Rotational Motion, Pendulum & Linearization

Rotational motion

Motion of a body described using angular displacement, angular velocity, and angular acceleration is called ______.

Newton's law for rotational motion

The rotational equivalent of Newton's second law states that the sum of moments acting on a body equals its moment of inertia multiplied by its angular acceleration; this is called ______.

ΣT = Iθ̈

The fundamental equation used to model rotational motion about a fixed axis is ______.

T

The symbol ______ represents torque or moment in a rotational mechanical system.

I

The symbol ______ represents the moment of inertia of a rotating body.

θ

The symbol ______ represents angular displacement.

θ̇

The symbol ______ represents angular velocity.

θ̈

The symbol ______ represents angular acceleration.

Moment of inertia, I

The rotational counterpart of mass that measures a body's resistance to angular acceleration is called ______.

Torque, T

The rotational counterpart of force that tends to produce angular acceleration is called ______.

θ̇ = dθ/dt

The relationship between angular displacement and angular velocity is ______.

θ̈ = d²θ/dt²

The relationship between angular displacement and angular acceleration is ______.

Fixed-point rotational equation

When a point in a rotating body is fixed with respect to an inertial reference frame, the rotational equation can be written as ΣT = Iθ̈; this is the ______.

Simple pendulum

A mass concentrated at the end of a massless connecting rod of length l that rotates about a pivot is modeled as a ______.

m

In the simple-pendulum model, the symbol ______ represents the concentrated mass at the end of the pendulum.

l

In the simple-pendulum model, the symbol ______ represents the distance from the pivot to the concentrated mass.

g

In the pendulum equations, the symbol ______ represents gravitational acceleration.

Tc

In the pendulum model, the externally applied control torque at the pivot is represented by ______.

I = ml²

For a point mass m located a distance l from the pivot, the moment of inertia about the pivot is ______.

mgl sin θ

The magnitude of the gravitational restoring torque acting on the simple pendulum is ______.

Tc - mgl sin θ = Iθ̈

The nonlinear equation of motion obtained by summing the applied control torque and gravitational torque for the simple pendulum is ______.

Tc - mgl sin θ = ml²θ̈

After substituting I = ml² into the pendulum equation of motion, the equation becomes ______.

θ̈ + (g/l)sin θ = Tc/(ml²)

The normalized nonlinear equation of motion for the controlled simple pendulum is ______.

Nonlinear equation

The pendulum equation θ̈ + (g/l)sin θ = Tc/(ml²) is classified as a ______ because of the sin θ term.

sin θ ≈ θ

For sufficiently small angular motion, the small-angle approximation used to linearize the pendulum model is ______.

Small-angle approximation

The approximation sin θ ≈ θ for sufficiently small θ is called the ______.

Linearization

The process of replacing a nonlinear relationship by an approximate linear relationship around an operating condition is called ______.

θ̈ + (g/l)θ = Tc/(ml²)

Using sin θ ≈ θ, the linearized equation of motion of the simple pendulum is ______.

Harmonic oscillator

With no applied torque, the linearized pendulum has the natural motion of a ______.

ωn

The symbol ______ conventionally represents the natural frequency of an oscillatory system.

ωn = √(g/l)

The natural frequency of the linearized simple pendulum is ______.

Natural frequency

The frequency at which the undriven linearized pendulum naturally oscillates is called its ______.

Θ(s)

The Laplace-domain representation of the pendulum angular displacement θ(t) is ______.

Tc(s)

The Laplace-domain representation of the applied control torque Tc(t) is ______.

Θ(s)/Tc(s)

The transfer function from applied control torque to pendulum angular displacement is represented by ______.

Θ(s)/Tc(s) = [1/(ml²)]/[s² + g/l]

The transfer function of the linearized simple pendulum from control torque Tc to angular displacement θ is ______.

1/(ml²)

In the linearized pendulum transfer function Θ(s)/Tc(s) = [1/(ml²)]/[s² + g/l], the numerator is ______.

s² + g/l

In the linearized pendulum transfer function, the denominator is ______.

Transfer function

A Laplace-domain relationship expressing the ratio of an output transform to an input transform under appropriate initial conditions is called a ______.

Step input

An input that changes suddenly from one constant level to another and remains there is called a ______.

Time response

The variation of a system output with time resulting from a specified input and initial conditions is called the ______.

Simulink

A block-diagram simulation environment used in the source to numerically simulate both linear and nonlinear system equations is ______.

Integrator, 1/s

In a Simulink dynamic model, the block used to mathematically integrate its input with respect to time is the ______.

Two integrators

Because angular acceleration θ̈ must be integrated once to obtain θ̇ and again to obtain θ, a second-order pendulum model requires ______ connected sequentially.

57.3

The approximate gain used in the Simulink diagrams to convert an angular quantity from radians to degrees is ______.

Radians to degrees

The gain 57.3 in the pendulum Simulink model performs the conversion from ______.

9.81 m/s²

The gravitational acceleration value used in the pendulum simulation is approximately ______.

θ̈ = -9.81θ + 1

For the linear pendulum example with m = 1 kg, l = 1 m, g = 9.81 m/s², and Tc = 1 N·m, the acceleration equation becomes ______.

θ̈ = -9.81 sin θ + 1

For the corresponding nonlinear pendulum example with m = 1 kg, l = 1 m, g = 9.81 m/s², and Tc = 1 N·m, the acceleration equation becomes ______.

Trigonometric function block

The Simulink block used to calculate sin θ in the nonlinear pendulum model is the ______.

Linear pendulum model

A pendulum model in which sin θ is replaced by θ using the small-angle approximation is called the ______.

Nonlinear pendulum model

A pendulum model that retains the sin θ gravitational term is called the ______.

Small θ

For ______, sin θ ≈ θ and the linear and nonlinear pendulum models produce very similar responses.

Large θ

As ______ increases, the approximation sin θ ≈ θ becomes less accurate and the linear and nonlinear responses diverge.

sin θ < θ

For positive θ away from zero, the source notes that gravitational restoring torque is reduced relative to the linearized prediction because ______.

Model validation

After designing a controller using a linearized model, testing its performance using a numerical simulation containing the significant nonlinearities is useful for ______.