Chapter 8: Frequency Domain Analysis


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1 Chapter 8: Frequency Domain Analysis Samantha Ramirez Preview Questions 1. What is the steadystate response of a linear system excited by a cyclic or oscillatory input? 2. How does one characterize the response at steadystate when the system is exposed to a consistent oscillatory input? 3. Is the time domain still appropriate for conducting our analyses of such systems? 4. What tools are useful for examining such dynamics? 1
2 Objectives and Outcomes Objectives 1. To analyze mechanical vibration systems including transmission and modal analysis, 2. To be able to analyze basic AC circuits, and 3. To conduct frequency response analysis. Outcomes: You will be able to 1. determine the steadystate response of a linear timeinvariant system to a sinusoidal input, 2. calculate the force or motion transmitted by a vibration isolation system, 3. conduct basic modal analysis of free vibration systems, 4. conduct basic analyses of AC circuits, 5. identify the characteristics for frequency responses of first and secondorder systems, and 6. compose Bode plots that visualize the frequency response of an oscillatory system Complex Numbers z = x + jy z ҧ = x jy z = x 2 + y 2 θ = tan 1 y x 2
3 5.2.2 Euler s Theorem Refer to the textbook ( for derivation of theorem and identities) Euler s Theorem Cosine Identity Sine Identity e jθ = cos θ j sin θ cos θ = ejθ + e jθ 2 sin θ = ejθ e jθ 2j Complex Algebra z = x + jy and w = u + jv z + w = x + u + j(y + v) z w = x u + j(y v) az = ax + jay jz = y + jx = z ( ) z = y jx = z (θ 90 ) j zw = xu yv + j(xv + yu) zw = z w (θ + φ) z n = z θ n = z n (nθ) z w = z w xu + yv yu xv θ φ = u 2 + j + y2 u 2 + y 2 z 1/n = z θ 1/n = z 1/n (θ/n) 3
4 Complex Variables and Functions Transfer function Ratio of polynomials in the sdomain Zeroes Roots of the numerator Poles Roots of the denominator s = σ + jω G s = K s + z 1 s + z 2 (s + z m ) s + p 1 s + p 2 (s + p n ) 8.2 Properties of Sinusoids Sinusoids of different amplitude Period of oscillation Sinusoids of different phase angle 4
5 The Sinusoidal Transfer Function y(s) u(s) = G s = K s + z 1 s + z 2 (s + z m ) s + p 1 s + p 2 (s + p n ) Partial fraction of sinusoidal response y s = G s Aω = a + s 2 +ω 2 s+jω തa s jω + b 1 s+p 1 + b 2 s+p b n s+p n y ss t = ae jωt + തae jωt If the response is stable, these terms lead to decaying exponentials and decaying sinusoids. a = A 2j G( jω) തa = A 2j G(jω) See textbook for derivation of these coefficients y ss t = ae jωt + തae jωt = A 2j G jω e jφ e jωt + A 2j G jω ejφ e jωt We use Euler s Theorem = G jω A ej(ωt+φ) e j(ωt+φ) 2j = Ysin(ωt + φ) = G jω Asin(ωt + φ) Magnitude and Phase Angle G jω y ss t = G jω Asin(ωt + φ) = Y sin(ωt + φ) Y = G jω A and φ = G(jω) = y(jω) u(jω) = output to input amplitude ratio G jω = y(jω) u(jω) = tan 1 I G(jω) R G(jω) I y(jω) = tan 1 R y(jω) tan 1 I u(jω) R u(jω) = phase shift of output with respect to input 5
6 Example 8.1 Find the steadystate response to a sinusoidal input displacement of the form y road = A sin ωt m/s where A=0.03 m and ω=10 rad/s. The system parameters for m, b, and k are 500 kg, 8000 Ns/m, and 34,000 N/m, respectively. The transfer function is given below. y(s) bs + k = G s = y road (s) ms 2 + bs + k Complex Operations in MATLAB Function abs(x) angle(x) conj(x) imag(x) real(x) Description Returns magnitude(s) of complex element(s) in X Returns phase angle(s) of complex element(s) in X Returns complex conjugate(s) of complex element(s) in X Returns imaginary part(s) of complex element(s) in X Returns real part(s) of complex element(s) in X 6
7 Example 8.2 >> m=500; b=8000; k=34000; w=10; A=0.03; >> G=(k+j*b*w)/((km*w^2)+j*b*w) G = i >> abs(g) ans = >> angle(g) ans = >> num=k+j*b*w; >> den=(km*w^2)+j*b*w den = e e+04i >> G=(num*conj(den))/(den*conj(den)) G = i >> magnitude=abs(num)/abs(den) magnitude = >> phase=angle(num)angle(den) phase = >> magnitude=sqrt(real(g)^2+imag(g)^2) magnitude = >> phase=atan(imag(g)/real(g)) phase = Mechanical Vibration 7
8 Transmissibility In vibration isolation systems, transmissibility is the amplitude ratio of the transmitted force (displacement) to the excitation force (displacement). TR = F out(jω) F in (jω) Example 8.3 Find the transmissibility if the foundation is forced by an excitation F in t = 5sin2t N. The first mass, damping constant, spring stiffness, and second mass are 2 kg, 2 Ns/m, 5 N/m, and 1 kg, respectively. 8
9 Motion Transmissibility Example 8.1 Input y road = 0.03 sin 10t m/s Output Response y ss t = sin 10t m/s Amplitude ratio of output to input G(jω) = Motion Transmissibility TR = = y(jω) y road (jω) = y(s) bs + k = G s = y road (s) ms 2 + bs + k G jω = y(jω) y road (jω) = k + bjω m jω 2 + bjω + k Resonant Frequency Occurs when a system s natural frequency is equal to the input frequency. Recall, from the prototypical secondorder system: ω n = k m ζ = b 2 km Normalized frequency is the input sinusoidal frequency divided by the natural frequency Resonant frequency for a prototypical secondorder system ω ω n ω r = ω n 1 2ζ 2 9
10 The QuarterCar Suspension Rewrite the motion transmissibility in terms of the damping ratio and normalized frequency. TR = k 2 + bω 2 k mω bω 2 Resonance for the QuarterCar Suspension ζ = b 2 km = 0.97 ω n = k = 8.25 rad/s m 10
11 Modal Analysis of Free Vibration Set & Tensor Notation Vector Vector in set form Matrix referenced as tensor Set of secondorder, free, undamped vibration equations 11
12 Converting System of FirstOrder D.E. to System of SecondOrder D.E. Recognizing that and we can reformulate the system of firstorder differential equations as Matrix Form of the Vibration Equations 12
13 Frequencies and Modes of Vibration In free vibration we assume Plug into M x ሷ + K x = 0 The eigenvalues are the squares of the frequencies of vibration and the eigenvectors are the mode shapes. Example 8.4 Find the natural frequencies of vibration and the respective mode shapes for the two DOF structure model and an axially vibrating beam where k 1 = 2k 2 and m 1 = 2m 2. For simplicity, k 2 = k and m 2 = m. 13
14 Example 8.4 det ω 2 I M 1 K = 0 Eigenvalues Eigenvector 1 Eigenvector 2 14
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