Assessing the ground vibrations produced by a heavy vehicle traversing a traffic obstacle

Science of the Total Environment 612 (2017) 1568–1576

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Science of the Total Environment journal homepage: www.elsevier.com/locate/scitotenv

Assessing the ground vibrations produced by a heavy vehicle traversing a traffic obstacle Loïc Ducarne* , Daniel Ainalis, Georges Kouroussis University of Mons — UMONS, Faculty of Engineering, Department of Theoretical Mechanics, Dynamics and Vibrations, Place du Parc 20, Mons B-7000, Belgium

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Two-step model to simulate ground vibrations of a truck driving over a speed hump. • Step 1 employs a multibody approach to model the truck and tyres. • Step 2 makes use of a finite element analysis to simulate ground wave propagation. • Three obstacle types examined: trapezoidal, half-sinusoidal, and Watts profile. • Sensitivity analysis on the influence of obstacle geometry, vehicle speed, etc.

A R T I C L E

I N F O

Article history: Received 28 June 2017 Received in revised form 21 August 2017 Accepted 21 August 2017 Available online xxxx Keywords: Traffic-induced ground vibrations Vehicle dynamics Tyre-road interaction Finite element analysis Wave propagation

A B S T R A C T Despite advancements in alternative transport networks, road transport remains the dominant mode in many modern and developing countries. The ground-borne motions produced by the passage of a heavy vehicle over a geometric obstacle (e.g. speed hump, train tracks) pose a fundamental problem in transport annoyance in urban areas. In order to predict the ground vibrations generated by the passage of a heavy vehicle over a geometric obstacle, a two-step numerical model is developed. The first step involves simulating the dynamic loads generated by the heavy vehicle using a multibody approach, which includes the tyre-obstacle-ground interaction. The second step involves the simulation of the ground wave propagation using a three dimensional finite element model. The simulation is able to be decoupled due to the large difference in stiffness between the vehicle’s tyres and the road. First, the two-step model is validated using an experimental case study available in the literature. A sensitivity analysis is then presented, examining the influence of various factors on the generated ground vibrations. Factors investigated include obstacle shape, obstacle dimensions, vehicle speed, and tyre stiffness. The developed model can be used as a tool in the early planning stages to predict the ground vibrations generated by the passage of a heavy vehicle over an obstacle in urban areas. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Despite the rapid development and improvement in recent decades into alternative transport networks, road freight remains vital

* Corresponding author. E-mail address: [email protected] (L. Ducarne).

http://dx.doi.org/10.1016/j.scitotenv.2017.08.226 0048-9697/© 2017 Elsevier B.V. All rights reserved.

to Europe. One of the principal advantages afforded by road transport is the flexibility in the selection of travel routes, and goods delivered. Apart from pollution issues, vehicles travelling over roads also generate speed-dependent ground-borne vibrations and noise (Iliou and Vogiatzis, 2005), which can significantly influence quality of life, and the vibration and acoustic environment (Vogiatzis, 2011). The ground-borne vibrations have the potential to cause significant negative environmental effects to residents, sensitive equipment, and

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Fig. 1. Schematic of a general heavy vehicle multibody model.

structures (Connolly et al., 2016). The health impact of environmental noise covers any type of road vehicles (Paviotti and Vogiatzis, 2012; Vogiatzis and Vanhonacker, 2016), but vibration problems are limited to specific vehicles and situations. Due to the limited operating speeds and weight per axle of road vehicles in comparison to rail transport, reduced levels of vibration are generally observed (Lu et al., in press; Connolly et al., 2015). However, when the road surface is not particularly smooth, comparable levels of ground-borne vibrations are experienced (Lak et al., 2011). The situation is exacerbated by the presence of localised transients, such as rail crossings, potholes, and speed humps (Hunaidi et al., 2000). While speed humps are designed to limit vehicle speed in urban areas, the large deflection can induce significant levels of vibration. Therefore, it is important to be able to predict the ground vibrations produced by singular defects. Hunaidi et al. (2000), Crispino and D’apuzzo (2001), and Watts and Krylov (2000) performed experimental surveys to study the effect of traffic-induced vibrations in urban areas and described the effect of vehicle characteristics, such as the mass or stiffness of the suspensions and tyres, on the generated ground vibrations. Watts (1992) established a series of empirical laws to estimate the level of ground vibrations caused by road traffic using transfer functions, based on experimental measurements. Numerical models have also been developed to simulate the effect of vehicle-hump interaction using different techniques, both for the source (Pyl et al., 2004b) and for the receiver, in direct proximity of the road (Lu et al., in press), the free field (Lombaert and Degrande, 2001), and the built environment (Pyl et al., 2004a) or underground spaces (Carels et al., 2012).

The research presented in this article is focused on the development, validation, and implementation of a two-step dynamic model to simulate the ground vibrations generated by the passage of a heavy vehicle over a speed hump (i.e. geometric obstacle). The first step of the model involves the development of a multibody model of a heavy vehicle, specifically a Volvo FL6 truck, in order to obtain the dynamic loads induced onto the pavement. The second step deals with the application of the computed dynamic loads into a ground model to simulate the propagation of vibration through the ground. The model provides a complete framework to evaluate heavy vehicle dynamics and the tyre-road interaction, along with the propagation of ground vibrations in the surrounding area. 2. Numerical model development This section describes the formulation, development, and implementation of the two-step numerical model. 2.1. Heavy vehicle model There exist various approaches to modelling vehicle dynamics. One popular approach is through the use of multibody systems, which allows for the representation of a mechanical system as a combination of bodies, and force elements. Each body is characterized by its inertial and geometric properties while force elements are used to define the interaction between them. Some common examples of force elements include springs, dampers and tyres. At any given moment, the position, speed or acceleration of each body can

Fig. 2. Schematic of the 7 degrees of freedom multibody model for the Volvo FL6 truck used in this study.

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L. Ducarne et al. / Science of the Total Environment 612 (2017) 1568–1576 Table 2 Material properties of each ground layer, from Lombaert and Degrande (2003).

Table 1 Parameters of the Volvo FL6 truck used in this study. Parameter

Value

Layer number

Thickness [m]

E [MPa]

m [–]

q [kg/m3 ]

b [–]

mCB ICB mr mf kp,r cp,r kp,f cp,f kt,r ct,r kt,f ct,f

9000 kg 35,000 kg m2 600 kg 400 kg 0.61 × 106 N/m 16,000 Ns/m 0.32 × 106 N/m 10,050 Ns/m 3.00 × 106 N/m 0 Ns/m 1.50 × 106 N/m 0 Ns/m

1 2 3 4 5

0.46 0.67 1.35 5.72 ∞

57 133 223 322 1288

1/3 1/3 1/3 1/2 1/2

1900 1900 1900 2000 2000

8 × 10 −4 6 × 10 −4 4 × 10 −4 4 × 10 −4 4 × 10 −4

be defined as a function of the configuration parameters qi as well as their first and second order time derivatives. Fig. 1 shows an example of a multibody model for a truck travelling on a road. This model is composed of 3 bodies: a car body and two axles. The number of configuration parameters is equal to the total number of Degrees of Freedom (DoF) of the system. The relative position and orientation of the local frame of a body i with respect to the local frame attached to a body j is given by the homogeneous matrix Ti,j :  Ti,j =

Ri,j {rj/i }i 000 1

 (1)

where Ri,j and rj/i are the rotational tensor and the coordinate vector of frame j with respect to frame i, respectively. The building and integration of the equations of motion of a multibody system, using the aforementioned approach, are performed using the C++ library EasyDyn (Verlinden et al., 2013). The vehicle model used in this work is based on the Volvo FL6 truck. Only plane movement in the vertical plane is considered (configuration parameters q2 , q4 , q6 , q7 , q10 , q12 , q13 , q16 , q18 from Fig. 1 are locked to 0) and the longitudinal displacement of the car body and both axles are considered equal (q1 = q11 = q17 in Fig. 1). This leads to a simpler, two-dimensional, 7 DoF model, depicted in Fig. 2.

h [mm]

Fig. 3. Conceptual illustration of a tyre modelled as an assembly of springs.

Since the speed of the vehicle is constant, the time derivatives of q0 , q4 and q6 are locked to their corresponding value. The geometric and inertial properties of the Volvo FL6 truck are obtained from Lombaert (2001), and given in Table 1. Unlike most models which use a lumped mass system (e.g. Lombaert and Degrande, 2003), a rolling tyre element is introduced to simulate the tyre-road interaction. The tyre is represented in EasyDyn as a force element whose value is established based on Fromm’s model (Gim, 1988). The vertical contact pressure follows a parabolic distribution along the circumference which leads to the equivalence between a tyre and an assembly of springs in the longitudinal, radial and lateral directions, illustrated in Fig. 3. With the multibody vehicle model defined, the next step is to define a road profile for the simulation. For this study, a flat road with a single geometric obstacle is used. The obstacle can easily be added as a shape function z = f(x). The function can have different shapes; e.g. trapezoidal (flat-top), half-sine or circular (Watts profile). For this work, three different types of humps are used (Fig. 4). A trapezoidal profile is used whose dimensions correspond to the obstacle used for the experimental validation. The same profile as well as a sine-shaped profile and a circular profile are used for the sensitivity analysis, whose dimensions are close to commonly used speed humps (Australian Standard, 2009). The model provides the time history of the contact forces between the tyres and the ground. These forces are the sum of the static weight of the vehicle, and the dynamic contribution caused by the interaction with the obstacle. 2.2. Ground model Soil is a complex, heterogeneous medium characterized by anisotropic and non-linear behaviour. According to Lombaert (2001), the non-linear behaviour can reasonably be neglected if the soil is dry and the shear strain is less than 10 −5 . For this study, time-domain analysis is employed, as the frequency-domain approach is not well suited for transient phenomena (Kouroussis et al., 2014b; Shih et al., 2016). Several methods can be used to model soil for the simulation of wave propagation. The Finite Element Method (FEM) is used due to its versatility compared to analytical and other numerical methods (Kouroussis et al., 2014a). It can be used to represent almost any kind of geometry and requires close to no modelling assumptions to be made. The only downside of FEM analysis is a higher computational time.

Flat-top (trapezoidal) Sine shaped Watts (circular)

100 54 0 -1.8

-0.85 -0.65

0 x [m]

0.65 0.85

Fig. 4. Different obstacle profiles used in the model.

1.8

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play an important role in the propagation of the waves as shown in the following equations:  cP =

 cS =

2G(1 − m) q(1 − 2m)

(2)

G q

(3)

where cP and cS are the respective velocities of the longitudinal pressure waves (P-waves) and shear waves (S-waves), G, m and q are the shear modulus, Poisson’s ratio and density of the material, respectively. These two waves are body waves and propagate to all directions from the source. There is also another type, Rayleigh waves, which only propagate along the surface of the ground. They travel slower than the aforementioned body waves but carry around 67% of the energy (Miller and Pursey, 1995), and are therefore the biggest cause of discomfort in areas close to a source of vibration (Hao et al., 2001). The most influential property, however, when it comes to vibration propagation, is the damping of the medium (Connolly et al., 2015). Soil damping can be implemented using a complex valued Young’s modulus and shear modulus, but this would lead to complex valued wave velocities as shown in Eqs. (2) and (3). This model of damping is not suited to time-domain analysis as it would result in a non-causal response (Shih et al., 2016). For time-domain finite element analysis, the Rayleigh damping model can be used. In this model, the damping matrix C is expressed as a linear combination of the mass matrix M and the stiffness matrix K.

C = aM + bK Fig. 5. Conceptual illustration outlining the steps required to develop the ground model (Kouroussis et al., 2014b).

Despite the heterogeneous character of the ground, which is composed of solid particles with water and air, it was modelled as an elastic isotropic half-space composed of several horizontal homogeneous layers with different material properties. These properties

(4)

This means that it is possible to express the loss factor g as the sum of stiffness-proportional and mass-proportional contributions:

g=

a + by y

(5)

face 2 face 5 7

8

8

5

7

face 4

6

5 4

face 6

3

6

3

4

1 2 face 1

1

face 3 2

(a)

(b)

Fig. 6. Comparison of infinite elements (a) and finite elements (b) (Kouroussis et al., 2014b).

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Table 3 Material properties of each road layer, from Lombaert and Degrande (2003). Layer number

Type

Thickness [m]

1 2 3

Asphalt 0.15 Crushed stone 0.19 Crushed concrete 0.25

E [MPa]

m [–]

q [kg/m3 ]

9150 500 200

1/3 1/2 1/2

2100 2000 1800

An exclusively stiffness-proportional viscous damping was used for the present model. This is achieved by setting a null a value (Shih et al., 2016). This offers the advantage of being easier to implement than actual Rayleigh damping while remaining usable for time-domain analyses (Kouroussis et al., 2013). Despite the heterogeneous character of the soil, it can reasonably be modelled by a superposition of five isotropic homogeneous layers. Each layer is characterized by its material properties, obtained from Lombaert and Degrande (2003). The value of the viscous damping b was estimated by dividing the corresponding hysteretic value g by the dominant frequency f of the ground vibrations in the experiment performed by Lombaert and Degrande (2003), as shown in the following equation (Gutowski and Dym, 1976):

(C3D8R) in order to generate the infinite elements (CIN3D8) at the border of the domain. 1 m long tetrahedral elements (C3D4) are used to mesh the internal part. Nevertheless, the interesting regions of the model (common surface with the road and data output points) are meshed with 0.3 m long elements for better accuracy. The road is meshed using 0.3 m long hexahedral elements. The simulations are performed using the dynamic implicit integration scheme. This implies a longer computational time than the explicit scheme, but better stability (Kouroussis et al., 2014b). The simulation duration is 5 s with a time increment of 0.005 s. Data output points are defined every meter from the center of the road, from 2 to 24 m, in order to obtain the time histories for the free field displacement, velocity and acceleration in all three directions x, y and z. Finally, moving loads are added to the model using a custom Fortran subroutine. For this, a surface corresponding to the path of the wheel-road contact patch must be defined. The location and magnitude of the contact pressure is then defined for every time increment to simulate the moving contact loads for both the rear and front wheels (Fig. 7). 3. Results

b=

g 2pf

(6)

The material properties of each layer are given in Table 2. In order to avoid wave reflection at the border of the domain, correct boundary conditions must be defined to account for the infinite nature of the medium (Kouroussis et al., 2014b). Fig. 5 describes the general steps to develop the ground model. Infinite elements (Fig. 6) are added at the border of a thin hemispherical shell, fractioned into the aforementioned layers, using an external script. The heavier internal part of the model is added afterwards and composed of the same layers. This approach maximizes computational efficiency when adding the infinite elements to the model. The road is then added on top of the model and all bodies are attached using a tie constraint to ensure that their common surfaces experience no displacement with respect to each other throughout the simulation. The road is also made of several layers, with the properties of each road layer shown in Table 3. The size of the finite elements was chosen in order to obtain decent accuracy while maintaining a reasonably short computational time. The thin shell is meshed with 0.8 m hexahedral elements

The simulation provides the time histories of the free field displacement, velocity and acceleration at points located every meter from the road in the perpendicular direction. Due to the presence of infinite elements, it is not possible to define appropriate boundary conditions at the border of the domain. A high-pass filter with a cut-off frequency of 0.5 Hz is used to eliminate the continuous displacement component due to the lack of proper boundary conditions. 3.1. Model validation The model is validated by comparing the predicted ground vibrations with an existing experimental case (Lombaert and Degrande, 2003). The case used a Volvo FL6 truck (see Table 1 for the vehicle characteristics) travelling at 30 km/h over a 54 mm high trapezoidal traffic plateau (see Fig. 4). The comparison between the time histories and frequency spectra of the free field velocities (tri-axial) at 8 m from the road is shown in Figs. 8 and 9. The passage of the front wheels is identified as the first significant vibration event while the vibration induced by the rear wheels occurs approximately 0.6 s later, which corresponds

loading definition

moving pressure loading

y

x z

infinite elements region of interest (finite element modelling)

Fig. 7. Definition of the moving contact loads on the road surface.

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10− 4

10− 3 Velocity [m/s]

Velocity [m/s]

5

0

−5

2 1 0

0

Time [s]

50 Frequency [Hz]

(a)

(a)

1

2

3

0

100

10− 4

10− 3 Velocity [m/s]

Velocity [m/s]

5

0

−5

2 1 0

0

1

2

3

0

50 Frequency [Hz]

Time [s]

(b)

(b) 10− 4

10− 3 Velocity [m/s]

Velocity [m/s]

5

0

−5

2 1 0

0

100

1

2

0

50 Frequency [Hz]

3

Time [s]

(c)

(c) Fig. 8. Numerical (solid blue line) and experimental (dash-dotted black line) time histories of the free field velocity at 8 m from the road in the longitudinal (a), lateral (b) and vertical (c) directions in the case of a Volvo FL6 truck travelling at 30 km/h over a trapezoidal traffic plateau. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to the spacing between the axles divided by the vehicle speed. The difference of magnitude between the two events is explained by differences in weight and tyre stiffness between the two axles. The time histories provided by the model fit well to the measured data. The predicted frequency are also in good agreement with the experimental results. The dominant frequency is well reproduced; however, a slight underestimation is noticeable between 0 and 20 Hz. 3.2. Sensitivity analysis With the two-step model validated, it can now be used to perform a sensitivity analysis, to study the influence of different parameters on the resulting ground vibrations. Several indicators can be used to characterize and compare ground vibrations (Kouroussis et al., 2014). The Peak Particle Velocity (PPV) is a simple but efficient indicator used to characterize and compare the magnitude of vibration signals. For example, it is frequently used to predict the probability of structural damage due to high magnitude ground vibrations (Hao et al., 2001). The PPV is also related to vibration-induced discomfort

100

Fig. 9. Numerical (solid blue line) and experimental (dash-dotted black line) frequency content of the free field velocity at 8 m from the road in the longitudinal (a), lateral (b) and vertical (c) directions in the case of a Volvo FL6 truck travelling at 30 km/h over a trapezoidal traffic plateau. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(Deutsches Institut für Normung, 1999b). The PPV is defined as the maximum absolute value of a signal over a given period of time:   PPV = max(v(t))

(7)

In order to evaluate the comfort (or lack thereof) of people in buildings, DIN 4150-2 recommends the use of the KBF indicator (Deutsches Institut für Normung, 1999a). At a given time, the magnitude of a vibration generally depends more on the vibrations occurring immediately before than on the previous ones. The KBF value corresponds to the RMS value of the vibration record multiplied by a decreasing exponential function so that, at any given moment, the most recent vibration events have a greater weight than the previous events, as shown in the following equation:  KBF (t) =

1 t



t 0

KB2 (n)e−

t−n t

dn

(8)

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1.5

Flat-top Half-sine Watts

6 4

PPV [m/s]

PPV [mm/s]

8

2 0 2

4

6

8 10 12 14 16 18 Distance from the road [m]

20

22

1 0.5

24

0

(a)

Flat-top Half-sine

KBF,max [mm/s]

2.5

1.5

L = Ldefault L = 2 Ldefault

Watts

Flat-top Half-sine Watts

2

(a)

1 0.5 0 4

6

8 10 12 14 16 18 Distance from the road [m]

20

22

24

(b) Fig. 10. Influence of profile shape on the ground vibrations in terms of (a) the PPV, and (b) the KBF (v = 30 km/h).

0.6 KB F,max [m/s]

2

0.4 0.2 0

where KB is the speed-proportional RMS value of the signal and t is equal to 0.125 s in the case of short signals. Both the PPV and KBF indicators are used to analyze the ground vibration records and investigate the influence of various parameters.

Flat-top Half-sine Watts

3.2.1. Influence of profile shape First, a comparison of the effect of obstacle shape on the generated ground vibrations was undertaken, shown in Fig. 10. As expected, the general trend has an exponential decrease as the distance from the source increases. The trapezoidal obstacle, despite the smaller dimensions, induces stronger ground vibrations than the other two obstacles. This is explained by its sharper geometry leading to more important dynamic interaction as the vehicle passes over it. The Watts and half-sine obstacles present similar levels of vibration, with a slightly smaller magnitude for the Watts obstacle. 3.2.2. Influence of the length of the profile The second parameter studied is the influence of the obstacle length on the generated ground vibrations. Two different lengths for each of the three obstacles are investigated (3.6 and 7.2 m for the sine-shaped and Watts profiles, and 1.7 and 3.4 m for the trapezoidal obstacle). The results shown in Fig. 11 show that for an obstacle twice as long (all other parameters kept constant), the PPV and KBF of the vibrations 8 m away from the road are approximately halved. This can be explained by the fact that a longer profile has a gentler slope, and since the variation in vertical position of the wheels occurs over a longer duration, the dynamic response of the suspension system and tyres have a smaller magnitude and so do the generated ground vibrations. 3.2.3. Influence of the height of the profile The influence of the obstacle height is also examined. The PPV and KBF of the ground vibrations for different obstacle heights are shown in Fig. 12 for the trapezoidal, half-sine, and Watts obstacles. The height of the obstacle has a similar influence on the vibrations as the length. However, the difference in magnitude of the ground

L = Ldefault L = 2 Ldefault

(b) Fig. 11. Influence of obstacle length (L) on the ground vibrations in terms of (a) the PPV, and (b) the KBF (v = 30 km/h, d = 8 m).

vibrations is less noticeable for the half-sine profile than for the circular and trapezoidal obstacles. This variation is explained by the difference in geometry at the edges of the profiles; as the height is doubled, the slope at the edges of the circular and trapezoidal profiles is approximately doubled as well. The initial slope of the sine-shaped profile is zero and increases progressively leading to a less severe dynamic interaction as the vehicle encounters the obstacle.

3.2.4. Influence of the speed of the vehicle Obviously, the speed of the vehicle has a major role in the dynamic interaction and therefore, influences on the level of ground vibrations. Fig. 13 shows the influence of vehicle speed for all three profiles. It appears that the influence of speed has a greater effect for the Watts profile. Again, this is because the major part of the dynamic interaction occurs at severe slope variations which are more emphasized for the Watts profile due to the angles at its edges. The influence of the tyre stiffness was also studied for the flat-top obstacle (Fig. 14) and while it is obvious that a 5-times greater tyre stiffness results in higher level ground vibrations, it is interesting to note that, from these results, increasing the stiffness of the tyres also increases the influence of the speed of the vehicle on the level of the ground vibrations generated.

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PPV [m/s]

3 2 1 0 Flat-top Half-sine h = 2 h default h = h default

Watts

(a)

(a) 0.8 KB F,max [m/s]

KB F,max [m/s]

1.5 1

0.6 0.4 0.2

0.5

0

0

Flat-top Half-sine

Flat-top

v = 60 km/h v = 30 km/h

Watts

Half-sine h = 2 h default h = h default

Watts

(b)

(b) Fig. 13. Influence of vehicle speed (v) on the ground vibrations in terms of (a) the PPV, and (b) the KBF (d = 8 m).

Fig. 12. Influence of obstacle height (h) on the ground vibrations in terms of (a) the PPV, and (b) the KBF (v = 30 km/h, d = 8 m).

PPV [mm/s]

This paper presented the development of a two-step numerical model to predict the ground vibrations generated by the passage of a heavy vehicle over a geometric obstacle (i.e. a speed hump). The first step used a multibody approach to model the heavy vehicle-tyreroad interaction for a Volvo FL6 truck. The second step employed a finite-infinite element model to simulate the propagation of ground vibrations due to the dynamic loads generated by the vehicle-tyreroad interaction. Validation of the model was undertaken using measured ground vibrations from an experimental study available in the literature. A sensitivity analysis was performed to investigate the influence of various speed-hump design factors on the generated ground vibrations. The ground vibrations were recorded perpendicular to the road, in front of the obstacle, every meter between 2 and 24 m. Three different speed hump types were investigated; trapezoidal, halfsinusoidal, and Watts (circular) profiles. The influence of the obstacle height and length were assessed, along with vehicle speed, and the tyre stiffness. The two-step numerical model can be used in the early planning stages to predict the ground vibrations generated by speed humps or other geometric obstacles in urban areas to ensure that not only no cosmetic or structural damage occurs, but that residents are not subjected to excessive or uncomfortable vibrations. In the future,

v = 30 km/h v = 60 km/h v = 30 km/h - k t x 5

30 20

v = 60 km/h - k t x 5

10 0 2

4

6

8 10 12 14 16 18 Distance from the road [m]

20

22

24

(a) 15 KBF,max [mm/s]

4. Conclusions

40

10

v = 30 km/h v = 60 km/h v = 30 km/h - k t x 5

5

v = 60 km/h - k t x 5

0 2

4

6

8 10 12 14 16 18 Distance from the road [m]

20

22

24

(b) Fig. 14. Influence of vehicle speed and tyre stiffness on the ground vibrations in terms of (a) the PPV, and (b) the KBF for the trapezoidal obstacle.

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