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Question: solution of mathematical model differential equation the solution to a...

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Solution of Mathematical Model (Differential Equation) The solution to a differential equation with no forcing (a homogeneous equation) consists only of the natural part, and the theory of ordinary differential equations shows that this solution is of the type x=X.sin(w . + φ) (3.10) where X is the amplitude, ω is the rotation frequency, and φ is the (initial) phase angle-all quantities being unknown. Eq. (3.10) between acceleration and displacement: f--ω . x. Comparing this relationship to Eq. (3.5) yields: es the following relationship The parameter ω-tin, which is based on the physical characteristics of the single- DOF mass-spring system, is named natural frequency, and plays an important role in defining the free undamped response of mechanical systems. Eqs. (3.10) and (3.11) indicate that the body motion, also known as natural or modal motion, is harmonic and has a frequency of oscillation equal to the natural frequencyThe second Eq. 3.11) indicates that, once the mathematical model was derived, the natural frequency is simply identified from the second-order differential equation. A similar approach is applied to the rotary system of Figure 3.11(b) by using e-e-sin(u, + φ), which enables finding a natural frequency: Time FIGURE 3.12 (3.12) The amplitude X and phase angle ф of Eq. (3.10) can be determined from two known initial conditions (the displacement xo and velocity vo at -0), which are Free Undamped Time Response Plot of Snge-DOF Mochanical System Bonus: (due 01/29) (3.13) Let X be the last nonzero digit of student ID Let l Let go :-- be the second last nonzero. The system of two equations with two unknowns (X and ф)--Eq. (3.13)--has the following solution: Plot x(t), X sin(wt + φ) using MATLAB (3.14) Submit the code and graph The time response of Eq. (3.10) is plotted in Figure 3.12, which highlights the sinu- soidal shape with its amplitude X, initial displacement xo (incidentally at 0 here). and the natural period T. which is connected to the natural frequency as
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