Find a single equivalent damping constant for the following cases

(i) when three dampers are parallel (ii) when three dampers are in series

(ii) when three dampers are connected to a rigid bar (as shown in figure below) and the equivalent damper is at site $ C_1$

Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 10M

Year: may 2016

$Ay_2 = \frac{Ao+AL}{2} = \frac{80+20}{2} = 50 mm^2$

$A_1 = \frac{80+50}{2} = 65 mm^2$

$A_2 = \frac{50+20}{2} = 35 mm^2$

$\hspace{1cm}L_1=L_2 = 30 mm$

Elemental stiffness matrix,

$\hspace{0.6cm}k_1=\frac{AE}{L}$ $\begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix} = 10^3$ $\begin{bmatrix} \ 455 & -455 \\ \ -455 & 455\\ \end{bmatrix}$

$\hspace{0.6cm}k_2=\frac{AE}{L}$ $\begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix} = 10^3$ $\begin{bmatrix} \ 245 & -245 \\ \ -245 &245\\ \end{bmatrix}$

Global stiffness matrix,

$[k]= 10^3\begin{bmatrix} \ 455 & -455 & 0 \\ \ -455 & 700 & -245 \\ \ 0 & -245 & 245 \\ \end{bmatrix}$

$\hspace{3cm}$[K][u]=[F]

$10^3$ $\begin{bmatrix} \ 455 & -455 & 0 \\ \ -455 & 700 & -245 \\ \ 0 & -245 & 245 \\ \end{bmatrix}$ $\begin{Bmatrix} \ u_1 \\ \ u_2 \\ \ u_3 \\ \end{Bmatrix}$= $\begin{Bmatrix} \ f_1 \\ \ f_2 \\ \ f_3 \\ \end{Bmatrix}$

Apply Boundary conditions ,

$\hspace{1cm}u_1 = 0 , f_3 = 0.5 \times10^3N$

$10^3$ $\begin{bmatrix} \ 455 & -455 & 0 \\ \ -455 & 700 & -245 \\ \ 0 & -245 & 245 \\ \end{bmatrix}$ $\begin{Bmatrix} \ u_1 \\ \ u_2 \\ \ u_3 \\ \end{Bmatrix}$= $\begin{Bmatrix} \ f_1 \\ \ 0 \\ \ 0.5 \\ \end{Bmatrix}\times 10^3$

$\hspace{2cm} \therefore 700u_2-245u_3=0$

$\hspace{2.2cm}-245u_2+245u_3 = 0.5$

$\hspace{2.6cm}\therefore u_2 = 0.0011mm$

$\hspace{2.9cm}u_3=0.00314mm$

Reaction , f_1 = 455u_1-455u_2

$\hspace{2.3cm}= 0- 455 \times (0.0011)$

$\hspace{2.3cm} = -0.5 KN$