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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!