Ljud i byggnad och samhälle (VTAF01)– SDOFMATHIAS BARBAGALLODIVISION OF ENGINEERING ACOUSTICS, LUND UNIVERSITY

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

… recap from last lecture

• Time & frequency domains

• Narrow band & Octaves & 1/3-octave

• Sound: pressure waves

– Sound pressure level (SPL / Lp) [dB]

Lp = 10 logp2

pref2 = 20 log

p

pref

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Outline

Introduction

MDOF

SDOF

Summary

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Learning outcomes

• Equations of motion of

– Single-degree-of-freedom systems (SDOF)

» Damped

» Undamped

– Multi-degree-of-freedom systems (MDOF)

• Concepts of

– Eigenfrequency

– Resonance

– Eigenmode

– Frequency response functions

• Vibration isolation

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Introduction

• A very broad definition…

– Acoustics: what can be heard…

– Vibrations: what can be felt…

• Coupled “problem”

– Hard to draw a line between both domains

• Nuisance to building users

‒ Comprise both noise and vibrations

Source: J. Negreira (2016)

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Structural dynamics – Introduction

• Types of systems

– Discrete: finite number of DOFs needed

» System of ordinary differential equations

» Depending on the number of DOFs:

– SDOF

– MDOF

– Continuous: infinite number of DOFs

» System of differential equations with partial derivatives

NOTE: Degrees of freedom (DOF): number of independent displacement components to define exact position of a systemNOTE2: The presented theory assumes linearity

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Outline

Introduction

MDOF

SDOF

Summary

Damped Undamped

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Mü(t) + Ku(t) = F(t)

Undamped SDOF – EOM

• Mass-spring-damper system – basic concept within acoustics and mechanics – useful tool to analyse resonant systems.

− u(t) obtained by solving the PDE together with the initial conditions

» Solution = Homogeneous + Particular

Inertialforce

Elasticforce

Appliedforce

F(t) = Fdriv·cos(t)

F (t) = Fdriv·cos(t) F (t) = 0

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Undamped SDOF – Solution (I)

• Eigenfrequency/Natural frequency: the frequency with which the system oscillates when it is left to free vibration after setting it into movement

‒ Expressed in angular frequency [rad/s] or Hertz [1/s=Hz]

• Homogeneous solution:

• Particular solution:

M

K0

M

Kf

2

10

)sin()cos()sin()cos()( 0

0

00000 t

vtutBtAtuh

)cos(

1

1)cos()(

2

0

0 tK

Ftutu driv

p

Initial conditions

If ≈ 0 Resonance

“Static solution”

Displacement response factor (Rd)

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Undamped SDOF – Solution (II)

)cos(

1

1)sin()cos()(

2

0

0

0

000 t

K

Ft

vtutu driv

total

Homogeneous

Particular

M

K0

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Mü(t) + Rů(t) + Ku(t) = F(t)

Damped SDOF – EOM

• Mass-spring-damper system (e.g. a floor)

− u(t) obtained by solving the PDE together with the initial conditions

» Solution = Homogeneous + Particular

Inertialforce

Elasticforce

Dampingforce

Appliedforce

NOTE: Damping is the energy dissipation of a vibrating system

F(t) = Fdriv·cos(t)

u(t)

F (t) = Fdriv·cos(t) F (t) = 0

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Damped SDOF – Eigenfrequency / damping

• Remember! The natural frequency is the frequency with which the system oscillates when it is left to free vibration after setting it into movement

‒ Undamped:

‒ Damped:

M

K0

2

0 1 d

NOTE: The natural frequency is notinfluenced very much by moderate viscous

damping (i.e. <0.2)

Various behaviours for realistic levels of damping

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Damped SDOF – Homogeneous solution (I)

• Solution yielded when F(t)=0

• Solved with help of the initial conditions (B1 and B2)

• Composed of:

‒ Decaying exponential part

‒ Harmonically oscillating part

NOTE: B1 and B2 calculated from the initial conditions

)cos()sin()( 212

212

00

tBtBeeAeAetu dd

ttiti

t

hdd

2MK

R2

02

1

d

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Homogeneous solution (II)

• Function of damping

– Responsible for the system’s energy loss

– Example

Without dampingWith damping

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Damped SDOF – Particular solution

• Solution showing the displacement under the driving force:

– For example: F(t) = Fdriv·cos(t)

• The solution has the form:

Which gives the solution

)cos()sin()( 21 tDtDtu p

driv

driv

FRMK

MKD

FRMK

RD

2222

2221

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Damped SDOF – Total solution

• Total solution = homogeneous + particular

‒ The homogeneous solution vanishes with increasing time. After some time: u(t )≈up(t )

)cos()sin()cos()sin()( 21212

0

tDtDtBtBetu dd

t

total

Homogeneous Particular

2

0 1 d

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Driving frequencies

• Ex:

– Without damping

– With damping

• Different driving freqs

0 0 0

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Low frequency excitation ( < 0 )

• The spring dominates

− Force and displacement in phase

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Excitation at resonance freq. ( = 0 )

• Damping dominates

− Phase difference = 90 or

• If no (or little) damping is present:

− The system collapses

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – High frequency excitation ( >0 )

• The mass dominates

• Force and displacement in counter phase:

- Phase difference = 180 or

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Linear dynamic response to harmonicexcitation

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Complex representation (Freq. domain)

u(ω) =Fdriv

K −Mω2 + Riω

• Euler’s formula:

• Then:

• Differenciating:

• Substituting in the EOM:

F t =Fdriv cos ωt = Re Fdriveiωt

u t =u0 cos ωt − φ = Re ueiφeiωt = Re u(𝜔)eiωt

eiφ = cos φ + i sin φ

ሶu t = Re iω ∙ u(ω)eiωt

ሷu t = Re −ω2 ∙ u(ω)eiωt

M ሷu t + R ሶu t + Ku t = Fdrivcos(ωt)

If the system is excited with 02=K/M

(K-M2)=M(02-2) Resonance

NOTE: This is the particular solutionin complex form for an undamped

SDOF system. In Acoustics, most of the times, we are interested in the

particular solution, which is the onenot vanishing as time goes by.

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Frequency response functions (FRF) – (I)

• In general, FRF = transfer function, i.e.:

‒ Contains system information

‒ Independent of outer conditions

• Different FRFs can be obtained depending on the measured quantity

Cdyn ω =u(ω)

Fdriv(ω)=

1

K − Mω2 + RiωKdyn ω = Cdyn ω −1 = −Mω2 + Riω + K

Measured quantity FRFAcceleration (a) Accelerance = Ndyn(𝜔) = a/F Dynamic Mass = Mdyn(𝜔) = F/a

Velocity (v) Mobility/admitance = Y(𝜔) = v/F Impedance = Z(𝜔) = F/v

Displacement (u) Receptance/compliance = Cdyn(𝜔)= u/F

Dynamic stiffness = Kdyn(𝜔) = F/u

𝐻𝑖𝑗 𝜔 =𝑠𝑖(𝜔)

𝑠𝑗(𝜔)=output

input

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Frequency response functions (FRF) – (I)

• In general, FRF = transfer function, i.e.:

‒ Contains system information

‒ Independent of outer conditions

𝐻𝑖𝑗 𝜔 =𝑠𝑖(𝜔)

𝑠𝑗(𝜔)=output

input

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Frequency response functions (FRF) – (II)

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Frequency response functions (FRF) – (III)

• Representation of FRFs: Bode plots

idyn AeC

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

SDOF – Frequency response functions (FRF) – (IV)

• Real and imaginary parts – the imaginary part has interesting information

Source: https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/The-FRF-and-its-Many-Forms

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Vibration isolation

)(~

)(~

m

driv

F

FT

u(t)

iRMK

Fu driv

)()(~

2

)(~)(~)(~

)(~

)(~

uiRuKFFF RKm

iRMK

iRK

F

F

F

F

u

m

driv

m

)()(~

)(~

)(~

)(~

2

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Helmholtz resonator (I)

Source: hyperphysics

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Helmholtz resonator (II)

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Helmholtz resonator (III)

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Helmholtz resonator (IV)

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Outline

Introduction

MDOF(just out of curiosity)

SDOF

Summary

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

MDOF – Multi-degree-of-freedom systems

• In reality, more DOFs are needed to define a system MDOFs

− Continuous systems often approximated by MDOFs

• Multi-degree-of-freedom system (Mass-spring-damper)

– Solution process: similar as in SDOFs (particular+homogeneous)

– ”The undamped modes form an orthogonal basis, i.e. they uncouple the system, allowing the solution to be expressed as a sum of the eigenmodes ofthe free-vibration SDOF system”

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

MDOF – Note on modal superposition

Source: http://signalysis.com

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Mode shapes – Example floor

NOTE: In floor vibrations, modes are superimposed on one another to give the overall response of the system. Fortunately it is generally sufficient to consider only the first 3 or 4

modes, since the higher modes are quickly extinguished by damping.

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

… resonance and modes are indeed “present” daily

Source: steelconstruction.info

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Resonance & Eigenmodes

Examples:

– Earthquake design

– Bridges (Tacoma & Spain)

– Modes of vibration: Plate

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Outline

Introduction

MDOF

SDOF

Summary

M. Barbagallo / Ljud i byggnad och samhälle / VTAF01 / 27 March 2019

Learning outcomes

• Equations of motion of

– Single-degree-of-freedom systems (SDOF)

» Damped

» Undamped

– Multi-degree-of-freedom systems (MDOF)

• Concepts of

– Eigenfrequency

– Resonance

– Eigenmode

– Frequency response functions

• Vibration isolation

Thank you for your attention!

mathias.barbagallo@gmail.com