Please show all of your work when answering this problem! I\'m having a hard tim

Question

Please show all of your work when answering this problem! I'm having a hard time understanding! Thank you in advance! 1. A mass of 1 kilogram is attached to a spring with a spring constant of 25 Newtons/meter. The spring is stretched 1=2 meter from equilibrium (in what we assume to be the positive-y direction, where y denotes the displacement from equilibrium in meters) and released from rest (so that the initial velocity of the mass is zero). The system is subject to damping with a damping force of 6 Newton-seconds/meter, and no external force is applied to the system. (Naturally, we assume that the modeling assumptions discussed in Section 6.5 hold for this system.) (a) Is this system underdamped, critically damped, or overdamped? (b) Determine y(t), which gives the displacement of the mass from equilibrium as a function of time.

Explanation / Answer

Consider a mass-spring system undergoing free vibration (i.e. without a forcing function) described by the equation: m u? + ? u' + k u = 0, The behavior of the system is determined by the magnitude of the damping coefficient ? relative to m and k. 1. Undamped system (when ? = 0) Displacement: u(t) = C1 cos ?0 t + C2 sin ?0 t Oscillation: Yes, periodic (at natural frequency ? 0 = Notes: Steady oscillation with constant amplitude R = 2. Underdamped system (when 0 < ?2 < 4mk) Displacement: u(t) = C1 e Oscillation: Yes, quasi-periodic (at quasi-frequency µ) Notes: Exponentially-decaying oscillation 3. Critically Damped system (when ?2 = 4mk) Displacement: u(t) = C1 e + C2 t e Oscillation: No ?t cos µ t + C2 e ? t sin µ t rt 4. Overdamped system (when ?2 > 4mk) r t r t Displacement: Oscillation: No u(t ) = C1e 1 + C2e 2

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.