Įquation ( 8.6) gives us the two natural frequencies for our two degrees of freedom system. This is known as the characteristic equation for this system and can be seen to be a quadratic equation for. This is a result from linear algebra and is required for equation ( 8.4) to have a non-trivial solution. This is called the trivial solution and is not of interest here. This implies that there is no motion (see equation ( 8.3)). Where and are the amplitudes of motion of masses 1 and 2 respectively. We assume the motion of the masses is of the form To solve ( 8.2) we look for solutions in which both masses are undergoing harmonic motion at the same frequency (known as s simple simultaneous harmonic motion). ![]() As a result it is not possible to solve either ( 8.1a) or ( 8.1b) independently. It will be convenient to write these equations in matrix form, introducing the concept of mass and stiffness matrices, asĪn important feature of these equations is that they are coupled, meaning that both and appear in each of these equations. So that the two equations of motion for the system are To make things easier, you may also assume if you wish that one of the coordinates is larger than the other, say assume, although it is not necessary to do so.įigure 8.5 show the FBD/MAD for each of the masses in the system.įigure 8.5: FBD/MAD’s for two degree of freedom system As with single degree of freedom systems, when drawing FBD/MAD’s we generally assume that all of the coordinates and their derivatives are positive to get the correct signs in the equations of motion. Here we will focus on the use of Newton’s Laws. There are many ways to find these equations, and most of these methods are beyond the scope of this course. There is usually one equation of motion per degree of freedom in the system. Since this is a two degree of freedom system, we will get two equations of motion. Having selected the coordinates ( and ) to describe the configuration of the system, the next step is to find the equations of motion in terms of the chosen coordinates (and their derivatives). As a result, we expect that if the masses are initially displaced with and released from rest, the resulting motion would be such that and the oscillations would occur at a frequency of Each mass in Figure 8.4 therefore is supported by two springs in parallel so the effective stiffness of each system is. Since one half of the middle spring appears in each system, the effective spring constant in each system is (remember that, other factors being equal, shorter springs are stiffer). The fixed boundary in Figure 8.4 has the same effect on the system as the stationary central point. Once the two degree of freedom system is understood, extensions to systems with more degrees of freedom is straightforward.įigure 8.4: Simple two degree of freedom system with central point fixed This will help to illustrate all of the important features of MDOF systems while keeping the development as simple as possible. We will begin our discussion of MDOF systems by considering two degree of freedom (TDOF) systems which are the simplest. There are many parallels between single and multiple degree of freedom systems. While there are numerous systems that can be reasonably modeled as having a single degree of freedom, there are many other systems that require a more detailed model.ĭiscrete vibrating systems are classified as either single degree of freedom (SDOF) or multiple degree of freedom (MDOF) systems. Up to now, all of the systems that we have considered have been single degree of freedom systems for which one coordinate is sufficient to completely specify the configuration of the system. Open Educational Resources Multiple Degree of Freedom Systems:įree Vibrations of Two Degree of Freedom Systems Application to Lateral Vibrations of Beams.Approximate Methods for Continuous Systems.Kinetic and Potential Energies in Multiple Degree of Freedom in Systems. ![]()
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