User:Eml4500.f08.jamama.justin/HW7

Meeting 41

$$\tilde{k}_{23}=\frac{6EI}{L^{2}}=\int_{0}^{l}{\frac{d^{2N_{2}}}{dx^{2}}(EI)\frac{d^{2N_{3}}}{dx^{2}}dx}$$

p. 36-2       P 38-2          HW7!!!! also on P 247 (second derivative of shape function

In general,  $$\tilde{k}_{ij}=\frac{6EI}{L^{2}}=\int_{0}^{l}{\frac{d^{2N_{i}}}{dx^{2}}(EI)\frac{d^{2N_{j}}}{dx^{2}}dx}$$ i,j=2,3,5,6

Elastodynamics (trusses, frames, 2-D, 3-D, elasticity, etc.)

P. 31-1: Model pb

Discrete PVW: $$\mathbf{\bar{w}}\bullet [\bar{M}\bar{\ddot{d}}+\bar{K}\bar{d}-\bar{F}]=0$$ for all w

(boundary conditions already applied) EQ 1: $$\Rightarrow \bar{M}\bar{\ddot{d}}+\bar{K}\bar{d}=\bar{F}(t)$$ $$\bar{d}(0)=\bar{d_{0}}$$ $$\bar{\dot{d}}(0)=\bar{V_{0}}$$

Complete ordinary diff eqs (ODE's) (second order in time) + initial conditions governing the elastodynamic of discretized cont pb (MDOF)

Solving equation 1

1) Consider unforced vibration problem:

$$\bar{M}\ddot{v}+\bar{K}v=0$$

nxn nx1 nxn nx1  nx1 (unforced)

Assume $$v(t)=(sin\omega t)\phi $$

nx1 nx1

$$\phi $$ is not time dependent

$$\ddot{v}=-\omega ^{2}sin\omega t\phi $$ $$-\omega ^{2}sin\omega t\bar{M}\phi +sin\omega t\bar{K\phi} =0$$ $$\Rightarrow \bar{K}\phi =\omega ^{2}\bar{M}\phi $$

Generalized igenvalue problem: In general form

$$Ax=\lambda Bx$$ where lambda is the eigen value

Standard eval problem: $$Ax=\lambda x$$

(B=I,identity matrix) $$\begin{bmatrix} 1 & & 0\\  &  ..& \\  0&  & 1 \end{bmatrix}$$

$$\lambda =\omega ^{2} $$ Eigen Value

$$(\lambda _{i},\phi _{i})$$ Eigen pairs

i=1,....,n

mode $$i\Rightarrow V_{i}(t)=(sin\omega _{i}t)\phi _{i}$$ where i=1,...,n (animation)

2) Model Superposition method: Orthogonal prop of eigenpairs:

$$\phi _{i}^{T}\bar{M}\phi _{j}=\delta _{ij}=\begin{cases} 1 & \text{ if } i=j \\ 0 & \text{ if } i\neq j \end{cases}$$

1xn nxn  nx1 (Kronecker delta)

Mass orthog. of Eigen Vector

Eq(2) p 41-2: Eq(1) p 41-3:

$$\bar{M}\phi _{j}=\lambda _{j}\bar{K}\phi _{j}$$ $$ \phi _{i}^{T}\bar{M}\phi _{j}=\lambda _{j}\phi _{i}^{T}\bar{K}\phi _{j}$$ where $$ \phi _{i}^{T}\bar{M}\phi _{j}= \delta _{ij}$$ $$\Rightarrow \phi _{i}^{T}\bar{K}\phi _{j}=\frac{d}{\lambda _{j}}\delta _{ij}$$

Eq 1 (P 41-2): $$\bar{d}(t)=\sum_{i=1}^{n}{\xi _{i}(t)\phi _{i}}$$

nx1 1x1   nx1

$$\bar{M}\sum_{j}^{}{\ddot{\xi}_{j}\phi _{j} }+\bar{K}\sum_{j}^{}{\ddot{\xi}_{j}\phi _{j} }=F$$

$$\ddot{\bar{d}}$$      $$\bar{d}$$

$$\sum_{j}^{}{\ddot{\xi }(\phi _{i}^{T}\bar{M}\phi _{j})}+\sum_{j}^{}{\xi _{j}(\phi _{i}^{T}\bar{K}\phi _{j})}=\phi _{i}^{T}F$$

$$\delta \ddot{y}$$ $$x_{i}\delta _{ij}$$

$$\Rightarrow \ddot{\xi} _{i}+\lambda _{i}\xi _{i}=\phi _{i}^{T}F $$ where i=1,....,n