crowd behavior
paerns are influenced by building functions, spatial arrangements, and unexpected
events [134]. Researchers initially combined single-human models into a crowd model,
considered the statistical laws of random factors, and used the Monte Carlo method for a
time-domain analysis to calculate the structural response [135]. Methods in the traffic
planning field, such as the discrete element theory (DET) [136] and the social force model
[137,138], have been used to simulate pathways of crowd activity. However, time-domain
load models exhibit natural disadvantages in stochastic analyses. The high computational
cost of the Monte Carlo method makes it difficult to use. Presently, the development of
time-domain crowd load models has stagnated.
Frequency-domain crowd load models represented by power spectrum models have
become a trend because of their advantages in stochastic analyses [108–
112,125,126,139,140]. Chen’s team contributed outstanding research results using this
method [111,112,125,126,140], and the proposed reaction spectrum method was adopted
for China’s building floor design standard [91]. However, for the current power spectrum
model of crowd load, the influence of various random crowd behaviors and the effect of
HSI need further consideration.
In addition, with emerging technologies, such as smartphones, big data, motion cap-
ture, and neural networks, human-induced load models based on big data have study and
application prospects [15,139–142].
2.3. Wind-Induced Vibration
Wind-induced vibration is the dynamic response of a structure to fluctuating wind.
Fluctuating wind loads are horizontal dynamic loads with random characteristics that act
on building structures. Because of the fluid–structure interaction, the structural dynamic
behavior caused by wind loads is complex and includes downwind and crosswind vibra-
tions (e.g., vortex-induced vibration).
Researchers have conducted measurements and formulated different wind speed
spectra, such as Karman, Davenport, Harris, Kaimal, and Simiu wind spectra, to analyze
downwind vibrations. Among these, the Davenport wind spectrum is based on the meas-
ured data of multiple regions and, therefore, has wide applicability [143]. Equation (3)
delineates the expression of the power spectrum 𝑆𝑛. By using the wind speed spec-
trum, pulsating wind speeds can be simulated through methods like linear filtering, har-
monic superposition, or other techniques. Subsequently, the pulsating wind pressure load
is derived from the correlation between wind pressure and speed.
𝑆𝑛=4𝐾𝑣
𝑥
1+𝑥
⁄ (3)
where 𝑥= 1200 𝑛𝑣
⁄; 𝑛 is frequency; 𝑣 is the mean hourly wind speed at the stand-
ard reference height of 10 m; 𝐾 is the drag coefficient for the surface.
The complexity of crosswind vibrations and wind loads in tall buildings surpasses
that of downwind vibrations, primarily due to factors like vortex-induced resonance. De-
veloping a mathematical model that accurately describes vortex-induced forces is an ef-
fective approach to address this problem, albeit challenging. The complexity of vortex-
induced vibrations stems from the shedding of wake vortices at different stages within
Buildings 2024, 14, 1592 9 of 28
the lock-in range, and various structural boundary conditions lead to the emergence of
different modes. Several simplified mathematical models for vortex-induced forces have
been proposed. For instance, the Ruscheweyh model describes vortex-induced forces as
harmonic forces [144] in Equation (4). However, the simple model lacks the representation
of key properties of vortex-induced vibrations, such as the locking phenomenon and self-
limiting amplitudes. Bishop [145], Tamura [146], Vickery [147], Ehsan [148], Larsen [149]
et al. proposed more intricate and refined expressions to depict the relationship between
frequency, amplitude, and phase of vortex-induced forces and structural parameters. The
key fluid parameters in these models must be assumed in advance or identified from
measured vortex-induced vibration responses of the structure. Currently, simulation
based on computational fluid dynamics is the main technique to calculate vortex-induced
forces [150,151].
𝐹=1
2𝜌𝑉𝐷𝐶sin𝜔𝑡+𝜑 (4)
where 𝐹 is the vortex-induced force; 𝜌 is the density of air; 𝑉 is the wind speed; 𝐷 is
the structural cross-section dimensions in crosswind direction; 𝐶 is the root mean square
of the lift coefficient; 𝜔 is the circular frequency; 𝑡 is the time; 𝜑 is the phase difference
between structural displacement response and vortex-induced force
Early research on the structural wind resistance design of high-rise buildings focused
mainly on structural safety and less on comfort [152]. With the development of urban con-
struction, the number of high-rise and super-high-rise structures is increasing, and wind-
induced vibration comfort has often determined the structural design of these buildings.
Studies of wind-induced vibration comfort have become increasingly critical.
Traditional methods for safety-based wind-resistant structural analysis are unsuita-
ble for comfort-based analysis, and the understanding of wind-induced comfort calcula-
tion and evaluation methods has not been unified. The evaluation index in China uses the
downwind and crosswind peak acceleration on top of structures [153,154], with the aim
of controlling the vibration at the peak position of the structural first-order mode. How-
ever, the tops of high-rise or super-high-rise buildings are often unmanned activity roof-
top platforms, masts for lightning protection, or antennae, where comfort problems do
not occur. Moreover, wind loads can excite high-order modes of the structure under spe-
cific circumstances, which may result in maximum vibration occurring in the middle
floors of the building. The ISO and some countries, such as Japan, stipulate that the max-
imum peak acceleration in the downwind and crosswind directions of all occupied floors
must be evaluated [155,156]. In addition, the maximum direction of the wind-induced vi-
bration requires further discussion. When the mass center of the structure does not coin-
cide with the stiffness center, the structure has torsional vibration modes, and the maxi-
mum direction of the horizontal vibration can form an angle with the downwind and
crosswind directions. Furthermore, under wind load excitation, vertical vibrations of the
floor slabs occurred with horizontal vibrations because of the smaller out-of-plane stiff-
ness of the slabs. Therefore, the vibration directions used for the evaluation, such as down-
wind, crosswind, maximum horizontal, vertical, or whole-body vibrations, require further
determination.
The recent SEG Plaza building vibration event (Shenzhen, China, 2021) brought
wind-induced comfort problems into public view and aracted the aention of research-
ers and engineers [157–160]. Experts investigated the event, and the cause was found to
be the coupling of the vortex-induced resonance of the mast and the change in the dy-
namic characteristics of the building and the mast. This event may become a landmark in
research on wind-induced vibration comfort. Discussing the vibration mechanism, struc-
tural analysis theory, evaluation methods, and other issues can significantly promote the
study of wind-induced vibration comfort and update structural design and evaluation
systems.
Buildings 2024, 14, 1592 10 of 28
2.4. Machinery-Induced Vibration
Machinery-induced vibration is the dynamic response of structures to dynamic ma-
chinery. According to the function and use scenario, the possible machinery includes liv-
ing equipment (elevators [161], water pumps, power transformers [162], and fans, among
others), industrial production equipment (ball mills, bridge cranes [163], and punch ma-
chinery, among others), and engineering construction equipment (dynamic compaction
machinery [164], rock excavation [165], and pile drivers, among others). Owing to the
shortage of urban land and intensive building planning, the distance between this dy-
namic machinery and the living environment is gradually reduced, even inside buildings.
In general, the load of dynamic machinery is not complex. However, different types of
dynamic machinery have different characteristics in the time and frequency domains, and
their corresponding load models and modeling techniques are also different. Based on the
generation mechanism of machinery-induced vibrations and load characteristics, dy-
namic machinery loads can be divided into three types, and commonly used load model-
ing techniques are listed in Table 3.
Table 3. Dynamic machinery load types and modeling techniques.
Type
No. Machinery Examples Generation
Mechanism Load Characteristic Modeling Techniques
I
elevator traction machinery,
power transformer, water
pump, fan, ball mill
reciprocating or rotat-
ing mechanisms
smooth stochastic process; fixed
frequency components; spec-
trum contains main and multi-
order harmonic frequencies
simple harmonic forces;
Fourier series model;
power spectrum model
II elevator car, bridge crane
machines move along
fixed tracks or routes on
the structure
moving load (related to mass,
speed, track irregularities, and
contact relationships)
moving force, multi-rigid
body model
III
punch machinery, dynamic
compaction machinery, pile
driver
impact behavior during
machinery operation pulse load impact force, impact pro-
cess numerical simulation
There are fewer research cases involving dynamic machinery loads and induced
structural dynamic response compared to the other categories of loads mentioned above,
such as vehicle-induced and human-induced loads. Moreover, because of the large num-
ber of machinery categories, most machinery categories have not been developed into ma-
ture load models. In engineering, measured vibration data at the vibration source of ma-
chinery are used directly as the input. With people paying more aention to comfort prob-
lems, it is necessary to establish standardized load models for common dynamic machin-
ery. In addition, structural calculation models, evaluation methods, and isolation
measures for machinery-induced vibrations require further investigation.
3. Comfort-Based Structural Analysis Method
With the development of numerical simulation technology and the upgrading of
computer performance, the FEM has become the mainstream method for dynamic analy-
sis in structural design. Compared with safety-based structural simplification models for
strong vibration analysis, comfort-based structural model simplification methods should
be different.
Strong vibration analysis focuses on the bearing capacity or deformation properties
of the main load-bearing skeleton of a structure under ultimate loads such as earthquakes.
The purpose is to avoid the loss of the seismic or gravity-bearing capacity of the entire
structure because of damage to parts of the structure or components. Instead, vibration
comfort problems are within the scope of weak vibration analysis. This focuses on the
serviceability of structures under micro-amplitude vibrations. Its purpose is to service the
Buildings 2024, 14, 1592 11 of 28
people in the structure and consider the comfort of the people in the vibration environ-
ment as a criterion. Under micro-amplitude vibrations, the structure has small defor-
mation and is in a state of linear elasticity. Therefore, the comfort-based model should
reflect the dynamic characteristics of the objective structure under micro-amplitude vibra-
tions and accurately calculate the acceleration of the floor where people are located. In
addition, the median perception threshold of fit persons for the weighted acceleration
peak magnitude is approximately 0.015 m/s2 [166,167], indicating that the accuracy of
comfort-based analysis should also achieve this magnitude.
Based on the above principles, He and Xie [168] proposed a sophisticated calculation
model for large-span railway station structures based on the evaluation of vibration ser-
viceability. Based on the strong vibration model of the structure, the method models the
non-structural components, considers the local construction and boundary conditions,
modifies the calculation parameters such as the live load coefficient and damping ratio,
and finally obtains a refined elasticity FEM of the structure during the normal service
state. Furthermore, by validating additional engineering structures, the modeling method
can be suitable for more structural comfort problems [43,161,162]. The following subsec-
tion presents the critical differences between the comfort-based structural analysis model
and the traditional strong vibration model.
3.1. Structural Floor Slab
The floor slab is the structural component where people are located and the source
of the human body receiving vibrations, and therefore, is the location that standards spec-
ify for evaluation. The dynamic characteristic of structural floor slabs is an emphasis in
modeling.
For a structural floor, a strong vibration model generally adopts the rigid floor as-
sumption and ignores its out-of-plane stiffness. Instead, in comfort-based structural anal-
ysis, the main structure, including the structural floor slabs, must be simplified as elastic
elements. Researchers have generally accepted this opinion and developed various elas-
ticity calculation models [163,169]. Application of conventional modeling techniques has
been shown to be adequate for typical floor slabs such as reinforced concrete slabs. How-
ever, novel slab materials or composite floor types have recently been developed with
excellent performance and complex construction [170–172]. For each of these applications,
it is necessary to study the dynamic behavior by developing suitable modeling techniques.
3.2. Non-Structural Components
Non-structural components are non-load-bearing components in the structural de-
sign, including flooring finishes, infill walls, and curtain walls, among others. Strong vi-
bration models generally ignore the effects of non-structural components on the global
stiffness and damping of the structure. The model simplified these components to addi-
tional mass or load composite values. Instead, when the structure is in a state of normal
use, some non-structural components can change the structural mass, stiffness distribu-
tion, and damping properties, which can significantly affect the dynamic characteristics
of the structure under micro-amplitude vibrations, thus affecting the dynamic response
[173]. The influence of various non-structural components on the dynamic characteristics
of structures has become a topic of interest [174]. Two types of non-structural components
with significant impacts are the non-structural floor layer and wall, which are explained
in the following.
• Non-structural layer of floors
The non-structural floor layer is the functional and decorative layer laid on the struc-
tural floor slab, such as a thermal insulation layer (foam board), levelling layer (cement
mortar), and decorative surface layer (tiles, timber, marble, rubber, and elevated flooring).
These non-structural layers may increase floor thickness by more than 50%. Research in-
dicates that an insulation layer can significantly increase the damping ratio of floors [168].
Buildings 2024, 14, 1592 12 of 28
Different materials and types of decorative surfaces have different effects on the dynamic
characteristics of floor slabs [174–177]. These effects should be considered in the calcula-
tions to modify the dynamic characteristics of the floors.
• Non-structural wall
Non-structural walls, such as infill and curtain walls, have no load-bearing functions
in their structural design. In comfort-based structural analysis, lightweight partition walls
can be equivalently modeled in terms of mass and stiffness; infill walls can transmit inter-
story vibrations, increase the stiffness and damping of floors, alter the modal shapes
[168,178], and increase the global lateral stiffness of the structure [179]; curtain walls have
weak confinement effects between layers in addition to the effect of mass [180]. At present,
the interaction mechanism between non-structural walls and the main structure remains
to be investigated. Infill walls are constructed directly on the floor beams. Curtain walls
are hung on the building facade using steel studs. The dynamic behavior of the connection
between non-structural walls and the main structure is central to such studies. The connection
stiffness and the vibration propagation law under weak vibration require further study.
In summary, the importance of non-structural components in comfort-based anal-
yses has gradually been accepted and considered in modeling. However, the refined mod-
eling for non-structural components increases the complexity of the numerical models
and decreases the calculation efficiency. Non-structural components are varied and com-
plex, and new component categories are constantly being developed and applied. Re-
search on various non-structural components requires further discussion. Quantifying the
effects of various non-structural components and developing simplified methods are prac-
tical approaches to consider non-structural components in structural design.
3.3. Local Construction and Boundary
Local constructions include window and door openings, structural joints, and bolted
joints, among others. The vibration characteristics of these constructions under micro-am-
plitude vibrations differ from those under strong vibration conditions. For instance, win-
dow and door openings can decrease the mass and stiffness of the infill wall and alter its
effect on the structural dynamic characteristics; structural joints under strong vibrations
divide the structure completely, whereas those under micro-amplitude vibrations can
transmit vibrations, and thus be simplified semi-rigid spring connections; trusses and
bolted connections under strong vibration are considered to be axially loaded members
and hinges, whereas the bolted nodes of trusses almost never rotate under micro-ampli-
tude vibrations, thus be modeled as beam members and rigid or semi-rigid, respectively.
Modeling of the boundary depends on the selection of the calculation domain. Based
on the vibration propagation features under different loads and structures, selecting the
calculation domain and reasonably addressing the boundary can improve calculation ef-
ficiency while balancing accuracy requirements. For instance, the following can be per-
formed: in vehicle-induced vibration, modeling the building structure and seing up
equivalent stiffness and damping at the base of the structure to consider the SSI [78]; in
human-induced vibration, modeling partial floors and constraining boundaries for calcu-
lations [169]; and in wind-induced vibration, modeling the superstructure, ignoring the
foundations, and constraining the model boom as the fixed end [157]. Current numerical
models may have redundant parts that affect the calculation efficiency, and the vibration
propagation mechanism requires further investigation.
3.4. Other Dynamic Parameters
This subsection adds other dynamic parameters that are not discussed above but
should not be ignored in comfort-based structural analyses.
Buildings 2024, 14, 1592 13 of 28
• Additional mass
Placing large-mass items on the floor, such as large decorations, fixed equipment, and
furniture, can change the mass distribution and decrease the fundamental frequency of
the floor [181].
• Dynamic elastic modulus
Elastic modulus can impact the natural frequency of structures, which is a key pa-
rameter in modeling [182,183]. In comfort-based structural analyses, the elastic modulus
of the material at small deformations, that is, the dynamic elastic modulus, should be
adopted. Compared to the static elastic modulus obtained by the axial compression ex-
periment of prisms, the dynamic elastic modulus was obtained by the ultrasonic pulse
method, representing the relationship between the stress and strain of the material under
dynamic loading. For instance, the dynamic elastic modulus of concrete is greater than its
static elastic modulus [184]. Bachmann et al. [12] proposed using the dynamic modulus of
elasticity to predict the fundamental frequency of the floors in calculating human-induced
vibration comfort. Therefore, the technical standard in China stipulates that for reinforced
concrete floor or steel-concrete composite floor, the elastic modulus used in the calculation
should be 1.2–1.35 times larger than the static elastic modulus to consider the dynamic
elastic modulus [91].
• Damping ratio
Damping reduces the structural response. Structural damping mechanisms are com-
plex, and the current theories are not fully comprehensive. In general structures, damping
is mainly related to the material properties and other damper mechanisms used in the
building. When a structure has great deformation transitioning from elasticity to plastic-
ity, its damping can increase significantly. Consequently, the damping ratio in comfort-
based structural analyses should be lower than in seismic analyses. Furthermore, even
when structural deformation remains within the elastic range, the dissipation of damping
energy during motion is influenced by the extent of deformation, thus exhibiting a rele-
vance of the value of the damping ratio to the vibration amplitude [185,186]. Although a
definitive relationship has yet to be established, it is generally observed that more minor
excitation results in smaller vibration amplitudes, corresponding to a lower damping ra-
tio. Therefore, in comfort-based structural analyses, the damping ratio under micro-am-
plitude vibration of the structure should be adopted, which is different from strong vibra-
tion analyses. Measured damping ratios of steel and concrete structures are generally
about 0.01–0.02 during the normal service stage (micro-amplitude vibration)
[168,182,187]. In addition, the wind-induced vibration of high-rise buildings is sensitive
to the damping ratio. Hu et al. [157] measured a super-high-rise building and observed a
sudden decrease in the damping ratio when wind-induced structural resonance occurred,
which may be related to the aerodynamic damping effect [188].
• Construction and measurement error
Construction errors during the building process are common, such as those involving
the properties of materials and the dimensions of beams, columns, and slabs. This results
in the dynamic characteristic of a completed structure that differs from the design expec-
tations. Construction errors are generally not considered in current structural modeling.
In addition, ambient temperature changes can influence the dynamic characteristics of
structures such as pedestrian bridges, resulting in fluctuations in the measured self-vibra-
tion frequency [189]. Wang et al. [190] monitored a reinforced concrete slab for over a year
and observed that possible temperature changes of 0~50 ℃ can lead to an error of more
than 15% in the self-vibration frequency of the tested slab. Errors in identifying structural
dynamic characteristics and response measurements can impact the validation of struc-
tural models. Therefore, when the dynamic characteristics of the model are not consistent
with the measurements, the effect of construction and measurement errors should be con-
sidered, in addition to the errors caused by modeling simplification and algorithms.
Buildings 2024, 14, 1592 14 of 28
In summary, the comfort-based structural analysis aims to accurately calculate the
vibration of the floors in the user activity area. Its principle is modeling the key members
(component, connection, and boundary, among others) in detail and simplifying others
based on the effect of each member under micro-amplitude vibration. However, this
method has not yet covered all types of vibration sources, structures, or construction de-
tails, and a systematic method for structural design has not yet been developed.
4. Vibration Comfort Evaluation Method
Vibration comfort is based on human feelings, which are influenced by the following
three categories of factors: (a) external vibration factors, including direction, frequency,
amplitude, duration, and load return period; (b) external environmental factors, including
structural-borne noise, airborne noise, raling, movement of furniture, and visual effects
(e.g., movement of hanging objects); and (c) receiver factors, including person type (gen-
der, age, height, and health, among others), body posture, experience, expectation, activ-
ity motivation (e.g., ease or difficulty of the task performed), economic or living situation,
and subjective ambiguity.
For these influencing factors, based on biodynamics, psychology, and fuzzy mathe-
matics methods, researchers have statistically analyzed a large amount of test data, and
their conclusions have been applied to comfort evaluations. The relationship between
comfort and the amplitude and frequency of vibration acceleration has been studied in
depth. The human body in buildings is sensitive to vibration in the 1–80 Hz range [191],
and the ISO provides frequency weighting methods for vibrations in different directions
and body postures [166,167]. Table 4 lists some of the comfort evaluation standards in
various countries with their indexes and partial limits.
Table 4. Comfort evaluation standards in various countries.
Standard
;
Country,
Region, or Organization
Vibration
Source Comfort Index Evaluation Objects and Limits (Part)
ISO 2631-1: 1997; ISO [166,167] - RMS; VDV uncomfortable: “RMS” 0.8 m/s2 to 1.6 m/s2
ISO 10137-2007; ISO [14]
vehicle; hu-
man; wind;
machinery
RMS; VDV; PA
1. vibration induced by vehicles, humans, or ma-
chinery: “RMS, VDV” base curve × multiplying
factor; 2. wind-induced vibration: “PA” fre-
quency-dependent curve
ISO 6897-1984; ISO [156] wind RMS frequency-dependent curve
GB 10070-88; China [192] vehicle; ma-
chinery VAL ( VLz) residential, cultural and educational building
(day): 70 dB
JGJ/T 441-2019; China [91] human; ma-
chinery PA building floor with walking loads primarily (resi-
dential, hospital, office, etc.): 0.05 m/s2
JGJ/T 170-2009; China [193] vehicle VAL (VLmax) residential, cultural and educational buildings
(day): 65 dB
JGJ 3-2010; China [153] wind PA concrete structure of tall buildings (residential,
apartment): 0.15 m/s2
JGJ 99-2015; China [154] wind PA steel structure of tall buildings (apartment): 0.20
m/s2
ATC Design Guide 1; USA [194] human RMS; PA
1. vertical vibration: “RMS” base curve × multi-
plying factor; 2. horizontal vibration (office, resi-
dential, etc.): “PA” 0.02 m/s2
AISC Design Guide 11; USA [195] human PA floor and pedestrian bridge: base curve × multi-
plying factor
EN 1990: 2002 [196] human PA vertical vibration (pedestrian bridge): 0.7 m/s2
BS 6472-1: 2008; UK [197] - VDV residential building (16 h day): adverse comment
possible 0.4 m/s1.75 to 0.8 m/s1.75
Buildings 2024, 14, 1592 15 of 28
AIJ ES001-V001; Japan [198]
vehicle; hu-
man; wind;
machinery
PA building: frequency-dependent curve
AIJ-GEH-2004; Japan [155] wind PA building: frequency-dependent curve
Note: RMS is weighted acceleration root mean square; VDV is vibration dose value; PA is peak ac-
celeration; VAL is weighted vibration acceleration level, and Z-vibration level VLz is weighted ver-
tical VAL, frequency maximum magnitude of vibration VLmax is maximum weighted vertical VAL at
the 1/3-octave center frequency.
From Table 4, the comfort indicators of the existing standards are not unified and use
different limits, which lack the basis for conversion. Some indicators are tailored to spe-
cific application conditions, yet even for the same comfort issue, different indicators may
exist. In engineering applications, using different standards can yield divergent assess-
ment outcomes. These comfort evaluation indices can be classified into two categories. For
vibrations with a single primary frequency, such as human-induced vibrations and wind-
induced vibrations, peak acceleration is generally adopted [14,91,153–155,194–196,198],
whereas broadband vibrations, such as vehicle-induced vibrations, are generally de-
scribed as weighted acceleration root mean square 𝑎, , vibration dose value
𝑎,[14] or Z-vibration level VLz [192], as expressed in Equations (5)–(7). Furthermore,
when assessing the vibration comfort problems of floors and pedestrian bridges caused
by humans, both natural frequency and peak acceleration are commonly employed [199].
For instance, the Chinese standard JGJ/T 441-2019 specifies that for the floor slabs in residential
or office buildings primarily excited by walking loads, the first-order vertical self-vibration
frequency should be over 3 Hz, and the limit of vertical peak acceleration is 0.05 m/s2 [91].
𝑎,=1
𝑇𝑎
𝑡d𝑡
⁄
(5)
𝑎,=1
𝑇𝑎
𝑡d𝑡
⁄
(6)
where 𝑎𝑡 is the weighted acceleration as a function of time; 𝑇 is the duration of the
measurement. 𝑉𝐿= 20lg 𝑎,
𝑎 (7)
where 𝑎=10
ms
⁄ is the reference acceleration.
Most existing evaluation standards do not consider or simplify the vibration duration
and load return period of the influencing factors. In ISO 10137-2007, different multiplying
factors are adopted for the “continuous vibration and intermient vibration” and the “im-
pulsive vibration excitation with several occurrences per day” [14], which demands more
stringent limits for the first type of vibration source. The ISO standard published in 1985,
ISO 2631-1:1985 Evaluation of human exposure to whole-body vibration-Part 1: General
requirements (Withdrawn) [200], specified the evaluation limits for exposure times from
1 min to 24 h and an equation for the accumulation of exposure times for intermient
vibration. As the exposure time increases, a person’s tolerance to vibration decreases, and
the demand for limits becomes more stringent. However, this part was simplified in the
revised version of ISO 2631-1:1997, and the definition of “fatigue-decreased proficiency”
was removed [166,167]. Despite this, the influence of the vibration duration and load re-
turn period on comfort evaluation should not be ignored, and further refined evaluation
methods are required.
In principle, the position of comfort evaluation should be located at the maximum
vibration of floors in the user activity area, which is related to the load characteristics, load
Buildings 2024, 14, 1592 16 of 28
locations, structure types, and building functions. However, the existing evaluation
method requires further improvement because of insufficient conclusions regarding the
location of the maximum vibration and the lag of the standards with respect to the latest
research. The shortcomings include the following: (a) in vehicle-induced vibration, the
spatial distribution of the response is uncertain, and the measured maximum locations
can be observed on the lower [201,202], middle [203], or top floors [204–206]. Therefore,
the influence of the structural global and floor local modes on the structural response re-
quires discussion. (b) Comprehensive evaluation of multi-directional vibration requires
consideration, such as the horizontal response induced by vehicles in high-rise buildings
increases with height [207], and horizontal wind loads may cause vertical floor vibration.
(c) The comfort evaluation limits for different types of structures should be unified. For
example, China’s wind vibration evaluation standards specify different peak acceleration
limits for concrete and steel structures [153,154]. In addition, the vibration duration and
load return period should be added to the comfort evaluation system.
Existing comfort tests have some shortcomings that require further refinements, such
as insufficient sample capacity, the difficulty in artificially simulating vibration to accu-
rately reflect the actual vibrations, consideration of the psychological state of the subject,
the difficulty in quantifying the influence of external environmental factors, and the cur-
rent lack of standardized test technology. With the widespread adoption of smartphones
and the Internet, human body movements and comfort feelings can by conveniently rec-
orded and then collected by uploading them to the cloud via the Internet. Deep learning
and artificial intelligence provide efficient algorithms to train collected big data of the im-
portant factors under various influential factors [208–210]. Researchers have begun ex-
ploring of these emerging technologies in studying comfort indexes. For instance, Chen et
al. [211] proposed a smartphone-based evaluation system for pedestrian-induced foot-
bridge vibration comfort; Cao et al. [15,212] designed a smartphone-based application and
an online big data approach to investigate vibration serviceability limits in real environ-
ments. In addition, the vibration acceleration of structures is an essential indicator for as-
sessing vibration comfort and obtaining reliable acceleration signal data. Various test tech-
niques developed for structural health monitoring could be used to enhance the comfort-
based vibration testing approach [213,214].
5. Mitigation Measures for Vibration Comfort
When the structural response exceeds standard limits and can cause discomfort for
building occupants, vibration mitigation measures must be implemented. Vibration con-
trol technology has mature theories and has evolved alongside the development of new
materials and processes. The vibration sources that cause comfort problems lead to struc-
tural responses with small amplitudes and possibly high frequencies, making not all tech-
niques or products for the earthquake-resistant design applicable. Depending on the pres-
ence or absence of external energy input, these measures can be categorized as active or
passive control technologies. In micro-amplitude vibration control for comfort, passive
control technology is primarily utilized. This section focuses on the commonly employed
measures to address comfort issues, including vibration isolation in railway tracks, vibra-
tion isolation barriers in soil, vibration isolation bearings for building structures, and
tuned mass dampers (TMD).
5.1. Vibration Isolation in Railway Tracks
Vibrations induced by vehicles and machinery are artificial vibration sources, mak-
ing isolating these sources an effective control strategy. In railway tracks, many systematic
vibration isolation applications have been implemented [215–217]. These isolators are typ-
ically installed at the rails, rail fastenings, rail slabs, or ballast beds to mitigate the propa-
gation of vehicle-induced vibrations to the surrounding environment. In recent years,
phononic crystal theory has been employed in the design of vibration control techniques.
Buildings 2024, 14, 1592 17 of 28
For instance, Hu et al. [218] developed a periodic layered slab track structure. The cur-
rently applied vibration isolators are typically effective in aenuating high-frequency vi-
brations but exhibit limited efficacy in addressing low-frequency components, sometimes
even amplifying such vibrations. Low-frequency vibrations demonstrate less aenuation
through soil over distances compared to high-frequency vibrations additionally, increas-
ing the risk of resonance with building structures possessing low-frequency natural fre-
quencies. Consequently, enhancing the capacity of track vibration isolators to mitigate
low-frequency vibrations has emerged as a primary imperative. Sung et al. [219] devel-
oped an additional anti-vibration sleeper track to reduce low-frequency vibrations. Qu et
al. [220] introduced a railway track concept inspired by chiral phononic crystals, offering
significantly improved low-frequency vibration isolation through an orthogonal polariza-
tion coupling mechanism within the chiral subunit cell. Although track vibration isolation
measures have been commonly implemented in urban rail transit systems, their adoption
in high-speed rail networks remains limited due to safety and durability concerns.
5.2. Vibration Isolation Barrier in Soil
Vibration isolation barriers are constructed within the soil to isolate propagation
paths. This strategy is particularly relevant in scenarios involving vibrations generated by
vehicles or machinery. Common types of vibration isolation barriers include in-filled
trenches [221–223], arrays of piles [224], and wave-impeding blocks (WIB) [225–227],
among others. Vibration waves exhibit phenomena such as reflection and diffraction upon
encountering these barriers. Consequently, the dimensions of the barrier, including width,
length, and depth, must be determined based on the wavelength and the dimensions of
the vibration source and the sensitive structure. In recent years, the application of phono-
nic crystal theory for analyzing wave behaviors in periodic structures has gained signifi-
cant traction. For instance, Pu et al. [228] revisited the surface wave manipulation in peri-
odic trench barriers from the perspective of complex band structures. A limitation of vi-
bration isolation barriers is the large engineering quantity for construction, especially
when targeting long wavelengths for isolation, which necessitates extensive lengths and
depths. For example, fillers in in-filled trenches may encounter contamination or consoli-
dation, diminishing the vibration mitigation effect.
5.3. Base Vibration Isolation Bearing
With the focus on comfort issues induced by railways and the construction of sensi-
tive building structures like over-track buildings, there is a growing need for vibration
isolation bearings to mitigate vehicle-induced vibrations. Traditional isolation bearings,
such as laminated rubber bearings, lead rubber bearings, and friction pendulum bearings,
have proven effective in mitigating horizontal ground motion caused by earthquakes.
However, vehicle-induced vibrations primarily occur in the vertical direction. The mech-
anism of isolation bearings involves shifting the natural frequency of the superstructure
to deviate from the primary frequency of ground motion [229]. Vehicle-induced excitation
typically spans a broad frequency range, posing the challenge of frequency shifting. Ini-
tially, researchers investigated the impact of traditional seismic isolation bearings on ver-
tical railway-induced vibration isolation [49,230,231]. Subsequently, the concept of inte-
grated control of engineering and seismic vibrations in China has led to a surge in research
interest in three-dimensional vibration isolation bearings (3D-VIB) that can effectively ad-
dress both seismic and vehicle-induced vibrations [232]. Building upon traditional seismic
bearings, researchers have proposed various innovative designs and configurations for
these bearings. For instance, Cao and Pan et al. [233,234] developed a 3D-VIB using a com-
bination of disc springs and single friction pendulum bearings and another design incor-
porating a thick laminated rubber bearing and a friction pendulum. Sheng et al. [235] de-
veloped a new 3D-VIB, which is a laminated rubber bearing with vertical through-holes
filled by a mixture of sand and rubber particles. Liang et al. [236] proposed a 3D-VIB com-
posed of rubber isolation bearings and disc springs. He et al. [237] proposed a hybrid
Buildings 2024, 14, 1592 18 of 28
vibration bearing consisting of lead rubber bearing and thick rubber bearing with decou-
pled horizontal and vertical behaviors. While the aforementioned innovative vibration
isolation bearings have demonstrated outstanding performance in both experimental and
simulation seings, their practical application effects in engineering remain to be vali-
dated. In addition, isolation bearings intended for vehicle-induced vibrations must permit
relative motion between the superstructure and substructure under daily micro-ampli-
tude vibrations. It inevitably decreases horizontal stiffness and potentially raises risks of
destabilization.
5.4. TMD in Building Structures
TMDs currently serve as a primary technology for mitigating wind-induced vibra-
tions in super tall buildings, human-induced vibrations in large-span floors and pedes-
trian bridges [238,239], and occasionally, vibrations caused by machinery within struc-
tures [240]. The fundamental configuration of TMDs consists of a mass, springs, and
dampers and is designed to absorb vibrations at specific frequencies. The target frequency
is generally the horizontal first-order frequency for tall buildings or the vertical first-order
frequency for floors and bridges. TMDs are installed at the location of the maximum am-
plitude of the mode shapes. The ratio between the TMD mass and the generalized mass
of the to-be-controlled mode defines the maximum vibration reduction effect. Therefore,
conventional TMDs often exhibit excessive mass, posing safety risks. Distributed tuned
mass dampers (MTMD) have been proposed to address this issue, and additionally im-
plemented for the control of multi-order structural modes [241]. With the development of
new materials and processes, novel TMDs have been proposed, including: (a) TMDs with
more complex DOFs, nonlinear stiffness components or novel dampers [242–245]; (b)
semi-active tuned mass dampers (SATMDs) [246,247] and active rotary inertia driver
(ARID) systems [248]; and (c) tuned mass damper inerters (TMDIs) [249]. Furthermore,
the fixed-point law in classical Den Hartog theory has been widely adopted for engineer-
ing designs. Using the minimum acceleration of the original structure as the optimization cri-
terion, Ikeda et al. [250] derived the optimal ratios of frequencies and damping ratios are
𝛾=1
1+𝜇+0.096 + 0.88𝜇−1.8𝜇𝜉+1.34 −2.9𝜇+3𝜇𝜉 (8)
𝜉=3𝜇1 + 0.49𝜇+3𝜇
8(1 + 𝜇)+(0.13 + 0.72𝜇+0.2𝜇)𝜉+(0.19 + 1.6𝜇−4𝜇)𝜉 (9)
where 𝛾 and 𝜉 are the optimal ratios of frequencies and damping ratios, respec-
tively; 𝜇 is the mass ratio; 𝜉 is the damping ratio of the structure.
Nevertheless, researchers have continued to investigate the location and parameters
of the novel TMDs to optimize the adsorption effect and robustness, such as H2 norm op-
timization [251]. In addition, despite the widespread adoption of TMD technology, its ef-
ficacy in vibration reduction occasionally falls short of engineering expectations due to
various factors: low sensitivity of TMDs, which cannot be activated in case of micro-am-
plitude vibrations; malfunctions in critical components such as damper oil leaks; inaccu-
rate prediction of structural frequency resulting in the deviation of the parameters of the
pre-customized TMD from the optimal values.
6. Conclusions
Compared to traditional seismic and wind resistance analyses, vibration comfort re-
mains a niche research direction. Nevertheless, it covers a broad scope and often involves
collaboration across multiple disciplines and fields. In recent years, research on vibration
comfort has progressed significantly, and calculation theories and evaluation methods
have been established. However, there are still some issues to be addressed. This paper dis-
cusses recent developments in this field and presents the existing shortcomings in research
Buildings 2024, 14, 1592 19 of 28
and applications. The following are suggested potential topics for future studies regarding the
load, structural analysis, evaluation methods, and vibration mitigation measures.
1. Standardized stochastic load models
The main load categories that can cause vibration comfort problems are vehicle-in-
duced, human-induced, fluctuating wind, and dynamic machinery loads. Current load
models for structural design are not sufficiently developed for adoption. For fluctuating
wind loads, mature power spectrum models have been established; for human-induced
loads, relevant studies are ongoing; however, for the various types of dynamic machinery
loads and the complex vibration source mechanism of vehicle-induced loads, widely ac-
cepted standardized stochastic load models are still lacking.
2. Simplified comfort-based modeling method for structural design
With increasing concern for vibration comfort problems, more building structures
require specialised structural design and optimisation based on comfort. However, the
current comfort-based modeling methods developed by researchers have not yet achieved
the balance of convenience, accuracy, and efficiency required for structural design. There-
fore, simplified comfort-based modeling methods for structural design have essential en-
gineering applications.
3. Comfort evaluation method considering duration and load return period
Existing vibration comfort evaluations only consider the amplitude, frequency, and
direction of vibration, and ignore other features, such as the vibration duration and load
return period. As the types and forms of vibration sources increase, evaluation methodol-
ogies require further refinement.
4. Application of novel vibration mitigation measures
With the development of new materials (e.g., polymer materials, magnetorheological
materials, and shape memory alloys), innovations in vibration isolator construction, and
proposed corresponding vibration isolation design theories, vibration reduction and iso-
lation measures have been upgraded and optimized. Researchers initially use theories, simu-
lations, or experiments to innovate novel technologies and finally promote their application.
However, many novel technologies have been devised without considering their implications
on the safety and durability of the original structure, which should be emphasized.
Structural vibration comfort is a complex aspect of building design that requires care-
ful consideration to ensure a comfortable environment for occupants. One possible ap-
proach is the installation of an alarm system that can monitor and notify managers and
occupants of excessive structural vibrations. This system can help detect vibration sources
within the building and effectively prevent secondary accidents, such as stampedes
caused by excessive panic in crowds. The latest vibration-based structural health moni-
toring techniques could be feasibly utilized in the design of the system [252,253]. In addi-
tion, big data and machine-learning techniques also provide new ideas for research on
structural vibration comfort and provide directions for future study. They can be useful
in studying the stochastic behavior of loads and structures and the subjective ambiguity
in comfort evaluation.
Author Contributions: Writing—original draft preparation, Y.H.; writing—review and editing, W.X.
and Y.H.; supervision, W.X.; funding acquisition, W.X. All authors have read and agreed to the pub-
lished version of the manuscript.
Funding: This research was funded by Natural Science Foundation of China, grant number
52278458.
Data Availability Statement: Data sharing is not applicable.
Conflicts of Interest: The authors declare no conflicts of interest.
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