User:Tmhoang81/Dynamic stability of discrete systems

Let us consider the discrete system, reference





Kinetic energy:
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$$   \displaystyle T = \frac{1}{2}m \dot x_1^2 + \frac{1}{2}m \dot x_2^2 $$     (1)
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Potential energy:
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$$   \displaystyle V = \frac{1}{2}k x_1^2 + \frac{1}{2}k_2 x_2^2 + \frac{1}{2}k_n (x_2-x_1)^2 $$     (2)
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The dissipation energy:
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$$   \displaystyle W = -c_1\dot x_1 x_1 - c_2(\dot x_2-\dot x_1)(x_2-x_1)-c_3\dot x_2 x_2 $$     (3)
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and then write the equations of motion using Lagrange's equations:
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$$   \displaystyle \frac{d}{dt}\frac{\partial T}{\partial \dot x_i} - \frac{\partial T}{\partial x_i} + \frac{\partial V}{\partial x_i} = Q_i, \qquad i=1,2 $$     (4)
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where generalzied forces:
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$$   \displaystyle Q_i=\frac{\partial W}{\partial x_i} $$     (5)
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Substitute (1),(2),(3) and (5) to (4) to get:
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$$   \displaystyle m\ddot x_1 + \left [ k x_1 - k_n (x_2-x_1) \right ] = -c_1\dot x_1 + c_2(\dot x_2-\dot x_1) $$     (6)
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$$   \displaystyle m\ddot x_2 + \left [ k_2 x_2 + k_n (x_2-x_1) \right ] = -c_2(\dot x_2-\dot x_1) - c_3\dot x_2 $$     (7)
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or
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$$   \displaystyle m\ddot x_1 + (c_1+c_2)\dot x_1 -c_2\dot x_2 + (k_n+k)x_1 - k_n x_2 = 0 $$     (8)
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$$   \displaystyle m\ddot x_2 - c_2\dot x_1 + (c_2+c_3)\dot x_2 - k_n x_1 + (k_n+k_2)x_2   = 0 $$     (9)
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Assume the solutions of (8) and (9) having the following normal modes:
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$$   \displaystyle x_1=A_1 e^{\omega t}, \qquad x_2=A_2 e^{\omega t}
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$$     (10)
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Plugging (10) to (8) and (9):
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$$   \displaystyle [k_n+k+(c_1+c_2)\omega+m\omega^2]A_1-(k_n+\omega c_2)A_2=0
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$$     (11)
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$$   \displaystyle -(k_n+\omega c_2)A_1+[k_n+k_2+(c_2+c_3)\omega+m\omega^2]A_2=0
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$$     (12)
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(11) and (12) have non-trial solutions if the determinant of the coefficients of $$\displaystyle A_i (i=1,2) $$ is zero:
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$$   \displaystyle [k_n+k+(c_1+c_2)\omega+m\omega^2][k_n+k_2+(c_2+c_3)\omega+m\omega^2]-(k_n+\omega c_2)^2=0 $$     (13)
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or rewrite this in the form of unknown $$\displaystyle \omega$$
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$$   \displaystyle m^2\omega^4+(c_1+2c_2+c_3)m\omega^3+[c_1c_2+c_2c_3+c_3c_1+(k+k_2+2k_n)m]\omega^2+[c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)]\omega+kk_2+(k+k_2)k_n=0 $$     (14)
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(8) and (9) have bounded solutions as time goes to infinity if $$\displaystyle Re(\omega)<0 $$. We will use this condition to impose conditions on the coefficients of equation (14). Before doing this, let us review a famous Hurwitz's theorem, reference,

An equation of order n
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$$   \displaystyle a_0x^n + a_1x^{n-1}+\cdots+a_{n-1}x+a_n=0 $$     (15)
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has all solutions (complex or real) with negative parts if the following conditions are satisfied:
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$$   \displaystyle \Delta_1=a_1>0, \quad \Delta_2=\begin{vmatrix} a_1 & a_0\\ a_3 & a_2 \end{vmatrix}>0, \quad \Delta_3=\begin{vmatrix} a_1 & a_0 & 0\\ a_3 & a_2 & a_1\\ a_5 & a_4 & a_3 \end{vmatrix}>0, \quad \cdots, \quad \Delta_n=\begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0\\ a_3 & a_2 & a_1 & a_0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ a_{2n-1} & a_{2n-2} & a_{2n-3} & a_{2n-4} & \cdots & a_n \end{vmatrix}=a_n\Delta_{n-1}>0 $$     (16)
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Apply this condition for the 4th order equation:
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$$   \displaystyle a_0x^4 + a_1x^3 + a_2x^2 + a_1x + a_4 = 0 $$     (17)
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we obtain:
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$$   \displaystyle a_1>0, \quad (a_1a_2-a_0a_3)>0, \quad a_3(a_1a_2-a_0a_3)-a_4a_1^2>0, \quad a_4>0 $$     (18)
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It is easy to prove that (18) is equivalent to the following condition:
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$$   \displaystyle a_1,a_3,a_4>0, \quad \quad a_3(a_1a_2-a_0a_3)-a_4a_1^2>0 $$     (19)
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Now require (19) must be true for coefficients of equation (14)
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$$   \displaystyle \begin{align} &m(c_1+2c_2+c_3)>0\\ &c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)>0 \\ &kk_2+(k+k_2)k_n>0 \end{align} $$     (20)
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and
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$$   \displaystyle \begin{Bmatrix} [c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)]\begin{pmatrix} m(c_1+2c_2+c_3)[c_1c_2+c_2c_3+c_3c_1+(k+k_2+2k_n)m] \\ -m^2[c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)] \end{pmatrix}\\ -m^2(c_1+2c_2+c_3)^2(kk_2+(k+k_2)k_n) \end{Bmatrix}>0 $$     (21)
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First we assume all dampers are positive and springs $$\displaystyle k_1,k_2$$ have positive stiffnesses while spring $$\displaystyle k_n$$ can have negative stiffness.
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$$   \displaystyle c_1,c_2,c_3>0 \quad \text{and} \quad k_1,k_2>0 $$     (22)
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With these conditions, the first inequality of (20) is satisfied. Now we will prove that if the third of (20) is also satisfied:
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$$  \displaystyle k_n>\frac{-kk_2}{k+k_2} $$     (23)
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then the second condtion of (20) and condition (21) automatically satisfied.

Indeed, (23) infers:
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$$   \displaystyle \begin{align} &k_n+k>\frac{-kk_2}{k+k_2}+k = \frac{k^2}{k+k_2}>0\\ &k_n+k_2>\frac{-kk_2}{k+k_2}+k_2 = \frac{k_2^2}{k+k_2}>0\\ &a_3=c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)>0\\ \end{align} $$     (24)
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Now, determine
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$$   \displaystyle \begin{align} a_1a_2-a_0a_3&=m(c_1+2c_2+c_3)[c_1c_2+c_2c_3+c_3c_1+(k+k_2+2k_n)m] - m^2[c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n)]\\ &= m(c_1+2c_2+c_3)[c_1c_2+c_2c_3+c_3c_1]+m^2[(c_1+2c_2+c_3)(k+k_2+2k_n)-c_1(k_2+k_n)-c_2(k+k_2)-c_3(k+k_n)]\\ &= m(c_1+2c_2+c_3)[c_1c_2+c_2c_3+c_3c_1]+m^2[(c_1(k+k_n)+c_2(k+k_2+4k_n)+c_3(k_2+k_n))\\ \end{align} $$     (25)
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to see that:


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$$   \displaystyle \begin{align} a_3(a_1a_2-a_0a_3)-a_1^2a_4&=a_3m(c_1+2c_2+c_3)[c_1c_2+c_2c_3+c_3c_1] + m^2\underbrace{\begin{Bmatrix} a_3[c_1(k+k_n)+c_2(k+k_2+4k_n)+c_3(k_2+k_n)]\\ -(c_1+2c_2+c_3)^2[kk_2+(k+k_2)k_n] \end{Bmatrix}}_{\mathbf P} \end{align} $$     (26)
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The first term (26) is obviously positive, the second term is:
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$$   \displaystyle \mathbf P = \begin{Bmatrix} \begin{bmatrix} c_1(k_2+k_n)+c_2(k+k_2)+c_3(k+k_n) \end{bmatrix} \begin{bmatrix} c_1(k+k_n)+c_2(k+k_2+4k_n)+c_3(k_2+k_n) \end{bmatrix}\\ -(c_1+2c_2+c_3)^2 \begin{bmatrix} kk_2+(k+k_2)k_n \end{bmatrix} \end{Bmatrix} $$   (27)
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After some operations to rewrite (27) in the following form, which is also positive:
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$$   \displaystyle \mathbf P = c_1^2k_n^2 + c_2^2(k-k_2)^2 + c_3^2k_n^2 + c_1c_2 \begin{bmatrix} (k_2+k_n-k)^2 + 3k_n^2 \end{bmatrix} + c_2c_3 \begin{bmatrix} (k+k_n-k_2)^2 + 3k_n^2 \end{bmatrix} + c_3c_1 \begin{bmatrix} (k-k_2)^2 + 2k_n^2 \end{bmatrix} >0 $$   (28)
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Finally we conlcude that for the discrete system above stable, the conditions are (22) and (23). These conditions also reveal that adding positive dampers does not change the stability requirements of the system.

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