Problem 40 - Classical Dynamics of Particles and Systems

Question

A double pendulum is attached to a cart of mass \(2 m\) that moves without friction on a horizontal surface. See Figure 7 -D. Each pendulum has length \(b\) and mass bob \(m\) Find the equations of motion.

Step-by-Step Solution

Answer

The equations of motion for the double pendulum attached to a cart system can be found by applying the Lagrangian method. First, assign variables to the position, velocity, and forces in the system. Next, calculate the total kinetic and potential energies of the system. Finally, apply the Lagrange equations of motion to obtain three equations, one for each generalized coordinate (x, θ1, θ2). Solving these equations will yield the desired equations of motion.

Step 1: Assign Variables

Let \(x\) be the horizontal position of the cart, and let the angles of the two pendulums with the vertical be \(\theta_{1}\) and \(\theta_{2}\). The lengths of the two pendulums are given as \(b\), and the mass of each pendulum bob is \(m\). We can find the coordinates of each pendulum bob as follows: Pendulum 1: \[x_1 = x + b\sin{\theta_1}\] \[y_1 = -b\cos{\theta_1}\] Pendulum 2: \[x_2 = x + b\sin{\theta_1} + b\sin{\theta_2}\] \[y_2 = -b\cos{\theta_1} - b\cos{\theta_2}\]

Step 2: Calculate Kinetic and Potential Energies

Now, we will calculate the kinetic and potential energies for each pendulum bob and the cart. Kinetic energy of the cart: \[ T_{cart} = \frac{1}{2} (2m) \dot{x}^2 \] Kinetic energy of the pendulum 1: \[ T_{1} = \frac{1}{2} m \left(\left(\frac{dx_1}{dt}\right)^2 + \left(\frac{dy_1}{dt}\right)^2\right) \] Kinetic energy of the pendulum 2: \[ T_{2} = \frac{1}{2} m \left(\left(\frac{dx_2}{dt}\right)^2 + \left(\frac{dy_2}{dt}\right)^2\right) \] Total kinetic energy of the system is the sum of the kinetic energies of the cart and the pendulum bobs: \[ T = T_{cart} + T_{1} + T_{2} \] Potential energy of the pendulum 1: \[ V_{1} = mgy_1 = -mgb\cos{\theta_1} \] Potential energy of the pendulum 2: \[ V_{2} = mgy_2 = -mg\left(b\cos{\theta_1} + b\cos{\theta_2}\right) \] Total potential energy of the system is the sum of the potential energies of the pendulum bobs: \[ V = V_{1} + V_{2} \]

Step 3: Apply Lagrangian Equation

Now we need to apply the Lagrangian equation which is given as: \[ L = T - V \] The Lagrange equations of motion are given by: \[\frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0 \] where \(q_i\) represents the generalized coordinates (\(x, \theta_1, \theta_2\)) and \(\dot{q}_i\) are their time derivatives. Applying the Lagrange equations of motion for the system, we will obtain three equations, one for each generalized coordinate: 1. Equation for \(x\): \[\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x} = 0\] 2. Equation for \(\theta_1\): \[\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}_1} - \frac{\partial L}{\partial \theta_1} = 0\] 3. Equation for \(\theta_2\): \[\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}_2} - \frac{\partial L}{\partial \theta_2} = 0\] Solving these equations will give us the equations of motion for the double pendulum attached to the cart system.

Key Concepts

Lagrangian Mechanics

Lagrangian Mechanics is a powerful method in classical mechanics that provides a framework to derive the equations of motion for a system. Instead of using forces and accelerations directly, as in Newtonian mechanics, Lagrangian mechanics utilizes energy concepts. The main idea is to use the differences between kinetic and potential energy to describe the system's dynamics.

In practice, the Lagrangian is defined as the difference between a system's total kinetic energy and potential energy. This is expressed as:

• The Lagrangian, denoted as \( L \), is: \( L = T - V \)
• Here, \( T \) represents the total kinetic energy, and \( V \) the total potential energy of the system.

By using the Euler-Lagrange equation, we can derive the equations of motion. This method works well for complex systems, like the double pendulum, because it elegantly handles constraints and provides clear equations for generalized coordinates without directly dealing with forces.

Double Pendulum

The double pendulum is a fascinating system in classical mechanics, known for its rich and complex dynamical behavior. It consists of two pendulums, one attached to the end of the other, and can exhibit chaotic motion.

In this system:

• Both pendulums are rigid rods of length \( b \), each with a mass \( m \) at the end, called the bob.
• The motion of the pendulum is influenced not just by gravity acting on each bob but by the coupling effects between the two pendulums.
• In our exercise, this double pendulum is mounted on a cart, adding another layer to its motion as the whole setup can move horizontally without friction.

The study of a double pendulum involves understanding its potential and kinetic energies, which allow the derivation of its complex motion equations under Lagrangian mechanics.Understanding how these energies shift as the pendulums swing is crucial to solving the system's equations.

Equations of Motion

Equations of Motion are fundamental in describing the dynamics of a mechanical system. In the context of Lagrangian mechanics, these equations are derived based on the system’s kinetic and potential energy relationships.

For the double pendulum on a cart:

• Three generalized coordinates are used: the position of the cart, \( x \), and the angles of the pendulums with respect to the vertical, \( \theta_1 \) and \( \theta_2 \).
• These coordinates allow us to express both kinetic and potential energies in terms of angles and linear position.
• The Lagrangian function \( L = T - V \) is constructed from these energies, where \( T \) is the total kinetic energy and \( V \) is the total potential energy.
• Applying the Euler-Lagrange equation to the Lagrangian provides three separate equations, one for each generalized coordinate.

Solving these differential equations yields the motion of the system, demonstrating how each part (the cart and each pendulum) interacts and moves over time. Understanding these equations is vital to analyzing the system's response to different initial conditions and understanding the pendulum's chaotic nature.