Dynamic Amplitude Response of an Undamped System under Harmonic Force

Topic: Harmonically Excited Vibration

Subject: Mechanical Vibration

Tool: Scilab & QCAD

By: Gani Comia, Mar. 2025

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

$$F(t) = F_0\,\cos(\omega\,t + \phi) \tag{1}$$

Where:
\(F(t)\) – 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

$$m\,{y''} + k\,y = F_0 \, \cos(\omega\,t) \tag{3}$$

Where:
\(m\) – 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

$$Y = \large \frac{\delta_{st}}{1 - \left(\frac{\omega}{\omega_n}\right)^2} \tag{4}$$

Where:
\(Y\) – 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.

Feel free to comment for inquiry, clarification, or suggestion for improvement. Drop your email to request the soft copy of the file.

Reference

1. Singiresu S. Rao. Mechanical Vibrations. 2^nd Ed. Addison-Wesley Publishing Company. 1990.