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please explain this equation the details and stats of how...

Question: please explain this equation the details and stats of how...

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Please explain this equation, the details and stats of how we arrived at this Differential Equation in the System Dynamics course

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: (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) produces the following relationship between acceleration and displacement:x. Comparing this relationship to Eq. (3.5) yields: The parameter 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 frequency wn The 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

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