Forced vibration

Forced vibration results when a periodic force F sin ωt is applied to the mass (m) of the structure, where ω is the angular velocity (the frequency) of the force and F is the magnitude of the force.

When the structure is in static equilibrium, the weight of the mass (mg) is equal to the spring force (Δk), as shown in the following graphic. The spring force is defined as the product of a stiffness constant, k, and the elongation of the spring at rest, Δ.

In a forced vibration, the structure vibrates with the frequency of the periodic force applied to the structure and magnitude that is dependent on the structure itself and not the applied force.

When the frequency of the periodic force is the same as the natural frequency of the structure, resonance occurs. This can cause problems as the amplitude of the motion can continue growing as long as the force is applied.

Undamped forced vibration

In the case of undamped forced vibration, the mass oscillates at the forced frequency around the point of equilibrium indefinitely as there is no energy dissipation.

In this case, the resultant force on the structure is the sum of the forces inherent to the system plus the periodic force and is equal to mass times acceleration:

F sinωt + mg - k (Δ + x) = m

where t the time, x the displacement, and the acceleration.

Recalling that at equilibrium the forces on the structure give mg = kΔ, the equation of motion for an undamped forced vibration can be written as

m + k x = F sinωt

In this case the force term has a nonzero value since the force applied to the structure has a magnitude different than zero.

Damped forced vibration

In the case of damped forced vibration, the mass oscillates around the point of equilibrium with a magnitude that declines to zero due to dissipation of energy at the frequency imposed by the periodic force.

Because a friction force is directly proportional to the velocity of the mass, the term for damping is obtained by multiplying the damping constant c by velocity . Damping is introduced as a negative value into the resultant force. The expression for the resultant force is then set equal to mass times acceleration:

F sinωt + mg - k (Δ + x) - c = m

Recalling that at equilibrium the forces on the structure give mg = kΔ, the equation of motion for a damped free vibration can be expressed as a second order differential equation:

m + c + k x = F sinωt

This is the general equation of motion for a single degree-of-freedom structure under forced vibration.

Learn more

Pre-test solution process

Correlation solution process

Look up more details

Vibration

Free vibration

Vibration and DOFs, modes, and mode shapes

Real structures and vibrational modes

Quick links

Command reference

Pre/Post video examples

Bulk Entry Descriptions

Simcenter 3D tutorials

Browse Simcenter 3D help by product area

Forced vibration , Simcenter 3D 2021.1 Series

© 2020 Siemens

window.mainLanguage="en_US"

window.delivId=""

window.projectId=""

MathJax.Hub.Config({ TeX: { extensions: ["autoload-all.js"] }, tex2jax: { displayMath: [ ] }, "SVG": { scale: 125 } });