VTAF 01 – Sound in Buildings and Environment

2.Simple-degree-of-freedom systems (SDOF)NIKOLAS  VARDAXISDIVISION  OF  ENGINEERING  ACOUSTICS,  LTH,  LUND  UNIVERSITY

2021.03.25

RECAP from first lecture

• Time & frequency domains

• Narrow band & Octaves & 1/3-octave

• Sound: pressure waves

– Sound pressure level (SPL, Lp) [dB]

L" = 10 logp*+

p,-.+ = 20 log

p*p,-.

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

– ”Unofficialy” vibrations exist between 0-20 Hz

• Nuisance to building users

‒ Comprise both noise and vibrations

‒ Exist both from indoor sources or outdoors.

Ph.D. thesis: J. Negreira (2016)

Ph.D. thesis: N.G. Vardaxis (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 (single-degree-of-freedom) system

– MDOF (multiple-degrees-of-freedom) system

– 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

Introduction: Linear systems

Ø What effect does an input signal have on an output signal?

Ø What effect does a force on a body have on its velocity?

Ø A way to answer it using theory of linear time-invariant systems

Introduction: Linear systemsØLinear time-invariant systems

– Mathematically: relation input/output described by linear differential equations.

– Characteristics:»Coefficients independent of time.

» Superposition principle.

» 𝑎 𝑡 →𝑐 𝑡 ; 𝑏 𝑡 →𝑑 𝑡  ⇒𝑎 𝑡 + 𝑏 𝑡 →𝑐 𝑡 + 𝑑(𝑡)

»Homogeneity principle: 𝛼𝑎(𝑡)→𝛼b(t)

» Frequency conserving:» a(t) comprises frequencies f1 and f2; » b(t) comprises f1 and f2.

Simplified model SDOF

Simplified model SDOF

• Mass – spring - damper system

Without dampingWith damping

Mü(t) + Ku(t) = F(t)

Undamped SDOF – EOM

• Mass – spring - damper system

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

» Solution = Homogeneous + Particular

Inertialforce

Elasticforce

Appliedforce

F(t) = Fdriv·cos(ωt)

F (t) = Fdriv·cos(ωt) 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:

MK

=0ω MKf ⋅=

π21

0

)sin()cos()sin()cos()( 00

00000 tvtutBtAtuh ω

ωωωω +=+=

)cos(

1

1)cos()(2

0

0 tKFtutu driv

p ω

ωω

ω ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅==

Initial conditions

If ω ≈ ω0 à Resonance

“Static solution”

Displacement response factor (Rd)

Undamped SDOF – Solution (II)

)cos(

1

1)sin()cos()(2

0

00

000 t

KFtvtutu driv

total ω

ωω

ωω

ω ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅++=

Homogeneous

Particular

MK

=0ω

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(ωt)

F (t) = Fdriv·cos(ωt) 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:

MK

=0ω

20 1 ζωω −=d

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

damping (i.e. <0.2)ζ

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

ttitit

hdd ωω

ωη

ωωωη

+=+=−−−

ζη 2==MKR 2

0 21 ⎟

⎠

⎞⎜⎝

⎛−=η

ωωd

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(ωt)

• The solution has the form:

Which gives the solution

)cos()sin()( 21 tDtDtup ωω +=

( ) ( )

( ) ( ) driv

driv

FRMK

MKD

FRMK

RD

⋅+−

−=

⋅+−

=

2222

2221

ωω

ωωω

ω

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 ωωωωω

η

+++=−

Homogeneous Particular

20 1 ζωω −=d

SDOF – Driving frequencies

• Ex:

– Without damping

– With damping

• Different driving freqs

0ωω >0ωω < 0ωω =

SDOF – Low frequency excitation ( ω < ω0 )

• The spring dominates

− Force and displacement in phase

SDOF – Excitation at resonance freq. ( ω = ω0 )

• Damping dominates− Phase difference = 90° or π

• If no (or little) damping is present: − The system collapses

SDOF – High frequency excitation ( ω >ω0 )

• The mass dominates

• Force and displacement in counter phase:- Phase difference = 180° or π

SDOF – Linear dynamic response to harmonic excitation

SDOF – Complex representation (Freq. domain)

u*(ω) =FA,BC

K −Mω+ +Riω

• Euler’s formula:

• Then:

• Differenciating:

• Substituting in the EOM:

F t =FA,BC cos ωt = Re FA,BCeBMN

                                                                     u t =uOcos ωt −φ = Re ueBQeBMN = Re u*(𝜔)eBMN

eBQ = cos φ + i sin φ  

u t = Re iω U u*(ω)eBMN

u t = Re −ω+ U u*(ω)eBMN

Mu t + Ru t + Ku t = FA,BCcos  (ωt)

If the system is excited with ω02=K/M à

(K-Mω2)=M(ω02-ω2) à 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

CAXY ω =u*(ω)

FA,BC(ω)=

1K− Mω+ + Riω

KAXY ω = CAXY ω Z[ = −Mω+ + 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

𝐻]^ 𝜔 =𝑠](𝜔)𝑠 (𝜔) =

outputinput

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

• Representation of FRFs: Bode plots

αidyn AeC =

Vibration isolation

)(~)(~

ωω

m

driv

FFT =

u(t)

ωωω

iRMKFu driv

⋅+⋅−=

)()(~ 2

)(~)(~)(~)(~)(~ ωωωωωω uiRuKFFF RKm ⋅+=+=

ωωω

ωω

ωω

iRMKiRK

FF

FF

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 à MDOFs− Continuous systems à 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

Outline

Introduction

MDOF

SDOF

Summary

Learning outcomes

• Equations of motion of– Single-degree-of-freedom systems (SDOF)

» Damped

» Undamped

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

• Concepts of– Eigenfrequency / Eigenmodes

– Resonance

– Frequency response functions (FRFs)

• Vibration isolation

• Study: T.E.Vigran, Building Acoustics - Ch. 1 +2

Reminder: free mobile apps for acoustics!• Sound level meter apps

– OpeNoise

» SPL, 1/3 oct. bands, Frequency Analyzer

– Noise Exposure (Buller)

»Representative dBA levels, Swedish Work Environment Authority

• Noise, ambient or masking sounds apps

– White Noise

»White, pink, blue noise, … , rain, thunder, seawaves, naturalsoundscapes… etc

Thank you for your attention!nikolas.vardaxis@construction.lth.se