2 Degree of Freedom System Vibration

Transfer Function at m1 (assuming no Force at m2)

$${\frac {v_1} {F_1}}=\frac {({{m_2s^3} +  {(k_2+k)s}})} {m_2m_1s^4+(m_2(k+k_1)+m_1(k+k_2))s^2 +    (k_2k_1+kk_1+k_2k)}   $$

Poles

$$ s = {\pm\sqrt{\frac {\left[{\pm\sqrt{\left[\frac {(k m_2 + k m_1 + k_2 m_1 + k_1 m_2)^2 - 4 m_2 m_1 (k k_2 + k k_1 + k_2 k_1)} {(m_2 m_1)}\right]} - \frac {k} {m_2} - \frac {k} {m_1} - \frac {k_2} {m_2} - \frac {k_1} {m_1}} \right]} {2}}} $$

Transfer Function at m2 (assuming no Force at m1)

$${\frac {v_2} {F_2}}=\frac {({{m_1s^3} +  {(k_1+k)s}})} {m_1m_2s^4+(m_1(k+k_2)+m_2(k+k_1))s^2 +    (k_1k_2+kk_2+k_1k)}   $$

Poles

$$ s = {\pm\sqrt{\frac {\left[{\pm\sqrt{\left[\frac {(k m_1 + k m_2 + k_1 m_2 + k_2 m_1)^2 - 4 m_1 m_2 (k k_1 + k k_2 + k_1 k_2)} {(m_1 m_2)}\right]} - \frac {k} {m_1} - \frac {k} {m_2} - \frac {k_1} {m_1} - \frac {k_2} {m_2}} \right]} {2}}} $$

Convective Heat Transfer over cylinder or tube or plate

Non Dimensional Numbers

\[ \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infty) x^3 = \mathrm{Gr}_{x}\mathrm{Pr} \]

Flow over Flat Plate

Local Friction Factor at x(laminar)

  

Average Friction Factor from 0 to x(laminar)

Local Nusselt number at x(laminar, Isothermal wall)

Average Nusselt number from 0 to x(laminar, Isothermal wall)

Local Friction Factor at x(Turbulent flow)

Local Nusselt Number at x(Turbulent flow, Isothermal wall)

Local Nusselt Number at x(laminar, uniform heat flux)

Local Nusselt Number at x(Turbulent, uniform heat flux)

Cross Flow Over Cylinder

Average Nusselt number(Churchill–Bernstein)

\[ \overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 \]

• There are other correlation for flow over bank of tubes which are not included here but can be found on various literature of heat and mass transfer textbook.

Internal Flow Thru Tube

Friction Factor(Darcy-Weisbach, from Moody Chart)

\[ f_D = 64 / \mathrm{Re}, \text{for laminar flow.} \]

\[ {1 \over \sqrt{f_D}}= -2.0 \log_{10} \left(\frac{\epsilon/D}{3.7} + {\frac{2.51}{\mathrm{Re} \sqrt{f_D } } } \right) , \text{for turbulent flow.} \]

Laminar Flow Fully Developed Friction factor and Nusselt Number 

 

Natural Convection

Vertical Plate

\[  \overline{\mathrm{Nu}}_L=\left[0.825+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.492}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]

Inclined Plate

\[  {replace\ \ \ g \ \rightarrow \ g\cos (\theta) \ in \ above \ Equation \ of \ vertical \ plate, \ \theta\le60.} \]

Horizontal Plate

Upper Surface of Hot Plate or Lower Surface of Cold Plate

\[ \overline{\mathrm{Nu}_L}=0.54{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^4\le{\mathrm{Ra}_L}\le10^7) \]

\[ \overline{\mathrm{Nu}_L}=0.15{\mathrm{Ra}_L}^\frac13\ \ \ \ (10^7\le{\mathrm{Ra}_L}\le10^{11}) \]

Lower Surface of Hot Plate or Upper Surface of Cold Plate

\[ \overline{\mathrm{Nu}_L}=0.27{\mathrm{Ra}_L}^\frac14\ \ \ \ (10^5\le{\mathrm{Ra}_L}\le10^{10}) \]

Horizontal Cylinder

\[  \overline{\mathrm{Nu}}_L=\left[0.6+\frac{0.387{\mathrm{Ra}}_L^\frac16}{\left(1+\left(\frac{0.559}{\mathrm{Pr}}\right)^\frac9{16}\right)^\frac8{27}}\right]^2 \]