Displacement Periodic Function

Mar 14, 2023, 16:45 IST

Displacement periodic function is a type of periodic function that represents the displacement of a body from its mean position over time. It is an important concept in physics and engineering, and is used to describe a wide range of physical phenomena. In this article, we will discuss what a displacement periodic function is, how it is defined, and some of its important properties. Let’s have a look.

Table of Content
Definition Examples of Displacement Periodic Functions Properties Applications Frequently Asked Questions (FAQs)

Definition

A body's displacement from its mean position over time is represented by a function known as a displacement periodic function. It repeats itself on a regular basis since it is periodic. A displacement periodic function can be described mathematically as:

f (t) = A sin(wt + Φ)

where A is the function's amplitude, is its angular frequency, t is its time dimension, and is its phase angle. Being a simple harmonic motion, which implies that the body's displacement is proportional to the sine of its location, the displacement is represented by the sine function. The body's starting position at time t=0 is represented by the phase angle ().

Amplitude

The biggest deviation of the body from its mean position is the amplitude of a displacement periodic function. In the function, the letter A stands in for it. How far the body deviates from its mean position is determined by the amplitude, a measurement of the function's strength. The largest angle the pendulum swings through, for instance, would be the amplitude in the case of a basic pendulum.

Angular Frequency

The displacement periodic function's angular frequency quantifies how frequently the function repeats. It is determined by: and is expressed in radians per second.

ω = 2π/T

where, T is the function's period. The length of time (T) required for one cycle of the function. Because the angular frequency and period are inversely related, a higher angular frequency corresponds to a shorter period.

Phase Angle

The initial position of the body at time t = 0 is represented by the phase angle() in the displacement periodic function. It establishes the function's beginning point and is expressed in radians. For instance, if the phase angle is 0, the body begins at time t=0 in its mean position. The body begins at its maximum displacement at time t = 0 if the phase angle is /2.

Examples of Displacement Periodic Functions

Simple Pendulum - A simple pendulum is a system in which a mass is connected by a string to a fixed point. A displacement periodic function is used to depict the displacement of the mass as it swings back and forth in a periodic motion. The pendulum's greatest swinging angle is represented by the amplitude of the function, and its period measures how long it takes for a swing to complete. The function's angular frequency is provided by:

ω = √g/l, where,

g is the acceleration brought on by gravity, and L is the length of the string,

Spring Mass System - System with a mass coupled to a spring is referred to as a mass-spring system. The displacement of the mass is represented by a displacement periodic function, and it oscillates back and forth in a periodic motion. The mass's largest deviation from its mean location is represented by the function's amplitude, while the mass's cycle time is represented by the function's period. The function's angular frequency is provided by:

ω = √K/m

where m is the system's mass and k is the spring constant.

Guitar String Oscillations - A displacement periodic function can be used to mimic a guitar string's oscillations. The function's amplitude is the string's maximum deflection from equilibrium, and its period is the length of time it takes for a cycle to complete. The function's angular frequency is provided by:

ω = πf

where f is the string's frequency. The tension, mass, and length of the string all affect its frequency, which in turn affects the pitch of the music it makes.

Properties

There are several important properties of displacement periodic functions that are useful in understanding their behavior. These properties include:

Periodicity - Displacement periodic functions are periodic, meaning that they repeat themselves at regular intervals. The period of the function is the time taken for the function to complete one cycle.
Symmetry - Displacement periodic functions are symmetric about their mean position. This means that the displacement of the body is the same distance above and below its mean position.
Phase Shift - Displacement periodic functions can be shifted in phase by adding a constant to the argument of the sine function. This has the effect of shifting the function horizontally on the time axis.
Superposition - Displacement periodic functions can be combined through the principle of superposition. This means that the displacement of a system made up of two or more periodic functions is the sum of the individual displacements.

Applications

Displacement periodic functions are used in a wide range of applications in physics and engineering. Some examples of these applications include:

Acoustics - The behavior of sound waves can be modeled using displacement periodic functions. This is important in the design of musical instruments and in the study of the propagation of sound.
Mechanical Engineering - Displacement periodic functions are used to model the behavior of mechanical systems, such as engines, turbines, and pumps. This is important in the design and optimization of these systems.
Electrical Engineering - Displacement periodic functions are used to model the behavior of electrical systems, such as oscillators and filters. This is important in the design of electronic circuits.
Robotics - Displacement periodic functions are used to model the motion of robotic systems, such as robot arms and legs. This is important in the design and control of these systems.