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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, 91, 92 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 g 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 = 2 = 0 and express yel, Ye2 as functions of mi, m2, l1, l2, k1, k2, g. YI Y2 www. mi Gravity m₂ Figure 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, 91, 92 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 g 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 = 2 = 0 and express yel, Ye2 as functions of mi, m2, l1, l2, k1, k2, g. YI Y2 www. mi Gravity m₂ Figure 2

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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Parts A and B

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, 92 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 g 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 = ÿ2 = 0 and express yel, Ye2 as functions of m1, m2, l1, l2, k1, k2, g. yı V2 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 = Ax, where y = [Ayel Aye2], [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 agree with intuition?
Transcribed Image Text: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, 92 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 g 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 = ÿ2 = 0 and express yel, Ye2 as functions of m1, m2, l1, l2, k1, k2, g. yı V2 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 = Ax, where y = [Ayel Aye2], [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 agree with intuition?
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