Dynamic Analysis of a Second-Order System with Harmonic Loading

differential equation

′′

(t)+2ζ

′

(t)+

u(t)

sin(ϖt)

system parameters

damping

0.11

stiffness

01.0

mass

10.0

load

01.0

load

0.016

initial conditions

initial

u(t)

+1.0

initial

′

(t)

+1.0

model information

frequency ratio ϖ/ ω n	0.32
natural frequency ω n	0.050	Hz
damped frequency ω d	0.050	Hz
natural period 2π/ ω n	19.869	sec
damped period 2π/ ω d	19.993	sec
damping coefficient c	00.702
magnification factor β	0001.108
static displacement Ϝ/k	0001.000
time constant τ	0003.162	sec

test cases

beating phenomenon

transient+steady state (combined)

time

200.0

(sec)

Argand

standard

This Demonstration gives a complete analysis of a second-order system with harmonic loading. The system's differential equation is

mu″+cu′+ku=f(t)

, where

f(t)=Fsin(ϖt)

is the mass of the system,

is the damping coefficient,

is the stiffness,

is the magnitude of the force, and

is the force frequency. The response

u(t)

is plotted as a function of time for the underdamped, critically damped, and overdamped cases. This Demonstration displays the transient response (the homogeneous part of the total solution), the steady state response (the particular part of the total solution), and the total response, which is the combination of the two. You can see the analytical solution for each case by moving the mouse over the response curve. Separate displays are given for the dynamic magnification factor and the phase of the response relative to the force. A number of pre-configured test cases can be chosen, to illustrate several important cases of system responses under different loading conditions.

Details

The equation of motion of a second-order linear system of mass

with harmonic applied loading is given by the differential equation

mu″+cu′+ku=Fsin(ϖt)

. There are 12 different analytical solutions depending on whether damping or loading is present and, if so, whether the system is underdamped, critically damped or overdamped.

The solution for each of the 12 cases was derived analytically and shown in the Demonstration, subject to the user's choice. Following are definitions of the relevant parameters. All units are in SI.

The damping ratio is

ξ=

, where

is the damping coefficient, such that

=2ωm

represents critical damping. The natural underdamped frequency is given by

ω=

, where

is the stiffness and

is the mass. The damped frequency of the system, defined for

ξ<1

, is

givenby

=ω

. The frequency ratio is

, where

is the forcing frequency. The dynamic magnification factor

is the ratio of the steady state response to the static response. The static response is given by

, where

is the force magnitude. The time constant is

τ=

ξω

and the damped period of oscillation is

2π

When the system is undamped and the load is harmonic, resonance occurs when

r=1

ϖ=ω

. When the system is underdamped and the load is again harmonic, practical resonance occurs when

ϖ=ω

1-2

and the corresponding maximum magnification factor is

β=

2ξ

. You can force the loading frequency to be equal to the natural frequency by clicking the button located to the right of the slider used to input the loading frequency. The forcing frequency

is expressed in Hz, but converted to radians per second internally.

This Demonstration also shows plots of the phase of the response

u(t)

relative to loading

sin(ϖt)

. The phase of the response lags behind loading by an angle

θ=-

-1

tan

2ξr

, which is plotted in the complex plane on an Argand diagram. The phase angle ranges from

0°

-180°

When the loading frequency

is set to zero, only

is used as the force

f(t)

. This allows a constant force loading,

. For example, by setting

ϖ=0

and

F=1

, a step response is obtained. To make the loading zero, the slider

is set equal to zero.

The Demonstration contains a number of pre-configured test cases to illustrate different loading conditions, such as beating phenomenon, resonance, practical response, impulse response, and step responses for different damping values.

References

[1] M. Paz and W. Leigh, ‪Structural Dynamics: Theory and Computation‬, 5th ed., Boston: Kluwer Academic Publishers, 2004.

[2] W. T. Thomson, ‪Theory of Vibration with Applications‬, Englewood Cliffs, NJ: Prentice-Hall, 1972.

[3] R. W. Clough and J. Penzien, Dynamics of Structures, New York: McGraw-Hill, 1975.

[4] R. K. Vierck, Vibration Analysis, Scranton, PA: International Textbook Company, 1967.

[5] A. A. Shabana, Theory of Vibration, Vol. 1, New York: Springer-Verlag, 1991.

[6] B. Morrill, Mechanical Vibrations, New York: Ronald Press, 1957.

External Links

Damped Simple Harmonic Motion (Wolfram MathWorld)

Critically Damped Simple Harmonic Motion (Wolfram MathWorld)

Underdamped Simple Harmonic Motion (Wolfram MathWorld)

Permanent Citation

Nasser M. Abbasi

"Dynamic Analysis of a Second-Order System with Harmonic Loading"
http://demonstrations.wolfram.com/DynamicAnalysisOfASecondOrderSystemWithHarmonicLoading/
Wolfram Demonstrations Project
Published: August 5, 2013