APPLICATION: Eigenvectors Modes of Vibration 12 m2. 31 Like solving linear syste

Question

APPLICATION: Eigenvectors Modes of Vibration 12 m2. 31 Like solving linear systems, solutions to eigenvalue problems crop up everywhere in application One of the easiest places to visualize eigenvectors is in systems of coupled oscillators. Simply p an oscillator is anything that goes back and forth, the current in a capacitor-inductor circuit mass on a spring, a suspension bridge or a water wave, to name just very few examples. Let's consider three masses connected together by a series of springs as shown in the fig above. We want to find equations for the locations (y) of each of the masses. Newton's Law sa that sum of the forces on a body is equal to its acceleration. SC F Hooke's law says that the force due to a spring is given by the spring constant k times how much spring is stretched. The force acts opposite to the direction of stretching or compression. From figure, if the springs are all unstretched when yi y2 y3 0 then we can write down equati for each mass kiyi k12(yi m131 SC F yi) k23(y2 k12(y2 SC F k23(y3 m333 Remember, the force on M1 due to spring k12 must be equal and opposite the force du spring k12 on M2. The structure of the equations is more easily seen when written in matrix form: (k1 k12) 12 m131 k12 (k12 k23) 23 m2y2 (k3 k23) m3g3 Now, to get to an eigenvalue problem, we need only guess at the form of the solution v y(t). Since this is a constant coefficient problem, a good guess would be to let

Explanation / Answer

2)

k1=1;

k12=1;

k23=1;

k3=1;

3)

The eigenvalues of

[cossinsincos][cossinsincos]

are cos±isin=e±icos±isin=e±i. This is an orthogonal matrix.

If a matrix with real entries is symmetric (equal to its own transpose) then its eigenvalues are real (and its eigenvectors are orthogonal). Every n×nn×n matrix whose entries are real has at least one real eigenvalue if nn is odd. That is because the characteristic polynomial has real coefficients so the complex conjugate of a root is another root, and you can't have an odd number of roots if they come in pairs of distinct entries.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

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