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
Outline Generalized SDOF Giacomo Boffj
Generalized SDOF Giacomo Boffj Sectjon 1 Introductory Remarks
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. Introductory Remarks Generalized SDOF Giacomo Boffj 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.
Introductory Remarks Generalized SDOF Giacomo Boffj 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.
Further Remarks on Rigid Assemblages Generalized SDOF Giacomo Boffj 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)
Further Remarks on Contjnuous Systems Generalized SDOF Giacomo Boffj 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 our 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.
Final Remarks on Generalised SDOF Systems Generalized SDOF Giacomo Boffj 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
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 = ∫ d 𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦 2 + 𝑧 2 ) d 𝑛 . 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 Assemblages of Rigid Bodies Generalized SDOF Giacomo Boffj planar, or bidimensional, rigid bodies, constrained to move in a plane,
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 = ∫ d 𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦 2 + 𝑧 2 ) d 𝑛 . 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 Assemblages of Rigid Bodies Generalized SDOF Giacomo Boffj planar, or bidimensional, rigid bodies, constrained to move in a plane, the fmexibility is concentrated in discrete elements, springs and dampers,
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 = ∫ d 𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦 2 + 𝑧 2 ) d 𝑛 . inertjal forces are distributed forces, actjng on each material point of each rigid body, their resultant can be described by Assemblages of Rigid Bodies Generalized SDOF Giacomo Boffj 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),
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 = ∫ d 𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦 2 + 𝑧 2 ) d 𝑛 . Assemblages of Rigid Bodies Generalized SDOF Giacomo Boffj 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
Assemblages of Rigid Bodies Generalized SDOF Giacomo Boffj 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 = ∫ d 𝑛 an inertjal couple, the product of the angular acceleratjon and the moment of inertja 𝐾 of the rigid body, 𝐾 = ∫(𝑦 2 + 𝑧 2 ) d 𝑛 .
̄ ̄ ̄ Rigid Bar Generalized SDOF Giacomo Boffj G x L Unit mass 𝑛 = constant , 𝑀, Length Centre of Mass 𝑦 𝐻 = 𝑀/2, 𝑛 = 𝑛𝑀, Total Mass 𝐾 = 𝑛 𝑀 2 𝑛 𝑀 3 Moment of Inertja 12 = 12
Rigid Rectangle Generalized SDOF Giacomo Boffj y G b a 𝛿 = constant , Unit mass Sides 𝑏, 𝑐 Centre of Mass 𝑦 𝐻 = 𝑏/2, 𝑧 𝐻 = 𝑐/2 Total Mass 𝑛 = 𝛿𝑏𝑐, 𝐾 = 𝑛𝑏 2 + 𝑐 2 = 𝛿𝑏 3 𝑐 + 𝑏𝑐 3 Moment of Inertja 12 12
Rigid Triangle Generalized SDOF Giacomo Boffj y b G a For a right triangle. 𝛿 = constant , Unit mass Sides 𝑏, 𝑐 Centre of Mass 𝑦 𝐻 = 𝑏/3, 𝑧 𝐻 = 𝑐/3 Total Mass 𝑛 = 𝛿𝑏𝑐/2, 𝐾 = 𝑛𝑏 2 + 𝑐 2 = 𝛿𝑏 3 𝑐 + 𝑏𝑐 3 Moment of Inertja 18 36
When 𝑏 = 𝑐 = 𝐸 = 2𝑆 the oval is a circle: 𝐾 = 𝑛 𝑆 2 = 𝛿 𝜌𝑆 4 𝑛 = 𝛿 𝜌𝑆 2 , . 2 2 Rigid Oval Generalized SDOF Giacomo Boffj Unit mass 𝛿 = constant , y 𝑏, 𝑐 Axes Centre of Mass 𝑦 𝐻 = 𝑧 𝐻 = 0 b x 𝑛 = 𝛿𝜌𝑏𝑐 Total Mass 4 , 𝐾 = 𝑛𝑏 2 + 𝑐 2 a Moment of Inertja 16
Rigid Oval Generalized SDOF Giacomo Boffj Unit mass 𝛿 = constant , y 𝑏, 𝑐 Axes Centre of Mass 𝑦 𝐻 = 𝑧 𝐻 = 0 b x 𝑛 = 𝛿𝜌𝑏𝑐 Total Mass 4 , 𝐾 = 𝑛𝑏 2 + 𝑐 2 a Moment of Inertja 16 When 𝑏 = 𝑐 = 𝐸 = 2𝑆 the oval is a circle: 𝐾 = 𝑛 𝑆 2 = 𝛿 𝜌𝑆 4 𝑛 = 𝛿 𝜌𝑆 2 , . 2 2
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