User:Egm6321.f09.Team06.sada/HW1

Problem 1
Derive the Equation 1 (Lecture 1) 1st order total time derivative of f where f is a function of (Y1(t),t).

Y1(t) is the Nominal motion

f(s,t)

s=X1 = axial co-ordinate

t=time

Evaluating at s=Y1(t), we have f(Y1(t),t)

To find the 1st derivative of function f(Y1(t),t)

$${d\over dt}f(Y^1(t),t)=$$$$\frac{\partial f}{\partial s}+\frac{\partial f}{\partial t}={\partial f\over\partial s}(Y^1(t),t)\dot{Y^1}(t)+\frac{\partial f}{\partial t}(Y^1(t),t)(1)$$

1st Derivative $$=\Bigg[{d\over dt}f(Y^1(t),t)={\partial f\over\partial s}(Y^1(t),t)\dot{Y^1}(t)+\frac{\partial f}{\partial t}(Y^1(t),t)\Bigg]$$

Problem 2
Derive the Equation 2 (Lecture 1) 2nd order total time derivative of function f.

1st Derivative $$=\Bigg[{d\over dt}f(Y^1(t),t)={\partial f\over\partial s}(Y^1(t),t)\dot{Y^1}(t)+\frac{\partial f}{\partial t}(Y^1(t),t)\Bigg]$$

$$={\partial f\over\partial s}(Y^1(t),t)\ddot{Y^1}(t)+\dot{Y^1}(t)\Bigg({\partial \over\partial s}{\partial f\over\partial s}(Y^1(t),t)\dot{Y^1}(t))+{\partial \over\partial t}{\partial f\over\partial s}(Y^1(t),t)\Bigg)+{\partial \over\partial s}{\partial f\over\partial t}(Y^1(t),t)\dot{Y^1}(t))+{\partial \over\partial t}{\partial f\over\partial t}(Y^1(t),t)$$

2nd Derivative $$=\Bigg[{d^2\over dt^2}f(Y^1(t),t)={\partial f\over\partial s}(Y^1(t),t)\ddot{Y^1}(t)+{\partial f\over\partial ss}(Y^1(t),t)\dot{Y^1}^2(t)+2{\partial f\over\partial st}(Y^1(t),t)\dot{Y^1}(t)+{\partial f\over\partial tt}(Y^1(t),t)\Bigg]$$