QuestionInstruction: The paper consists of THREE questions. Answer question ONE (1) and any one from the rest questions 1 Find the equivalent stiffness for the following structure shown in Fig Q1. Neglect the mass of the bars. (10 marks)
Fig Q1 2 Determine the displacement $x$ velocity $x$ and acceleration $x \ddot{x}$ of a spring-mass system with $\omega_{\mathrm{n}}=10 \mathrm{rad} / \mathrm{s}$ for the initial conditions $\mathrm{x}_{0}=0.05 \mathrm{~m}$ and $\mathrm{x}_{0}=1 \mathrm{~m} / \mathrm{s}$ . ( 10 marks )
3 A circular cylinder of radius $r$ and mass $m$ is connected by spring stiffness $k$ to a rigid support as shown in Fig Q2. Find the natural frequency of free oscillations using energy method (10 marks)
Fig Q3

Studdy Solution

STEP 1

1. The problem involves a mechanical system with springs.
2. We need to find the equivalent stiffness of the system.
3. The system consists of springs arranged in both series and parallel configurations.
4. The mass of the bars is neglected.
5. The equivalent stiffness is calculated by combining the stiffness of springs in series and parallel.

STEP 2

1. Identify the configuration of the springs.
2. Calculate the equivalent stiffness for springs in series.
3. Calculate the equivalent stiffness for springs in parallel.
4. Combine the results to find the total equivalent stiffness of the system.

STEP 3

Identify the configuration of the springs.
- $K_1$ and $K_4$ are in parallel. - $K_2$ and $K_3$ are in series. - $K_5$ is connected vertically to the bottom bar.

STEP 4

Calculate the equivalent stiffness for springs in series.
For springs in series, the equivalent stiffness $K_{\text{series}}$ is given by:
$\frac{1}{K_{\text{series}}} = \frac{1}{K_2} + \frac{1}{K_3}$
Solve for $K_{\text{series}}$ :
$K_{\text{series}} = \frac{K_2 \cdot K_3}{K_2 + K_3}$

STEP 5

Calculate the equivalent stiffness for springs in parallel.
For springs in parallel, the equivalent stiffness $K_{\text{parallel}}$ is given by:
$K_{\text{parallel}} = K_1 + K_4$

STEP 6

Combine the results to find the total equivalent stiffness of the system.
The total equivalent stiffness $K_{\text{eq}}$ is the combination of the parallel and series arrangements, and considering $K_5$ connected vertically:
$K_{\text{eq}} = K_{\text{parallel}} + K_{\text{series}} + K_5$
Substitute the expressions from previous steps:
$K_{\text{eq}} = (K_1 + K_4) + \left(\frac{K_2 \cdot K_3}{K_2 + K_3}\right) + K_5$
This is the equivalent stiffness of the system.

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