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Dynamics of Structures 2017-2018 2. 1DOF system

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2.Vibrations: single degree of freedom system

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Dynamics of structures

Arnaud Deraemaeker ([email protected])

*One degree of freedom systems in real lifeHypothesisExamples

*Response of a single degree of freedom (sdof) systemFree responseForced responseInfluence of the damping

*Reduction to a sdof systemEquivalent stiffnessEquivalent massEquivalent dampingValidity of the reduction

Outline of the chapter

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One degree of freedom systems in real life

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Reduction of a system to a one dof system

Hypothesis :

-Rigid body-Motion in one direction-Linked to a reference point through a flexible element (with negligible mass)

Valid under certains conditions

-Specific direction of excitation and motion-Limited frequency band

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Rigid vs flexible body

In statics: flexibility = ability to deform under applied load, function of the stiffness

In dynamics, flexibility is a function of:- the stiffness- the mass- the frequency of excitation- the damping in the system

Freq < 20 Hz(earthquake)

Rigid vs flexible body : illustration


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Freq > 100 Hz


Not a rigid body

Massless spring

Can be reduced to a one dof system


Cannot be reduced to a one dof system


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Example 1:

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Example 2:

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Example 3:

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The motion is the rotation instead of the displacement


Example 4:

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Example 5:

Offshore platform

Response of a single degree of freedom (sdof) system

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Conservative system: equation of motion

Newton law:

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The displacement x is defined with respect to the equilibrium position of the mass subjected to gravity. The effect of gravity should therefore not be takeninto account in the equation of motion of the system.

Conservative system: equation of motion

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What about the effect of gravity ?


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Equation of motion: general solution

Characteristic equation:

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•In the absence of external excitation force, the motion is oscillatory. The natural angular frequency is defined by the values of k and m•The motion is initialized by imposing initial conditions on the displacement and the velocity

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*


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Alternative representation:

The motion can be described by a cosine function with a phase. The phase is a function of the initial conditions.


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*

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Equation of motion: particular solution

Consider the continuous inverse Fourier transform of the excitation f

and isolate a single frequency component of the form

We now compute the particular solution of the equation of motion for this single frequency harmonic excitation


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Positive if <n

Negative if n

Infinite if n


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Bode diagram (linear frequency axis)


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Dynamic amplification factor


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Bode diagram (logarithmic frequency axis) see MATH-H-201 (ULB) 25


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Displacement in the time domain

Equation of motion: back to the time domain

Continuous Fourier

transform

Inverse continuous

Fourier transform


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What is resonance ?

Amplification of given frequencies

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How resonance can lead to failure


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How resonance can lead to failure

Wine glass

Damped equation of motion

Damping force :

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viscous damping


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Damped equation of motion: general solution

Characteristic equation:

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*General solution

damping coefficient

Initial conditions:

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*


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Number of oscillations after which the vibration amplitude is reduced by one half

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Critical damping

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Critical damping

Impulse response

Equivalent to initial velocity at t

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Impulse response

For an initial velocity, the response of the system is:

with

For a unit impulse , we define the impulse response h(t)

Damped equation of motion : particular solution

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Maximum

Max ampl.For small

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Bode diagram

Displacement

Velocity

Acceleration

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Nyquist plot

The plot enhances strongly the frequencies around resonance


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We have and for so that we can write

The contribution of one impulse to the response of the system is given by :

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Time domain response using Duhamel’s integral

f(t) is decomposed into a series of short impulses at time

The total contribution is therefore:

(h(t) is the impulse response)

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Time domain response using Duhamel’s integral

For a single degree of freedom, we have :

Special case : harmonic excitation

The Fourier transform of h(t) is the transfer function


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Base excitation

Excitation

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Illustration

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Reduction to a sdof system

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Reduction to a sdof system

-Equivalent stiffness?-Equivalent mass ?-Validity (frequency band) ?

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Equivalent stiffness

Particular case: the system is a beam subject to

- traction- bending- torsion

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General principle :

F in the direction of excitation(and motion) of the mass

Equivalent stiffness: bar in traction

Displacement of the attachmentpoint (in the direction of excitation):

Constitutive equation: Stress is constant

Static equilibrium

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Direction of excitation

Attachment point


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Equivalent stiffness: bar in torsion

Circular sectionConstitutive equation :

Rotation at the attachment point: 51


Attachmentpoint

Static equilibrium

J=rotational intertia

Equivalent stiffness : cantilever beam in bending

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Attachment point

From tables


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Equivalent stiffness : of beams in bending

Equivalent stiffness of beams in bending can be computed using formulas fromresistance of materials or using tables :

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Equivalent stiffness: portal frame


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Equivalent stiffness: portal frame

From tables (internet)

Equivalent stiffness: energy approach

Strain energy in a mass-spring system :

Bar in traction:

Bar in torsion:

Beam in bending:

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Equivalent stiffness : general case

General case:- simplified model- numerical approximation (finite elements)

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Equivalent mass : energy method

Kinetic energy of a mass-spring system

Kinetic energy of the spring

Additional mass

Example 1 :

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Equivalent mass : energy method

Kinetic energy of a mass-spring system

Kinetic energy of the bar

Additional mass ma

Example 2 :

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Equivalent damping : Types of damping

Damping = dissipation of energy

[Vibration Problems in Structures, CEB, 1991]


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Hysteresis loop due to damping

When dissipation is present, the stress is not in phase with the strain, which results in a hysteresis loop

The mechanical energy dissipated in one cycle per unit volume is given by the area inside the loop

[Vibration Problems in Structures, CEB, 1991]

T=period

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Damping factor

The damping factor of a material is proportional to the ratio of energy dissipated in one cycle to the maximum strain potential energy

The damping factor of the structure is given by (V is the volume of the structure) :

For a homogeneous structure, we have


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Damping factor of a SDOF system with viscous damping

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Damping factor of a SDOF system with viscous damping

At resonance :

at resonance


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Equivalent viscous damping in real structures

Example of energy dissipation in a real structure

Frequency dependent damping coefficient -> difficult to use for time domain computations

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Equivalent viscous damping in real structures

The real damping is replaced by an equivalent viscous dampingThe energy dissipated for this equivalent system is equal to the real dissipation only for n, where , for lightly damped systems, the effect of damping is the most important


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Hysteretic damping : equivalent damping

In many engineering applications,S is found to be constant with the frequency

Hysteretic damping-> depends on frequency

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Coulomb friction damping : equivalent damping

Coulomb friction damping-> Depends on frequency and amplitude


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Measurement of damping

Logarithmic decrement method Half-power bandwidth

Estimation of in the time domain Estimation of in the frequency domain

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Logarithmic decrement method


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Half-power bandwidth

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Typical values of damping

- bare structure- materials

- joints

-non-structural elements-Footbridges : pavement, railings-Building floors : partition walls, flooring, furniture, ceilings ..

-energy radiation in soil

Contributions to damping

Damping coefficients are usually derived from practice if no measurement is possible


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Frequency limits for one dof systems

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Continuous model

SDOF model 1 SDOF model 2

m=100 kg, E =210 GPa, =7800 kg/m3, L=1m;A = 0.0004 m2

Frequency limits of SDOF systems for real continuous systems

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k =EA/L = 8.4 107 N/mM = 100 kg

fn=145.87 Hz (mass M)

fn=145.11 Hz (mass M+ma)ma = AL/3 = 1.04 kg <<M

SDOF hypothesis not valid for frequencies > 2000 Hz


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Frequency limits for one dof systems

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Continuous model

SDOF model 1 SDOF model 2

m=100 kg, E =210 GPa, =78000 kg/m3, L=1m;A = 0.0004 m2

Frequency limits of SDOF systems for real continuous systems

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k =EA/L = 8.4 107 N/mM = 100 kg

fn=145.87 Hz (mass M)

fn=138.83 Hz (mass M+ma)ma = AL/3 = 10.4 kg <M

SDOF hypothesis not valid for frequencies > 500 Hz

ma should be taken into account


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Summary

In particular cases, a real system can be reduced to a sdof system.

This requires computing the equivalent mass, stiffness and damping of the system

The validity of the model is limited and it should not be used outside of a certain frequency band, and for motions and excitations in other directions.

The tools to compute the response of single degree of freedom systems have been described in this chapter, both in the frequency and in the time domain

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Application example: the accelerometer

In the frequency domain :


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Application example: the accelerometer

Piezoelectric accelerometer

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High resonant frequencyLow sensitivity

Low resonant frequencyHigh sensitivity

Application example : the accelerometer

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