Generalized Single Degree of Freedom Systems PVD, Generalized - - PowerPoint PPT Presentation

Generalized Single Degree of Freedom Systems

PVD, Generalized Parameters, Rayleigh Quotjent Giacomo Boffj

htup://intranet.dica.polimi.it/people/boffj‐giacomo Dipartjmento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano

March 24, 2020

Generalized SDOF Giacomo Boffj

Outline

Generalized SDOF Giacomo Boffj

Sectjon 1 Introductory Remarks

Generalized SDOF Giacomo Boffj

Introductory Remarks

Untjl now our SDOF’s were described as composed by a single mass connected to a fjxed reference by means of a spring and a damper. While the mass‐spring is a useful representatjon, many difgerent, more complex systems can be studied as SDOF systems, either exactly or under some simplifying assumptjon.

1 SDOF rigid body assemblages, where the fmexibility is concentrated in a number of

springs and dampers, can be studied, e.g., using the Principle of Virtual Displacements and the D’Alembert Principle.

2 simple structural systems can be studied, in an approximate manner, assuming a

fjxed patuern of displacements, whose amplitude (the single degree of freedom) varies with tjme.

Generalized SDOF Giacomo Boffj

Introductory Remarks

Untjl now our SDOF’s were described as composed by a single mass connected to a fjxed reference by means of a spring and a damper. While the mass‐spring is a useful representatjon, many difgerent, more complex systems can be studied as SDOF systems, either exactly or under some simplifying assumptjon.

1 SDOF rigid body assemblages, where the fmexibility is concentrated in a number of

springs and dampers, can be studied, e.g., using the Principle of Virtual Displacements and the D’Alembert Principle.

2 simple structural systems can be studied, in an approximate manner, assuming a

fjxed patuern of displacements, whose amplitude (the single degree of freedom) varies with tjme.

Generalized SDOF Giacomo Boffj

Further Remarks on Rigid Assemblages

Today we restrict our consideratjon to plane, 2‐D systems. In rigid body assemblages the limitatjon to a single shape of displacement is a consequence of the confjguratjon of the system, i.e., the dispositjon of supports and internal hinges. When the equatjon of motjon is writuen in terms of a single parameter and its tjme derivatjves, the terms that fjgure as coeffjcients in the equatjon of motjon can be regarded as the generalised propertjes of the assemblage: generalised mass, damping and stjfgness on lefu hand, generalised loading on right hand. 𝑛⋆ ̈ 𝑦 + 𝑑⋆ ̇ 𝑦 + 𝑙⋆𝑦 = 𝑞⋆(𝑢)

Generalized SDOF Giacomo Boffj

Further Remarks on Contjnuous Systems

Contjnuous systems have an infjnite variety of deformatjon patuerns. By restrictjng the deformatjon to a single shape of varying amplitude, we introduce an infjnity of internal contstraints that limit the infjnite variety of deformatjon patuerns, but under this assumptjon the system confjguratjon is mathematjcally described by a single parameter, so that

• ur model can be analysed in exactly the same way as a strict SDOF system,

we can compute the generalised mass, damping, stjfgness propertjes of the SDOF model of the contjnuous system.

Generalized SDOF Giacomo Boffj

Final Remarks on Generalised SDOF Systems

From the previous comments, it should be apparent that everything we have seen regarding the behaviour and the integratjon of the equatjon of motjon of proper SDOF systems applies to rigid body assemblages and to SDOF models of fmexible systems, provided that we have the means for determining the generalised propertjes of the dynamical systems under investjgatjon.

Generalized SDOF Giacomo Boffj

Sectjon 2 Assemblage of Rigid Bodies

Generalized SDOF Giacomo Boffj

Assemblages of Rigid Bodies

planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers, rigid bodies are connected to a fjxed reference and to each other by means of springs, dampers and smooth, bilateral constraints (read hinges, double pendulums and rollers), inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by

an inertjal force applied to the centre of mass of the body, the product of the acceleratjon vector of the centre of mass itself and the total mass of the rigid body, 𝑁 = ∫ d𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦2 + 𝑧2)d𝑛.

Generalized SDOF Giacomo Boffj

Assemblages of Rigid Bodies

planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers, rigid bodies are connected to a fjxed reference and to each other by means of springs, dampers and smooth, bilateral constraints (read hinges, double pendulums and rollers), inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by

an inertjal force applied to the centre of mass of the body, the product of the acceleratjon vector of the centre of mass itself and the total mass of the rigid body, 𝑁 = ∫ d𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦2 + 𝑧2)d𝑛.

Generalized SDOF Giacomo Boffj

Assemblages of Rigid Bodies

planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers, rigid bodies are connected to a fjxed reference and to each other by means of springs, dampers and smooth, bilateral constraints (read hinges, double pendulums and rollers), inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by

an inertjal force applied to the centre of mass of the body, the product of the acceleratjon vector of the centre of mass itself and the total mass of the rigid body, 𝑁 = ∫ d𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦2 + 𝑧2)d𝑛.

Generalized SDOF Giacomo Boffj

Assemblages of Rigid Bodies

planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers, rigid bodies are connected to a fjxed reference and to each other by means of springs, dampers and smooth, bilateral constraints (read hinges, double pendulums and rollers), inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by

an inertjal force applied to the centre of mass of the body, the product of the acceleratjon vector of the centre of mass itself and the total mass of the rigid body, 𝑁 = ∫ d𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦2 + 𝑧2)d𝑛.

Generalized SDOF Giacomo Boffj

Assemblages of Rigid Bodies

planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers, rigid bodies are connected to a fjxed reference and to each other by means of springs, dampers and smooth, bilateral constraints (read hinges, double pendulums and rollers), inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by

an inertjal force applied to the centre of mass of the body, the product of the acceleratjon vector of the centre of mass itself and the total mass of the rigid body, 𝑁 = ∫ d𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦2 + 𝑧2)d𝑛.

Generalized SDOF Giacomo Boffj

Rigid Bar

x G L

Unit mass ̄ 𝑛 = constant, Length 𝑀, Centre of Mass 𝑦𝐻 = 𝑀/2, Total Mass 𝑛 = ̄ 𝑛𝑀, Moment of Inertja 𝐾 = 𝑛 𝑀2 12 = ̄ 𝑛 𝑀3 12

Generalized SDOF Giacomo Boffj

Rigid Rectangle

G y a b

Unit mass 𝛿 = constant, Sides 𝑏, 𝑐 Centre of Mass 𝑦𝐻 = 𝑏/2, 𝑧𝐻 = 𝑐/2 Total Mass 𝑛 = 𝛿𝑏𝑐, Moment of Inertja 𝐾 = 𝑛𝑏2 + 𝑐2 12 = 𝛿𝑏3𝑐 + 𝑏𝑐3 12

Generalized SDOF Giacomo Boffj

Rigid Triangle

For a right triangle.

y G a b

Unit mass 𝛿 = constant, Sides 𝑏, 𝑐 Centre of Mass 𝑦𝐻 = 𝑏/3, 𝑧𝐻 = 𝑐/3 Total Mass 𝑛 = 𝛿𝑏𝑐/2, Moment of Inertja 𝐾 = 𝑛𝑏2 + 𝑐2 18 = 𝛿𝑏3𝑐 + 𝑏𝑐3 36

Generalized SDOF Giacomo Boffj

Rigid Oval

x y a b

Unit mass 𝛿 = constant, Axes 𝑏, 𝑐 Centre of Mass 𝑦𝐻 = 𝑧𝐻 = 0 Total Mass 𝑛 = 𝛿𝜌𝑏𝑐 4 , Moment of Inertja 𝐾 = 𝑛𝑏2 + 𝑐2 16 When 𝑏 = 𝑐 = 𝐸 = 2𝑆 the oval is a circle: 𝑛 = 𝛿 𝜌𝑆2, 𝐾 = 𝑛 𝑆2 2 = 𝛿 𝜌𝑆4 2 .

Generalized SDOF Giacomo Boffj

Rigid Oval

x y a b

Unit mass 𝛿 = constant, Axes 𝑏, 𝑐 Centre of Mass 𝑦𝐻 = 𝑧𝐻 = 0 Total Mass 𝑛 = 𝛿𝜌𝑏𝑐 4 , Moment of Inertja 𝐾 = 𝑛𝑏2 + 𝑐2 16 When 𝑏 = 𝑐 = 𝐸 = 2𝑆 the oval is a circle: 𝑛 = 𝛿 𝜌𝑆2, 𝐾 = 𝑛 𝑆2 2 = 𝛿 𝜌𝑆4 2 .

Generalized SDOF Giacomo Boffj

trabacolo1

c k c k

2 2 1 1

N m , J

2 2

p(x,t) = P x/a f(t) a 2 a a a a a

The mass of the lefu bar is 𝑛1 = ̄ 𝑛 4𝑏 and its moment of inertja is 𝐾1 = 𝑛1

(4𝑏)2 12

= 4𝑏2𝑛1/3. The maximum value of the external load is 𝑄

max = 𝑄 4𝑏/𝑏 = 4𝑄 and the resultant of

triangular load is 𝑆 = 4𝑄 × 4𝑏/2 = 8𝑄𝑏

Generalized SDOF Giacomo Boffj

Forces and Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝑣 = 7𝑏 − 4𝑏 cos 𝜄1 − 3𝑏 cos 𝜄2, 𝜀𝑣 = 4𝑏 sin 𝜄1𝜀𝜄1 + 3𝑏 sin 𝜄2𝜀𝜄2 𝜀𝜄1 = 𝜀𝑎/(4𝑏), 𝜀𝜄2 = 𝜀𝑎/(3𝑏) sin 𝜄1 ≈ 𝑎/(4𝑏), sin 𝜄2 ≈ 𝑎/(3𝑏) 𝜀𝑣 =

1 4𝑏 + 1 3𝑏 𝑎 𝜀𝑎 = 7 12𝑏𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Forces and Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝑣 = 7𝑏 − 4𝑏 cos 𝜄1 − 3𝑏 cos 𝜄2, 𝜀𝑣 = 4𝑏 sin 𝜄1𝜀𝜄1 + 3𝑏 sin 𝜄2𝜀𝜄2 𝜀𝜄1 = 𝜀𝑎/(4𝑏), 𝜀𝜄2 = 𝜀𝑎/(3𝑏) sin 𝜄1 ≈ 𝑎/(4𝑏), sin 𝜄2 ≈ 𝑎/(3𝑏) 𝜀𝑣 =

1 4𝑏 + 1 3𝑏 𝑎 𝜀𝑎 = 7 12𝑏𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝜀𝑋

I = −𝑛1

̈ 𝑎 2 𝜀𝑎 2 − 𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4𝑏 − 𝑛2 2 ̈ 𝑎 3 2𝜀𝑎 3 − 𝐾2 ̈ 𝑎 3𝑏 𝜀𝑎 3𝑏 = − 𝑛1 4 + 4𝑛2 9 + 𝐾1 16𝑏2 + 𝐾2 9𝑏2 ̈ 𝑎 𝜀𝑎 𝜀𝑋

D = −𝑑1

̇ 𝑎 4 𝜀𝑎 4 − −𝑑2𝑎 𝜀𝑎 = − (𝑑2 + 𝑑1/16) ̇ 𝑎 𝜀𝑎 𝜀𝑋

S = −𝑙1

3𝑎 4 3𝜀𝑎 4 − 𝑙2 𝑎 3 𝜀𝑎 3 = − 9𝑙1 16 + 𝑙2 9 𝑎 𝜀𝑎 𝜀𝑋

Ext = 8𝑄𝑏 𝑔(𝑢)2𝜀𝑎

3 + 𝑂 7 12𝑏 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝜀𝑋

I = −𝑛1

̈ 𝑎 2 𝜀𝑎 2 − 𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4𝑏 − 𝑛2 2 ̈ 𝑎 3 2𝜀𝑎 3 − 𝐾2 ̈ 𝑎 3𝑏 𝜀𝑎 3𝑏 = − 𝑛1 4 + 4𝑛2 9 + 𝐾1 16𝑏2 + 𝐾2 9𝑏2 ̈ 𝑎 𝜀𝑎 𝜀𝑋

D = −𝑑1

̇ 𝑎 4 𝜀𝑎 4 − −𝑑2𝑎 𝜀𝑎 = − (𝑑2 + 𝑑1/16) ̇ 𝑎 𝜀𝑎 𝜀𝑋

S = −𝑙1

3𝑎 4 3𝜀𝑎 4 − 𝑙2 𝑎 3 𝜀𝑎 3 = − 9𝑙1 16 + 𝑙2 9 𝑎 𝜀𝑎 𝜀𝑋

Ext = 8𝑄𝑏 𝑔(𝑢)2𝜀𝑎

3 + 𝑂 7 12𝑏 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝜀𝑋

I = −𝑛1

̈ 𝑎 2 𝜀𝑎 2 − 𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4𝑏 − 𝑛2 2 ̈ 𝑎 3 2𝜀𝑎 3 − 𝐾2 ̈ 𝑎 3𝑏 𝜀𝑎 3𝑏 = − 𝑛1 4 + 4𝑛2 9 + 𝐾1 16𝑏2 + 𝐾2 9𝑏2 ̈ 𝑎 𝜀𝑎 𝜀𝑋

D = −𝑑1

̇ 𝑎 4 𝜀𝑎 4 − −𝑑2𝑎 𝜀𝑎 = − (𝑑2 + 𝑑1/16) ̇ 𝑎 𝜀𝑎 𝜀𝑋

S = −𝑙1

3𝑎 4 3𝜀𝑎 4 − 𝑙2 𝑎 3 𝜀𝑎 3 = − 9𝑙1 16 + 𝑙2 9 𝑎 𝜀𝑎 𝜀𝑋

Ext = 8𝑄𝑏 𝑔(𝑢)2𝜀𝑎

3 + 𝑂 7 12𝑏 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

𝑑1 ̇ 𝑎 4 𝑛1 ̈ 𝑎 2 3𝑙1𝑎 4

𝑑2 ̇ 𝑎

2𝑛2 ̈ 𝑎 3 𝑙𝑎 3

𝑂 𝑎(𝑢)

𝐾2 ̈ 𝑎 3𝑏

8𝑄𝑏 𝑔(𝑢)

𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4 𝜀𝑎 2

3

𝜀𝑎 4

𝜀𝑎 2

𝜀𝑎 3 𝜀𝑎 3

𝜀𝑣 𝜀𝜄2 = 𝜀𝑎/(3𝑏) 𝜀𝜄1 = 𝜀𝑎/(4𝑏)

𝜀𝑋

I = −𝑛1

̈ 𝑎 2 𝜀𝑎 2 − 𝐾1 ̈ 𝑎 4𝑏 𝜀𝑎 4𝑏 − 𝑛2 2 ̈ 𝑎 3 2𝜀𝑎 3 − 𝐾2 ̈ 𝑎 3𝑏 𝜀𝑎 3𝑏 = − 𝑛1 4 + 4𝑛2 9 + 𝐾1 16𝑏2 + 𝐾2 9𝑏2 ̈ 𝑎 𝜀𝑎 𝜀𝑋

D = −𝑑1

̇ 𝑎 4 𝜀𝑎 4 − −𝑑2𝑎 𝜀𝑎 = − (𝑑2 + 𝑑1/16) ̇ 𝑎 𝜀𝑎 𝜀𝑋

S = −𝑙1

3𝑎 4 3𝜀𝑎 4 − 𝑙2 𝑎 3 𝜀𝑎 3 = − 9𝑙1 16 + 𝑙2 9 𝑎 𝜀𝑎 𝜀𝑋

Ext = 8𝑄𝑏 𝑔(𝑢)2𝜀𝑎

3 + 𝑂 7 12𝑏 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

For a rigid body in conditjon of equilibrium the total virtual work must be equal to zero 𝜀𝑋

I + 𝜀𝑋 D + 𝜀𝑋 S + 𝜀𝑋 Ext = 0

Substjtutjng our expressions of the virtual work contributjons and simplifying 𝜀𝑎, the equatjon of equilibrium is 𝑛1 4 + 4𝑛2 9 + 𝐾1 16𝑏2 + 𝐾2 9𝑏2 ̈ 𝑎+ + (𝑑2 + 𝑑1/16) ̇ 𝑎 + 9𝑙1 16 + 𝑙2 9 𝑎 = 8𝑄𝑏 𝑔(𝑢)2 3 + 𝑂 7 12𝑏𝑎

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

Collectjng 𝑎 and its tjme derivatjves give us 𝑛⋆ ̈ 𝑎 + 𝑑⋆ ̇ 𝑎 + 𝑙⋆𝑎 = 𝑞⋆𝑔(𝑢) introducing the so called generalised propertjes, in our example it is 𝑛⋆ =

1 4𝑛1 + 4 99𝑛2 + 1 16𝑏2 𝐾1 + 1 9𝑏2 𝐾2,

𝑑⋆ =

1 16𝑑1 + 𝑑2,

𝑙⋆ =

9 16𝑙1 + 1 9𝑙2 − 7 12𝑏𝑂,

𝑞⋆ =

16 3 𝑄𝑏.

It is worth writjng down the ex‐ pression of 𝑙⋆: 𝑙⋆ = 9𝑙1 16 + 𝑙2 9 − 7 12𝑏𝑂

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

Collectjng 𝑎 and its tjme derivatjves give us 𝑛⋆ ̈ 𝑎 + 𝑑⋆ ̇ 𝑎 + 𝑙⋆𝑎 = 𝑞⋆𝑔(𝑢) introducing the so called generalised propertjes, in our example it is 𝑛⋆ =

1 4𝑛1 + 4 99𝑛2 + 1 16𝑏2 𝐾1 + 1 9𝑏2 𝐾2,

𝑑⋆ =

1 16𝑑1 + 𝑑2,

𝑙⋆ =

9 16𝑙1 + 1 9𝑙2 − 7 12𝑏𝑂,

𝑞⋆ =

16 3 𝑄𝑏.

It is worth writjng down the ex‐ pression of 𝑙⋆: 𝑙⋆ = 9𝑙1 16 + 𝑙2 9 − 7 12𝑏𝑂

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

Collectjng 𝑎 and its tjme derivatjves give us 𝑛⋆ ̈ 𝑎 + 𝑑⋆ ̇ 𝑎 + 𝑙⋆𝑎 = 𝑞⋆𝑔(𝑢) introducing the so called generalised propertjes, in our example it is 𝑛⋆ =

1 4𝑛1 + 4 99𝑛2 + 1 16𝑏2 𝐾1 + 1 9𝑏2 𝐾2,

𝑑⋆ =

1 16𝑑1 + 𝑑2,

𝑙⋆ =

9 16𝑙1 + 1 9𝑙2 − 7 12𝑏𝑂,

𝑞⋆ =

16 3 𝑄𝑏.

It is worth writjng down the ex‐ pression of 𝑙⋆: 𝑙⋆ = 9𝑙1 16 + 𝑙2 9 − 7 12𝑏𝑂 Geometrical stjfgness

Generalized SDOF Giacomo Boffj

Sectjon 3 Contjnuous Systems

Generalized SDOF Giacomo Boffj

Let’s start with an example...

Consider a cantjlever, with varying propertjes ̄ 𝑛 and 𝐹𝐾, subjected to a dynamic load that is functjon of both tjme 𝑢 and positjon 𝑦, 𝑞 = 𝑞(𝑦, 𝑢).

𝑦 ̄ 𝑛 = ̄ 𝑛(𝑦) 𝑂 𝐹𝐾 = 𝐹𝐾(𝑦) 𝑤(𝑦, 𝑢) 𝑞(𝑦, 𝑢) 𝐼

Even the transverse displacements 𝑤 will be functjon of tjme and positjon, 𝑤 = 𝑤(𝑦, 𝑢) and because the inertjal forces depend on ̈ 𝑤 = 𝜖2𝑤/𝜖𝑢2 and the elastjc forces on 𝑤″ = 𝜖2𝑤/𝜖𝑦2 the equatjon of dynamic equilibrium must be writuen in terms of a partjal derivatjves difgerentjal equatjon.

Generalized SDOF Giacomo Boffj

Let’s start with an example...

Consider a cantjlever, with varying propertjes ̄ 𝑛 and 𝐹𝐾, subjected to a dynamic load that is functjon of both tjme 𝑢 and positjon 𝑦, 𝑞 = 𝑞(𝑦, 𝑢).

𝑦 ̄ 𝑛 = ̄ 𝑛(𝑦) 𝑂 𝐹𝐾 = 𝐹𝐾(𝑦) 𝑤(𝑦, 𝑢) 𝑞(𝑦, 𝑢) 𝐼

Even the transverse displacements 𝑤 will be functjon of tjme and positjon, 𝑤 = 𝑤(𝑦, 𝑢) and because the inertjal forces depend on ̈ 𝑤 = 𝜖2𝑤/𝜖𝑢2 and the elastjc forces on 𝑤″ = 𝜖2𝑤/𝜖𝑦2 the equatjon of dynamic equilibrium must be writuen in terms of a partjal derivatjves difgerentjal equatjon.

Generalized SDOF Giacomo Boffj

... and an hypothesis

To study the previous problem, we introduce an approximate model by the following hypothesis, 𝑤(𝑦, 𝑢) = Ψ(𝑦) 𝑎(𝑢), that is, the hypothesis of separatjon of variables Note that Ψ(𝑦), the shape functjon, is adimensional, while 𝑎(𝑢) is dimensionally a generalised displacement, usually chosen to characterise the structural behaviour. In our example we can use the displacement of the tjp of the chimney, thus implying that Ψ(𝐼) = 1 because 𝑎(𝑢) = 𝑤(𝐼, 𝑢) and 𝑤(𝐼, 𝑢) = Ψ(𝐼) 𝑎(𝑢)

Generalized SDOF Giacomo Boffj

... and an hypothesis

To study the previous problem, we introduce an approximate model by the following hypothesis, 𝑤(𝑦, 𝑢) = Ψ(𝑦) 𝑎(𝑢), that is, the hypothesis of separatjon of variables Note that Ψ(𝑦), the shape functjon, is adimensional, while 𝑎(𝑢) is dimensionally a generalised displacement, usually chosen to characterise the structural behaviour. In our example we can use the displacement of the tjp of the chimney, thus implying that Ψ(𝐼) = 1 because 𝑎(𝑢) = 𝑤(𝐼, 𝑢) and 𝑤(𝐼, 𝑢) = Ψ(𝐼) 𝑎(𝑢)

Generalized SDOF Giacomo Boffj

... and an hypothesis

To study the previous problem, we introduce an approximate model by the following hypothesis, 𝑤(𝑦, 𝑢) = Ψ(𝑦) 𝑎(𝑢), that is, the hypothesis of separatjon of variables Note that Ψ(𝑦), the shape functjon, is adimensional, while 𝑎(𝑢) is dimensionally a generalised displacement, usually chosen to characterise the structural behaviour. In our example we can use the displacement of the tjp of the chimney, thus implying that Ψ(𝐼) = 1 because 𝑎(𝑢) = 𝑤(𝐼, 𝑢) and 𝑤(𝐼, 𝑢) = Ψ(𝐼) 𝑎(𝑢)

Generalized SDOF Giacomo Boffj

Principle of Virtual Displacements

For a fmexible system, the PoVD states that, at equilibrium, 𝜀𝑋

E = 𝜀𝑋 I.

The virtual work of external forces can be easily computed, the virtual work of internal forces is usually approximated by the virtual work done by bending moments, that is 𝜀𝑋

I ≈ 𝑁 𝜀𝜓

where 𝜓 is the curvature and 𝜀𝜓 the virtual increment of curvature.

Generalized SDOF Giacomo Boffj

𝜀𝑋

E

The external forces are 𝑞(𝑦, 𝑢), 𝑂 and the forces of inertja 𝑔

I; we have, by separatjon

• f variables, that 𝜀𝑤 = Ψ(𝑦)𝜀𝑎 and we can write

𝜀𝑋

p = 𝐼

𝑞(𝑦, 𝑢)𝜀𝑤 d𝑦 =

𝐼

𝑞(𝑦, 𝑢)Ψ(𝑦) d𝑦 𝜀𝑎 = 𝑞⋆(𝑢) 𝜀𝑎 𝜀𝑋

Inertja = 𝐼

− ̄ 𝑛(𝑦) ̈ 𝑤𝜀𝑤 d𝑦 =

𝐼

− ̄ 𝑛(𝑦) Ψ(𝑦) ̈ 𝑎 (Ψ(𝑦) 𝜀𝑎) d𝑦 =

𝐼

− ̄ 𝑛(𝑦)Ψ2(𝑦) d𝑦 ̈ 𝑎(𝑢) 𝜀𝑎 = 𝑛⋆ ̈ 𝑎 𝜀𝑎. The virtual work done by the axial force deserves a separate treatment...

Generalized SDOF Giacomo Boffj

𝜀𝑋

N

The virtual work of 𝑂 is 𝜀𝑋

N = 𝑂𝜀𝑣 where 𝜀𝑣 is the variatjon of the vertjcal displacement of the top of

the chimney. We start computjng the vertjcal displacement of the top of the chimney in terms of the rotatjon of the axis line, 𝜚 ≈ Ψ′(𝑦)𝑎(𝑢), 𝑣(𝑢) = 𝐼 −

𝐼

cos 𝜚 d𝑦 =

𝐼

(1 − cos 𝜚) d𝑦, substjtutjng the well known approximatjon 𝑑𝑝𝑡𝜚 ≈ 1 −

𝜚2 2 in the above equatjon we have

𝑣(𝑢) =

𝐼

𝜚2 2 d𝑦 =

𝐼

Ψ′2(𝑦)𝑎2(𝑢) 2 d𝑦 ⇒ ⇒ 𝜀𝑣 =

𝐼

Ψ′2(𝑦)𝑎(𝑢)𝜀𝑎 d𝑦 =

𝐼

Ψ′2(𝑦) d𝑦 𝑎𝜀𝑎 and 𝜀𝑋

N = 𝐼

Ψ′2(𝑦) d𝑦 𝑂 𝑎 𝜀𝑎 = 𝑙⋆

𝐻 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

𝜀𝑋

Int

Approximatjng the internal work with the work done by bending moments, for an infjnitesimal slice of beam we write d𝑋

Int = 1

2𝑁𝑤"(𝑦, 𝑢) d𝑦 = 1 2𝑁Ψ"(𝑦)𝑎(𝑢) d𝑦 with 𝑁 = 𝐹𝐾(𝑦)𝑤"(𝑦) 𝜀(d𝑋

Int) = 𝐹𝐾(𝑦)Ψ"2(𝑦)𝑎(𝑢)𝜀𝑎 d𝑦

integratjng 𝜀𝑋

Int = 𝐼

𝐹𝐾(𝑦)Ψ"2(𝑦) d𝑦 𝑎𝜀𝑎 = 𝑙⋆ 𝑎 𝜀𝑎

Generalized SDOF Giacomo Boffj

Remarks

the shape functjon must respect the geometrical boundary conditjons of the problem, i.e., both Ψ1 = 𝑦2 and Ψ2 = 1 − cos 𝜌𝑦 2𝐼 are accetuable shape functjons for our example, as Ψ1(0) = Ψ2(0) = 0 and Ψ′

1(0) = Ψ′ 2(0) = 0

betuer results are obtained when the second derivatjve of the shape functjon at least resembles the typical distributjon of bending moments in our problem, so that between Ψ′′

1 = constant

and Ψ2" = 𝜌2 4𝐼2 cos 𝜌𝑦 2𝐼 the second choice is preferable.

Generalized SDOF Giacomo Boffj

Remarks

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 vi"/Z(t) v/Z(t) x/H f1=1-cos(pi*x/2) f2=x2

the shape functjon must respect the geometrical boundary conditjons of the problem, i.e., both Ψ1 = 𝑦2 and Ψ2 = 1 − cos 𝜌𝑦 2𝐼 are accetuable shape functjons for our example, as Ψ1(0) = Ψ2(0) = 0 and Ψ′

1(0) = Ψ′ 2(0) = 0

betuer results are obtained when the second derivatjve of the shape functjon at least resembles the typical distributjon of bending moments in our problem, so that between Ψ′′

1 = constant

and Ψ2" = 𝜌2 4𝐼2 cos 𝜌𝑦 2𝐼 the second choice is preferable.

Generalized SDOF Giacomo Boffj

Remarks

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 vi"/Z(t) v/Z(t) x/H f1=1-cos(pi*x/2) f2=x2 f1" f2"

the shape functjon must respect the geometrical boundary conditjons of the problem, i.e., both Ψ1 = 𝑦2 and Ψ2 = 1 − cos 𝜌𝑦 2𝐼 are accetuable shape functjons for our example, as Ψ1(0) = Ψ2(0) = 0 and Ψ′

1(0) = Ψ′ 2(0) = 0

betuer results are obtained when the second derivatjve of the shape functjon at least resembles the typical distributjon of bending moments in our problem, so that between Ψ′′

1 = constant

and Ψ2" = 𝜌2 4𝐼2 cos 𝜌𝑦 2𝐼 the second choice is preferable.

Generalized SDOF Giacomo Boffj

Remarks

the shape functjon must respect the geometrical boundary conditjons of the problem, i.e., both Ψ1 = 𝑦2 and Ψ2 = 1 − cos 𝜌𝑦 2𝐼 are accetuable shape functjons for our example, as Ψ1(0) = Ψ2(0) = 0 and Ψ′

1(0) = Ψ′ 2(0) = 0

betuer results are obtained when the second derivatjve of the shape functjon at least resembles the typical distributjon of bending moments in our problem, so that between Ψ′′

1 = constant

and Ψ2" = 𝜌2 4𝐼2 cos 𝜌𝑦 2𝐼 the second choice is preferable.

Generalized SDOF Giacomo Boffj

Example

Using Ψ(𝑦) = 1 − cos

𝜌𝑦 2𝐼, with

̄ 𝑛 = constant and 𝐹𝐾 = constant, with a load characteristjc of seismic excitatjon, 𝑞(𝑢) = − ̄ 𝑛 ̈ 𝑤𝑕(𝑢), 𝑛⋆ = ̄ 𝑛

𝐼

(1 − cos 𝜌𝑦 2𝐼)2 d𝑦 = ̄ 𝑛(3 2 − 4 𝜌)𝐼 𝑙⋆ = 𝐹𝐾 𝜌4 16𝐼4

𝐼

cos2 𝜌𝑦 2𝐼 d𝑦 = 𝜌4 32 𝐹𝐾 𝐼3 𝑙⋆

𝐻 = 𝑂 𝜌2

4𝐼2

𝐼

sin2 𝜌𝑦 2𝐼 d𝑦 = 𝜌2 8𝐼𝑂 𝑞⋆

𝑕 = − ̄

𝑛 ̈ 𝑤𝑕(𝑢)

𝐼

1 − cos 𝜌𝑦 2𝐼 d𝑦 = − 1 − 2 𝜌 ̄ 𝑛𝐼 ̈ 𝑤𝑕(𝑢)

Generalized SDOF Giacomo Boffj

Sectjon 4 Vibratjon Analysis by Rayleigh’s Method

Generalized SDOF Giacomo Boffj

Vibratjon Analysis

The process of estjmatjng the vibratjon characteristjcs of a complex system is known as vibratjon analysis. We can use our previous results for fmexible systems, based on the SDOF model, to give an estjmate of the natural frequency 𝜕2 = 𝑙⋆/𝑛⋆ A difgerent approach, proposed by Lord Rayleigh, starts from difgerent premises to give the same results but the Rayleigh’s Quotjent method is important because it ofgers a betuer understanding of the vibratjonal behaviour, eventually leading to successive refjnements of the fjrst estjmate of 𝜕2.

Generalized SDOF Giacomo Boffj

Vibratjon Analysis

The process of estjmatjng the vibratjon characteristjcs of a complex system is known as vibratjon analysis. We can use our previous results for fmexible systems, based on the SDOF model, to give an estjmate of the natural frequency 𝜕2 = 𝑙⋆/𝑛⋆ A difgerent approach, proposed by Lord Rayleigh, starts from difgerent premises to give the same results but the Rayleigh’s Quotjent method is important because it ofgers a betuer understanding of the vibratjonal behaviour, eventually leading to successive refjnements of the fjrst estjmate of 𝜕2.

Generalized SDOF Giacomo Boffj

Vibratjon Analysis

The process of estjmatjng the vibratjon characteristjcs of a complex system is known as vibratjon analysis. We can use our previous results for fmexible systems, based on the SDOF model, to give an estjmate of the natural frequency 𝜕2 = 𝑙⋆/𝑛⋆ A difgerent approach, proposed by Lord Rayleigh, starts from difgerent premises to give the same results but the Rayleigh’s Quotjent method is important because it ofgers a betuer understanding of the vibratjonal behaviour, eventually leading to successive refjnements of the fjrst estjmate of 𝜕2.

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

Our focus will be on the free vibratjon of a fmexible, undamped system. inspired by the free vibratjons of a proper SDOF we write 𝑎(𝑢) = 𝑎0 sin 𝜕𝑢 and 𝑤(𝑦, 𝑢) = 𝑎0Ψ(𝑦) sin 𝜕𝑢, ̇ 𝑤(𝑦, 𝑢) = 𝜕 𝑎0Ψ(𝑦) cos 𝜕𝑢. the displacement and the velocity are in quadrature: when 𝑤 is at its maximum ̇ 𝑤 = 0, hence 𝑊 = 𝑊

max, 𝑈 = 0 and when ̇

𝑤 is at its maximum it is 𝑤 = 0, hence 𝑊 = 0, 𝑈 = 𝑈

max,

disregarding damping, the energy of the system is constant during free vibratjons, 𝑊

max + 0 = 0 + 𝑈 max

⇒ 𝑊

max = 𝑈 max

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

Our focus will be on the free vibratjon of a fmexible, undamped system. inspired by the free vibratjons of a proper SDOF we write 𝑎(𝑢) = 𝑎0 sin 𝜕𝑢 and 𝑤(𝑦, 𝑢) = 𝑎0Ψ(𝑦) sin 𝜕𝑢, ̇ 𝑤(𝑦, 𝑢) = 𝜕 𝑎0Ψ(𝑦) cos 𝜕𝑢. the displacement and the velocity are in quadrature: when 𝑤 is at its maximum ̇ 𝑤 = 0, hence 𝑊 = 𝑊

max, 𝑈 = 0 and when ̇

𝑤 is at its maximum it is 𝑤 = 0, hence 𝑊 = 0, 𝑈 = 𝑈

max,

disregarding damping, the energy of the system is constant during free vibratjons, 𝑊

max + 0 = 0 + 𝑈 max

⇒ 𝑊

max = 𝑈 max

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

Our focus will be on the free vibratjon of a fmexible, undamped system. inspired by the free vibratjons of a proper SDOF we write 𝑎(𝑢) = 𝑎0 sin 𝜕𝑢 and 𝑤(𝑦, 𝑢) = 𝑎0Ψ(𝑦) sin 𝜕𝑢, ̇ 𝑤(𝑦, 𝑢) = 𝜕 𝑎0Ψ(𝑦) cos 𝜕𝑢. the displacement and the velocity are in quadrature: when 𝑤 is at its maximum ̇ 𝑤 = 0, hence 𝑊 = 𝑊

max, 𝑈 = 0 and when ̇

𝑤 is at its maximum it is 𝑤 = 0, hence 𝑊 = 0, 𝑈 = 𝑈

max,

disregarding damping, the energy of the system is constant during free vibratjons, 𝑊

max + 0 = 0 + 𝑈 max

⇒ 𝑊

max = 𝑈 max

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

Our focus will be on the free vibratjon of a fmexible, undamped system. inspired by the free vibratjons of a proper SDOF we write 𝑎(𝑢) = 𝑎0 sin 𝜕𝑢 and 𝑤(𝑦, 𝑢) = 𝑎0Ψ(𝑦) sin 𝜕𝑢, ̇ 𝑤(𝑦, 𝑢) = 𝜕 𝑎0Ψ(𝑦) cos 𝜕𝑢. the displacement and the velocity are in quadrature: when 𝑤 is at its maximum ̇ 𝑤 = 0, hence 𝑊 = 𝑊

max, 𝑈 = 0 and when ̇

𝑤 is at its maximum it is 𝑤 = 0, hence 𝑊 = 0, 𝑈 = 𝑈

max,

disregarding damping, the energy of the system is constant during free vibratjons, 𝑊

max + 0 = 0 + 𝑈 max

⇒ 𝑊

max = 𝑈 max

Generalized SDOF Giacomo Boffj

Rayleigh’ s Quotjent Method

Now we write the expressions for 𝑊

max and 𝑈 max,

𝑊

max = 1

2𝑎2

𝑇

𝐹𝐾(𝑦)Ψ′′2(𝑦) d𝑦, 𝑈

max = 1

2𝜕2𝑎2

𝑇

̄ 𝑛(𝑦)Ψ2(𝑦) d𝑦, equatjng the two expressions and solving for 𝜕2 we have 𝜕2 = ∫

𝑇 𝐹𝐾(𝑦)Ψ′′2(𝑦) d𝑦

∫

𝑇

̄ 𝑛(𝑦)Ψ2(𝑦) d𝑦 . Recognizing the expressions we found for 𝑙⋆ and 𝑛⋆ we could questjon the utjlity of Rayleigh’s Quotjent...

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

in Rayleigh’s method we know the specifjc tjme dependency of the inertjal forces 𝑔

I = − ̄

𝑛(𝑦) ̈ 𝑤 = ̄ 𝑛(𝑦)𝜕2𝑎0Ψ(𝑦) sin 𝜕𝑢 𝑔

I has the same shape we use for displacements.

if Ψ were the real shape assumed by the structure in free vibratjons, the displacements 𝑤 due to a loading 𝑔

I = 𝜕2 ̄

𝑛(𝑦)Ψ(𝑦)𝑎0 should be proportjonal to Ψ(𝑦) through a constant factor, with equilibrium respected in every point of the structure during free vibratjons. startjng from a shape functjon Ψ0(𝑦), a new shape functjon Ψ1 can be determined normalizing the displacements due to the inertjal forces associated with Ψ0(𝑦), 𝑔

I =

̄ 𝑛(𝑦)Ψ0(𝑦), we are going to demonstrate that the new shape functjon is a betuer approximatjon of the true mode shape

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

in Rayleigh’s method we know the specifjc tjme dependency of the inertjal forces 𝑔

I = − ̄

𝑛(𝑦) ̈ 𝑤 = ̄ 𝑛(𝑦)𝜕2𝑎0Ψ(𝑦) sin 𝜕𝑢 𝑔

I has the same shape we use for displacements.

if Ψ were the real shape assumed by the structure in free vibratjons, the displacements 𝑤 due to a loading 𝑔

I = 𝜕2 ̄

𝑛(𝑦)Ψ(𝑦)𝑎0 should be proportjonal to Ψ(𝑦) through a constant factor, with equilibrium respected in every point of the structure during free vibratjons. startjng from a shape functjon Ψ0(𝑦), a new shape functjon Ψ1 can be determined normalizing the displacements due to the inertjal forces associated with Ψ0(𝑦), 𝑔

I =

̄ 𝑛(𝑦)Ψ0(𝑦), we are going to demonstrate that the new shape functjon is a betuer approximatjon of the true mode shape

Generalized SDOF Giacomo Boffj

Rayleigh’s Quotjent Method

in Rayleigh’s method we know the specifjc tjme dependency of the inertjal forces 𝑔

I = − ̄

𝑛(𝑦) ̈ 𝑤 = ̄ 𝑛(𝑦)𝜕2𝑎0Ψ(𝑦) sin 𝜕𝑢 𝑔

I has the same shape we use for displacements.

if Ψ were the real shape assumed by the structure in free vibratjons, the displacements 𝑤 due to a loading 𝑔

I = 𝜕2 ̄

𝑛(𝑦)Ψ(𝑦)𝑎0 should be proportjonal to Ψ(𝑦) through a constant factor, with equilibrium respected in every point of the structure during free vibratjons. startjng from a shape functjon Ψ0(𝑦), a new shape functjon Ψ1 can be determined normalizing the displacements due to the inertjal forces associated with Ψ0(𝑦), 𝑔

I =

̄ 𝑛(𝑦)Ψ0(𝑦), we are going to demonstrate that the new shape functjon is a betuer approximatjon of the true mode shape

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes

Given difgerent shape functjons Ψ𝑗 and considering the true shape of free vibratjon Ψ, in the former cases equilibrium is not respected by the structure itself. To keep inertja induced deformatjon proportjonal to Ψ𝑗 we must consider the presence of additjonal elastjc constraints. This leads to the following consideratjons the frequency of vibratjon of a structure with additjonal constraints is higher than the true natural frequency, the criterium to discriminate between difgerent shape functjons is: betuer shape functjons give lower estjmates of the natural frequency, the true natural frequency being a lower bound of all estjmates.

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes

Given difgerent shape functjons Ψ𝑗 and considering the true shape of free vibratjon Ψ, in the former cases equilibrium is not respected by the structure itself. To keep inertja induced deformatjon proportjonal to Ψ𝑗 we must consider the presence of additjonal elastjc constraints. This leads to the following consideratjons the frequency of vibratjon of a structure with additjonal constraints is higher than the true natural frequency, the criterium to discriminate between difgerent shape functjons is: betuer shape functjons give lower estjmates of the natural frequency, the true natural frequency being a lower bound of all estjmates.

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes

Given difgerent shape functjons Ψ𝑗 and considering the true shape of free vibratjon Ψ, in the former cases equilibrium is not respected by the structure itself. To keep inertja induced deformatjon proportjonal to Ψ𝑗 we must consider the presence of additjonal elastjc constraints. This leads to the following consideratjons the frequency of vibratjon of a structure with additjonal constraints is higher than the true natural frequency, the criterium to discriminate between difgerent shape functjons is: betuer shape functjons give lower estjmates of the natural frequency, the true natural frequency being a lower bound of all estjmates.

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes

Given difgerent shape functjons Ψ𝑗 and considering the true shape of free vibratjon Ψ, in the former cases equilibrium is not respected by the structure itself. To keep inertja induced deformatjon proportjonal to Ψ𝑗 we must consider the presence of additjonal elastjc constraints. This leads to the following consideratjons the frequency of vibratjon of a structure with additjonal constraints is higher than the true natural frequency, the criterium to discriminate between difgerent shape functjons is: betuer shape functjons give lower estjmates of the natural frequency, the true natural frequency being a lower bound of all estjmates.

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes 2

In general the selectjon of trial shapes goes through two steps,

1 the analyst considers the fmexibilitjes of difgerent parts of the structure and the

presence of symmetries to devise an approximate shape,

2 the structure is loaded with constant loads directed as the assumed

displacements, the displacements are computed and used as the shape functjon,

• f course a litule practjce helps a lot in the the choice of a proper patuern of loading...

Generalized SDOF Giacomo Boffj

Selectjon of mode shapes 3

𝑞 = 𝑛(𝑦) 𝑄 = 𝑁 𝑞 = 𝑛(𝑦) 𝑞 = 𝑛(𝑦) 𝑞 = 𝑛(𝑦)

(𝑏) (𝑐) (𝑑) (𝑒)

Generalized SDOF Giacomo Boffj

Refjnement 𝑆00

Choose a trial functjon Ψ(0)(𝑦) and write 𝑤(0) = Ψ(0)(𝑦)𝑎(0) sin 𝜕𝑢 𝑊

max = 1

2𝑎(0)2 𝐹𝐾Ψ(0)′′2 d𝑦 𝑈

max = 1

2𝜕2𝑎(0)2 ̄ 𝑛Ψ(0)2 d𝑦

• ur fjrst estjmate 𝑆00 of 𝜕2 is

𝜕2 = ∫ 𝐹𝐾Ψ(0)′′2 d𝑦 ∫ ̄ 𝑛Ψ(0)2 d𝑦 .

Generalized SDOF Giacomo Boffj

Refjnement 𝑆01

We try to give a betuer estjmate of 𝑊

max computjng the external work done by the

inertjal forces, 𝑞(0) = 𝜕2 ̄ 𝑛(𝑦)𝑤(0) = 𝑎(0)𝜕2Ψ(0)(𝑦) the defmectjons due to 𝑞(0) are 𝑤(1) = 𝜕2 𝑤(1) 𝜕2 = 𝜕2Ψ(1) 𝑎(1) 𝜕2 = 𝜕2Ψ(1) ̄ 𝑎(1), where we write ̄ 𝑎(1) because we need to keep the unknown 𝜕2 in evidence. The maximum strain energy is 𝑊

max = 1

2 𝑞(0)𝑤(1) d𝑦 = 1 2𝜕4𝑎(0) ̄ 𝑎(1) ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦 Equatjng to our previus estjmate of 𝑈

max we fjnd the 𝑆01 estjmate

𝜕2 = 𝑎(0) ̄ 𝑎(1) ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(0) d𝑦 ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦

Generalized SDOF Giacomo Boffj

Refjnement 𝑆11

With litule additjonal efgort it is possible to compute 𝑈

max from 𝑤(1):

𝑈

max = 1

2𝜕2 ̄ 𝑛(𝑦)𝑤(1)2 d𝑦 = 1 2𝜕6 ̄ 𝑎(1)2 ̄ 𝑛(𝑦)Ψ(1)2 d𝑦 equatjng to our last approximatjon for 𝑊

max we have the 𝑆11 approximatjon to the

frequency of vibratjon, 𝜕2 = 𝑎(0) ̄ 𝑎(1) ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦 ∫ ̄ 𝑛(𝑦)Ψ(1)Ψ(1) d𝑦. Of course the procedure can be extended to compute betuer and betuer estjmates of 𝜕2 but usually the refjnements are not extended beyond 𝑆11, because it would be contradictory with the quick estjmate nature of the Rayleigh’s Quotjent method and also because 𝑆11 estjmates are usually very good ones. Nevertheless, we recognize the possibility of itereatjvely computjng betuer and betuer estjmates opens a world of new opportunitjes.

Generalized SDOF Giacomo Boffj

Refjnement 𝑆11

With litule additjonal efgort it is possible to compute 𝑈

max from 𝑤(1):

𝑈

max = 1

2𝜕2 ̄ 𝑛(𝑦)𝑤(1)2 d𝑦 = 1 2𝜕6 ̄ 𝑎(1)2 ̄ 𝑛(𝑦)Ψ(1)2 d𝑦 equatjng to our last approximatjon for 𝑊

max we have the 𝑆11 approximatjon to the

frequency of vibratjon, 𝜕2 = 𝑎(0) ̄ 𝑎(1) ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦 ∫ ̄ 𝑛(𝑦)Ψ(1)Ψ(1) d𝑦. Of course the procedure can be extended to compute betuer and betuer estjmates of 𝜕2 but usually the refjnements are not extended beyond 𝑆11, because it would be contradictory with the quick estjmate nature of the Rayleigh’s Quotjent method and also because 𝑆11 estjmates are usually very good ones. Nevertheless, we recognize the possibility of itereatjvely computjng betuer and betuer estjmates opens a world of new opportunitjes.

Generalized SDOF Giacomo Boffj

Refjnement 𝑆11

With litule additjonal efgort it is possible to compute 𝑈

max from 𝑤(1):

𝑈

max = 1

2𝜕2 ̄ 𝑛(𝑦)𝑤(1)2 d𝑦 = 1 2𝜕6 ̄ 𝑎(1)2 ̄ 𝑛(𝑦)Ψ(1)2 d𝑦 equatjng to our last approximatjon for 𝑊

max we have the 𝑆11 approximatjon to the

frequency of vibratjon, 𝜕2 = 𝑎(0) ̄ 𝑎(1) ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦 ∫ ̄ 𝑛(𝑦)Ψ(1)Ψ(1) d𝑦. Of course the procedure can be extended to compute betuer and betuer estjmates of 𝜕2 but usually the refjnements are not extended beyond 𝑆11, because it would be contradictory with the quick estjmate nature of the Rayleigh’s Quotjent method and also because 𝑆11 estjmates are usually very good ones. Nevertheless, we recognize the possibility of itereatjvely computjng betuer and betuer estjmates opens a world of new opportunitjes.

Generalized SDOF Giacomo Boffj

Refjnement 𝑆11

With litule additjonal efgort it is possible to compute 𝑈

max from 𝑤(1):

𝑈

max = 1

2𝜕2 ̄ 𝑛(𝑦)𝑤(1)2 d𝑦 = 1 2𝜕6 ̄ 𝑎(1)2 ̄ 𝑛(𝑦)Ψ(1)2 d𝑦 equatjng to our last approximatjon for 𝑊

max we have the 𝑆11 approximatjon to the

frequency of vibratjon, 𝜕2 = 𝑎(0) ̄ 𝑎(1) ∫ ̄ 𝑛(𝑦)Ψ(0)Ψ(1) d𝑦 ∫ ̄ 𝑛(𝑦)Ψ(1)Ψ(1) d𝑦. Of course the procedure can be extended to compute betuer and betuer estjmates of 𝜕2 but usually the refjnements are not extended beyond 𝑆11, because it would be contradictory with the quick estjmate nature of the Rayleigh’s Quotjent method and also because 𝑆11 estjmates are usually very good ones. Nevertheless, we recognize the possibility of itereatjvely computjng betuer and betuer estjmates opens a world of new opportunitjes.

Generalized SDOF Giacomo Boffj

Refjnement Example

𝑛 1.5𝑛 2𝑛 𝑙 2𝑙 3𝑙 Ψ(0) 1 11/15 6/15 1 1 1 1 1.5 2 Ψ(1)

𝑞(0) 𝜕2𝑛

𝑈 = 1 2𝜕2 × 4.5 × 𝑛 𝑎2 𝑊 = 1 2 × 1 × 3𝑙 𝑎2 𝜕2 = 3 9/2 𝑙 𝑛 = 2 3 𝑙 𝑛 𝑤(1) = 15 4 𝑛 𝑙 𝜕2Ψ(1) ̄ 𝑎(1) = 15 4 𝑛 𝑙 𝑊(1) = 1 2𝑛 15 4 𝑛 𝑙 𝜕4 (1 + 33/30 + 4/5) = 1 2𝑛 15 4 𝑛 𝑙 𝜕4 87 30 𝜕2 =

9 2𝑛

𝑛 87

8 𝑛 𝑙

= 12 29 𝑙 𝑛 = 0.4138 𝑙 𝑛