Generalized SDOF’s Giacomo Boffi Introductory Remarks Assemblage of Generalized Single Degree of Freedom Rigid Bodies Continuous Systems Systems Vibration Analysis by Rayleigh’s Method Giacomo Boffi Selection of Mode Shapes Refinement of Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano Rayleigh’s Estimates April 14, 2015
Outline Generalized SDOF’s Giacomo Boffi Introductory Remarks Introductory Remarks Assemblage of Rigid Bodies Continuous Assemblage of Rigid Bodies Systems Vibration Analysis by Rayleigh’s Method Continuous Systems Selection of Mode Shapes Refinement of Vibration Analysis by Rayleigh’s Method Rayleigh’s Estimates Selection of Mode Shapes Refinement of Rayleigh’s Estimates
Introductory Remarks Generalized SDOF’s Giacomo Boffi Until now our SDOF ’s were described as composed by a Introductory Remarks single mass connected to a fixed reference by means of a Assemblage of spring and a damper. Rigid Bodies While the mass-spring is a useful representation, many Continuous Systems different, more complex systems can be studied as SDOF Vibration Analysis systems, either exactly or under some simplifying by Rayleigh’s Method assumption. Selection of Mode Shapes Refinement of Rayleigh’s Estimates
Introductory Remarks Generalized SDOF’s Giacomo Boffi Until now our SDOF ’s were described as composed by a Introductory Remarks single mass connected to a fixed reference by means of a Assemblage of spring and a damper. Rigid Bodies While the mass-spring is a useful representation, many Continuous Systems different, more complex systems can be studied as SDOF Vibration Analysis systems, either exactly or under some simplifying by Rayleigh’s Method assumption. Selection of Mode Shapes 1. SDOF rigid body assemblages, where flexibility is Refinement of concentrated in a number of springs and dampers, can Rayleigh’s Estimates 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 fixed pattern of displacements, whose amplitude (the single degree of freedom) varies with time.
Further Remarks on Rigid Assemblages Generalized SDOF’s Giacomo Boffi Introductory Remarks Today we restrict our consideration to plane, 2-D systems. Assemblage of Rigid Bodies In rigid body assemblages the limitation to a single shape of Continuous displacement is a consequence of the configuration of the Systems system, i.e., the disposition of supports and internal hinges. Vibration Analysis by Rayleigh’s When the equation of motion is written in terms of a single Method parameter and its time derivatives, the terms that figure as Selection of Mode Shapes coefficients in the equation of motion can be regarded as Refinement of the generalised properties of the assemblage: generalised Rayleigh’s Estimates mass, damping and stiffness on left hand, generalised loading on right hand. x + c ⋆ ˙ m ⋆ ¨ x + k ⋆ x = p ⋆ ( t )
Further Remarks on Continuous Systems Generalized SDOF’s Giacomo Boffi Introductory Remarks Continuous systems have an infinite variety of deformation Assemblage of Rigid Bodies patterns. Continuous By restricting the deformation to a single shape of varying Systems amplitude, we introduce an infinity of internal contstraints Vibration Analysis by Rayleigh’s that limit the infinite variety of deformation patterns, but Method under this assumption the system configuration is Selection of Mode Shapes mathematically described by a single parameter, so that Refinement of Rayleigh’s ◮ our model can be analysed in exactly the same way as Estimates a strict SDOF system, ◮ we can compute the generalised mass, damping, stiffness properties of the SDOF system.
Final Remarks on Generalised SDOF Systems Generalized SDOF’s Giacomo Boffi Introductory Remarks Assemblage of Rigid Bodies Continuous From the previous comments, it should be apparent that Systems everything we have seen regarding the behaviour and the Vibration Analysis by Rayleigh’s integration of the equation of motion of proper SDOF Method systems applies to rigid body assemblages and to SDOF Selection of Mode Shapes models of flexible systems, provided that we have the Refinement of Rayleigh’s means for determining the generalised properties of the Estimates dynamical systems under investigation.
Assemblages of Rigid Bodies Generalized SDOF’s Giacomo Boffi ◮ planar, or bidimensional, rigid bodies, constrained to Introductory move in a plane, Remarks Assemblage of Rigid Bodies Continuous Systems Vibration Analysis by Rayleigh’s Method Selection of Mode Shapes Refinement of Rayleigh’s Estimates
Assemblages of Rigid Bodies Generalized SDOF’s Giacomo Boffi ◮ planar, or bidimensional, rigid bodies, constrained to Introductory move in a plane, Remarks Assemblage of ◮ the flexibility is concentrated in discrete elements, Rigid Bodies springs and dampers, Continuous Systems Vibration Analysis by Rayleigh’s Method Selection of Mode Shapes Refinement of Rayleigh’s Estimates
Assemblages of Rigid Bodies Generalized SDOF’s Giacomo Boffi ◮ planar, or bidimensional, rigid bodies, constrained to Introductory move in a plane, Remarks Assemblage of ◮ the flexibility is concentrated in discrete elements, Rigid Bodies springs and dampers, Continuous Systems ◮ rigid bodies are connected to a fixed reference and to Vibration Analysis each other by means of springs, dampers and smooth, by Rayleigh’s Method bilateral constraints (read hinges, double pendulums Selection of Mode and rollers), Shapes Refinement of Rayleigh’s Estimates
Assemblages of Rigid Bodies Generalized SDOF’s Giacomo Boffi ◮ planar, or bidimensional, rigid bodies, constrained to Introductory move in a plane, Remarks Assemblage of ◮ the flexibility is concentrated in discrete elements, Rigid Bodies springs and dampers, Continuous Systems ◮ rigid bodies are connected to a fixed reference and to Vibration Analysis each other by means of springs, dampers and smooth, by Rayleigh’s Method bilateral constraints (read hinges, double pendulums Selection of Mode and rollers), Shapes ◮ inertial forces are distributed forces, acting on each Refinement of Rayleigh’s material point of each rigid body, their resultant can be Estimates described by
Assemblages of Rigid Bodies Generalized SDOF’s Giacomo Boffi ◮ planar, or bidimensional, rigid bodies, constrained to Introductory move in a plane, Remarks Assemblage of ◮ the flexibility is concentrated in discrete elements, Rigid Bodies springs and dampers, Continuous Systems ◮ rigid bodies are connected to a fixed reference and to Vibration Analysis each other by means of springs, dampers and smooth, by Rayleigh’s Method bilateral constraints (read hinges, double pendulums Selection of Mode and rollers), Shapes ◮ inertial forces are distributed forces, acting on each Refinement of Rayleigh’s material point of each rigid body, their resultant can be Estimates described by ◮ a force applied to the centre of mass of the body, proportional to acceleration vector and total mass � M = d m ◮ a couple, proportional to angular acceleration and the moment of inertia J of the rigid body, ( x 2 + y 2 ) d m . � J =
Rigid Bar Generalized SDOF’s Giacomo Boffi Introductory Remarks Assemblage of G Rigid Bodies Continuous x Systems L Vibration Analysis by Rayleigh’s Method Unit mass m = constant , ¯ Selection of Mode Shapes Length L, Refinement of Rayleigh’s Centre of Mass x G = L/ 2 , Estimates Total Mass m = ¯ mL, J = mL 2 mL 3 Moment of Inertia 12 = ¯ 12
Rigid Rectangle Generalized SDOF’s Giacomo Boffi Introductory Remarks y Assemblage of Rigid Bodies G b Continuous Systems Vibration Analysis by Rayleigh’s a Method Selection of Mode Shapes Unit mass γ = constant , Refinement of Rayleigh’s Sides a, b Estimates Centre of Mass x G = a/ 2 , y G = b/ 2 Total Mass m = γab, J = ma 2 + b 2 = γ a 3 b + ab 3 Moment of Inertia 12 12
Rigid Triangle Generalized SDOF’s Giacomo Boffi Introductory Remarks y Assemblage of Rigid Bodies b Continuous G Systems Vibration Analysis by Rayleigh’s a For a right triangle. Method Selection of Mode Shapes Unit mass γ = constant , Refinement of Rayleigh’s Sides a, b Estimates Centre of Mass x G = a/ 3 , y G = b/ 3 Total Mass m = γab/ 2 , J = ma 2 + b 2 = γ a 3 b + ab 3 Moment of Inertia 18 36
Rigid Oval Generalized SDOF’s Giacomo Boffi When a = b = D = 2 R the oval is a circle. Introductory y Remarks Assemblage of Rigid Bodies b Continuous Systems x Vibration Analysis by Rayleigh’s Method a Selection of Mode Shapes Refinement of Unit mass γ = constant , Rayleigh’s Estimates Axes a, b Centre of Mass x G = y G = 0 m = γ πab Total Mass 4 , J = ma 2 + b 2 Moment of Inertia 16
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