Lesson 17 (10/02/26): Mechanical Vibrations (3.4)

Lesson 17 (10/02/26): Mechanical Vibrations (3.4)

Newton's 2^nd Law:

\[ F_{\text{net}} = ma \] \[ F_{\text{ext}} - F_{\text{spring}} - F_{\text{friction}} = mx'' \] \[ F_{\text{spring}} = kx, \quad k \text{ is constant} \] \[ F_{\text{friction}} = cv = cx', \quad c \text{ is constant} \]

Equation:

\[ mx'' + cx' + kx = F_{\text{ext}} \]

Constant coeff, linear, \(2^{\text{nd}}\) order ODE.

Review: Const. Coeff, Linear, Homogeneous ODE

\[ a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y' + a_0 y = 0 \]

Ch. equation:

\[ a_n r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0 = 0 \]

Suppose \((r - r_0)^k\) is a factor of ch. equation, \(r_0\) is Real.

it gives us \(k\) L.I. solutions

\[ e^{r_0 x}, \, x e^{r_0 x}, \, x^2 e^{r_0 x}, \, \dots, \, x^{k-1} e^{r_0 x} \]

\((a r^2 + b r + c)^k\) is a factor with \(a r^2 + b r + c\) having complex roots \(\alpha \pm i \beta\)

this gives us \(2k\) L.I. solutions

\[ e^{\alpha x} \cos \beta x, \, e^{\alpha x} \sin \beta x \] \[ x e^{\alpha x} \cos \beta x, \, x e^{\alpha x} \sin \beta x, \, x^2 e^{\alpha x} \cos \beta x, \, x^2 e^{\alpha x} \sin \beta x \] \[ \dots, \, x^{k-1} e^{\alpha x} \cos \beta x, \, x^{k-1} e^{\alpha x} \sin \beta x \]

Example: \( (r-7)(r+9)^4 (r^2-4r+29)^3=0 \) is the ch. equation for 11th order linear constant coeff. homogeneous ODE.

find its general solution.

\( r-7 \rightsquigarrow \) gives one solution \( e^{7x} \)

\( (r+9)^4 \rightsquigarrow \) gives \( 4 \) solutions \( e^{-9x}, \; x e^{-9x}, \; x^2 e^{-9x}, \; x^3 e^{-9x} \)

\[ r^2 - 4r + 29 = 0 \rightsquigarrow r = \frac{4 \pm \sqrt{16-116}}{2} = \frac{4 \pm 10i}{2} = 2 \pm 5i \]

\( (r^2 - 4r + 29)^3 \rightsquigarrow \) gives \( 6 \) solutions \( \left[ e^{(2+5i)x} = e^{2x}[\cos 5x + i \sin 5x] \right] \)

\[ e^{2x} \cos 5x, \quad e^{2x} \sin 5x \]

\[ x e^{2x} \cos 5x, \quad x e^{2x} \sin 5x \]

\[ x^2 e^{2x} \cos 5x, \quad x^2 e^{2x} \sin 5x. \]

General solution: Linear combination of all these \( 11 \) solutions

Going back to solving \( mx'' + cx' + kx = F_{\text{ext}} \)

Case ①: No friction, No external force

\[ mx'' + \underset{\text{friction}}{\overset{0}{\cancel{cx'}}} + kx = \overset{0}{\cancel{F_{\text{ext}}}} \] \[ mx'' + kx = 0, \quad \text{Let } \omega_0^2 = \frac{k}{m} \] \[ x'' + \omega_0^2 x = 0 \]

ch. equation:

\[ r^2 + \omega_0^2 = 0 \implies r = \pm \omega_0 i \]

two solutions: \( \cos\omega_0 t, \sin\omega_0 t \)

General solution:

\[ x(t) = \underbrace{c_1 \cos\omega_0 t + c_2 \sin\omega_0 t}_{\text{for any } c_1, c_2 \text{ this is sinusoidal}} \]

there is Damping.

\[ x(t) = C_1 \cos\omega_0 t + C_2 \sin\omega_0 t \]

Lets find Amplitude & time period/frequency using trig identities

\[ \cos\phi = \frac{\text{adj}}{\text{hyp}} = \frac{C_1}{\sqrt{C_1^2 + C_2^2}} \]

\[ \sin\phi = \frac{\text{opp}}{\text{hyp}} = \frac{C_2}{\sqrt{C_1^2 + C_2^2}} \]

\[ x(t) = \left(\sqrt{C_1^2 + C_2^2}\right) \left[ \underbrace{\frac{C_1}{\sqrt{C_1^2 + C_2^2}}}_{\cos\phi} \cos\omega_0 t + \underbrace{\frac{C_2}{\sqrt{C_1^2 + C_2^2}}}_{\sin\phi} \sin\omega_0 t \right] \]

\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]

\[ x(t) = \overbrace{\left(\sqrt{C_1^2 + C_2^2}\right)}^{A} \cos(\omega_0 t - \phi) \]

\[ m x'' + c x' + k x = f_{\text{ext}} \]

if \( f_{\text{ext}} = 0 \) & there is no friction (undamped) \( \rightsquigarrow \)

\[ x'' + \omega_0^2 x = 0 \] \[ \omega_0 = \sqrt{\frac{k}{m}} \]

Solution:

\[ x(t) = A \cos(\omega_0 t - \phi) \]

\( A \) is amplitude

\( \phi \) is phase shift

\[ T = \text{Period} = \frac{2\pi}{\omega_0} \] \[ f = \text{frequency} = \frac{1}{\text{Period}} = \frac{\omega_0}{2\pi} \]

Suppose there is friction, still no ext. force } Damped case.

Equation:

\[ mx'' + cx' + kx = 0 \] \[ x'' + \frac{c}{m}x' + \omega_0^2 x = 0 \qquad \left(\omega_0^2 = \frac{k}{m}\right) \]

Ch. equation:

\[ r^2 + \frac{c}{m}r + \omega_0^2 = 0 \]

Roots:

\[ \frac{-\frac{c}{m} \pm \sqrt{\left(\frac{c}{m}\right)^2 - 4\omega_0^2}}{2} = \frac{-c \pm \sqrt{c^2 - 4m^2\omega_0^2}}{2m} \]

1. \( c^2 - 4m^2\omega_0^2 > 0 \leadsto \) two distinct real roots
2. \( c^2 - 4m^2\omega_0^2 = 0 \leadsto \) one real root, repeated
3. \( c^2 - 4m^2\omega_0^2 < 0 \leadsto \) c'x roots

(i) \( c^2 - 4m^2\omega_0^2 > 0 \)

Two roots:

\[ r_1 = \frac{-c + \sqrt{c^2 - 4m^2\omega_0^2}}{2m}, \quad r_2 = \frac{-c - \sqrt{c^2 - 4m^2\omega_0^2}}{2m} \] \[ c^2 - 4m^2\omega_0^2 < c^2 \] \[ \Rightarrow -c + \sqrt{c^2 - 4m^2\omega_0^2} < -c + \sqrt{c^2} = 0 \] \[ \Rightarrow r_1 < 0 \quad \text{also} \quad r_2 < 0 \]

Gen. Solution:

\[ x(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} \] \[ \lim_{t \to \infty} x(t) = 0 \quad \text{bcf} \quad \lim_{t \to \infty} e^{rt} = 0 \quad \text{for } r < 0. \]

\[ c^2 > 4m^2\omega_0^2 \]

"Over" Damped.

goes to 0 the fastest

(ii) \( c^2 = 4\omega_0^2 m^2 \)

\( \Rightarrow \) only one root \( = r = -\frac{c}{2m} \), repeated

General solution: \( x(t) = c_1 e^{rx} + c_2 x e^{rx}, \quad r = -\frac{c}{2m} < 0 \)

\[ \lim_{t \to \infty} e^{rx} = 0, \quad \lim_{t \to \infty} x e^{rx} = 0 \]

e.g. \( x e^{-3x} = \frac{x}{e^{3x}} \)

but \( x e^{rx} \) goes to \( 0 \) slower than \( e^{rx} \)

this case is called "Critically" Damped

\[ c^2 - 4\omega_0^2 m^2 > 0 \quad \text{"Over Damped"} \]

\[ c^2 - 4\omega_0^2 m^2 = 0 \quad \text{"Critically Damped"} \]

(iii) Roots: \( \alpha \pm i\beta, \quad \alpha < 0 \)

Gen. solution:

\[ C_1 e^{\alpha t} \cos\beta t + C_2 e^{\alpha t} \sin\beta t \] \[ e^{\alpha t} \left[ C_1 \cos\beta t + C_2 \sin\beta t \right] \]

Where \( e^{\alpha t} \) goes to \( 0 \) as \( t \to \infty \), and the term \( [C_1 \cos\beta t + C_2 \sin\beta t] \) is sinusoidal.