With advancements in computers, several solvers, including online examples, have been developed. There are many ways to solve the type of ODE that is used to describe an SDOF system. Solving the ODE for a Single Degree of Freedom System The initial displacement can be set to zero, or some other value, usually depending on the engineering preference or the types of values that the engineer will need in subsequent calculations.īy solving this ODE, a plot of the system as a function of the time can be generated, which will show the position of the mass oscillating with the oscillations becoming less and less until the mass stops moving. As stated earlier, the initial velocity is usually zero, but it is not completely uncommon to have some initial velocity of the system.
Where t 0 is the initial time, x 0 is the initial displacement, and v 0 is the initial velocity. To solve this ODE, it is necessary to specify the initial conditions, which are usually as follows: If things like gravity and friction are to be included, they can be incorporated into the values of c v and k. This equation can be simplified as follows: Equation of motion for a Single Degree of Freedom SystemĪ mass suspended by a spring is a single degree of freedom system (SDOF).Īn SDOF system can be described by a second-order, non-homogenous, ordinary differential equation (ODE). This equation is referred to as the equation of motion of the system. This equation can be rearranged in countless ways, including to describe the motion of an SDOF system.
Where F is a force acting on some mass, m, causing some acceleration, a. The way the system moves can be described using an equation of motion which satisfies Newton’s Second Law. In this system, things like gravity and friction are often ignored, although both can be included. The damper, however, will reduce the magnitude of this oscillation until the mass is no longer moving. The system starts with some initial displacement and velocity, although usually, the initial velocity is zero.Īfter the mass is released, it will oscillate or vibrate in the x-direction as the spring is stretched and contracts. A mass is attached to a spring and a damper, which are both fixed at the opposite end. Describing a Single Degree of Freedom SystemĪn SDOF is often described using a damped spring-mass system in the x-direction. The simplest SDOF system often describes a system that moves only linearly in the x- or y-direction.
But beginning with a system that is constrained to moving in only one of these six ways provides a solid foundation for subsequent modeling of systems that move in multiple ways. To put this in perspective, an airplane moves in six degrees of freedom: it moves forward and backward in the x-direction, right and left in the y-direction, up and down in the z-direction, rolls about the x-axis, pitches about the y-axis, and yaws about the z-axis. This means that the system will only move in the x-, y-, or z-direction, or that the system will only rotate about the x-, y-, or z-axis. In the most general terms, an SDOF describes the motion of a system that is constrained to only a single linear or angular direction.
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