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The treatment of large structural systems may be simplified by dividing the system into

smaller systems called components. The components are related through the

displacement, and force conditions at their junction points. Each component is represented

by mode shapes (or functions).

Construction

smaller systems called components. The components are related through the

displacement, and force conditions at their junction points. Each component is represented

by mode shapes (or functions).

- 1. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 1 of 11 Component Mode Synthesis (CMS) Analysis by Julio C. Banks The treatment of large structural systems may be simplified by dividing the system into smaller systems called components. The components are related through the displacement, and force conditions at their junction points. Each component is represented by mode shapes (or functions). The sum of the component mode shape functions allows the satisfaction of the displacement and force conditions at the junctions [1]. ≔ α 1 ≔ m 1 ＝ α ― L2 L1 ≔ L1 1 ≔ L2 ⋅ α L1 Component 1: ≔ M , 1 1 ― 1 5 ≔ M , 1 2 ― 1 6 ≔ M , 2 1 M , 1 2 ≔ M , 2 2 ― 1 7 Component 2: ≔ M , 3 3 ⋅ 1.0 α ≔ M , 3 4 ⋅ ― 1 2 α ≔ M , 3 5 ⋅ ― 1 5 α ≔ M , 4 3 M , 3 4 ≔ M , 4 4 ⋅ ― 1 3 α ≔ M , 4 5 ⋅ ― 1 6 α ≔ M , 5 3 M , 3 5 ≔ M , 5 4 M , 4 5 ≔ M , 5 5 ⋅ ― 1 9 α Component 3: ≔ M , 6 6 α Julio C. Banks
- 2. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 2 of 11 ≔ M , 6 6 α = M 0.2000 0.1667 0.0000 0.0000 0.0000 0.0000 0.1667 0.1429 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.5000 0.2000 0.0000 0.0000 0.0000 0.5000 0.3333 0.1667 0.0000 0.0000 0.0000 0.2000 0.1667 0.1111 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ m L1 ≔ E 1 ≔ I 1 ≔ K , 1 1 4 ≔ K , 1 2 6 ≔ K , 2 1 K , 1 2 ≔ K , 2 2 12 ≔ K , 5 5 ⋅ 28.8 ― 1 α 3 ≔ K , 6 6 0 = K 4.00 6.00 0.00 0.00 0.00 0.00 6.00 12.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 28.80 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ―― ⋅ E I L1 3 ＝ ⋅ 1 1 0 0 0 1 0 0 1 1 1 0 2 3 0 -― 1 α -― 4 α 0 2 6 0 0 ― 12 α 2 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ p1 p2 p3 p4 P5 P6 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 Apply the nonzero diagonal criterion to select the independent (generalized) coordinates, p. ＝ ⋅ 1 0 0 0 0 1 1 1 2 0 -― 1 α -― 4 α 2 0 0 ― 12 α ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ p1 p3 p4 P5 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ - 1 1 0 0 3 0 6 0 ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ p2 p6 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Julio C. Banks
- 3. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 3 of 11 Or ＝ ⋅ S p ⋅ Q q where ＝ S 1 0 0 0 0 1 1 1 2 0 -― 1 α -― 4 α 2 0 0 ― 12 α 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and ＝ Q -1 -1 0 0 -3 0 -6 0 ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ Let ＝ p2 q1 and ＝ p6 q2 therefore, ＝ ⋅ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 2 0 0 -― 1 α -― 4 α 0 2 0 0 0 ― 12 α 2 0 0 0 0 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ p1 p2 p3 p4 P5 P6 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ -1 -1 1 0 0 0 -3 0 -6 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ q1 q2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ or ＝ ⋅ S' p ⋅ Q' q where ≔ S' 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 2 0 0 -― 1 α -― 4 α 0 2 0 0 0 ― 12 α 2 0 0 0 0 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and ≔ Q' -1 -1 1 0 0 0 -3 0 -6 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Therefore, ＝ p ⋅ T q Where ≔ T = ⋅ ( (S') )-1 Q' -1.000 -1.000 1.000 0.000 -2.000 2.500 2.333 -2.667 -0.333 0.167 0.000 1.000 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Julio C. Banks
- 4. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 4 of 11 ≔ A = ⋅ ⋅ T T M T 1.18 -1.48 -1.48 3.18 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ B = ⋅ ⋅ T T K T 7.200 -3.600 -3.600 4.800 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ C = ⋅ A -1 B 11.40 -2.80 4.19 0.20 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ λ = sort( (eigenvals( (C) )) ) 1.37 10.23 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ ω = → ― ‾‾ λ 1.172 3.198 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 Calculate the eigenvectors ≔ Nm = length( (λ) ) 2 ≔ i ‥ 1 Nm ≔ Φ ⟨ ⟨i⟩ ⟩ eigenvec⎛ ⎝ , C λ i ⎞ ⎠ The mode shapes in normal coordinates is = Φ 0.2693 -0.9225 0.9631 -0.3859 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Normalized mode shapes ≔ Φn = Vnorm ( (Φ) ) 0.280 1.000 1.000 0.418 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ First and Second Mode Shapes Reference "Theory of Vibration with Applications, 5th Ed.", Thomson, W. T., and Marie Dillon Dahleh. Prentice Hall. ISBN 0-13-651068-X, Pp. 341 through 346. Julio C. Banks MSME Thesis - "Component Synthesis Methods for Vibrating Systems". Tufts University, Medford Massachusetts, May 1984. Appendix A Julio C. Banks
- 5. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 5 of 11 Appendix A ＝ ⋅ 1 1 0 0 0 1 0 0 1 1 1 0 2 3 0 -1 -4 0 2 6 0 0 12 0 ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ p1 p2 p3 p4 P5 P6 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 Since the total number of coordinates used are six and there are four constraint equations, the number of generalized coordinates for the system is two (i.e., there are four superfluous coordinates corresponding to the four constraint equations. We can thus choose any two (the first example uses the nonzero diagonal criterion) of the generalized coordinates, q. Let p1 = q1, and p6 = q6 be the generalized coordinates, and express p1 ..p6 in terms of q1, and q6 according to the following steps: ＝ ⋅ 1 0 0 0 0 1 1 1 3 0 -― 1 α -― 4 α 6 0 0 ― 12 α 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ p1 p3 p4 P5 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ - 1 1 0 0 2 0 2 0 ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ q2 q6 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ or ＝ ⋅ S p ⋅ Q q Let ＝ p1 q1 where ＝ S 1 0 0 0 0 1 1 1 3 0 -― 1 α -― 4 α 6 0 0 ― 12 α 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and ＝ Q -1 -1 0 0 -2 0 -2 0 ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ and ＝ p6 q2 ＝ ⋅ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 3 0 0 -― 1 α -― 4 α 0 6 0 0 0 ― 12 α 2 0 0 0 0 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ p1 p2 p3 p4 P5 P6 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ 1 0 -1 -1 0 0 -2 0 -2 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ q1 q2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ or ＝ ⋅ S' p ⋅ Q' q Julio C. Banks
- 6. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 6 of 11 For = α 1.00 Where ≔ S' 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 3 0 -― 1 α -― 4 α 0 0 6 0 0 ― 12 α 2 0 0 0 0 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and ≔ Q' 1 0 -1 -1 0 0 -2 0 -2 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ＝ p ⋅ T. q Where ≔ T = ⋅ ( (S') )-1 Q' 1.00 0.00 -1.00 -1.00 2.00 4.50 -2.33 -5.00 0.33 0.50 0.00 1.00 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≔ A = ⋅ ⋅ T T M T 1.1774 2.6614 2.6614 7.3206 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ B = ⋅ ⋅ T T K T 7.200 10.800 10.800 19.200 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ C = ⋅ A -1 B 15.60 18.20 -4.19 -3.99 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ λ = sort( (eigenvals( (C) )) ) 1.37 10.23 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ ω = → ― ‾‾ λ 1.172 3.198 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 Example Julio C. Banks
- 7. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 7 of 11 Example ≡ gc ⋅ g ―― lb lbf ≔ γ ⋅ 0.280 ―― lbf in 3 ＝ γ ⋅ ρ ― g gc ⇒ ≔ ρ = ⋅ γ ― gc g 0.280 ―― lb in 3 ≔ Do ⋅ 1.315 in ≔ Di ⋅ 1.049 in ≔ L1 ⋅ 10 in ≔ E ⋅ ⋅ 30 10 6 psi Area: ≔ A = ⋅ ― π 4 ⎛ ⎝ - Do 2 Di 2 ⎞ ⎠ 0.4939 in 2 Moment of Inertia: ≔ I = ⋅ ― π 64 ⎛ ⎝ - Do 4 Di 4 ⎞ ⎠ ⎛ ⎝ ⋅ 8.734 10 -2⎞ ⎠ in 4 Thickness: ≔ t = ――― - Do Di 2 0.003 m ≔ V = ⋅ A L1 4.94 in 3 ≔ m = ⋅ ρ A 0.1383 ― lb in ≔ ω' = ⋅ ω ⎛ ⎜ ⎜ ⎝ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 ⎞ ⎟ ⎟ ⎠ 1.172 ⋮ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 ≔ f = ―― ω' ⋅ 2 π 159.6 ⋮ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Hz FEM Validation: ANSYS FEA ≔ f' ⋅ 159.2 433.0 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Hz ≔ error = → ― ―― - f' f f -0.24 -0.55 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ % 1 CAEFEM FEA ≔ f' ⋅ 165 470 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Hz ≔ error = → ― ―― - f' f f 3.4 8.0 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ % 1 Results Commentary ANSYS model uses 2D Beam Elements, while CAEFEM model utilized 3D Beam Elements (That is, ANSYS will most closely follow the closed-form solution since the latter is 2D. On the other hand, CAEFEM model had to be restrained in the out-of-plane dimension in order to emulate a 2D plane frame (CAEFEM has 3D beam elements only). Julio C. Banks
- 8. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 8 of 11 Results Commentary ANSYS model uses 2D Beam Elements, while CAEFEM model utilized 3D Beam Elements (That is, ANSYS will most closely follow the closed-form solution since the latter is 2D. On the other hand, CAEFEM model had to be restrained in the out-of-plane dimension in order to emulate a 2D plane frame (CAEFEM has 3D beam elements only). In general, the natural frequencies can be expressed as a function of . The choices ＝ α ― L2 L1 of dependent, and independent coordinates follows those chosen in the reference. The results are identical. The mass matrix: The Stiffness matrix: ≔ M( (α) ) ― 1 5 ― 1 6 0 0 0 0 ― 1 6 ― 1 7 0 0 0 0 0 0 α ⋅ ― 1 2 α ⋅ ― 1 5 α 0 0 0 ⋅ ― 1 2 α ⋅ ― 1 3 α ⋅ ― 1 6 α 0 0 0 ⋅ ― 1 5 α ⋅ ― 1 6 α ⋅ ― 1 9 α 0 0 0 0 0 0 α ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≔ K( (α) ) 4 6 0 0 0 0 6 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ―― 28.8 α 3 0 0 0 0 0 0 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Recall ≔ S'( (α) ) 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 2 0 0 -― 1 α -― 4 α 0 2 0 0 0 ― 12 α 2 0 0 0 0 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and ≔ Q' -1 -1 1 0 0 0 -3 0 -6 0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ＝ p ⋅ T q Where ≔ T( (α) ) ⋅ ( (S'( (α) )) )-1 Q' = T( (α) ) -1.000 -1.000 1.000 0.000 -2.000 2.500 2.333 -2.667 -0.333 0.167 0.000 1.000 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ≔ α 1 Julio C. Banks
- 9. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 9 of 11 ≔ α 1 ≔ A( (α) ) ⋅ ⋅ T T( (α) ) M( (α) ) T( (α) ) ≔ B( (α) ) ⋅ ⋅ T T( (α) ) K( (α) ) T( (α) ) = A( (α) ) 1.1774 -1.4840 -1.4840 3.1753 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = B( (α) ) 7.2000 -3.6000 -3.6000 4.8000 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ C( (α) ) ⋅ A( (α) ) -1 B( (α) ) = C( (α) ) 11.40 -2.80 4.19 0.20 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ λ = sort( (eigenvals( (C( (α) )) )) ) 1.37 10.23 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ ω = ⋅ ⎛ ⎝ → ― ‾‾ λ ⎞ ⎠ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 1.172 ⋮ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ‾‾‾‾‾‾ ――― ⋅ E I ⋅ m L1 4 ≔ f = ―― ω ⋅ 2 π 159.6 435.4 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Hz Calculate the eigenvectors ≔ Nm = length( (λ) ) 2 ≔ i ‥ 1 Nm ≔ Φ ⟨ ⟨i⟩ ⟩ eigenvec⎛ ⎝ , C( (α) ) λ i ⎞ ⎠ The mode shapes in normal coordinates is = Φ 0.2693 -0.9225 0.9631 -0.3859 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ≔ D = ⋅ T( (α) ) Φ -1.232 1.308 0.269 -0.923 1.869 0.880 -1.940 -1.123 ⎡ ⎢ ⎢ ⎢ ⎢ ⎤ ⎥ ⎥ ⎥ ⎥ Julio C. Banks
- 10. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 10 of 11 The mode shapes in physical coordinates is ≔ D = ⋅ T( (α) ) Φ -1.232 1.308 0.269 -0.923 1.869 0.880 -1.940 -1.123 0.071 0.243 0.963 -0.386 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Normalized mode shapes ≔ Dn = Vnorm ( (D) ) 0.635 1.000 -0.139 -0.705 -0.964 0.673 1.000 -0.859 -0.036 0.186 -0.496 -0.295 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Julio C. Banks
- 11. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 11 of 11 Appendix B Define two (2) procedures to unit-normalize the columns of a matrix. The first algorithm determines the maximum magnitude in each column. The second algorithm, expands the first algorithm to the unit-normalization phase of the solution ≡ Vmax ( (v) ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ | | | | | | | | | | | | | | | | | | | | “Determine the maximum-magnitude” “element in each column of a matrix” ← Nr rows( (v) ) ← Nc cols( (v) ) for ∊ | | | | | | | | | | | | | j ‥ 1 Nc ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ← max v , 1 j for ∊ | | | | | | | i ‥ 2 Nr ‖ ‖ ‖ ‖ ‖ | | | | | if > | | v , i j | | | |max| | ‖ ‖ ‖ ← max v , i j ← z j max return z ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ≡ Vnorm ( (v) ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ | | | | | | | | | | | | | | | | | | | | | | | “Determine the maximum-magnitude” “element in each column of a matrix” ← Nr rows( (v) ) ← Nc cols( (v) ) for ∊ | | | | | | | | | | | | | | | | j ‥ 1 Nc ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ← max v , 1 j for ∊ | | | | | | | i ‥ 2 Nr ‖ ‖ ‖ ‖ ‖ | | | | | if > | | v , i j | | | |max| | ‖ ‖ ‖ ← max v , i j ← z j max ← Vn ⟨ ⟨j⟩ ⟩ ― v ⟨ ⟨j⟩ ⟩ z j return Vn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Julio C. Banks

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