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求解运动方程，其中AC、BC杆通过铰接形式连接，CB杆中点有一质量为m的质量点，B点通过刚度为k的弹簧连接地面

To solve the equation of motion for the given system, we need to analyze the mechanical system depicted in the image. The system consists of a beam \(ACB\) with segments \(AC\) and \(CB\) connected by a hinge at \(C\). There is a mass \(m\) attached at the midpoint \(m\) of segment \(CB\), and segment \(B\) is connected to the ground via a spring with stiffness \(k\).

### Step-by-Step Approach:

1. **Define the Coordinates:**
   - Let the displacement of point \(B\) be \(x_B(t)\).
   - The displacement of the mass \(m\) at the midpoint can be described as \(x_m(t)\).

2. **Displacement Relations:**
   - Assuming small displacements and the beam being relatively rigid, the displacement \(x_m(t)\) at the midpoint \(m\) can be related to \(x_B(t)\) as \(x_m(t) = \frac{x_B(t)}{2}\).

3. **Kinetic Energy (T):**
   - The kinetic energy is due to the mass \(m\) at point \(m\).
   \[
   T = \frac{1}{2} m \left(\frac{\dot{x}_B(t)}{2}\right)^2 = \frac{1}{8} m \dot{x}_B^2(t)
   \]

4. **Potential Energy (V):**
   - The potential energy includes the elastic potential energy stored in the spring.
   \[
   V = \frac{1}{2} k x_B^2(t)
   \]

5. **Lagrangian (L):**
   - The Lagrangian \(L\) is given by \(L = T - V\):
   \[
   L = \frac{1}{8} m \dot{x}_B^2(t) - \frac{1}{2} k x_B^2(t)
   \]

6. **Equation of Motion using the Euler-Lagrange Equation:**
   - The Euler-Lagrange equation is:
   \[
   \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}_B} \right) - \frac{\partial L}{\partial x_B} = 0
   \]
   - Calculating the derivatives:
     \[
     \frac{\partial L}{\partial \dot{x}_B} = \frac{1}{4} m \dot{x}_B(t)
     \]
     \[
     \frac{\partial L}{\partial x_B} = - k x_B(t)
     \]
     \[
     \frac{d}{dt} \left( \frac{1}{4} m \dot{x}_B(t) \right) = \frac{1}{4} m \ddot{x}_B(t)
     \]
   - Substituting into the Euler-Lagrange equation:
     \[
     \frac{1}{4} m \ddot{x}_B(t) + k x_B(t) = 0
     \]
     \[
     m \ddot{x}_B(t) + 4k x_B(t) = 0
     \]

### Final Equation of Motion:
The equation of motion for the system is:
\[
m \ddot{x}_B(t) + 4k x_B(t) = 0
\]
This is a second-order differential equation that describes simple harmonic motion, with the angular frequency \(\omega\) given by:
\[
\omega = \sqrt{\frac{4k}{m}}
\]

Would you like further details or have any questions?

Here are some related questions to consider:
1. How does the stiffness \(k\) of the spring affect the frequency of oscillation?
2. What would be the effect of a damping force in this system?
3. How would the equation of motion change if the mass \(m\) was located at a different point on the beam?
4. How can the natural frequency be determined experimentally?
5. What assumptions are made in this analysis, and how might they affect the accuracy of the model?

**Tip:** When analyzing mechanical systems with springs and masses, always consider energy methods (like the Lagrangian approach) as they can provide a clear path to the equation of motion.

求解运动方程，其中AC、BC杆通过铰接形式连接，CB杆中点有一质量为m的质量点，B点通过刚度为k的弹簧连接地面。

Learn how to solve the equation of motion for a mechanical system consisting of a hinged beam with a mass attached at the midpoint and connected to a spring. This problem involves the use of Lagrangian mechanics, kinetic and potential energy calculations, and is suitable for undergraduate students.

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