Prediction of structure borne sound in

(1)

Prediction of structure borne sound in

buildings close to subways

Moa Wijkmark

Master of Science Thesis

Stockholm, Sweden 2014

(2)

(3)

Prediction of structure borne sound in buildings

close to subways

Moa Wijkmark

ISSN 1651-7668 TRITA-AVE 2015:03 Stockholm 2014

Master of Science Thesis Royal Institute of Technology School of Engineering Sciences

Department of Aeronautical and Vehicle Engineering

The Marcus Wallenberg Laboratory for Sound and Vibration Research

(4)

(5)

Abstract

A model predicting vibrations in buildings caused by trains and subways have been developed and tested. The load bearing elements of the building are represented by a two dimensional beam structure. Two general situations were considered; a column of inner walls, with floors extending in two directions, and a column of outer walls, with floor extending only in one direction. The model was constructed using a MATLAB- based software which is based on a variant of the Dynamic Stiffness Method. Both longitudinal waves and bending waves are included in all elements. The influence of the ground is included by calculating the ground impedance. The results from the modeling were compared to measurements on an existing building.

It was shown that bending waves in the walls, caused by horizontal ground movement, will have substantial impact on the results in the lowest levels of the building, whereas the behavior of the upper part of the building is dominated by the longitudinal waves, caused by vertical ground movement. The analysis of the ground impedance showed that buildings on mud will be mostly sensitive to horizontal ground vibrations, since vertical vibrations couples poorly to the building.

The measurements showed that solid concrete floors in a normal sized apartment building will behave like infinite plates at frequencies above 100 Hz, contrary to previous research where an increase due to resonances was assumed. It was also concluded that transversal waves in the walls can safely be excluded from the model, due to the comparatively low vibration levels.

(6)

Sammanfattning

En beräkningsmodell för att uppskatta vibrationer i byggnader orsakade av tåg och tunnelbanor har utvecklats och testats. De bärande elementen i byggnaden representeras i modellen av en tvådimensionell balkstruktur, vilket ger en kort beräkningstid jämfört med de annars vanligt förekommande FE-modellerna. Två generella fall undersöktes; en kolumn av inre väggar, med bjälklag på båda sidor, och en kolumn av yttre väggar med bjälklag bara på ena sidan. Modellerna konstruerades med hjälp av ett MATLAB-baserat program som grundar sig på en variant av dynamiska styvhetsmetoden. Både longitudinalvågor och böjvågor inkluderas i samtliga element.

Markens egenskaper tas hänsyn till genom att beräkna markimpedansen. Resultaten från modellen jämfördes med mätningar på en befintlig byggnad.

It was shown that bending waves in the walls, caused by horizontal ground movement, will have substantial impact on the results in the lowest levels of the building, whereas the behavior of the upper part of the building is dominated by the longitudinal waves, caused by vertical ground movement. The analysis of the ground impedance showed that buildings on mud will be mostly sensitive to horizontal ground vibrations, since vertical vibrations couples poorly to the building.

The measurements showed that solid concrete floors will behave like infinite plates at frequencies above 100 Hz, contrary to previous research where an increase due to resonances was assumed.

It was also concluded that transversal waves in the walls can safely be excluded from the model, due to the comparatively low vibration levels.

(7)

1. Introduction

Structure borne sound due to rail bound traffic is an increasing concern in modern cities. Due to lack of space, places that earlier would have been considered unsuitable in view of disturbances are now being utilized, and buildings are placed ever closer to subway tunnels and railways. With the increasing vibration levels in the ground, the problem of accurately predicting the effects on buildings is actualized. Depending on the type of ground and where the train is located relative to the building, different types of ground waves will be generated and the way the building is

affected will vary. A building located on solid rock above a tunnel will be hit mainly by vertical vibrations, whereas if the building is moved sideways the horizontal velocity component will become more important. A train running on softer ground will generate Rayleigh waves, with both horizontal and vertical velocity components. Today, the knowledge about the coupling between ground and building in these different situations, as well as the behavior of the building itself, is limited. Especially when building on softer ground, such as clay soil, the assessment of the risk for disturbances is more or less guesswork, causing a high risk for unforeseen problems as well as the opposite; that costly precautions are being prescribed unnecessarily.

The goal of the thesis was to find a prediction method which should be quick and easy to use, and at the same time versatile and adaptable to the conditions at the current building site. The work was initialized by an acoustic consultant company, seeing a future use of the model as a tool to assess the risk for problems as well as identifying where in the building they will likely

manifest, and thereby determine if and what action should be taken.

1.1 Earlier work

That vibrations caused by rail traffic is a live and growing research field is apparent considering the number of research and thesis projects started the last few years. Despite this, precious little research concerning simple predictive models of the type sought in this thesis has been found.

Most existing models are either based on the Finite Element Method (FEM), making the simulations computationally heavy, or rely on empirical conclusions often valid only in very specific cases. One other more promising approach, with a few existing examples in the literature, is to generalize the geometry of the building as a tree-like, two dimensional, structure and study the wave propagation.

1

(10)

Figure 1. Geometry of generic building used in some simple modeling approaches.

Ljunggren [1] treated the problem of a building placed directly on rock; i.e. the ground has much higher impedance than the building and can be considered as a constant velocity source. He argued that since the wave speed in rock is much higher than the speed of any wave type in a concrete plate, all types of waves could be induced equally easy. However, since the power transmitted into the building by different wave types is dependent on the velocity component squared times the wave speed, the power carried by a longitudinal or transversal wave will be much larger than the contribution of a bending wave. Therefore only purely vertical excitation, causing longitudinal waves in the walls, is considered in the model he presents. Ljunggren’s model consists of an infinite cascade of rods, representing the walls, alternated with infinite beams extending symmetrically in two directions, representing the floors. Consequently no resonance effects are included, neither in the floors or coupled to the height of the building.

Researchers at Tuft University [2, 3] developed and tested a similar model. In their model the load bearing columns of a building is represented by vertical rods, carrying longitudinal waves, and the floors are modeled as infinite plates conducting energy away. A dynamic stiffness (DS) representation of the system is constructed, where the contribution of the rods are given by their local DS-matrices and the floors are included as mass terms calculated using the driving point impedances of the plates. Experiments on a scale model building with simple geometry showed a good match between the predicted and the measured velocity levels [3]. It was also shown that resonances observed in a single floor element was not present in the measurements on the complete building, leading to the conclusion that the floors of a building can be treated as infinite plates.

Hassan [4] built upon Ljunggren’s work by making the height of the building finite, and thereby including the reflections at the roof. Furthermore, he includes the ground impedance by

3

1 2

𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔

x y

z

2

(11)

calculating the impedance given by a line of circular indenters representing a wall. Hassan uses the rod/beam-model to calculate the vertical velocity in the junction at each level of the building, and argues that the average velocity level in the floor will be 8 dB higher than the level at the junction, as long as the internal losses is much lower than the boundary losses and all parts of the building are of equal thickness.

1.2 Present work

Based on the research presented above a similar model was constructed using a MATLAB- program based on a variant of the Dynamic Stiffness Method (DSM), developed by Svante Finnveden at the MWL in Stockholm [5].

Compared to previous work and other existing modeling techniques the main merits of the model presented in this thesis are

- Both horizontal and vertical ground vibrations are included - A finite ground impedance is included

- A simulation is run in a matter of seconds even on a simple laptop - The geometry can easily be adapted to the situation at hand - Floors can be modeled as either infinite or with a chosen length

3

(12)

2. Dynamic Stiffness Method

The Dynamic Stiffness Method (DSM) is, just like the more commonly known Finite Element Method (FEM), a way of modeling complex structures by the use of building blocks of simpler geometry with a known behavior. In a DS model the deformation of the elements are described by wave solutions, whereas the FEM uses polynomials. This means that in a DS model much fewer (larger) elements can be used than in a FE model. Also, as opposed to the FEM the exactness of the solution using the DSM is more or less independent of the number of elements used. As long as a part of the structure is continuous, with no material or shape changes, it can be modeled using a single element. This makes it possible to simulate large structures, such as buildings, in a matter of minutes even on a simple laptop.

2.1 Calculating the local dynamic stiffness matrix

The goal when formulating a DS model of a system is to describe the relationship between the displacements of the nodes and the forces acting on those nodes, based on a prescribed behavior in between them. There are several ways to do this, some more straightforward than others.

Either way, the starting point will always be to decide which types of waves are present and to formulate the equation of motion governing the behavior of the deformation caused by those waves. The derivation given below is based on the method presented by Richards and Leung [6].

Figure 2. Beam element with two degrees of freedom at each end.

A beam element, carrying bending waves, with two degrees of freedom at each node, one translational and one rotational, will be used as an example. The displacement, 𝑢 , of the beam is described by

𝑑⁴𝑢

𝑑𝑥⁴− 𝜆⁴𝑢 = 0 ⁽¹⁾

Node 1 Node 2

x, 𝜉𝜉 y

z

𝜉𝜉 = 0 𝜉𝜉 = 1

𝑞𝑞₂, 𝑄𝑄₂

𝑞𝑞₁, 𝑄𝑄₁ 𝑞𝑞₃, 𝑄𝑄₃

𝑞𝑞₄, 𝑄𝑄₄

4

(13)

where 𝜆⁴ = ^𝜔²_𝐸𝐼^{𝜌𝐴𝑤𝑤}⁴, 𝐴 is the cross-sectional area, 𝑙 is the length of the beam, 𝐸𝐸𝐼 is the bending stiffness, 𝜌𝜌 is the density and 𝜔 the angular frequency. The general solution to equation (1) can be written

𝑢(𝜉𝜉) = 𝐴₁cos(𝜆𝜉𝜉) + 𝐴₂sin(𝜆𝜉𝜉) + 𝐴₃cosh(𝜆𝜉𝜉) + 𝐴₄sinh (𝜆𝜉𝜉) ⁽²⁾ where 𝜉𝜉 = ^𝑥_𝑤𝑤 and 𝑙 is the length of the element. Assuming that the end displacement are known, and named according to Figure 2, the amplitudes 𝐴1−4 can be calculated and an expression for the displacement as a function of the end displacement found

{𝑢} = � 4 x 4 matrix containing

sin, cos, sinh and cosh expressions� � 𝑞𝑞1

𝑞𝑞₂ 𝑞𝑞₃ 𝑞𝑞₄

� ⁽³⁾

Using this relation together with the boundary conditions, linking the displacements at the nodes to the forces acting on those,

𝑄𝑄1 = 𝐸𝐸𝐼𝑑³𝑢(0)

𝑑𝑥³ , 𝑄𝑄2 = −𝐸𝐸𝐼𝑑²𝑢(0)

𝑑𝑥² , 𝑄𝑄3 = −𝐸𝐸𝐼𝑑³𝑢(1)

𝑑𝑥³ , 𝑄𝑄4 =𝐸𝐸𝐼𝑑²𝑢(1)

𝑑𝑥² ⁽⁴⁾

gives a relation between the assumed node displacements 𝑞𝑞1−4 and the assumed forces 𝑄𝑄1−4

[𝐷]{𝑞𝑞} = {𝑄𝑄} ⁽⁵⁾

where the matrix 𝐷 is the dynamic stiffness matrix giving the method its name. In this case, using the simple beam element, it becomes

𝐷 =𝐸𝐸𝐼 𝑙³

⎣⎢

⎢⎡ 𝐹¹ −𝐹₄𝑙 𝐹₅ 𝐹₃𝑙

−𝐹₄𝑙 𝐹5

𝐹₃𝑙

𝐹₂𝑙² −𝐹₃𝑙 𝐹₁𝑙²

−𝐹₃𝑙 𝐹₆ 𝐹₄𝑙 𝐹1𝑙² 𝐹4𝑙 𝐹2𝑙² ⎦⎥⎥⎤ 𝐹1 = −𝜆 (sinh 𝜆 − sin 𝜆)/𝛿

𝐹₂ = −𝜆 (cosh 𝜆 sin 𝜆 − sinh 𝜆 cos 𝜆)/𝛿 𝐹3 = −𝜆²(cosh 𝜆 − cos 𝜆)/𝛿

𝐹4 = 𝜆²(sinh 𝜆 sin 𝜆)/𝛿 𝐹₅ = 𝜆³(sinh 𝜆 + sin 𝜆)/𝛿

𝐹6 = −𝜆³(cosh 𝜆 sin 𝜆 + sinh 𝜆 cos 𝜆)/𝛿 𝛿 = cosh 𝜆 cos 𝜆 − 1

(6)

The DS-matrix of a more complex beam, carrying longitudinal- and torsional waves as well as bending waves, is given by Richards and Leung [6].

For beams that are short, _𝜆^𝑤𝑤 ≪ 1, the functions in equation (6) need to be calculated by series expansions and the matrix D will then be as in an FE model.

5

(14)

2.2 Assembling the global DS-matrix

When the local dynamic stiffness matrix of each element is known, these elements needs to be locked in space and related to each other in order to create a model of a bigger system. A structure with three elements, as shown in Figure 3, will be used as an example.

Figure 3. System with three beam elements, each node having two degrees of freedom.

Node 1 Node 2

2

3

1 Red = global dof Blue = local dof

Green = element numbering

Node 1 Node 2 Node 3

Node 4

6

(15)

First, the local coordinate system of each element needs to be related to the global coordinate system. This is done by the use of a rotational matrix, 𝑛^(𝑖)(𝜙, 𝜃, 𝜓), which gives the relationship between the two coordinate systems as

𝑞𝑞𝑔𝑔𝑤𝑤𝑔𝑔𝑏𝑤𝑤𝑤𝑤(𝑖) = 𝑛^(𝑖)𝑞𝑞𝑤𝑤𝑔𝑔𝑐𝑤𝑤𝑤𝑤(𝑖) (7)

Using this relationship the transformation of the dynamic stiffness matrix can in turn be expressed as

𝐷𝑔𝑔𝑤𝑤𝑔𝑔𝑏𝑤𝑤𝑤𝑤(𝑖) = 𝑛^(𝑖)−1𝐷^{𝑤𝑤𝑔𝑔𝑐𝑤𝑤𝑤𝑤}𝑛^(𝑖)= 𝑛^(𝑖)𝑇𝐷^{𝑤𝑤𝑔𝑔𝑐𝑤𝑤𝑤𝑤}𝑛^(𝑖)

Defining the Euler angles (𝜙, 𝜃, 𝜓) according to the x-convention [7], the rotational matrix for element 1 and 2 is given by setting 𝜙 = 90, 𝜃 = 0 and 𝜓 = 0, resulting in

𝑛^(1,2) = �

−10 00

10 00

00

−11 00 00

� ⁽⁸⁾

The local coordinate system of element 3 coincides with the global coordinate system, meaning that all rotational angles are equal to zero and the dynamic stiffness matrix remains unchanged.

After this coordinate transformation the elements are locked in space, but are still uncoupled. In order to lock the elements to each other one more transformation is needed; the local nodes must be related to the global nodes. This is achieved via a transformation matrix 𝐵^(𝑖) coupling the local degrees of freedoms to the appropriate global degree of freedom as,

𝑞𝑞𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤(𝑖)= 𝐵^(𝑖)𝑞𝑞𝑔𝑔𝑤𝑤𝑔𝑔𝑏𝑤𝑤𝑤𝑤(𝑖)

In this case, with four local degrees of freedom per element and a total of eight global degrees of freedom, 𝐵^(𝑖) will have four columns and eight rows. Taking element three as an example, where local node one equals global node two and local node two equals global node four, the

transformation matrix becomes

𝐵⁽³⁾ =

⎣⎢

⎢⎢

⎢⎡0 01 00 00 0

00 01 00 00

00 00 00 10

00 00 00 01⎦⎥⎥⎥⎥⎥⎥⎤

(9)

giving the element’s contribution to the total dynamic stiffness matrix as 𝐷𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤(3)= 𝐵^(3)𝑇𝐷𝑔𝑔𝑤𝑤𝑔𝑔𝑏𝑤𝑤𝑤𝑤(3)𝐵⁽³⁾

The total dynamic stiffness matrix describing the complete coupled system is then given by 𝐷^{𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤} = ∑3 𝐷𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤(𝑖)

𝑖=1 , and the behavior of the whole system can finally be expressed as

𝐷^{𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤}𝑞𝑞^{𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤} = 𝑄𝑄^{𝑡𝑔𝑔𝑡𝑤𝑤𝑤𝑤} ⁽¹⁰⁾

7

(16)

Using equation (10) the reaction of the system due to any force or displacement, acting on any node, can now easily be calculated using Matlab or any other calculation software.

2.3 Method used in the simulations

The program used in the simulations is based on a variant of the method described above, where instead of the force balance approach a variational principle is used when determining the local dynamic stiffness matrices. The resulting matrices are identical to those derived according to the scheme above, but the mathematics needed to get there are much more involved. The main merits of the variational approach is an easier handling of the coupling between elements, as well as a reduction of eventual errors introduced when complex elements such as plates are used. This is described in detail in an article by Finnveden [5]. The shape functions, equation (6), cannot be used for beams that are more than approximately five wavelengths long, as the hyperbolic terms will then be too large compared to the trigonometric terms. Therefore, Finnveden [5] scales the shape functions so that they are useful for arbitrary length beams.

8

(17)

3. Modeling

Similarly to the models by Hassan, Ljunggren and Brett et al, the main idea behind the models presented in this thesis is to represent the building by a two dimensional beam structure. By omitting one dimension, some information will clearly be lost, but since the main interest lies in general trends and phenomena much knowledge can still be gained. In order to keep the

approximations at a minimum, a simple type of building was studied; a house with bearing walls, with both walls and floors made of solid concrete and the same planning on every level of the building. Assuming that all levels of the building look the same, two general situations are possible: the wall being situated somewhere inside the building, with floors extending

symmetrically in two directions, or the wall being part of the outer façade of the building, with floors extending only in one direction. Consequently two different models were constructed and analyzed, one representing each situation. In both cases all parts of the building were assumed to be equally thick and be made of the same material; 200 mm solid concrete. All elements were modeled as Timoshenko beams, with the same length for all wall and floor elements respectively.

The only connection to the ground was assumed to be the lowest wall element, allowing excitation in form of vertical or/and horizontal velocity. Based on previous experience of train induced ground vibrations the analysis was limited to the frequency range 0-400 Hz. To start with, the case of a building placed upon solid rock, so that the ground acts as a constant velocity source, was considered.

The program used to perform the calculations is a MATLAB-based script developed by Svante Finnveden [5] at KTH Royal Institute of Technology in Stockholm. It allows the user to decide the geometry of the structure, the properties of each element, the number and types of degrees of freedom as well as the type and magnitude of the excitation.

3.1 Comparison to similar models

As a first test of the program a building with the same properties as the one described by Hassan [4] was modeled; a three floor structure with one wall and infinite floors extending symmetrically, the third floor being the roof of the building. All elements were modeled as Timoshenko beams with properties and geometry according to the table found in Figure 4 below.

9

(18)

Figure 4. Structure modeled by Hassan.

The model was excited by a prescribed vertical velocity, with a magnitude of 1 m/s at all

frequencies, at the first node of the lowest vertical element. Figure 5 shows the resulting vertical velocity level in the middle node of each floor compared to the input velocity at the ground. As seen in the figure the results are equal to those given by Hassan.

50 100 150 200 250

-10 -5 0 5 10 15

Frequency [Hz]

Velocity [dB rel 1 m/s]

Vertical velocity at the middle node of each floor

Floor 1 Floor 2 Floor 3

1 2 3

𝑣𝑣 = 1m/s

Material Concrete

Young’s modulus, 𝐸𝐸 26 GPa Poisson’s ratio, 𝜈𝜈 0.3

Density, 𝜌𝜌 2400 kg/m³ Loss factor, 𝜂𝜂 0.015 Cross section 0.2 x 0.2 m² Length, wall element 2.7 m

Length, floor element 500 m (“infinite”)

10

(19)

Figure 5. Vertical velocity in the middle node of a three floor building with infinite floors. Comparison between the model presented by Hassan [4], lower graph, and a model built using the program written by Finnveden [5], upper graph.

3.2 Inner wall

Secondly, the model was extended to include horizontal excitation as well as vertical, and the number of levels increased. The results from a model with five levels, with global nodes and element numbers according to Figure 6, will be presented.

Three types of excitation were investigated; pure vertical motion, pure horizontal motion and both at the same time. In all cases the magnitude of the velocity was set to 1 m/s for all

frequencies between 0-400 Hz, and applied to node number one. In order to study the effect of floor resonances, two different models were analyzed, one were the floor elements was made long enough for the wave to be effectively dead when reaching the farther end (500 m) giving no resonances, and one with 10 m long floors. All elements were modeled as Timoshenko beams made of solid concrete, with geometry and material parameters according to Table 1.

Since the ultimate goal of the model was to study the risk for structure borne sound, the analysis was focused on the mean value of the velocity component perpendicular to the surface of each element. That is, the movement resulting from bending waves. In the case of infinite floor elements the average of the first 10 meters of the element was used, whereas the average of the whole element was used when analyzing the finite elements (both walls and floors).

11

(20)

Figure 6. Node- and element numbering in the symmetric structure.

Finite floor element Infinite floor element Wall element

Length 𝐿_𝑥 = 10 m 𝐿_𝑥= 500 m 𝐿_𝑦 = 2.7 m

Width x Height 0.2 x 0.2 m² Density, 𝝆 2400 kg/m³ Young’s modulus, E 26 GPa Poisson’s ratio, 𝝊 0.15 Loss factor, 𝜼 0.05

Table 1. Properties of symmetric model.

3.2.1 Horizontal excitation

If the ground moves horizontally, perpendicular to the long side of the wall, a bending wave will be initialized in the wall which in turn creates bending waves and longitudinal waves in the floors.

To start with the model with almost infinite (500 m) floor elements is analyzed. Figure 7 shows the average velocity level perpendicular to the surface of the elements in the three lowest levels of the building. As seen the losses at each junction is high, with a decrease of around 10 dB per

12

(21)

junction, effectively limiting the risk for structure borne sound to the first couple of levels of the building.

Figure 7. Average velocity perpendicular to the surface of floors and walls, given by a horisontal force at the ground node.

If the floor elements are made finite, in this case 10 m long, resonances will occur, increasing the magnitude of the vibrations at certain frequencies. Figure 8 shows the average velocity level in the elements for that case, whereas Figure 9 displays a comparison to the levels obtained in the model with infinite floor elements.

0 50 100 150 200 250 300 350 400

-60 -50 -40 -30 -20 -10 0 10

Frequency [Hz]

o so te a t

Element 1 Element 2 Element 4 Element 5 Element 7 Element 8

13

(22)

Figure 8. Average velocity level normal to surface of elements in structure with finite floor elements (10 m).

Figure 9. Difference in average velocity level between structure with infinite floor elements (500 m), solid lines, and structure with finite floor elements (10 m), dotted lines.

3.2.2 Vertical excitation

If the building instead is subject to purely vertical ground vibrations, energy will travel upwards in the building as longitudinal waves, inducing bending waves in the floors. Since the building is completely symmetrical the movement of the junctions will be purely vertical, giving no bending waves in the walls and thus no movement perpendicular to the surface of those elements.

Therefore only floor elements were analyzed for this case. Figure 10 shows the average velocity level in each floor when the floor elements are considered infinite (500 m). As seen in the figure

0 50 100 150 200 250 300 350 400

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20

Frequency [Hz]

0 50 100 150 200 250 300 350 400

-60 -50 -40 -30 -20 -10 0 10 20

Frequency [Hz]

Element 4 Element 4 Element 5 Element 5

14

(23)

the transmission losses at each junction is much smaller than in the case with bending waves, increasing the risk for disturbances higher up in the building. Also, due to vertical resonances, the vibration levels will in some frequency regions even be higher upwards in the building. The first resonance peak, at 44 Hz, yielding high vibration levels in the upper part of the building is due to the height of the building corresponding to one fourth of the wave length of the quasi-

longitudinal waves.

Figure 10. Average velocity level in the floors given by a vertical excitation force at the ground node.

Figure 11 shows the average velocity levels if the floor elements are made finite (10 m). As seen in Figure 12 the levels are increased at most frequencies compared to the model where the floors are considered infinite. Figure 13 shows the difference between the level at the junction and the mean level of the whole element¹. According to Hassan [4] the mean level in the elements should be around 8 dB higher than level at the junction. As seen in the figure the model gives an

increased level for frequencies below around 50 Hz, whereas the level is higher at the junction for higher frequencies. This behavior was, to some extent, observed in the measurements as well, as is shown in Chapter 5. As seen in Figure 14 the average velocity of the entire element in the model with finite floors agrees quite well with the velocity level at the junctions in the model with infinite floor elements. Consequently, in order to simplify the calculations it might be possible to use the values at the junctions to represent the average of the floors, at least if only higher frequencies (above 50 Hz) are of interest.

1 This comparison was not made in the case of horizontal excitation, since there is no vertical movement in the junctions in that case.

0 50 100 150 200 250 300 350 400

-25 -20 -15 -10 -5 0 5

Frequency [Hz]

j g

Element 2 Element 5 Element 8 Element 11 Element 14

15

(24)

Figure 11. Average velocity in the floors in the case of finite floor elements (10 m long).

Figure 12. Difference in average velocity level in floors between structure with infinite floor elements, solid lines, and structure with finite floor elements, dotted lines.

0 50 100 150 200 250 300 350 400

-25 -20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Element 2 Element 5 Element 8

0 50 100 150 200 250 300 350 400

-20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Element 2, infinite Element 2, finite

0 50 100 150 200 250 300 350 400

-25 -20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Element 8, infinite Element 8, finite

16

(25)

Figure 13. Difference in level between the junctions, solid lines, and the average of the whole element (10 m), dotted lines.

Figure 14. Difference between the velocity of the junctions in the model with infinite floors and the average velocity of the whole element in the model with finite floors.

3.2.3 Horizontal + vertical excitation

Lastly, the interaction of the two types of excitation was investigated. The models were excited by prescribing equal velocity (1 m/s) both horizontally and vertically. Comparing the vibration levels due to pure horizontal excitation (Figure 7) to those given by pure vertical excitation (Figure 10), it can be observed that the vertical excitation gives much higher levels except at low frequencies and in the lowest wall element. Consequently, the response of a combined excitation will be dominated by the vibrations given by the vertical component. As seen in Figure 15 the behavior is as expected more or less equal to the case with pure vertical excitation, the only exception being the elements closest to the ground.

0 50 100 150 200 250 300 350 400

-20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Nod 2 Element 2

0 50 100 150 200 250 300 350 400

-25 -20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Nod 8 Element 8

0 50 100 150 200 250 300 350 400

-20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Node 2, infinite floors Average of element 2, finite floors

0 50 100 150 200 250 300 350 400

-25 -20 -15 -10 -5 0 5 10 15 20

Frequency [Hz]

Average of element 8, finite floors Node 8, infinite floors

17

(26)

Figure 15. Average velocity level measured perpendicular to the surface given by a vertical and horizontal force of equal magnitudes.

3.3 Outer wall

If instead a wall which is part of the outer façade of the building is considered, the situation gets more complex. Since the junctions are no longer symmetrical, as seen from the ground, an incoming longitudinal wave will cause a reflected and a transmitted bending wave as well as longitudinal waves, and vice versa. Consequently, both wave types will be present in all elements regardless of the direction of the ground vibrations. In view of structure borne sound this means that the walls will carry bending waves, and thus radiate sound, even in the case of pure vertical excitation

The node and element numbering of this model are given in Figure 16, and the properties of the elements in Table 2. Due to numerical issues the length of the floor elements in the model with infinite floors was set to 750 m, instead of 500 m as used in the previous model. All elements were again modeled as Timoshenko beams and the analysis performed in the frequency region 0- 400 Hz. The model was excited by prescribing the velocity in node number 1, and the elements on the left hand side of the model considered in the analysis.

0 50 100 150 200 250 300 350 400

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5

Frequency [Hz]

18

(27)

Figure 16. Asymmetrical building structure.

Finite floor element Infinite floor element Wall element

Length 𝐿_𝑥 = 10 m 𝐿_𝑥= 750 m 𝐿𝑦 = 2.7 m

Width x Height 0.2 x 0.2 m² Density, 𝝆 2400 kg/m³ Young’s modulus, E 26 GPa Poisson’s ratio, 𝝊 0.15 Loss factor, 𝜼 0.05

Table 2. Properties of elements in asymmetrical model.

3.3.1 Horizontal excitation

Figure 17 shows the average velocity level perpendicular to the surface of the elements in the model with infinite floors due to a horizontal excitation at the ground node. Comparing this to the results from the symmetric model (Figure 7) the differences are negligible, which indicates that the amount of energy being transformed to longitudinal waves is small.

19

(28)

Figure 17. Average velocity level perpendicular to the surface of walls and floors given by a horizontal force at the ground node.

3.3.2 Vertical excitation

In the case of vertical ground movement, the situation becomes more interesting. As seen in Figure 18 the coupling between longitudinal waves and bending waves is now much stronger, resulting in considerable sound radiation from the walls as well as the floors. At frequencies where the bending wave resonances matches the vertical resonances, in this case around 175 Hz, the walls will move just as much as the floors.

0 50 100 150 200 250 300 350 400

-50 -40 -30 -20 -10 0 10

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

20

(29)

Figure 18. Average velocity level perpendicular to the surface of walls and floors given by a vertical force at the ground node.

3.3.3 Horizontal + vertical excitation

Exciting the structure with equal parts vertical and horizontal velocity gives, as seen in Figure 19, results again in a clear domination of the vertical ground motion, whereas the bending waves induced by the horizontal motion mainly contributes at the resonances. Compared to the symmetrical building the relative impact of the horizontal motion is slightly higher.

0 50 100 150 200 250 300 350 400

-50 -40 -30 -20 -10 0 10

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

21

(30)

Figure 19. Average velocity level perpendicular to the surface of walls and floors given by a vertical and a horizontal force of equal magnitude.

3.4 Finite ground impedance

In order to determine the amount of energy being transmitted to the building when the ground is too soft to be considered a constant velocity source, the impedances of both the ground and the building are needed. The impedance of the building was derived by simply exciting the model with a given force and observing the resulting velocity. The impedance of the ground was calculated by representing the wall resting on the ground by a two dimensional stamp being pressed into the ground, and calculating the resulting deformation. In the DS-model there are two modes of excitation vertical and horizontal velocity, consequently two different impedances are needed, one representing each form of excitation. The derivation in the case of a vertical force is given below.

3.4.1 Calculations

To start with the coupled problem is separated in two parts according to Figure 20, and seen as a superposition of a velocity source (the free ground velocity) and a contact force.

0 50 100 150 200 250 300 350 400

-40 -30 -20 -10 0 10 20

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

22

(31)

Figure 20. Velocities and forces used in ground impedance calculations.

Using this representation the force balance at the surface gives

𝑓𝑓0 = 𝑍𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤∙ �𝑣𝑣𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 + 𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ⁽¹¹⁾ and

𝑓𝑓0 = −𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔∙ 𝑣𝑣𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 (12)

where 𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 is the free velocity of the ground, 𝑣𝑣𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 is the internal velocity of the wall and 𝑍𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 and 𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 are the impedances of the wall and the ground respectively. Combining equations (11) and (12) the total velocity at the surface in the connected system can in turn be written as

𝑣𝑣 =𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔∙ 𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔

𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔+ 𝑍𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 (13)

The only unknown quantity in equation (13) is 𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔, since the impedance of the wall is given by the DS-model and 𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 is assumed to be known.

The total displacement field in the interior of a solid can be written [8] as 𝐺 �∇²𝑢 + 1

1 − 2𝜐 𝑔𝑟𝑎𝑑�𝑑𝑖𝑣𝑣(𝑢)�� = 𝜌𝜌𝜕²𝑢

𝜕𝑡² (14)

where 𝑢 is the displacement, 𝐺 is the shear modulus, 𝜈𝜈 is Poisson’s ratio and 𝜌𝜌 is the density.

Considering a two dimensional space, with positive coordinate directions according to Figure 1, the displacement field, 𝑢 , will have two components 𝑢𝑥 and 𝑢𝑦. The displacement field can be

𝒇𝒇

_𝟎𝟎

𝑣𝑣_{𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤}

𝑣𝑣

𝑣𝑣𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔

23

(32)

split in two parts; a rotational free part represented by a scalar potential 𝜙(𝑥, 𝑦) and a divergence free part represented by a vector potential 𝜓(𝑥, 𝑦) [8], giving

𝑢 = �𝑢𝑥, 𝑢𝑦� = 𝑔𝑟𝑎𝑑(𝜙) + 𝑟𝑜𝑡(𝜓𝑧) ⁽¹⁵⁾ 𝑢_𝑥 =𝜕𝜙

𝜕𝑥 +

𝜕𝜓𝑧

𝜕𝑦 ⁽¹⁶⁾

𝑢_𝑦 = 𝜕𝜙

𝜕𝑦 −

𝜕𝜓𝑧

𝜕𝑥 ⁽¹⁷⁾

Combining Equations (14), (16) and (17) gives 𝐸𝐸(1 − 𝜐)

(1 + 𝜐)(1 − 2𝜐) ∇²𝜙 − 𝜌𝜌𝜕²𝜙

𝜕𝑡² = 0 (18)

𝐺∇²𝜓_𝑧− 𝜌𝜌𝜕²𝜓_𝑧

𝜕𝑡² = 0 (19)

which for a time dependence of 𝑒^{−𝑖𝜔𝑡} can be rewritten as

∇²𝜙 + 𝑘_𝐿²𝜙 = 0 (20)

∇²𝜓𝑧+ 𝑘𝑇2𝜓𝑧= 0 (21)

where 𝑘𝐿 is the wave number of the longitudinal wave and 𝑘𝑇 is the wave number of the transversal wave. By Fourier transforming 𝜙 and 𝜓 according to

𝜙, 𝜓𝑧(𝑥, 𝑦) = 1

2𝜋 � 𝜙^∞ �, 𝜓�(𝜅, 𝑦)𝑒𝑧 ^𝑖𝜅𝑥𝑑𝜅

−∞

𝜙�, 𝜓�(𝜅, 𝑦) = � 𝜙, 𝜓_𝑧 ^∞ 𝑧(𝑥, 𝑦)𝑒^{−𝑖𝜅𝑥}𝑑𝑥

−∞

(22)

and thereby stepping from the xy-plane to a domain where the x-dependence is replaced by a wave solution, we can rewrite Equations (20) and (21) as

𝜕²𝜙�

𝜕𝑦² + 𝛼²𝜙� = 0 ⁽²³⁾

𝜕²𝜓�_𝑧

𝜕𝑦² + 𝛽²𝜓� = 0 𝑧

where 𝛼² = 𝑘_𝐿²− 𝜅² and 𝛽² = 𝑘_𝑇² − 𝜅². Assuming solutions on the form

𝜙� = 𝜙�𝑒^𝑖𝛼𝑦 (24)

𝜓� = 𝜓𝑧 �𝑒𝑧 ^𝑖𝛽𝑦

the system can now be solved using the boundary conditions at the surface. In the case of a constant, vertical force of magnitude 𝑓𝑓0 acting on the surface between 𝑥 = −ℎ and 𝑥 = ℎ, see Figure 21, these becomes

𝜎𝑦 = 𝜆 �𝜕𝑢_𝑦

𝜕𝑦 +

𝜕𝑢𝑥

𝜕𝑥 � + 2𝜇

𝜕𝑢_𝑦

𝜕𝑦 = �−𝑓𝑓₀, −ℎ ≤ 𝑥 ≤ ℎ

0, 𝑒𝑙𝑠𝑒 (25)

24

(33)

𝜏𝑦𝑥 = 𝜇 �𝜕𝑢_𝑥

𝜕𝑦 +

𝜕𝑢_𝑦

𝜕𝑥 ��_𝑦=0 = 0

Figure 21. Ground impedance, vertical force.

Since the calculations are performed in the y𝜅-domain, the force is Fourier transformed as well giving

𝑓𝑓(𝜅, 𝑦) = � 𝑓𝑓^ℎ ₀𝑒^{−𝑖𝜅𝑥}

−ℎ 𝑑𝑥�

𝑦=0

=2𝑓𝑓0sin(𝜅ℎ)

𝜅 ⁽²⁶⁾

By combining Equations (16), (17), (22) and (23)-(26) the system can now be solved. This was done by using the calculation software Maple and Matlab. Analytical expressions for 𝑢𝑥 and 𝑢𝑦, with the integrals introduced by the Fourier transform left unsolved, were calculated in Maple giving

𝑢𝑥= − 1 2𝜋 �

𝑖2𝑓𝑓₀sin(𝜅ℎ)(2𝛼𝛽 − 𝛽²+ 𝜅²)

𝛼²𝛽²𝜆 + 2𝛼²𝛽²𝜇 − 𝛼²𝜅²𝜆 − 2𝛼²𝜅²𝜇 + 4𝛼𝛽𝜅²𝜇 + 𝛽²𝜅²𝜆 − 𝜅⁴𝜆

∞

−∞

𝑑𝜅

𝑢_𝑦= 1 2𝜋 �

𝑖2𝑓𝑓0sin(𝜅ℎ)𝛼( 𝛽²+ 𝜅²)

𝜅(𝛼²𝛽²𝜆 + 2𝛼²𝛽²𝜇 − 𝛼²𝜅²𝜆 − 2𝛼²𝜅²𝜇 + 4𝛼𝛽𝜅²𝜇 + 𝛽²𝜅²𝜆 − 𝜅⁴𝜆)

∞

−∞

𝑑𝜅

(27)

These expressions were then copied to Matlab where the integrals were solved numerically. In order to get the integrals to converge, damping was introduced by letting the wave number of the transverse wave be complex.

The ground impedance was then calculated as

𝑍𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔= 𝑓𝑓0

−𝑖𝜔𝑢 ⁽²⁸⁾

Since the force in this case is acting solely in the vertical direction only the vertical component of the impedance is non-zero. Similarly, the horizontal component will be given by repeating the derivation above with a horizontal force.

2h

𝑓𝑓₀

x y

25

(34)

3.4.2 Results

The absolute value of the impedance calculated using material properties according to Table 3, is shown in Figure 22 and Figure 23. As a comparison the vertical ground impedance given by Hassan [4], slightly rewritten to give the impedance per unit length from one indenter, is plotted as well. The expression given by Hassan, rewritten to represent the same situation as in the calculations above, becomes

𝑍𝑦,𝐻𝑤𝑤𝑠𝑠𝑤𝑤𝑔𝑔 = 𝜋𝐸𝐸

𝑖𝜔(1 − 𝜈𝜈²)2 ⁽²⁹⁾

Soft ground Stiff ground

Young’s modulus, 𝑬𝒈𝒓𝒐𝒖𝒏𝒅 70 MPa 70 GPa

Density, 𝝆𝒈𝒓𝒐𝒖𝒏𝒅 1900 kg/m³ 2700 kg/m³

Poisson’s ratio, 𝝊 0.3 0.3

Table 3. Material properties used when calculating ground impedances of a general soft and stiff ground respectively.

Figure 22. Absolute value of impedance of soft ground.

0 50 100 150 200 250 300 350 400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 10⁷

Frequency [Hz]

Impedance [N/(m/s)]

Horizontal Vertical

Vertical, Hassan

26

(35)

Figure 23. Absolute value of impedance of stiff ground.

The impedance of a symmetrical five floor building in node number one, modeled according to the description in section 3.2, calculated by exciting the building with a force of unit magnitude in node number one, is shown in Figure 24. The excitation is either a purely vertical or horizontal one.

0 50 100 150 200 250 300 350 400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 10¹⁰

Frequency [Hz]

Impedance [N/(m/s)]

Horizontal Vertical

Vertical, Hassan

27

(36)

Figure 24. Impedance of symmetric 5 level building with infinite (500 m) floors. Blue lines = horizontal, red lines = vertical.

The resulting velocity in the node coupled to the ground, calculated using Equation (13) and the impedances given above, in the case of soft and stiff ground respectively, are shown in Figure 25 and Figure 26.

0 50 100 150 200 250 300 350 400

-6 -4 -2 0 2 4 6 8 10 12x 10⁵

Frequency [Hz]

Impedance [N/(m/s)]

Re(Zx) Im(Zx) Re(Zy) Im(Zy)

28

(37)

Figure 25. Difference between the free ground velocity and the velocity of the node coupling the building to the ground in the case of a building placed on soft ground.

Figure 26. Difference between the free ground velocity and the velocity of the node coupling the building to the ground in the case of a building placed on stiff ground.

0 50 100 150 200 250 300 350 400

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

Frequency [Hz]

Free ground velocity With building, horisontal With building, vertical

0 50 100 150 200 250 300 350 400

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01

Frequency [Hz]

Free ground velocity With building, horisontal With building, vertical

29

(38)

Using this new nodal velocity to excite the model, the interaction between ground and building can now be included in the analysis. Figure 27 shows the resulting mean velocity in the elements of the symmetrical five level structure when assuming the building is placed on stiff ground and the free ground velocity has a magnitude of 1 m/s both horizontally and vertically. As expected, considering the high impedance, the difference to the previous calculations when the ground is represented as a constant velocity source (Figure 15), is minimal.

Figure 28 shows the resulting velocity when the building is instead placed on soft ground. If the vibration levels in the walls (element 1 and 4) in this case is compared to the results in Figure 15, the differences are still small, which is expected considering the still effective coupling of the horizontal ground vibrations (Figure 25). Looking at the floors instead (element 2, 5 and 8), the vibration levels are now lower than before, a result of the weakened coupling between the vertical ground vibrations and the building (Figure 25).

Consequently, as the vertical coupling between ground and building is weakened and the bending waves caused by horizontal ground movement dampens quickly upwards in the building, the potential problem with structure borne sound in a building on soft ground will be limited to the first couple of floors of the building.

However, one should bear in mind that the results given above could be quite far from the situation in a real building. For instance, in the calculations the connection between ground and building is assumed to be completely stiff. Also, no bottom plate is included giving that the only connection between ground and building is assumed to be the walls, which are treated as simply supported. In reality none of this will be strictly true, but the results should nonetheless give a hint about the differences in response between softer and harder ground types.

30

(39)

Figure 27. Average velocity of elements in the symmetrical five level structure assuming stiff ground.

Figure 28. Average velocity of elements in symmetrical five level structure assuming soft ground.

0 50 100 150 200 250 300 350 400

-35 -30 -25 -20 -15 -10 -5 0 5

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

0 50 100 150 200 250 300 350 400

-35 -30 -25 -20 -15 -10 -5 0 5

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

31

(40)

3.5 Vibrations to sound

In order to estimate the resulting sound level in a room the contributions from each vibrating surface; two walls, the floor and the roof, are added. Two general situations are assumed to be possible, that the room is situated along the outer perimeter of the building and thereby have one outer wall and one inner wall, or the room being somewhere inside the building having two inner walls.

3.5.1 Radiation efficiency/radiation factor

The radiation efficiency for each element was calculated using the expressions presented by Leppington et al. [10], giving approximate values in each third octave band. Since the formulas are based on the assumption that each frequency band contains at least 2-3 modes, i.e. only valid if the structure is large compared to the wavelength, the real dimensions of the plates were up scaled by a factor of five in order to get a reasonable results. Considering that half of the

information regarding the shape of the deformations is lost when gong from three dimensions to two, the results from a model like this will never be more than a rough estimate anyway, and the results from the larger plates should at least give a rough idea of the real situation.

Equation (30) is used for frequencies below the coincidence frequency, Equation (31) for the frequency band containing the coincidence frequency and Equation (32) for frequencies above the coincidence frequency.

𝜎 = 𝑈𝑐₀

4𝜋²�𝑓𝑓𝑓𝑓𝑐𝑆�𝜇²− 1�ln �𝜇 + 1 𝜇 − 1� +

2𝜇

𝜇²− 1� 𝑓𝑓 < 𝑓𝑓^𝑐 ⁽³⁰⁾

𝜎 = �2𝜋𝑓𝑓

𝑐₀ �𝑙_𝑥�0.5 − 0.15𝑙_𝑥

𝑙_𝑦� 𝑓𝑓 = 𝑓𝑓_𝑐 (31)

𝜎 = 1

�1 − 𝑓𝑓𝑓𝑓^𝑐

𝑓𝑓 > 𝑓𝑓𝑐 (32)

In the expressions 𝑙𝑥 och 𝑙𝑦 are the dimensions of the radiating surface, where the shorter edge is taken as 𝑙_𝑥. 𝑈 is the perimeter of the radiating surface, 𝑆 is the surface area, 𝑓𝑓 is the center

frequency of the frequency band of interest, 𝜇 = �^𝑓_𝑓^𝑐 and the critical frequency

𝑓𝑓_𝑐 = ^𝑐_2𝜋⁰²�^𝑚_𝐵^′ where 𝑚^′ is the surface density and 𝐵 is the bending stiffness. The expressions are originally valid for simply supported panels placed in a flat infinite baffle, radiating in free space.

In order to compensate for the fact that the baffle in this case is at a right angle the radiation efficiency below coincidence is doubled [11].

32

(41)

Figure 29 shows the resulting radiation factor when using Equation (30)-(32), with a correction factor of 2 due to right angled baffles, to calculate the radiation factor in room with floor/roof dimensions 5 ∙ 6 m and wall dimensions 2.7 ∙ 6 m.

Figure 29. Radiation factor calculated using Leppington’s formulas.

3.5.2 Radiated sound power

The models presented above gives the vibration levels in the floors due to one wall. However, in any real situation there will always be at least two walls contributing. Depending on where the room in question is located, either a combination of both models or a doubling of the effect from one model can be used.

When the total vibration level is known the sound power radiated from each element can then be calculate according to

𝐿𝑤𝑤 = 10 ∙ log �𝑣𝑣�²𝜌𝜌𝑐0𝑆𝜎

𝑤𝑔𝑔𝑒𝑓 � ⁽³³⁾

where 𝜎 is the radiation factor, 𝑤𝑔𝑔𝑒𝑓 = 10⁻¹², and 𝑣𝑣�² is the mean value of the vibrations perpendicular to the surface. The contributions from all surfaces in the room are then added as

𝐿𝑤𝑤,𝑡𝑔𝑔𝑡 = 10 ∙ log �10^𝐿^{𝑤,𝑓𝑙𝑜𝑜𝑟}^/10+ 10^𝐿^{𝑤,𝑟𝑜𝑜𝑓}^/10+ � 10^𝐿^{𝑤,𝑤𝑎𝑙𝑙 𝑖}^/10

4 𝑖=1

� ⁽³⁴⁾

0 50 100 150 200 250 300 350 400

0 0.5 1 1.5 2 2.5 3

Frequency [Hz]

Radiation factor

Floor Wall

33

Prediction of structure borne sound in

Prediction of structure borne sound in

buildings close to subways

Moa Wijkmark

Master of Science Thesis

Stockholm, Sweden 2014

Prediction of structure borne sound in buildings

close to subways

Moa Wijkmark

Abstract

Sammanfattning

Contents

1. Introduction

x y

z

2. Dynamic Stiffness Method

2

3

1

Red = global dof Blue = local dof

Green = element numbering

3. Modeling

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

𝒇𝒇

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]

Frequency [Hz]

V el oc it y [ dB r el 1 m /s ]