台灣師範大學機電科技學系 C. R. Yang, NTNU MT -1- Chapter 14 Random Vibration 14

台灣師範大學機電科技學系 C. R. Yang, NTNU MT -2- Chapter Outline 14.1IntroductionIntroduction 14.2Random Variables and Random ProcessesRandom Variables and Random Processes 14.3Probability DistributionProbability Distribution 14.4Mean Value and Standard DeviationMean Value and Standard Deviation 14.5Joint Probability Distribution of Several Random VariablesJoint Probability Distribution of Several Random Variables 14.6Correlation Functions of a Random ProcessCorrelation Functions of a Random Process 14.7Stationary Random ProcessStationary Random Process 14.8Gaussian Random ProcessGaussian Random Process 14.9Fourier AnalysisFourier Analysis 14.10Power Spectral DensityPower Spectral Density 14.11Wide-Band and Narrow-Band ProcessesWide-Band and Narrow-Band Processes 14.12Response of a Single DOF systemResponse of a Single DOF system 14.13Response Due to Stationary Random ExcitationsResponse Due to Stationary Random Excitations 14.14Response of a Multi-DOF SystemResponse of a Multi-DOF System

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Introduction 14.1

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Introduction Random processes has parameters that cannot be precisely predicted. E.g. pressure fluctuation on the surface of a flying aircraft

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Random Variables and Random Processes 14.2

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Random Variables and Random Processes Any quantity whose magnitude cannot be precisely predicted is known as a random variable (R.V) Experiments conducted to find the value of the random variable will give an outcome that is not a function of any parameter If n experiments are conducted, the n outcomes form the sample space of the random variable. Random processes produces outcomes that is a function of some parameters. If n experiments are conducted, the n sample functions form the ensemble of the random variable.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Probability Distribution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Probability Distribution Consider a random variable x.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Probability Distribution Consider a random time function as shown:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Probability Distribution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Mean Value and Standard Deviation 14.4

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Mean Value and Standard Deviation Expected value of f(x) =μ f The positive square root of σ(x) is the standard deviation of x.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Mean Value and Standard Deviation Example 14.1 Probabilistic Characteristics of Eccentricity of a Rotor The eccentricity of a rotor (x), due to manufacturing errors, is found to have the following distribution where k is a constant. Find the mean, standard deviation and the mean square value of the eccentricity and the probability of realizing x less than or equal to 2mm.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Mean Value and Standard Deviation Example 14.1 Probabilistic Characteristics of Eccentricity of a Rotor Solution Normalize the probability density function: Mean value of x: Standard deviation of x:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Mean Value and Standard Deviation Example 14.1 Probabilistic Characteristics of Eccentricity of a Rotor Solution The mean square value of x is

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Joint Probability Distribution of Several RV 14.5

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Joint Probability Distribution of Several RV Joint behavior of 2 or more RV is determined by joint probability distribution function Joint pdf of single RV is called univariate distributions Joint pdf of 2 RVs is called bivariate distributions Joint pdf of more than one RV is called multivariate distributions Bivariate density function of RV x 1 and x 2 :

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Joint Probability Distribution of Several RV Joint pdf of x 1 and x 2 : Marginal density functions:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Joint Probability Distribution of Several RV Variances of x and y:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Joint Probability Distribution of Several RV Correlation coefficient between x and y:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Correlation Functions of a Random Process 14.6

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Correlation Functions of a Random Process Form products of RV x 1, x 2, … Average the products over the set of all possibilities to obtain a sequence of functions: These functions are called correlation functions

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Correlation Functions of a Random Process E[x 1 x 2 ] is also known as the autocorrelation function, designated as R(t 1,t 2 ) Experimentally we can find R(t 1,t 2 ) by multiplying x (i) (t 1 ) and x (i) (t 2 ) and averaging over the ensemble:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Correlation Functions of a Random Process

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process 14.7

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process Probability distribution remain invariant under shift of time scale Pdf p(x 1 ) becomes universal density function p(x) independent of time Joint density function p(x 1,x 2 ) becomes p(t,t+τ) Expected value of stationary random processes Autocorrelation function depend only on the separation time τ where τ=t 2 -t 1

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process R(0)=E[x 2 ] If the process has zero mean and is extremely irregular as shown, R(τ) will be small.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process If x(t)≈x(t+τ), R(τ) will be constant.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process If x(t) is stationary, its mean and standard deviations will be independent of t:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process R(τ) is an even function of τ. When τ  ∞, ρ  0, R(τ  ∞)  μ 2 A typical autocorrelation function is shown:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Stationary Random Process Ergodic Process We can obtain all the probability info from a single sample function and assume it applies to the entire ensemble. x (i) (t) represents the temporal average of x(t)

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Gaussian Random Process 14.8

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Gaussian Random Process Most commonly used distribution for modeling physical random processes The forms of its probability distribution are invariant wrt linear operations Standard normal variable:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Gaussian Random Process The graph of a Gaussian probability density function is as shown:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Gaussian Random Process Some typical values are shown below:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis 14.9

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Series Any periodic function x(t) of period τ can be express as a complex Fourier series Multiply both side with e -imω 0 t and integrating:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Series x(t) can be expressed as a sum of infinite number of harmonics Difference between any 2 consecutive frequencies: If x(t) is real, the integrand of c n is the complex conjugate of that of c -n

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Series Mean square value of x(t):

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Find the complex Fourier series expansion of the functions shown below:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Solution The equation can be reduced to

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Solution Note that

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.2 Complex Fourier Series Expansion Solution Frequency spectrum is as shown:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Integral A non periodic function as shown can be treated as a periodic function with τ  ∞

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Integral As τ  ∞,

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Integral Integral Fourier Transform pair Mean square value of x(t):

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Fourier Integral This is known as Parseval ’ s formula for nonperiodic functions.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.3 Fourier Transform of a Triangular Pulse Find the Fourier transform of the triangular pulse shown below.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.3 Fourier Transform of a Triangular Pulse Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.3 Fourier Transform of a Triangular Pulse Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Fourier Analysis Example 14.3 Fourier Transform of a Triangular Pulse Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Power Spectral Density 14.10

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Power Spectral Density Power spectral density S(ω) is the Fourier transform of R(τ)/2π If the mean is zero, R(0) is the average energy

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Power Spectral Density S(-ω)=S(ω) Only positive frequencies are counted in an equivalent one-sided spectrum W x (f)

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Power Spectral Density

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes 14.11

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes Wide-band random process: E.g. pressure fluctuations on surface of rocket Narrow-band random process: A process whose power spectral density is constant over a frequency range is called white noise.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes Ideal white noise – band of frequencies is infinitely wide Band-limited white noise – band of frequencies has finite cut off frequencies. Mean square value is the total area under the spectrum: 2S 0 (ω 2 – ω 1 )

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes Example 14.4 Autocorrelation and Mean Square Value of a Stationary Process The power spectral density of a stationary random process x(t) is shown below. Find its autocorrelation function and the mean square value.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes Example 14.4 Autocorrelation and Mean Square Value of a Stationary Process Solution We have

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Wide-Band and Narrow-Band Processes Example 14.4 Autocorrelation and Mean Square Value of a Stationary Process Solution Mean Square Value

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System 14.12

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System Impulse Response Approach Let the forcing function be a series of impulses of varying magnitude as shown:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System Impulse Response Approach y(t)=h(t-τ) is the impulse response function Total response can be found by superposing the responses. Response to total excitation:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System Frequency Response Approach Transient function:, H(ω) is the complex frequency response function. Total response of the system:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Single DOF System Frequency Response Approach Since h(t- )=0 when t t, Change the variable from to θ=t-, Both the superposition integral and the Fourier integral can be used to find system response

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations 14.13

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations When excitation is a stationary random process, the response is also a stationary random process

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Impulse Response Approach Autocorrelation

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Frequency Response Approach Power Spectral Density

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Frequency Response Approach

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Frequency Response Approach H(-ω) is the complex conjugate of H(ω). Mean Square Response:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.5 Mean Square Value of Response A single DOF system is subjected to a force whose spectral density is a white noise S x (ω)=S 0. Find the following: a) Complex frequency response function of the system b) Power spectral density of the response c) Mean square value of the response

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.5 Mean Square Value of Response Solution a)Substitute input as e iωt and corresponding response as y(t)=H(ω)e iωt

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.5 Mean Square Value of Response Solution b) We have c) Mean square value

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building A single-storey building is modeled by 4 identical columns of Young’s modulus E and height h and a rigid floor of weight W. The columns act as cantilevers fixed at the ground. The damping in the structure can be approximated by a constant spectrum S 0. If each column has a tubular cross section with mean diameter d and wall thickness t=d/10, find the mean diameter of the columns such that the standard deviation of the displacement of the floor relative to the ground does not exceed a specified value δ.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building Solution Model the building as a single DOF system.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building Solution When the base moves, equation of motion:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response Due to Stationary Random Excitations Example 14.6 Design of the Columns of a Building Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System 14.14

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Equation of motion: Physical and generalized coordinates are related as:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Physical and generalized forces are related as:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Mean square value of physical displacement:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System For stationary random process,

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Mean square value of x i (t)

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System For lightly damped systems,

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake A 3-storey building is subjected to an earthquake. The ground acceleration during the earthquake can be assumed to be a stationary random process with a power spectral density S(y)=0.05(m 2 /s 4 )/(rad/s). Assuming a modal damping ratio of 0.02 in each mode, determine the mean square values of the responses of the various floors of the building frame under the earthquake.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution Compute eigenvalues and eigenvectors using k=10 6 N/m and m=1000kg

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution Relative displacements of the floors: z i (t)=x i (t)-y(t), i=1,2,3 Equation of motion: where [Z] denotes the modal matrix.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution Assume damping ratio ζ i = 0.02 Uncoupled equations of motion:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution Mean square values

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.7 Response of a Building Frame Under an Earthquake Solution Mean square values of relative displacements of various floors of the building frame:

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.8 Probability of Relative Displacement Exceeding a Specified Value Find the probability of the magnitude of the relative displacement of the various floors exceeding 1,2,3, and 4 standard deviations of the corresponding relative displacement for the building frame of Example 14.7

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.8 Probability of Relative Displacement Exceeding a Specified Value Solution Assume ground acceleration to be normally distributed random process with zero mean. Relative displacements of various floors can also be assumed to be normally distributed.

台灣師範大學機電科技學系 C. R. Yang, NTNU MT Response of a Multi-DOF System Example 14.8 Probability of Relative Displacement Exceeding a Specified Value Solution