SDOF linear oscillator G. Boffi Response to Periodic Loading Response to Impulsive Loading SDOF linear oscillator Response to General Dynamic Response to Periodic and Non-periodic Loadings Loadings Giacomo Boffi Dipartimento di Ingegneria Strutturale, Politecnico di Milano March ��, ���� . . . . . .
SDOF linear Outline oscillator G. Boffi Response to Periodic Loading Response to Impulsive Loading Response to Response to Periodic Loading General Dynamic Loadings Response to Impulsive Loading Response to General Dynamic Loadings . . . . . .
SDOF linear Response to Periodic Loading oscillator G. Boffi Response to Response to Periodic Loading Periodic Loading Introduction Introduction Fourier Series Representation Fourier Series Representation Fourier Series of the Response Fourier Series of the Response An example Response to An example Impulsive Loading Response to Response to Impulsive Loading General Dynamic Loadings Introduction Response to Half-Sine Wave Impulse Response for Rectangular and Triangular Impulses Shock or response spectra Approximate Analysis of Response Peak Response to General Dynamic Loadings Response to infinitesimal impulse Numerical integration of Duhamel integral . . . . . .
SDOF linear Introduction oscillator G. Boffi A periodic loading is characterized by the identity Response to Periodic Loading Introduction p ( t ) = p ( t + T ) Fourier Series Representation Fourier Series of the Response An example where T is the period of the loading, and ω � = � π T is its Response to Impulsive Loading principal frequency . Response to General Dynamic Loadings p p ( t ) p ( t + T ) t T . . . . . .
SDOF linear Introduction oscillator G. Boffi Periodic loadings can be expressed as an infinite series of Response to Periodic Loading harmonic functions using Fourier theorem, e.g., an Introduction Fourier Series Representation antisymmetric loading is Fourier Series of the Response An example p ( t ) = p (− t ) = ∑ ∞ j = � p j sin j ω � t = ∑ ∞ j = � p j sin ω j t . Response to Impulsive Loading The steady-state response of a SDOF system for a harmonic Response to General Dynamic loading ∆ p j ( t ) = p j sin ω j t is known; with β j = ω j /ω n it is: Loadings p j x j , s-s = k D ( β j , ζ ) sin ( ω j t − θ ( β j , ζ )) . In general, it is possible to sum all steady-state responses, the infinite series giving the SDOF response to p ( t ) . Due to the asymptotic behaviour of D ( β ; ζ ) ( D goes to zero for large, increasing β ) it is apparent that a good approximation to the steady-state response can be obtained using a limited number of low-frequency terms. . . . . . .
SDOF linear Introduction oscillator G. Boffi Periodic loadings can be expressed as an infinite series of Response to Periodic Loading harmonic functions using Fourier theorem, e.g., an Introduction Fourier Series Representation antisymmetric loading is Fourier Series of the Response An example p ( t ) = p (− t ) = ∑ ∞ j = � p j sin j ω � t = ∑ ∞ j = � p j sin ω j t . Response to Impulsive Loading The steady-state response of a SDOF system for a harmonic Response to General Dynamic loading ∆ p j ( t ) = p j sin ω j t is known; with β j = ω j /ω n it is: Loadings p j x j , s-s = k D ( β j , ζ ) sin ( ω j t − θ ( β j , ζ )) . In general, it is possible to sum all steady-state responses, the infinite series giving the SDOF response to p ( t ) . Due to the asymptotic behaviour of D ( β ; ζ ) ( D goes to zero for large, increasing β ) it is apparent that a good approximation to the steady-state response can be obtained using a limited number of low-frequency terms. . . . . . .
SDOF linear Introduction oscillator G. Boffi Periodic loadings can be expressed as an infinite series of Response to Periodic Loading harmonic functions using Fourier theorem, e.g., an Introduction Fourier Series Representation antisymmetric loading is Fourier Series of the Response An example p ( t ) = p (− t ) = ∑ ∞ j = � p j sin j ω � t = ∑ ∞ j = � p j sin ω j t . Response to Impulsive Loading The steady-state response of a SDOF system for a harmonic Response to General Dynamic loading ∆ p j ( t ) = p j sin ω j t is known; with β j = ω j /ω n it is: Loadings p j x j , s-s = k D ( β j , ζ ) sin ( ω j t − θ ( β j , ζ )) . In general, it is possible to sum all steady-state responses, the infinite series giving the SDOF response to p ( t ) . Due to the asymptotic behaviour of D ( β ; ζ ) ( D goes to zero for large, increasing β ) it is apparent that a good approximation to the steady-state response can be obtained using a limited number of low-frequency terms. . . . . . .
SDOF linear Fourier Series oscillator G. Boffi Using Fourier theorem any practical periodic loading can be Response to Periodic Loading expressed as a series of harmonic loading terms. Introduction Consider a loading of period T p , its Fourier series is given by Fourier Series Representation Fourier Series of the Response An example ∞ ∞ ∑ ∑ ω j = j ω � = j � π Response to p ( t ) = a � + a j cos ω j t + b j sin ω j t , , Impulsive Loading T p j = � j = � Response to General Dynamic where the harmonic amplitude coefficients have Loadings expressions: ∫ T p ∫ T p a � = � a j = � p ( t ) dt , p ( t ) cos ω j t dt , T p T p � � ∫ T p b j = � p ( t ) sin ω j t dt , T p � ∫ T p ∫ T p o a j cos � ω j t dt = T p as, by orthogonality, � a j , o p ( t ) cos ω j dt = etc etc. . . . . . .
SDOF linear Fourier Series oscillator G. Boffi Using Fourier theorem any practical periodic loading can be Response to Periodic Loading expressed as a series of harmonic loading terms. Introduction Consider a loading of period T p , its Fourier series is given by Fourier Series Representation Fourier Series of the Response An example ∞ ∞ ∑ ∑ ω j = j ω � = j � π Response to p ( t ) = a � + a j cos ω j t + b j sin ω j t , , Impulsive Loading T p j = � j = � Response to General Dynamic where the harmonic amplitude coefficients have Loadings expressions: ∫ T p ∫ T p a � = � a j = � p ( t ) dt , p ( t ) cos ω j t dt , T p T p � � ∫ T p b j = � p ( t ) sin ω j t dt , T p � ∫ T p ∫ T p o a j cos � ω j t dt = T p as, by orthogonality, � a j , o p ( t ) cos ω j dt = etc etc. . . . . . .
SDOF linear Fourier Series oscillator G. Boffi Using Fourier theorem any practical periodic loading can be Response to Periodic Loading expressed as a series of harmonic loading terms. Introduction Consider a loading of period T p , its Fourier series is given by Fourier Series Representation Fourier Series of the Response An example ∞ ∞ ∑ ∑ ω j = j ω � = j � π Response to p ( t ) = a � + a j cos ω j t + b j sin ω j t , , Impulsive Loading T p j = � j = � Response to General Dynamic where the harmonic amplitude coefficients have Loadings expressions: ∫ T p ∫ T p a � = � a j = � p ( t ) dt , p ( t ) cos ω j t dt , T p T p � � ∫ T p b j = � p ( t ) sin ω j t dt , T p � ∫ T p ∫ T p o a j cos � ω j t dt = T p as, by orthogonality, � a j , o p ( t ) cos ω j dt = etc etc. . . . . . .
SDOF linear Fourier Coefficients oscillator G. Boffi If p ( t ) has not an analytical representation and must be measured Response to experimentally or computed numerically, we may assume that it is Periodic Loading possible Introduction Fourier Series Representation ( a ) to divide the period in N equal parts ∆ t = T p / N , Fourier Series of the Response An example ( b ) measure or compute p ( t ) at a discrete set of instants Response to t � , t � , . . . , t N , with t m = m ∆ t , Impulsive Loading obtaining a discrete set of values p m , m = � , . . . , N (note that Response to General Dynamic p � = p N by periodicity). Loadings Using the trapezoidal rule of integration, with p � = p N we can write, for example, the cosine-wave amplitude coefficients, N a j ≅ � ∆ t ∑ p m cos ω j t m T p m = � N N = � ∑ p m cos ( j ω � m ∆ t ) = � ∑ p m cos jm � π . N N N m = � m = � It’s worth to note that the discrete function cos jm � π is periodic N with period N . . . . . . .
SDOF linear Fourier Coefficients oscillator G. Boffi If p ( t ) has not an analytical representation and must be measured Response to experimentally or computed numerically, we may assume that it is Periodic Loading possible Introduction Fourier Series Representation ( a ) to divide the period in N equal parts ∆ t = T p / N , Fourier Series of the Response An example ( b ) measure or compute p ( t ) at a discrete set of instants Response to t � , t � , . . . , t N , with t m = m ∆ t , Impulsive Loading obtaining a discrete set of values p m , m = � , . . . , N (note that Response to General Dynamic p � = p N by periodicity). Loadings Using the trapezoidal rule of integration, with p � = p N we can write, for example, the cosine-wave amplitude coefficients, N a j ≅ � ∆ t ∑ p m cos ω j t m T p m = � N N = � ∑ p m cos ( j ω � m ∆ t ) = � ∑ p m cos jm � π . N N N m = � m = � It’s worth to note that the discrete function cos jm � π is periodic N with period N . . . . . . .
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