Ljud i byggnad och samhälle (VTAF01) · • Multi-degree-of-freedom system (Mass-spring-damper)...

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

https://www.youtube.com/watch?v=r18Gi8lSkfM

https://www.youtube.com/watch?v=r18Gi8lSkfM


Outline

Introduction

MDOF

SDOF

Summary


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


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)


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


Outline

Introduction

MDOF

SDOF

Summary

Damped Undamped


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


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)


Undamped SDOF – Solution (II)

)cos(

1

1)sin()cos()(

2

0

0

0

000 t

K

Ft

vtutu driv

total

Homogeneous

Particular

M

K0


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


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


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


SDOF – Homogeneous solution (II)

• Function of damping

– Responsible for the system’s energy loss

– Example

Without dampingWith damping


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


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


SDOF – Driving frequencies

• Ex:

– Without damping

– With damping

• Different driving freqs

0 0 0


SDOF – Low frequency excitation ( < 0 )

• The spring dominates

− Force and displacement in phase


SDOF – Excitation at resonance freq. ( = 0 )

• Damping dominates

− Phase difference = 90 or

• If no (or little) damping is present:

− The system collapses


SDOF – High frequency excitation ( >0 )

• The mass dominates

• Force and displacement in counter phase:

- Phase difference = 180 or


SDOF – Linear dynamic response to harmonicexcitation


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.


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


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


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



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

• Representation of FRFs: Bode plots

idyn AeC


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

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



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


Helmholtz resonator (I)

Source: hyperphysics


Helmholtz resonator (II)


Helmholtz resonator (III)


Helmholtz resonator (IV)


Outline

Introduction

MDOF(just out of curiosity)

SDOF

Summary


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”


MDOF – Note on modal superposition

Source: http://signalysis.com


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.


… resonance and modes are indeed “present” daily

Source: steelconstruction.info


Resonance & Eigenmodes

Examples:

– Earthquake design

– Bridges (Tacoma & Spain)

– Modes of vibration: Plate

http://www.bbc.co.uk/news/world-asia-pacific-12726775

http://www.youtube.com/watch?v=LMTyIRBXeyE&feature=related

https://www.youtube.com/watch?v=uRmPMdw1FD0

http://www.youtube.com/watch?v=IuyQSZX8HMI


Outline

Introduction

MDOF

SDOF

Summary


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!

[email protected]

Ljud i byggnad och samhälle (VTAF01) · • Multi-degree-of-freedom system (Mass-spring-damper)...

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