Degrees of Freedom
In mathematics, degrees of freedom refer to the number of independent parameters or variables that can be varied in a given system or statistical model. It represents the flexibility or variability within the system and is crucial for determining the distribution of sample statistics and making inferences about population parameters. In essence, it quantifies the amount of information available for making estimations or predictions.
4 Key excerpts on "Degrees of Freedom"
- Nicoleta Gaciu(Author)
- 2020(Publication Date)
- SAGE Publications Ltd(Publisher)
...The Degrees of Freedom (df) concept refers to the number of choices one can make in repeated random samples that constitute the distribution, and it is equal to df = n – 1 for each sample. The concept of Degrees of Freedom is important in many inferential statistics tests. For example, it refers to the number of scores in a frequency distribution that are free to vary. Suppose we had a distribution of only three scores, a, b and c, and the sum of these scores is 10. We can have several different values that add up to 10: 2 2 and 6 1 3 and 6 3 5 and 2 Once any two of these values are fixed, then the third one cannot vary. Hence, we say that there are two Degrees of Freedom in this distribution – any two of the values are free to vary. There is a family of t -distributions, and there is one distribution for each number of Degrees of Freedom. The size of the sample indicates which t -distribution to use. For example, if the sample size n = 10, then df = 10 – 1 = 9, and we check the row labelled 9 in the statistical Table 7.1. We use the Degrees of Freedom alongside the desired confidence level (which will be presented in Chapter 8) to decide whether to reject or not a null hypothesis. For example, the critical t -value for 9 Degrees of Freedom and a probability of 95 per cent is equal to the value (1.833) found at the intersection between the ninth row and the third column (t 0.95). Table 7.1 t t Figure 7.9 shows the t -distribution with 9 Degrees of Freedom (dashed curve) and the normal distribution (solid curve). The area under the curve equals 1.00, the same as for the normal standard distribution, and the shape of the distribution depends on the Degrees of Freedom; the t -distribution has fatter tails than the normal distribution...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...7 Systems with More than One Degree of Freedom A degree of freedom is a coordinate, either translational or rotational, required to describe the motion of a mass in a system. We could have, for example, the problem of describing the general motion of a rigid body constrained to move on a plane. A complete specification of its motion would require two orthogonal translational coordinates to locate its center of mass plus a single rotational coordinate to specify its orientation on the plane. There are therefore three Degrees of Freedom for the body. If we released the constraint that it must move on a plane, we would need to deal with a rigid body moving in three‐dimensional space and would find ourselves with six Degrees of Freedom (three translations and three rotations). Alternatively, we might be dealing with a system that has multiple bodies, each with a single degree of freedom so that we need as many Degrees of Freedom as there are bodies. In any case, we will now be dealing with NDOF systems – systems with N Degrees of Freedom. We start with a discussion of systems with two Degrees of Freedom. These systems are relatively easy to consider and have all of the properties of systems with any number of Degrees of Freedom. 7.1 2DOF Undamped Free Vibrations – Modeling We start with a 2DOF system without damping as shown in Figure 7.1. There are two masses, and, that are constrained to have one‐dimensional, vertical, motion only. We can define the two Degrees of Freedom as and, the vertical motions away from the equilibrium positions of the masses. The effects of gravity can be ignored because they are completely canceled out by the preloads in the springs when the system is in equilibrium. As with 1DOF systems, we are only concerned with things that change when we move away from equilibrium and gravitational forces and preloads in springs don't change. Figure 7.2 shows free body diagrams for the system...
eBook - ePub- J. P. Den Hartog(Author)
- 2013(Publication Date)
- Dover Publications(Publisher)
...CHAPTER 2 THE SINGLE-DEGREE-OF-FREEDOM SYSTEM 2.1. Degrees of Freedom. A mechanical system is said to have one degree of freedom if its geometrical position can be expressed at any instant by one number only. Take, for example, a piston moving in a cylinder; its position can be specified at any time by giving the distance from the cylinder end, and thus we have a system of one degree of freedom. A crank shaft in rigid bearings is another example. Here the position of the system is completely specified by the angle between any one crank and the vertical plane. A weight suspended from a spring in such a manner that it is constrained in guides to move in the up-and-down direction only is the classical single-degree-of-freedom vibrational system (Fig. 2.3). F IG. 2.1. Two Degrees of Freedom. Generally if it takes n numbers to specify the position of a mechanical system, that system is said to have n Degrees of Freedom. A disk moving in its plane without restraint has three Degrees of Freedom: the x- and y -displacements of the center of gravity and the angle of rotation about the center of gravity. A cylinder rolling down an inclined plane has one degree of freedom; if, on the other hand, it descends partly rolling and partly sliding, it has two Degrees of Freedom, the translation and the rotation. A rigid body moving freely through space has six Degrees of Freedom, three translations and three rotations. Consequently it takes six numbers or "coordinates" to express its position. These coordinates are usually denoted as x, y, z, φ, Ψ, χ. A system of two rigid bodies connected by springs or other ties in such a manner that each body can move only along a straight line and cannot rotate has two Degrees of Freedom (Fig. 2.1). The two quantities determining the position of such a system can be chosen rather arbitrarily. For instance, we may call the distance from a fixed point O to the first body x 1 and the distance from O to the second body x 2...
eBook - ePub- James C. Anderson, Farzad Naeim(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
...Chapter 2 Single-Degree-of-Freedom Systems As mentioned in Chapter 1, in many cases an approximate analysis involving only a limited number of Degrees of Freedom will provide sufficient accuracy for evaluating the dynamic response of a structural system. The single-degree-of-freedom (SDOF) system will represent the simplest solution to the dynamic problem. Therefore, initial consideration will be given to a system having only a single degree of freedom that defines the motion of all components of the system. For many systems, this will depend on the assemblage of the members and the location of external supports or internal hinges. In order to develop an SDOF model of the actual structure, it is necessary to reduce the continuous system to an equivalent discrete system having a displaced shape that is defined in terms of a single displacement coordinate. The resulting representation of the actual structure is often referred to as the discretized model. 2.1 Reduction of Degrees of Freedom In order to reduce the Degrees of Freedom, one of the following methods is often used: 1. Lumped parameters 2. Assumed deflection pattern 3. A deflection pattern based on the static deflected shape A common practice is to lump the participating mass of the structure at one or more discrete locations. For example, in the case of a simply supported beam, the participating mass may be located at the center of the simple span with the stiffness represented by a weightless beam with distributed elasticity. Alternatively, the same beam may have the elasticity represented by a concentrated resistance (spring) at the center of the span connecting two rigid segments having distributed mass. For many simple dynamic systems, it may be possible to accurately estimate the displaced shape...
