Lab5A Vibration Theory

MAE 244 Introduction to Vibration (written by Julio A Noriega, slightly modified) Lab 5-A s

Introduction to Vibration Theory

Vibration is a terminology used to describe a broad range of phenomena, both natural and man-made, ranging from the oscillating motion of the atoms, the swaying back and forth of the water in a beach, to the beautiful violin sounds and the rattling of the steering wheel of a car in motion. Our bodies are an embodiment of vibratory phenomena. We can not even say ‘vibration’ properly without the tip of the tongue oscillating.

Vibration prevails in man-made devices, machines, and transportation systems such as automobiles, airplanes and satellites as every successful engineering design must address vibration problems. The vibration of mechanical systems may be caused by sudden or continuos disturbances, such as aerodynamics forces on airplanes, oscillations of the internal combustion engines in automobiles, etc. Qualitatively speaking, the energy contained in the disturbance is transmitted to the mechanical systems and finds its way to propagate throughout the system. The energy carried by mechanical vibrations from one part of the system may reverse its transmission path and reverse its flow path.

The designer must evaluate whether any component of the mechanical system break during the initial violent disturbance stages, subsequent steady-state vibration, degradations of system performance during the post-disturbance period, and potential fatigue failure due to prolonged vibrations.

Undamped Free Vibrations for Single-Degree-of-Freedom SystemsThe simplest vibratory system consists of an elastic member and a mass element,

as represented in Fig. 1a. This is a single-degree-of-freedom system since it can move in only one coordinate; that is, it requires only coordinate x to define its configuration. Since there is no external force to drive the system, the motion is designated as a free vibration. It is also undamped, as no condition is present which would inhibit the motion.

Now consider a free-body diagram of the mass in Fig.1a, with the massless spring elongated from its rest, or equilibrium position. The mass of the object is m and the stiffness of the spring is k. Assuming that the mass moves on a frictionless surface along the x direction, the only force acting on the mass in the x direction is the spring force. As long as the motion of the spring does not exceed its linear range, the force in the x direction equal the product of mass and acceleration:

(1)

1

Fig 1a Single-degree-of-freedom systems


(2)

One of the goals of vibration analysis is to be able to predict the response, or motion, of a vibration system. Thus it is desirable to calculate the solution to equation (2), which can be written as:

(3)

Where the natural frequency is defined as:

Equation (3) is a homogeneous linear differential equation with constant coefficient. Since it is of second order, the solution must contain two arbitrary constants.

(4)

The arbitrary constant A and B can be determined from the initial conditions of the motion

When t = 0

(5) (6)

Substituting Eqs. 5 and 6 into Eq. 4, its time derivative will evaluate the arbitrary constants as

(7)

(8)So the solution becomes

(9)

Using trigonometric relations we can transform to

(10)

2


where X represents the amplitude of the displacement and is the phase angle as defined by:

(11)

(12)

The motion represented by Eqs. 9 and 10 is said to be harmonic, because of its sinusoidal form. The motion is repeated (see Fig.2), with the time for one cycle being defined by the value of t equal to 2. Thus the period , or the time for one cycle, is given by

= 2/ (13)

The reciprocal of expresses the frequency f in cycles per unit time. Thus f = /2 (14)

Because the solution is a circular function. The term is designated as the circular frequency. It is measured in radians per second.

Damped Free-Vibration for Single-Degree-of-Freedom SystemThe vibrations considered in the preceding section were self-sustaining and would

not increase, diminish, or change in character with time. That is, there was not source which would excite the system and hence increase the amplitude of the vibration, nor was there any form of resistance that would dissipate energy and reduce the oscillation in any way. A consideration of practical cases, however, would reveal that this condition is not realistic, since all vibrations gradually lose amplitude and eventually cease altogether, unless, they are maintained by some external source. Since the amplitude of a free vibration slowly dies away, something must cause energy to be removed from the system. The vibration is said to be damped, and the means of energy removal is called damper.

(15)

3

Figure 2 undamped free vibrations for single-degree-of-freedom systems [1]


where the damping constant c is the resistance developed per unit velocity.

Coulomb or dry-friction damping is encountered when bodies slide on dry surfaces

(16)

Hysteresis damping which is also called solid or structural damping is due to internal friction of the material.

Free Vibration with Viscous Damping

Damping that produces a damping force proportional to the mass's velocity is commonly referred to as "viscous damping" as shown in Figure 3.

Using Newton’s Second Law.

(17)

which can be rearranged as

(18)

The solution of this equation (18) is

(19)

To simplify the solutions coming up, we define the critical damping constant cc as,

(20)

4

Figure 3 Free vibration systems with viscous damping


or

(21)

For a damped system the ratio of the damping constant to the critical value is a dimensionless parameter which represents a meaningful measure of the amount of damping present in the system. This ratio is called the damping factor, . It is defined by

(22)

whence

(23)

Three are three main forms of damping [1]:1). If c2 - 4mk < 0, the system is termed “underdamped”. The roots of the characteristic equation are complex conjugates, corresponding to oscillatory motion with an exponential decay in amplitude.2). If c2 - 4mk =0, the system is termed “critically-damped”. The roots of the characteristic equation are repeated, corresponding to simple decaying motion with at most one overshoot of the system's resting position.3). If c2 - 4mk > 0, the system is termed “overdamped”. The roots of the characteristic equation are purely real and distinct, corresponding to simple exponentially decaying motion.

Underdamped System

When c2 - 4mk < 0 ( 1), the solution of the equation (18) can be written as

(24)

Alternatively, the solution may be expressed by the equivalent form:

(25)

where

5


(26)

d represents the damped circular frequency. Note that d will equal to when the damping of the system is zero (i.e. undamped).

Equation (25) demonstrates that the displacement amplitude decays exponentially (Fig.4), i.e. the natural logarithm of the amplitude ratio for any two displacements separated in time by a ratio is a constant. In this curve, the logarithmic decrement is expressed by:

(27)

Where xj is the amplitude after j cycles; xj+1 is the amplitude after j+1 cycles.

(28)

(29)

The damped period is:

6

Figure 4 Vibration decay with time in the underdamped system [1]

http://www.efunda.com/formulae/vibrations/sdof_free_undamped.cfm


System OverviewThe experimental dynamic system comprises the three subsystems is shown in

Figure 5. The first of these is the electromechanical apparatus which consists of the rectilinear mechanism, its actuator and sensors. The design features a brushless DC servo motor, precision rack and pinion drive (actuator, force generator), high resolution encoders (sensor), adjustable mass carriages (system mass) and reconfigurable system type (variable damping coefficient dashpot, springs with different stiffness).

The next subsystem is the real time controller unit which contains the digital signal processor (DSP) based real-time controller, sevo/actuator interfaces, servo amplifiers and auxiliary supplies. The controller also supports such functions as data acquisition, driving function shape generation, system health and safety check.

The third subsystem is the executive program which runs on a PC. This menu –driven program is the user’s interface to the system and supports driving function specification, input shape definition, data acquisition, plotting, system execution commands and more.

Figure 5 The experimental dynamic system

Ref [1] http://www.efunda.com/formulae/vibrations/sdof_free_damped.cfm

7

Documents

Lab5A Vibration Theory