2. Consider a mechanical system consisting of two metal plates and two springs, connected vertically as shown in Figure 2. Let m₁, m2 repre- sent the masses of the metal plates, y₁, y2 represent their positions, and l1.l2 represent the natural lengths of the springs. Suppose the spring forces are nonlinear functions of the actual lengths of the springs, given by F1 (31) = k₁(y₁ - ₁)³ and F2(y2 - y₁) = k2(y2 - Y1 - 2)³, respectively. Let 9 denote the gravitational constant. (a) Using Newton's second law, derive the equations of motion gov- erning the system (i.e., differential equations in ÿ₁ and 2). (b) Let yel. Ye2 represent the equilibrium point of the system, i.e., the positions y₁, y2 when the system is at rest. Determine Yel, Ye2 by setting ₁ = 0 and express Yel, Ye2 as functions of m1, m2, l1, l2, k1, k2, g. = Y2 ww mi k₂ Gravity m₂ Figure 2 (c) Let m₁ = m₂ = l₁ = l₂ = k₁ = k2 = 1 and g = 9.81. Determine how small changes in m₁, m2 affect the equilibrium point ye1, Ye2 via linearization. Write your result in the form of y = Ar, where y = [Aye1 Aye2] ¹, [Am₁ Am₂], and A denotes small change. , X = (d) If you want to slightly increase ye2 without affecting yel, how would you change m₁, m₂? Answer this question based on your analysis from (c) and based on your physical intuition. Does analysis
2. Consider a mechanical system consisting of two metal plates and two springs, connected vertically as shown in Figure 2. Let m₁, m2 repre- sent the masses of the metal plates, y₁, y2 represent their positions, and l1.l2 represent the natural lengths of the springs. Suppose the spring forces are nonlinear functions of the actual lengths of the springs, given by F1 (31) = k₁(y₁ - ₁)³ and F2(y2 - y₁) = k2(y2 - Y1 - 2)³, respectively. Let 9 denote the gravitational constant. (a) Using Newton's second law, derive the equations of motion gov- erning the system (i.e., differential equations in ÿ₁ and 2). (b) Let yel. Ye2 represent the equilibrium point of the system, i.e., the positions y₁, y2 when the system is at rest. Determine Yel, Ye2 by setting ₁ = 0 and express Yel, Ye2 as functions of m1, m2, l1, l2, k1, k2, g. = Y2 ww mi k₂ Gravity m₂ Figure 2 (c) Let m₁ = m₂ = l₁ = l₂ = k₁ = k2 = 1 and g = 9.81. Determine how small changes in m₁, m2 affect the equilibrium point ye1, Ye2 via linearization. Write your result in the form of y = Ar, where y = [Aye1 Aye2] ¹, [Am₁ Am₂], and A denotes small change. , X = (d) If you want to slightly increase ye2 without affecting yel, how would you change m₁, m₂? Answer this question based on your analysis from (c) and based on your physical intuition. Does analysis
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Parts A and B please. Thanks
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