1607161

Centro de Investigacion

Assignment 2: Frequency Response Function of a SDOF System

Sergio Ortega Cheno February 17 of 2015, Structural Dynamics, Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León

Abstract The following document corresponds to the assignment number two of the Structural Dynamics class, which consists in the analysis of a Single Degree of

Freedom system. The analysis corresponds to the obtainment of the time response of damped and undamped systems and to the characterization of the dynamics of the system behavior, including the Bode diagram, which consists in the magnitude and phase angle analysis, plotting both of them against the frequency ratio, and the obtainment of the dynamic parameter of receptance. Characteristics of the system, as mass, stiffness and damping coefficient are varied to show how the receptance graphic of the system changes its behavior, and identify which and how the frequency zone is affected.

I. INTRODUCTION

The single degree of freedom systems are the simplest

vibratory systems, and can be described as a single mass

connected to a spring modeled as a linear spring, which

provides a restoring force, and a damper modeled as a

viscous damper, which provides a damping force

proportional to a relative displacement and acting in the

direction against the velocity vector. The system will vibrate

when an external force is applied to the mass, which would

only be able to travel along the spring elongation direction;

this is why it’s called a single degree of freedom.

A Mass-Spring-Damping system is represented as

follows:

The system’s equation of motion is given by:

Where m is the mass, is the acceleration, c is the

damping coefficient, is the velocity, k is the stiffness of

the spring and x is the displacement of the mass. This

equation can also be expressed in terms of the natural

frequency and viscous damping ratio. Then the motion of

the system will be given by:

Where is the natural frequency of the system and is

the viscous damping ratio.

In order to characterize the dynamics of a system, the

frequency response is need to be obtained, which is the

quantitative measure of the output spectrum in response to

an external force “F(t)” acting as a stimulus of the system’s

behavior. The measure of magnitude and phase of the output

as a function of frequency describe the frequency response,

which is represented in a Bode Diagram.

One of the dynamic parameters in order to study how

mass, stiffness and damping coefficients affect the dynamic

of our system is the receptance, which is the displacement

per unit of force, i.e. the receptance will be given by the

magnitude of the steady state amplitude.

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For a SDOF system, the steady state amplitude is given by:

In order to plot the receptance behavior of the system and

analyze the way it’s being modified by the variation of its

properties it is necessary to monitor it against a frequency

vector given by:

Where w is the excitation frequency and wn is the natural

frequency of the system.

II. BACKGROUND

In order to analyze a system’s behavior, a program was

developed, where the variables that interact in the system

are the following:

√

⁄

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Free Vibration

The diagram represents a

M-K-C system, which is

submitted to an external

force, where the force is

acting as a unitary impulse

F(t) =1, so the system will

be allowed to vibrate freely

after submitted to the force.

The equation of motion of

the system will be given by:

The time response of the system will depend on the

amount of damping . There are 4 different cases and

behaviors that the system can experiment, given by the

value of the damping, these are:

√

Forced Vibration For the second case, the force will be given as

leaving the equation of motion of the system as:

This represents a system submitted to harmonic forced

vibration. The total response of the system will be given by

the sum of the free response and steady state forced

vibration.

This response will be analyzed, varying the parameters

of mass, stiffness and damping coefficient in order to

analyze the behavior of the receptance of the system. The

zones of low frequency, resonance zone and high frequency

will be affected depending on the parameter that is varying.

After declaring the variables that will be involved in our

system we proceed to develop a program to obtain the

dynamic parameters of the system. The method that we will

use is the transitory excitement, which requires applying the

Frequency Response method as well, in order to obtain the

FRF’s. The excitation consisted in an impulsive force

applied directly to the system plotting the time response. In

agree with the fundamental relation of the Frequency

Response Method, the relation between the Fourier

transform of the response X(ω) and the excitation F(ω) is

the Frequency Response Function H(ω), also called the

transfer function of the system.

The measured response can be the displacement,

velocity or acceleration, where the Frequency Respond

Function will be the Receptance, Mobility or Accelerance,

respectively. In this case the Frequency Response Function

that will be analyzed is the Receptance.

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In order to obtain the behavior of the system when

referring to receptance, this parameter must be plot against

logarithmic scale of frequencies (

) for the x-axes.

III. RESULTS

Undamped system (

Figure 1

Under damped system (

Figure 2

Critically damped system (

Figure 3

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Over damped system

Figure 4

Bode Diagram for a MKC system

Magnitude

Figure 5.a

Phase

Figure 5.b

Receptance plot with a Mass Variation (m = 1:1:10)

Figure 6

Receptance plot with a Stiffness Variation (k = 1:1:10)

Figure 7

Receptance plot with a Damping coefficient variation (c = 0.1:0.1:1.5)

Figure 8

IV. DISCUSSION In Figure 1 the displacement behavior of an undamped system

is plotted in function of a defined period of time. The graphic

represent five different lines, which represent the displacement

behavior of the same system but with a different value for its

natural frequency. The number of oscillation per second of the

system will increase as the natural frequency increases, but the

amplitude of the oscillation would not be affected, as this

characteristic of the system can only be affected by the damping

ratio of the system, which is zero in this case. As we can see, any

of the displacement representation reaches a steady state, that

because of the lack of damping.

In Figure 2, 3 and 4, the displacement behavior of

underdamped, critically damped and overdamped systems is

represented, respectively. Five lines are plotted, each of them

represent the behavior of the system for a different established

natural frequency. In the all of the cases, as the natural frequency

of the system increases, the system will reach its steady state in a

smaller period of time, i.e. the amplitude of the displacement will

decrease faster as the natural frequency of the system reduces.

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As the settling time of any damped system is given by:

√

The Bode Diagram presented above in Figure 5, is

serving to characterize the frequency response of our

analyzed system. Figure 5.a corresponds to the magnitude

of the function, where the module of the transfer function is

plotted in decibels in function of the frequency in

logarithmic scale. As we can see, the magnitude of the

frequency response is reaching its highest value in a certain

value of frequency, this will be the frequency ratio of the

system at the system will enter in resonance. When varying

the damping ratio we can see how the magnitude of the

receptance is decreasing as the damping ratio increases in

the resonance zone.

On the other hand, as the magnitude plot, the phase

angle of the frequency response is plotted in function of the

angular frequency in a logarithmic scale (Figure 5b). This

plot allows us to analyze the displacement in phase of a

systems behavior at the output with respect to the input. As

seen in the figure the behavior of the system at the output

with respect to the input decreases its variation as the

damping ratio increases.

The behavior of the Frequency Respond Function of the

displacement of a system can be controlled by three

different parameters, which are the mass for the high

frequency zone control, the stiffness for the low frequency

zone control and the damping ratio for the resonance zone

control. Varying these parameters we can manipulate the

zone we wish to. As we can see in Figure 6, the resonance

and low frequency zones do not vary much, but the high

frequency zone does, as the mass increases it value, the

high frequency zone grows as well. In Figure 7, we can

observe how the low frequency zone of the system

receptance behavior increases as the stiffness coefficient

increase its value. Finally, analyzing Figure 8, we can

observe how the low and high frequency zone do not vary at

all, as the mass and stiffness remain constant, but how the

magnitude of receptance decrease its highest value as the

damping coefficient increases.

V. CONCLUSIONS

The purpose of this report was to analyze the time and

frequency responses of a given system, which, in this case,

was a single degree of freedom system. The functions

obtained and plotted can be used in vibration analyses and

modal testing, of different structures and elements, as

buildings when exposed to vibrations during earthquakes

and airplane wings when exposed to the aerodynamic forces

when in flight. The purpose of time response functions is to

obtain the period of time that will take to a system to reach a

steady state after submitted to an impulse type force. The

purpose of the analysis of the frequency response function is

to identify natural frequencies, damping ratios and mode

shapes of the structures. The natural frequency, i.e. the

frequency at which any structure will oscillate if disturbed

form its rest position and then allowed to vibrate freely, is

an important fact when analyzing the structure, as resonance

occurs when the applied force or base excitation frequency

coincides with the natural frequency structural natural

frequency.

With the graphics obtained in the Bode Diagram and

receptance behavior of the system we can establish the

frequencies at which resonance will occur. During resonant

vibration, the response displacement may increase until the

structure or element analyzed experiences buckling, yielding

or some other failure in the mechanism. This is why it is so

important to have in consideration this zone.

REFERENCIAS

[1] Ledezma, D. , Structural Dynamics SDOF Systems Universidad Autónoma de Nuevo León. [2] Irving, T., An introduction to Frequency Response Functions [online], Consult Date: February 16, 2015 http://www.vibrationdata.com/tutorials/frf.pdf [3] Roy R. Craig, Andrew J. Kurdila “Fundamentals of Structural Dynamics”, 2nd Edition, July 2006

VI. ANNEX Transmissibility Derivation