Basics of vibration, calculation method of equation of motion, and usage examples

Understanding the basics of vibration and the calculation method of the equation of motion can be very useful in many fields, from engineering to the natural sciences.

This guide will explain these concepts simply and include some usage examples to make these ideas clearer.

Let’s dive in and explore these key aspects of physics and engineering.

What is Vibration?

Vibration refers to the oscillatory motion that occurs when a system moves back and forth around a point of equilibrium.

In simpler terms, it means something moves or shakes repeatedly in a specific pattern.

Vibrations occur in many everyday objects, such as the strings of a guitar, car engines, or even in an earthquake.

When you pluck a guitar string, it vibrates to produce sound.

These vibrations are essential in countless applications and can be both useful and problematic.

Types of Vibration

Free Vibration

Free vibration occurs when a mechanical system is temporarily disturbed from its equilibrium position and then allowed to vibrate without any continuous external force applied.

A common example of free vibration is a swing moving back and forth after someone pushes it and lets go.

Forced Vibration

Forced vibration happens when a system is continuously subjected to an external force.

This external influence keeps the system vibrating.

A good example is when you turn on a washing machine and it shakes due to the force of the motor.

Damped Vibration

Damped vibration includes a resistance, like friction, that gradually reduces the motion or energy over time.

The vibrations eventually diminish until the system returns to a state of rest.

Imagine a pendulum in air.

Over time, air resistance slows it down until it stops completely.

Equation of Motion

To understand how systems vibrate, scientists and engineers use what’s known as the equation of motion.

This equation is a second-order differential equation essential for describing the dynamics of mechanical systems.

Newton’s Second Law

The fundamental basis of the equation of motion is Newton’s Second Law of Motion: F = ma, where F is force, m is mass, and a is acceleration.

This law explains how the movement changes when a force is applied.

To use this in vibration analysis, you replace F with the forces acting on the system, such as spring forces and damping forces.

Formulating the Equation

A typical linear equation of motion for a single-degree-of-freedom (SDOF) vibrating system is:
\[ m\ddot{x} + c\dot{x} + kx = F(t) \]
– \( m \) is the mass of the system.
– \( \ddot{x} \) is the acceleration or the second derivative of displacement.
– \( c \) is the damping constant.
– \( \dot{x} \) is the velocity or the first derivative of displacement.
– \( k \) is the stiffness constant.
– \( x \) is the displacement.
– \( F(t) \) is an external force as a function of time.

This equation helps us determine how the system moves over time considering mass, damping, stiffness, and external forces.

Solving the Equation of Motion

Depending on the complexity of the system, different methods exist to solve the equation of motion.

Analytical Method

For simpler systems, like those involving one mass or spring, analytical solutions can be derived using mathematical methods such as calculus.

These solutions provide a precise formula describing the motion over time.

Numerical Method

In more complex systems, numerical methods are often used.

These involve computer simulations and algorithms, like the finite element method, to generate approximate solutions.

These methods are useful when precise analytical solutions are difficult to obtain.

Practical Examples of Vibration and Motion Equations

Understanding vibrations and solving the equation of motion have practical applications in many fields.

Automobiles

Automobiles use vibrations in analyzing the suspension system, which affects ride quality and handling.

Engine vibrations are examined for both performance and comfort, and the motion equations are applied to model how suspension parts react to road conditions.

Buildings

Skyscrapers and bridges are designed to withstand wind and earthquakes using principles of vibration and motion equations.

Engineers must calculate if a structure can handle vibrations, ensuring safety and integrity.

Electronics

In electronics, vibrations play a role in device performance.

For example, in a smartphone, the haptic feedback relies on controlled vibrations to let users feel a response.

In each of these scenarios, the equation of motion helps predict how structures and machines will react to internal and external forces, thereby guiding design and safety decisions.

Conclusion

Vibration and the related equation of motion are core concepts that affect everything from the mundane to the massive.

Whether it’s the gentle vibration of a musical instrument or the calculations that keep a skyscraper secure in gusty winds, understanding these principles is crucial.

While the concepts may seem complex at first, breaking them down into their fundamental parts makes them accessible and useful.

These basics serve as a foundation for further study and application in various fields, enabling the design of safer, more efficient, and effective systems.