Topic: Harmonically Excited Vibration
Subject: Mechanical Vibration
Tool: Scilab & QCAD
The undamped system under harmonic force is a dynamic system subjected to external force or excitation. This excitation is called the forcing or excitation function. The excitation function is usually time-dependent and may be harmonic or nonharmonic.
This article presents a dynamic response of a single degree
of freedom system under the harmonic motions of the form
$$F(t) = F_0\,\cos(\omega\,t + \phi) \tag{1}$$
\(F(t)\) – excitation of forcing function
\(F_0\) – constant static force
\(\omega\) – excitation angular frequency
\(t\) - time
\(\phi\) – phase angle of harmonic excitation
If the force \(F(t)\) acts on an undamped spring-mass system
with \(\phi = 0\), the equation of motion is
$$m\,{y''} + k\,y = F(t) \tag{2}$$
Or
$$m\,{y''} + k\,y = F_0 \, \cos(\omega\,t) \tag{3}$$
\(m\) – mass
\(k\) – spring constant
\({y''}\) – acceleration of mass
\(y\) – displacement from rest position
The dynamic amplitude of the undamped spring-mass system is a
vibration response which refers to the maximum displacement under dynamic
conditions subjected to an oscillatory force or motion. This can be expressed
as
$$Y = \large \frac{\delta_{st}}{1 - \left(\frac{\omega}{\omega_n}\right)^2} \tag{4}$$
\(Y\) – dynamic amplitude
\(\delta_{st}\) – static deflection
\(\omega\) – excitation angular frequency
\(\omega_n\) – natural angular frequency
The static deflection and natural angular frequency are defined as
follows
$$\delta_{st} = \frac{F_0}{k}\;, \quad \omega_n = \sqrt{\frac{k}{m}} \tag{5}$$
The response of the system can be
identified to be of three types.
Case 1. When \(\mathbf{ 0 < {\omega}/{\omega_n}
< 1 }\), the harmonic response of the system \(y(t)\) is said to be lagging the external force.
Case 2. When \(\mathbf{ \omega / \omega_n > 1 }\), the response of the system to a harmonic force of very high frequency is close to zero. The response of the system is said to be leading the force.
Case 3. When \(\mathbf{ \omega / \omega_n = 1 }\), the response is called resonance and the amplitude of \(y(t)\) becomes infinite. This phenomenon can result on large displacements and stresses.
- Application
Consider a sample application illustrated in Figure 1, let us analyze its dynamic amplitude response in terms of excitation frequency based on the three types of responses.
Figure 1. Undamped System under Harmonic Force – Sample Problem
Below is the calculation of dynamic amplitude for the illustrated application with the use of Scilab script.
- Scilab Script
// Copyright (C) 2025 - Gani Comia // Date of creation: 20 Mar 2025 // Dynamic Amplitude from Undamped System under Harmonic Force clear;clc; // (1) dynamic amplitude function function Y = dynAmp(w) Y = delta./(1 - (w./wn).^2) endfunction // (2) primary parameters m = 100/9.8 // kg, mass k = 2000 // N/m, spring stiffness or constant F0 = 25 // N, static force (constant) delta = F0./k // m, static amplitude or deflection // (3) secondary parameters wn = sqrt(k./m) // Hz, natural angular frequency mprintf("Natural frequency, w_n = %3.1f",wn) // (4) frequency domain w_l = linspace(0,wn-0.05,100) // left side of natural frequency w_r = linspace(wn+0.05,60,100) // right side of natural frequency // (5) calculation of dynamic amplitude Y_l = dynAmp(w_l) Y_r = dynAmp(w_r) clf; // (6) figure properties fig = gcf() fig.figure_size = [700,700] // (7) plot of results plot(w_l,Y_l,"b-","linewidth",3.5) plot(w_r,Y_r,"m-","linewidth",3.5) plot([wn wn],[-1,1],"r--","linewidth",2) note1 = "Vibration Response of an Undamped System w/ Harmonic Force" title(note1,"fontsize",3.75) ylabel("$\Large\text{Dynamic Amplitude (m),}\;\;\mathbf{Y(\omega)}$") xlabel("$\Large\text{Mechanical Frequency (Hz),}\;\;\mathbf{\omega}$") xgrid(3,0) // (8) additional plot information note2 = "$\LARGE m\ddot{y}(t) + k y(t) = F_0\,cos(\omega\,t)$" xstring(34,0.425,note2) note3 = "$\LARGE Y(\omega\;<\;\omega_n)$" note4 = "$\LARGE Y(\omega\;>\;\omega_n)$" note5 = "$\LARGE Y(\omega\;=\;\omega_n)$" legend(note3,note4,note5,with_box=%F) note6 = "$\Large \text{Resonance}$" xstring(16,0.4,note6,-90) xstring(13.5,-0.8,note6,-90) // (9) axes properties ax = gca() ax.x_location = 'origin' ax.data_bounds = [0 -1; 60 1];
- Scilab Output (Figure 2)
Figure 2. Vibration Response to Undamped System with Harmonic Excitation
Figure 2 shows the dynamic amplitude in three types of responses to harmonic excitation. For the illustrated problem, a suitable forcing frequency \(\omega\) can be chosen based on the requirements. Resonance frequency shows an infinite displacement.
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Reference
- Singiresu S. Rao. Mechanical Vibrations. 2nd Ed. Addison-Wesley Publishing Company. 1990.
