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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\beta and the external driving force *Acos(αt)A \cos(\alpha t)* oscillates with time (AA: drive amplitude, α\alpha:drive frequency). Define the differential equations in python for this forced, damped Pendulum. The solution of this differential equation, y=[θ,ω]T y=[\theta, \omega]^T, represents the state vector of the system, which includes the angle *θ\theta* and the angular velocity *ω\omega*. θ\theta is measured in radians and ω\omega in radians per second.
Plain-text mathematical notation (without MathML)
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.
Original LaTeX notation
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.

step background

Background A single pendulum consists of: - A mass mm attached to the end of a string or rod of length LL. - The pivot point is frictionless. - There is a damping force proportional to the angular velocity with a damping coefficient β\beta. - An external driving force Acos(αt)A \cos(\alpha t) oscillates with time. (a) Identify the Forces and Torques The forces acting on the pendulum bob are: - Gravitational force mgmg acting downward. - Tension in the string (which does not do work as it acts along the string). - Damping force proportional to the angular velocity −βdθdt-\beta \frac{d\theta}{dt}. - External driving force Acos(αt)A \cos(\alpha t). The torque τ\tau around the pivot due to the gravitational force, damping force, and driving force is given by: τ=−mgLsin(θ)−βLdθdt+Acos(αt) \tau = -mgL \sin(\theta) - \beta L \frac{d\theta}{dt} + A \cos(\alpha t) The moment of inertia II for a point mass mm at a distance LL from the pivot is: I=mL2 I = mL^2 (b) Apply Newton's Second Law for Rotation Using Newton's second law for rotation τ=Iα\tau = I \alpha: −mgLsin(θ)−βLdθdt+Acos(αt)=mL2d2θdt2 -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 mL2mL^2: −gLsin(θ)−βmLdθdt+AmL2cos(αt)=d2θdt2 -\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: d2θdt2+βmLdθdt+gLsin(θ)=AmL2cos(αt) \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: dθdt=ω \frac{d\theta}{dt}= \omega Then, the first-order ODEs for this damped, forced single pendulum system becomes: dωdt=−gLsin(θ)−βmLω+AmL2cos(αt) \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: $$ \mathbf{y} = \begin{pmatrix} \theta \\ \omega \end{pmatrix} $$
Plain-text mathematical notation (without MathML)
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:

$$
\mathbf{y} = \begin{pmatrix} \theta \\ \omega \end{pmatrix}
$$
Original LaTeX notation
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:

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

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Source and history

upstream_id / Original problem ID
78.1
task_id
fbb65551-2f68-5fcd-af48-e106fe7a12f3
task_key
dev--831ff049-2748-5e6d-ba9b-f8252555cd23--78~2e1
task_revision_id
5
config / split
/ dev

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