SciCode / 78.1 / The motion of a forced, damped single pendulum can be described by the…

Background

A single pendulum consists of:
- A mass m attached to the end of a string or rod of length L.
- The pivot point is frictionless.
- There is a damping force proportional to the angular velocity with a damping coefficient β.
- An external driving force Acos(αt) oscillates with time.

(a) Identify the Forces and Torques

The forces acting on the pendulum bob are:
- Gravitational force mg acting downward.
- Tension in the string (which does not do work as it acts along the string).
- Damping force proportional to the angular velocity −β(dθ)/(dt).
- External driving force Acos(αt).

The torque τ around the pivot due to the gravitational force, damping force, and driving force is given by:

τ=−mgLsin(θ)−βL(dθ)/(dt)+Acos(αt)

The moment of inertia I for a point mass m at a distance L from the pivot is:

I=mL²

(b) Apply Newton's Second Law for Rotation

Using Newton's second law for rotation τ=Iα:

−mgLsin(θ)−βL(dθ)/(dt)+Acos(αt)=mL²(d²θ)/(dt²)

(c) Simplify the Equation

Dividing both sides by mL²:

−(g)/(L)sin(θ)−(β)/(mL)(dθ)/(dt)+(A)/(mL²)cos(αt)=(d²θ)/(dt²)

Rearranging the terms:

(d²θ)/(dt²)+(β)/(mL)(dθ)/(dt)+(g)/(L)sin(θ)=(A)/(mL²)cos(αt)

(d) Converting to First-Order ODEs

To use numerical methods like RK4, we need to convert this second-order ODE into a system of first-order ODEs by defining:

(dθ)/(dt)=ω


Then, the first-order ODEs for this damped, forced single pendulum system becomes:

(dω)/(dt)=−(g)/(L)sin(θ)−(β)/(mL)ω+(A)/(mL²)cos(αt)

(e) Define The State Vector

The state vector of this forced, damped pendulum is:

$$
\mathbf{y} = \begin{pmatrix} \theta \\ \omega \end{pmatrix}
$$

Background

A single pendulum consists of:
- A mass $m$ attached to the end of a string or rod of length $L$.
- The pivot point is frictionless.
- There is a damping force proportional to the angular velocity with a damping coefficient $\beta$.
- An external driving force $A \cos(\alpha t)$ oscillates with time.

(a) Identify the Forces and Torques

The forces acting on the pendulum bob are:
- Gravitational force $mg$ acting downward.
- Tension in the string (which does not do work as it acts along the string).
- Damping force proportional to the angular velocity $-\beta \frac{d\theta}{dt}$.
- External driving force $A \cos(\alpha t)$.

The torque $\tau$ around the pivot due to the gravitational force, damping force, and driving force is given by:

$$
\tau = -mgL \sin(\theta) - \beta L \frac{d\theta}{dt} + A \cos(\alpha t)
$$

The moment of inertia $I$ for a point mass $m$ at a distance $L$ from the pivot is:

$$
I = mL^2
$$

(b) Apply Newton's Second Law for Rotation

Using Newton's second law for rotation $\tau = I \alpha$:

$$
-mgL \sin(\theta) - \beta L \frac{d\theta}{dt} + A \cos(\alpha t) = mL^2 \frac{d^2\theta}{dt^2}
$$

(c) Simplify the Equation

Dividing both sides by $mL^2$:

$$
-\frac{g}{L} \sin(\theta) - \frac{\beta}{mL} \frac{d\theta}{dt} + \frac{A}{mL^2} \cos(\alpha t) = \frac{d^2\theta}{dt^2}
$$

Rearranging the terms:

$$
\frac{d^2\theta}{dt^2} + \frac{\beta}{mL} \frac{d\theta}{dt} + \frac{g}{L} \sin(\theta) = \frac{A}{mL^2} \cos(\alpha t)
$$

(d) Converting to First-Order ODEs

To use numerical methods like RK4, we need to convert this second-order ODE into a system of first-order ODEs by defining:

$$
\frac{d\theta}{dt}= \omega
$$


Then, the first-order ODEs for this damped, forced single pendulum system becomes:

$$
\frac{d\omega}{dt} = -\frac{g}{L} \sin(\theta) - \frac{\beta}{mL} \omega+ \frac{A}{mL^2} \cos(\alpha t)
$$

(e) Define The State Vector

The state vector of this forced, damped pendulum is:

$$
\mathbf{y} = \begin{pmatrix} \theta \\ \omega \end{pmatrix}
$$

The motion of a forced, damped single pendulum can be described by the second-order differential equation. Assuming the damping force is proportional to the angular velocity with coefficient β and the external driving force *Acos(αt)* oscillates with time (A: drive amplitude, α:drive frequency). Define the differential equations in python for this forced, damped Pendulum. The solution of this differential equation, y=[θ,ω]^(T), represents the state vector of the system, which includes the angle *θ* and the angular velocity *ω*. θ is measured in radians and ω in radians per second.

The motion of a forced, damped single pendulum can be described by the second-order differential equation. Assuming the damping force is proportional to the angular velocity with coefficient $\beta$ and the external driving force *$A \cos(\alpha t)$* oscillates with time ($A$: drive amplitude, $\alpha$:drive frequency). Define the differential equations in python for this forced, damped Pendulum. The solution of this differential equation, $ y=[\theta, \omega]^T$, represents the state vector of the system, which includes the angle *$\theta$* and the angular velocity *$\omega$*. $\theta$ is measured in radians and $\omega$ in radians per second.