Forced response of system without damping

The differential equation of forced vibration system without damping with one degree of freedom, which is subject to jednofrekventnoj excited states:

 
If ω is different from ωn, then a method of undetermined coefficients to obtain the particular solution of the equation:

When a homogeneous solution of differential equations defined by expression 3.7 added the particular solutions and incorporate the initial conditions we obtain a function of forced vibration system with one degree of freedom:

Figure 3.3 - Response system without damping. The diagram shows a homogeneous, particular and total solution, and it is obtained as the sum of the homogeneous and particular.

Differential equation of forced vibration

Differential equations that describe the forced vibration system with one degree of freedom can be divided into :1) The free body diagram ,2 ) Method of equivalent systems .Excitation forces are non-conservative . Operation which cause non-conservative forces in the system can be calculated as follows :

The work that we carry out non-conservative external forces can be included in the analysis . Application of this method leads to the angular momentum of the system which describes the motion snaps degree of freedom . The method of equivalent systems can also be used to obtain the differential equations describing the motion of a single degree of freedom .

If the differential equation which describes the forced vibration ekvivalnentnih system with one degree of freedom dispensation equivalent weight gain :
 

 

The solution of the previous differential equation consists of a homogeneous part of the solutions that we get when the right-hand side of the equation equal to zero and paratikularnog solutions . Homogeneous solution for the value of the damping ratio ζ < 1 reads as follows : = " read " p = " p " >

Forms of the particular solution depends on the Feq ( t ) . If ζ > 0 then the value of the homogeneous riješnja tends to zero as the time interval increases . The presence of a homogeneous solution in the main leads to the solution of the initial tranzijetnih gbianja that very short runs . Stationary response system after tranzijetnog motion significantly decreases . Only the particular solution contributes to the steady response of a linear system .

The harmonic excitation system with one degree of freedom

Forced vibration of single degree of freedom arise when work is done on the system until the system vibrates . An example of forced vibration displacements are the foundation during an earthquake or movement caused by being balanced reciprocal components .
The following figure shows the equivalent system of forced vibration with one degree of freedom. This is a generalized coordinate system shift .

 Figure 3.1 - Forced vibration of single degree of freedom

Figure 3.2 . Free body diagram : a) the external forces that function at the system , b) the internal forces

Applying Newton 's law to diagram free system , we get a differential equation that describes the leaf spring system .

Excitation of such a system is periodic , if there is T such that :

for all t . Examples of periodic excitation is shown in Figure 3.1 . Frequency periodic excitation can be calculated using the following formula :

Single-frequency excitation has the form:

where Fo is the amplitude of excitation , omega is the frequency , phase and dogs . The frequency of excitation is independent of the size of the natural frequencies

Free vibration of single degree of freedom systems

Free vibrations are oscillations about a system’s equilibrium position that occur in the absence of an external excitation. Free vibrations are result of a kinetic energy imparted to the system or of a displacement from the equilibrium position that leads to a difference in potential energy from the system’s equilibrium position.

Let’s consider a model of one degree of freedom system which is shown in Fig 2.1. This system consists of a block of mass m and the spring with stiffens k. When block is displaced a distance a from its equilibrium position, a potential energy ka^2/2 is developed in the spring. When the system is released from equilibrium, the spring force draws the block toward the system’s equilibrium position, with the potential energy being converted to kinetic energy. When the block reaches its equilibrium position the kinetic energy reaches a maximum and the motion continues. The kinetic energy is converted to potential energy until the spring is compressed a distance a. This process of transfer of potential energy to kinetic energy and vice versa is continual in the absence of non-conservative forces.

Remainder:

            Conservative force is a force with the property that the work done in moving a particle between two points is independent of the taken path. For example when you lift a book, the work that you do against gravity in lifting is stored and is available for kinetic energy of the book once you let go. Some conservative forces are: Gravity, Elastic (Hooke’s Law), electric etc.

            Non-conservative force is a force with the property that the work done in moving a particle between two points is dependent on the taken path. For example friction is a non-conservative force. When you move an object the work that you do “against friction” is apparently lost-it is certainly not available to the object as kinetic energy. In general any friction type force, like air resistance is non-conservative force.

            Examples of free vibrations of systems that can be modeled using one degree of freedom include the oscillations of a pendulum about a vertical equilibrium position, the motion of a recoil mechanism of a firearm once it has been fired, and the motion of a vehicle suspension system after the vehicle encounters a pothole.

            Vibrations of the system with one degree of freedom can be described with the second order differential equation. The independent variable is time, while the dependent variable is the chosen generalized coordinate. The chosen generalized coordinate represents the displacement of a particle in the system or an angular displacement and is measured from the system’s equilibrium position.        

            We can derive differential equation by applying one of two methods and these methods are:

1)      The Free Body Diagram Method or

2)      Equivalent System Method

The general solution of the differential equation is a linear combination of two linearly independent solutions. The arbitrary constants, called constants of integration, are uniquely determined on application of two initial conditions. The necessary initial conditions are values of the generalized coordinate and its first time derivative at specified time, usually t = 0.

     The form of the solution of the differential equation depends on system parameters. That means that solution of system with no damping will be different from the solution with viscous damping.