SDOF linear oscillator G. Boffi Response to Impulsive Loading Review SDOF linear oscillator Step-by-step Methods Response to Impulsive Loads & Step by Step Methods Examples of SbS Methods Giacomo Boffi Dipar�mento di Ingegneria Stru�urale, Politecnico di Milano March 17, 2017
SDOF linear Outline oscillator G. Boffi Response to Impulsive Loading Review Response to Impulsive Loading Step-by-step Methods Examples of SbS Methods Review of Numerical Methods Step-by-step Methods Examples of SbS Methods
SDOF linear Response to Impulsive Loadings oscillator G. Boffi Response to Impulsive Loading Response to Impulsive Loading Introduc�on Response to Half-Sine Wave Introduc�on Impulse Response for Rectangular and Response to Half-Sine Wave Impulse Triangular Impulses Shock or response spectra Response for Rectangular and Triangular Impulses Approximate Analysis of Response Peak Shock or response spectra Review Approximate Analysis of Response Peak Step-by-step Methods Examples of SbS Methods Review of Numerical Methods Step-by-step Methods Examples of SbS Methods
Impulsive or shock loads are of great importance for the design of certain classes of structural systems, e.g., vehicles or cranes. Damping has much less importance in controlling the maximum response to impulsive loadings because the maximum response is reached in a very short �me, before the damping forces can dissipate a significant por�on of the energy input into the system. For this reason, in the following we’ll consider only the undamped response to impulsive loads. SDOF linear Nature of Impulsive Loadings oscillator G. Boffi Response to p(t) An impulsive load is characterized Impulsive Loading Introduc�on ▶ by a single principal impulse, and Response to Half-Sine Wave Impulse Response for Rectangular and ▶ by a rela�vely short dura�on. Triangular Impulses Shock or response spectra t Approximate Analysis of Response Peak Review Step-by-step Methods Examples of SbS Methods
SDOF linear Nature of Impulsive Loadings oscillator G. Boffi Response to p(t) An impulsive load is characterized Impulsive Loading Introduc�on ▶ by a single principal impulse, and Response to Half-Sine Wave Impulse Response for Rectangular and ▶ by a rela�vely short dura�on. Triangular Impulses Shock or response spectra t Approximate Analysis of Response Peak ▶ Impulsive or shock loads are of great importance for the design of Review certain classes of structural systems, e.g., vehicles or cranes. Step-by-step Methods ▶ Damping has much less importance in controlling the maximum Examples of SbS response to impulsive loadings because the maximum response is Methods reached in a very short �me, before the damping forces can dissipate a significant por�on of the energy input into the system. ▶ For this reason, in the following we’ll consider only the undamped response to impulsive loads.
The peak response is the maximum of the absolute value of the response ra�o, R max max R t . If t 0 T n necessarily R max happens a�er the end of the loading, and its value can be determined studying the free vibra�ons of the dynamic system. On the other hand, if the excita�on lasts enough to have at least a local extreme (maximum or minimum) during the excita�on we have to consider the more difficult problem of completely determining the response during the applica�on of the impulsive loading. SDOF linear Defini�on of Peak Response oscillator G. Boffi When dealing with the response to an impulsive loading of dura�on t 0 Response to Impulsive Loading of a SDOF system, with natural period of vibra�on T n we are mostly Introduc�on interested in the peak response of the system. Response to Half-Sine Wave Impulse Response for Rectangular and Triangular Impulses Shock or response spectra Approximate Analysis of Response Peak Review Step-by-step Methods Examples of SbS Methods
If t 0 T n necessarily R max happens a�er the end of the loading, and its value can be determined studying the free vibra�ons of the dynamic system. On the other hand, if the excita�on lasts enough to have at least a local extreme (maximum or minimum) during the excita�on we have to consider the more difficult problem of completely determining the response during the applica�on of the impulsive loading. SDOF linear Defini�on of Peak Response oscillator G. Boffi When dealing with the response to an impulsive loading of dura�on t 0 Response to Impulsive Loading of a SDOF system, with natural period of vibra�on T n we are mostly Introduc�on interested in the peak response of the system. Response to Half-Sine Wave Impulse Response for Rectangular and The peak response is the maximum of the absolute value of Triangular Impulses Shock or response spectra the response ra�o, R max = max {| R ( t ) |} . Approximate Analysis of Response Peak Review Step-by-step Methods Examples of SbS Methods
SDOF linear Defini�on of Peak Response oscillator G. Boffi When dealing with the response to an impulsive loading of dura�on t 0 Response to Impulsive Loading of a SDOF system, with natural period of vibra�on T n we are mostly Introduc�on interested in the peak response of the system. Response to Half-Sine Wave Impulse Response for Rectangular and The peak response is the maximum of the absolute value of Triangular Impulses Shock or response spectra the response ra�o, R max = max {| R ( t ) |} . Approximate Analysis of Response Peak Review Step-by-step ▶ If t 0 ≪ T n necessarily R max happens a�er the end of the loading, Methods Examples of SbS and its value can be determined studying the free vibra�ons of Methods the dynamic system. ▶ On the other hand, if the excita�on lasts enough to have at least a local extreme (maximum or minimum) during the excita�on we have to consider the more difficult problem of completely determining the response during the applica�on of the impulsive loading.
2 where 2 t 0 is the frequency associated with the load. Note that t 0 SDOF linear Half-sine Wave Impulse oscillator G. Boffi The sine-wave impulse has expression Response to Impulsive Loading { Introduc�on p 0 sin π t t 0 = p 0 sin ω t for 0 < t < t 0 , Response to Half-Sine Wave Impulse p ( t ) = Response for Rectangular and 0 otherwise. Triangular Impulses Shock or response spectra Approximate Analysis of Response Peak Review Step-by-step Methods Examples of SbS Methods
2 where 2 t 0 is the frequency associated with the load. Note that t 0 SDOF linear Half-sine Wave Impulse oscillator G. Boffi The sine-wave impulse has expression Response to Impulsive Loading { Introduc�on p 0 sin π t t 0 = p 0 sin ω t for 0 < t < t 0 , Response to Half-Sine Wave Impulse p ( t ) = Response for Rectangular and 0 otherwise. Triangular Impulses Shock or response spectra Approximate Analysis of Response Peak Review Step-by-step p 0 Methods Examples of SbS Methods p(t) 0.5 p 0 0 0.0 0.5 t 0 t 0 time
SDOF linear Half-sine Wave Impulse oscillator G. Boffi The sine-wave impulse has expression Response to Impulsive Loading { Introduc�on p 0 sin π t t 0 = p 0 sin ω t for 0 < t < t 0 , Response to Half-Sine Wave Impulse p ( t ) = Response for Rectangular and 0 otherwise. Triangular Impulses Shock or response spectra Approximate Analysis of Response Peak Review Step-by-step p 0 Methods Examples of SbS Methods where ω = 2 π 2 t 0 is the frequency p(t) 0.5 p 0 associated with the load. Note that ω t 0 = π. 0 0.0 0.5 t 0 t 0 time
SDOF linear Response to sine-wave impulse oscillator G. Boffi Response to Impulsive Loading Consider an undamped SDOF ini�ally at rest, with natural period T n , Introduc�on excited by a half-sine impulse of dura�on t 0 . Response to Half-Sine Wave Impulse The frequency ra�o is β = T n / 2 t 0 and the response ra�o in the interval Response for Rectangular and Triangular Impulses Shock or response spectra 0 < t < t 0 is Approximate Analysis of Response Peak 1 − β 2 ( sin ω t − β sin ω t 1 [ NB: ω Review R ( t ) = β ) . β = ω n ] Step-by-step Methods Examples of SbS It is ( 1 − β 2 ) R ( t 0 ) = − β sin π / β and ( 1 − β 2 ) ˙ β ) , R ( t 0 ) = − ω ( 1 + cos π / Methods consequently for t o ≤ t the response ra�o is − β ( ( 1 + cos π β ) sin ω n ( t − t 0 ) + sin π ) R ( t ) = β cos ω n ( t − t 0 ) 1 − β 2
The next slide regards the characteris�cs of these roots. SDOF linear Maximum response to sine impulse oscillator G. Boffi We have an extreme, and a possible peak value, for 0 ≤ t ≤ t 0 if Response to Impulsive Loading 1 − β 2 ( cos ω t − cos ω t ω ˙ R ( t ) = β ) = 0 . Introduc�on Response to Half-Sine Wave Impulse Response for Rectangular and Triangular Impulses That implies that cos ω t = cos ω t / β = cos − ω t / β , whose roots are Shock or response spectra Approximate Analysis of Response Peak ω t = ∓ ω t /β + 2 n π, n = 0 , ∓ 1 , ∓ 2 , ∓ 3 , . . . . Review Step-by-step It is convenient to subs�tute ω t = πα , where α = t / t 0 : Methods Examples of SbS ( ) ∓ a Methods π a = π β + 2 n n = 0 , ∓ 1 , ∓ 2 , . . . , 0 ≤ a ≤ 1 . , Eventually solving for α one has α = 2 n β β ∓ 1 , n = 0 , ∓ 1 , ∓ 2 , . . . , 0 < α < 1 .
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