This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MOTION IN A STABILITY REGION (PART I) 4 MOTION IN A STABILITY REGION (PART I) When motion is confined to one independent degreeoffreedom, the linearized equation that governs the motion is of the form (4 – 1) f kx x c x m = + + & & & In this section, we analyze Eq. (4 – 1) in more detail than in Chapter 1. The system’s free motion ( f = 0) is analyzed first and then its forced motion. The analysis performed for the free motion is called a transient analysis . The transient analysis is independent of the nonhomogeneous term f that appears on the right side of the differential equation. The nonhomogeneous term is often a force but it can also arise as a result of prescribing the displacement at a point in the system. The right side of the differential equation is generally called the excitation . It is shown how the time dependence of an excitation affects a system’s time response. We start with the constant excitation. Static loads, weight forces, and prescribed displacements are the most frequent examples. The constant excitation change’s a MAE 461: DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION (PART I) system’s equilibrium position. Next, we consider the harmonic excitation. Systems that contain unbalanced rotating elements, like a washing machine, milling machines, and rotating shafts, are systems that are acted on by harmonic excitations. The harmonic excitation causes a system to undergo harmonic motion. 1. Free Undamped Motion Figure 4 – 1: The massspringdamper system First consider the free undamped system (See Fig. 4 – 1). Letting f = 0 and c = 0 in Eq. (4 – 1) yields (4 – 2) = + kx x m & & Equation (4 – 2) is a homogeneous constant coefficient linear differential equation. As with any constantcoefficient linear differential equation, the solution is a combination of complex exponential functions. We start by looking at the single complex exponential function (4 – 3) st e x = MAE 461: DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION (PART I) where s is a complex number that needs to be determined. Substitute Eq. (4 – 3) and its second time derivative into Eq. (4 – 2) to get ) ( 2 = + st st ke e s m Dividing by st e (4 – 3) 2 = + k ms The values of s for which x = satisfies the differential equation are st e (4 – 4 a , b ) m k i s n n = ± = ω ω , where . 1 − = i The two solutions are (4 – 5) t i t i n n e x e x ω ω − = = 2 1 These two solutions may seem a bit odd; after all they’re complex. The complex solutions are actually just building blocks from which the real solution is constructed. Recall that and in Eq. (4 – 5) are complex harmonic functions of the form (See Fig. 4 – 2) t i n e ω t i n e ω − (4 – 6) t i t e t i t e n n t i n n t i n n ω ω ω ω ω ω sin cos sin cos − = + = − MAE 461: DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION (PART I) Figue 4 – 2: e st for pure imaginary s The real solution is a linear combination of the two complex solutions. From Eq. (4 – 5) and Eq. (4 – complex solutions....
View
Full
Document
This note was uploaded on 09/17/2009 for the course MAE 469 taught by Professor Silverberg during the Fall '08 term at N.C. State.
 Fall '08
 Silverberg
 Controls

Click to edit the document details