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SDOF linear oscillator Giacomo Boffi Response to Periodic Loading Fourier Transform SDOF linear oscillator The Discrete Fourier Transform Response to Periodic and Non-periodic Loadings Response to General Dynamic Loadings Giacomo Boffi Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano March 25, 2014

Outline SDOF linear oscillator Giacomo Boffi Response to Periodic Loading Fourier Transform The Discrete Response to Periodic Loading Fourier Transform Response to General Dynamic Fourier Transform Loadings The Discrete Fourier Transform Response to General Dynamic Loadings

Response to Periodic Loading SDOF linear oscillator Giacomo Boffi Response to Periodic Loading Response to Periodic Loading Introduction Fourier Series Introduction Representation Fourier Series of the Response Fourier Series Representation An example An example Fourier Series of the Response Fourier Transform An example The Discrete An example Fourier Transform Response to General Dynamic Loadings Fourier Transform The Discrete Fourier Transform Response to General Dynamic Loadings

Introduction SDOF linear oscillator Giacomo Boffi A periodic loading is characterized by the identity Response to Periodic Loading p ( t ) = p ( t + T ) Introduction Fourier Series Representation Fourier Series of the where T is the period of the loading, and ω 1 = 2 π T is its Response An example principal frequency . An example Fourier Transform The Discrete Fourier Transform p Response to General Dynamic Loadings p ( t ) p ( t + T ) t T

Introduction SDOF linear oscillator Giacomo Boffi A periodic loading is characterized by the identity Response to Periodic Loading p ( t ) = p ( t + T ) Introduction Fourier Series Representation Fourier Series of the where T is the period of the loading, and ω 1 = 2 π T is its Response An example principal frequency . An example Fourier Transform The Discrete Fourier Transform p Response to General Dynamic Loadings p ( t ) p ( t + T ) t T Note that a function with period T / n is also periodic with period T .

Introduction SDOF linear oscillator Giacomo Boffi Periodic loadings can be expressed as an infinite series of Response to harmonic functions using the Fourier theorem, e.g., for an Periodic Loading Introduction antisymmetric loading you can write Fourier Series Representation Fourier Series of the p ( t ) = − p ( − t ) = � ∞ j = 1 p j sin j ω 1 t = � ∞ Response j = 1 p j sin ω j t . An example An example The steady-state response of a SDOF system for a Fourier Transform The Discrete harmonic loading ∆ p j ( t ) = p j sin ω j t is known; with Fourier Transform β j = ω j /ω n it is: Response to General Dynamic x j , s-s = p j Loadings k D ( β j , ζ ) sin ( ω j t − θ ( β j , ζ )) .

Introduction SDOF linear oscillator Giacomo Boffi Periodic loadings can be expressed as an infinite series of Response to harmonic functions using the Fourier theorem, e.g., for an Periodic Loading Introduction antisymmetric loading you can write Fourier Series Representation Fourier Series of the p ( t ) = − p ( − t ) = � ∞ j = 1 p j sin j ω 1 t = � ∞ Response j = 1 p j sin ω j t . An example An example The steady-state response of a SDOF system for a Fourier Transform The Discrete harmonic loading ∆ p j ( t ) = p j sin ω j t is known; with Fourier Transform β j = ω j /ω n it is: Response to General Dynamic x j , s-s = p j Loadings 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 ) .

Introduction SDOF linear oscillator Giacomo Boffi Periodic loadings can be expressed as an infinite series of Response to harmonic functions using the Fourier theorem, e.g., for an Periodic Loading Introduction antisymmetric loading you can write Fourier Series Representation Fourier Series of the p ( t ) = − p ( − t ) = � ∞ j = 1 p j sin j ω 1 t = � ∞ Response j = 1 p j sin ω j t . An example An example The steady-state response of a SDOF system for a Fourier Transform The Discrete harmonic loading ∆ p j ( t ) = p j sin ω j t is known; with Fourier Transform β j = ω j /ω n it is: Response to General Dynamic x j , s-s = p j Loadings 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.

Fourier Series SDOF linear oscillator Giacomo Boffi Using Fourier theorem any practical periodic loading can be Response to Periodic Loading expressed as a series of harmonic loading terms. Introduction Fourier Series Representation Fourier Series of the Response An example An example Fourier Transform The Discrete Fourier Transform Response to General Dynamic Loadings

Fourier Series SDOF linear oscillator Giacomo 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 ∞ ∞ ω j = j ω 1 = j 2 π An example � � p ( t ) = a 0 + a j cos ω j t + b j sin ω j t , , An example T p Fourier Transform j = 1 j = 1 The Discrete Fourier Transform Response to General Dynamic Loadings

Fourier Series SDOF linear oscillator Giacomo 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 ∞ ∞ ω j = j ω 1 = j 2 π An example � � p ( t ) = a 0 + a j cos ω j t + b j sin ω j t , , An example T p Fourier Transform j = 1 j = 1 The Discrete where the harmonic amplitude coefficients have Fourier Transform expressions: Response to General Dynamic Loadings � T p � T p a 0 = 1 a j = 2 p ( t ) d t , p ( t ) cos ω j t d t , T p T p 0 0 � T p b j = 2 p ( t ) sin ω j t d t , T p 0 as, by orthogonality, � T p � T p a j cos 2 ω j t d t = T p 2 a j , etc etc. p ( t ) cos ω j d t = o o

Fourier Coefficients SDOF linear oscillator Giacomo Boffi If p ( t ) has not an analytical representation and must be measured experimentally or computed numerically, we may Response to Periodic Loading assume that it is possible Introduction Fourier Series ( a ) to divide the period in N equal parts ∆ t = T p / N , Representation Fourier Series of the Response ( b ) measure or compute p ( t ) at a discrete set of instants An example An example t 1 , t 2 , . . . , t N , with t m = m ∆ t , Fourier Transform obtaining a discrete set of values p m , m = 1 , . . . , N (note that The Discrete Fourier Transform p 0 = p N by periodicity). Response to General Dynamic Loadings

Fourier Coefficients SDOF linear oscillator Giacomo Boffi If p ( t ) has not an analytical representation and must be measured experimentally or computed numerically, we may Response to Periodic Loading assume that it is possible Introduction Fourier Series ( a ) to divide the period in N equal parts ∆ t = T p / N , Representation Fourier Series of the Response ( b ) measure or compute p ( t ) at a discrete set of instants An example An example t 1 , t 2 , . . . , t N , with t m = m ∆ t , Fourier Transform obtaining a discrete set of values p m , m = 1 , . . . , N (note that The Discrete Fourier Transform p 0 = p N by periodicity). Response to Using the trapezoidal rule of integration, with p 0 = p N we can General Dynamic write, for example, the cosine-wave amplitude coefficients, Loadings N a j ≅ 2 ∆ t � p m cos ω j t m T p m = 1 N N = 2 p m cos ( j ω 1 m ∆ t ) = 2 p m cos jm 2 π � � . N N N m = 1 m = 1

Fourier Coefficients SDOF linear oscillator Giacomo Boffi If p ( t ) has not an analytical representation and must be measured experimentally or computed numerically, we may Response to Periodic Loading assume that it is possible Introduction Fourier Series ( a ) to divide the period in N equal parts ∆ t = T p / N , Representation Fourier Series of the Response ( b ) measure or compute p ( t ) at a discrete set of instants An example An example t 1 , t 2 , . . . , t N , with t m = m ∆ t , Fourier Transform obtaining a discrete set of values p m , m = 1 , . . . , N (note that The Discrete Fourier Transform p 0 = p N by periodicity). Response to Using the trapezoidal rule of integration, with p 0 = p N we can General Dynamic write, for example, the cosine-wave amplitude coefficients, Loadings N a j ≅ 2 ∆ t � p m cos ω j t m T p m = 1 N N = 2 p m cos ( j ω 1 m ∆ t ) = 2 p m cos jm 2 π � � . N N N m = 1 m = 1 It’s worth to note that the discrete function cos jm 2 π is periodic N with period N .