Question

Use Lagrange's equations to derive the equations of motion for the triple pendulum whose bobs are subjected to horizontal forces \(F_{1}, F_{2}\) and \(F_{3}\), respectively.

Short answer

Answer: The equations of motion for a triple pendulum with bobs subjected to horizontal forces using Lagrange's equations are: \(\frac{d^2\theta_1}{dt^2} = f_1(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\) \(\frac{d^2\theta_2}{dt^2} = f_2(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\) \(\frac{d^2\theta_3}{dt^2} = f_3(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\)

Step-by-step solution

Identify generalized coordinates

In this case, the triple pendulum can be described by three angles: \(\theta_1\), \(\theta_2\), and \(\theta_3\). These angles represent the positions of each bob in respect to the vertical reference.

Write positions of bobs in terms of angles

Let's assume that each pendulum has the same length, L. The positions of the bobs can be described as follows in cartesian coordinates: \(x_1 = L\sin(\theta_1)\) \(y_1 = -L\cos(\theta_1)\) \(x_2 = x_1+L\sin(\theta_2)\) \(y_2 = y_1-L\cos(\theta_2)\) \(x_3 = x_2+L\sin(\theta_3)\) \(y_3 = y_2-L\cos(\theta_3)\)

Compute the kinetic and potential energy of the system

We now compute the kinetic (T) and potential energy (V) of the three bobs: \(T_1 = \frac{1}{2}m_1(\dot{x}_1^2 +\dot{y}_1^2)\) \(T_2 = \frac{1}{2}m_2(\dot{x}_2^2 +\dot{y}_2^2)\) \(T_3 = \frac{1}{2}m_3(\dot{x}_3^2 +\dot{y}_3^2)\) \(T = T_1 + T_2 + T_3\) \(V_1 = m_1gy_1\) \(V_2 = m_2gy_2\) \(V_3 = m_3gy_3\) \(V = V_1 + V_2 + V_3\)

Write the Lagrangian of the system

The Lagrangian (L) of the system can be written as: \(L = T - V\) Now, we need to consider the horizontal forces acting on the bobs: \(F_1 = F_1\cos(\theta_1)\) \(F_2 = F_2\cos(\theta_2)\) \(F_3 = F_3\cos(\theta_3)\) \(L = T-V-\sum_{i=1}^3 F_i\cos(\theta_i)dx_i\)

Derive the equations of motion using Lagrange's equations

The Lagrange's equation is given by: \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta_i}}\right)-\frac{\partial L}{\partial \theta_i}=0\) Where \(i\) represents the angle \(\theta_1\), \(\theta_2\), and \(\theta_3\) for each pendulum bob. Applying Lagrange's equations for each angle and simplifying the expressions, we obtain the following equations of motion: \(\frac{d^2\theta_1}{dt^2} = f_1(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\) \(\frac{d^2\theta_2}{dt^2} = f_2(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\) \(\frac{d^2\theta_3}{dt^2} = f_3(\theta_1, \theta_2, \theta_3, \dot{\theta}_1, \dot{\theta}_2, \dot{\theta}_3, F_1, F_2, F_3)\) These equations represent the equations of motion for the triple pendulum subjected to horizontal forces \(F_1, F_2,\) and \(F_3\).

Key concepts

Equations of Motion

Understanding the equations of motion for a triple pendulum involves diving into the dynamic behavior of this complex system. The pendulum in question is a sequential arrangement of three pendulums, each connected to the mass (or bob) of the one above. This setup leads to a system where the motion of each pendulum affects the others, creating a coupled motion scenario.

When horizontal forces are applied to the bobs, as specified in the exercise, the dynamics become even more intricate. These forces introduce external influences that cause the system to deviate from its natural oscillatory motion. To capture the behavior mathematically, Lagrange's equations are employed, which are formulated based on the principle of least action. This principle states that the actual path taken by a system is the one that minimizes the action, a quantity that is integrated over time and involves both kinetic and potential energy of the system.

In the context of the triple pendulum, the equations of motion are differential equations that describe how the angles \(\theta_1\), \(\theta_2\), and \(\theta_3\) change over time. These angles are crucial as they dictate the position and velocity of each bob. Solving these equations provides insights into the time-evolution of the system, which is essential for understanding and predicting its behavior under the influence of the applied forces.

Kinetic and Potential Energy

In any physical system, kinetic and potential energy play pivotal roles in determining dynamics, and this holds true for the triple pendulum as well. Kinetic energy, represented as \(T\), is the energy of motion. For our pendulums, this is the sum of the energies associated with the linear velocities of the bobs, which are derived from their respective angular velocities.

Potential energy, denoted as \(V\), is the energy associated with the position of an object within a field, such as the gravitational field in this case. It is dependent on the height of the bobs along the vertical axis. The potential energies of the bobs are summed to obtain the total potential energy of the system.

The relationship between kinetic and potential energy in the triple pendulum is dynamic and cyclical. As the pendulums swing, kinetic energy is converted into potential energy and vice versa. This conservation and interplay of energies is at the heart of the oscillatory motion that is characteristic of pendulums. It's crucial to accurately calculate both forms of energy to create the Lagrangian of the system, which is used in Lagrange's equations to derive the equations of motion.

Generalized Coordinates

Generalized coordinates are a set of parameters that define the configuration of a mechanical system relative to a reference frame. For the triple pendulum, we use three angles \(\theta_1\), \(\theta_2\), and \(\theta_3\) as generalized coordinates, which describe the orientation of each pendulum in the system.

The beauty of using generalized coordinates lies in their ability to simplify complex systems by reducing the number of variables required to describe the system's state. This simplification allows for a more manageable analysis of the mechanics involved, especially when dealing with coupled systems like our triple pendulum. By choosing variables that are independent of one another and directly related to the system's constraints, we formulate the problem in a way that leads to clearer insights into the physical behavior of the system.

In classical mechanics, these coordinates play a central role in the application of Lagrange's equations to derive the equations of motion. They serve as the foundation for understanding the system's evolution over time, providing a comprehensive depiction of how each component of the pendulum moves in response to internal and external forces.