Project 2 Linearization of a Nonlinear Mechanical System An undamped, one degree-of-freedom mechanical system with forces dependent on position is modeled by a second order differential equation тx" + f(x) — 0, (1) where x denotes the position coordinate of a particle of mass m and - f(x) denotes the force acting on the mass. We assume that f(0) the equivalent dynamical system. Iff(x) has two derivatives at x = 0, then its second degree Taylor formula with remainder at x = 0 is = 0 so x = 0i + 0j is a critical point of f(x) = f'(0)x + f"(z)x², (2) where 0 < z < x and we have used the fact that f(0) = 0. If the operating range of the system is such that f"(z)x²/2 is always negligible com- pared to the linear term f'(0)x, then the nonlinear equation (1) may be approximated by its linearization, mx" + f'(0)x = 0. (3) Under these conditions Eq. (3) may provide valuable information about the motion of the physical system. There are instances, however, where nonlinear behaviorof f(x), represented by the remainder term f"(z)x²/2 in Eq. (2), is not negligible relative to f'(0)x and Eq. (3) is a poor approximation to the actual system. In such cases, it is necessary to study the nonlinear system directly. In this project, we explore some of these questions in the context of a simple nonlinear system consisting of a mass attached to a pair of identical springs and confined to motion in the horizontal direction on a frictionless surface, as shown in Figure 4.P.2. The springs are assumed to obey Hooke's law, F,(Ay) = -kAy where Ay is the change in length of each spring from its equilibrium length L. When the mass is at its equilibrium position x = Thus in the rest state both springs are either elongated and under tension (h > 0) or at their natural rest length and under zero tension (h = 0). %3| 0, both springs are assumed to have length L +h with h > 0. х т L+ Ay FIGURE 4.P..2 Horizontal motion of a mass attached to two identical springs.
Project 2 Linearization of a Nonlinear Mechanical System An undamped, one degree-of-freedom mechanical system with forces dependent on position is modeled by a second order differential equation тx" + f(x) — 0, (1) where x denotes the position coordinate of a particle of mass m and - f(x) denotes the force acting on the mass. We assume that f(0) the equivalent dynamical system. Iff(x) has two derivatives at x = 0, then its second degree Taylor formula with remainder at x = 0 is = 0 so x = 0i + 0j is a critical point of f(x) = f'(0)x + f"(z)x², (2) where 0 < z < x and we have used the fact that f(0) = 0. If the operating range of the system is such that f"(z)x²/2 is always negligible com- pared to the linear term f'(0)x, then the nonlinear equation (1) may be approximated by its linearization, mx" + f'(0)x = 0. (3) Under these conditions Eq. (3) may provide valuable information about the motion of the physical system. There are instances, however, where nonlinear behaviorof f(x), represented by the remainder term f"(z)x²/2 in Eq. (2), is not negligible relative to f'(0)x and Eq. (3) is a poor approximation to the actual system. In such cases, it is necessary to study the nonlinear system directly. In this project, we explore some of these questions in the context of a simple nonlinear system consisting of a mass attached to a pair of identical springs and confined to motion in the horizontal direction on a frictionless surface, as shown in Figure 4.P.2. The springs are assumed to obey Hooke's law, F,(Ay) = -kAy where Ay is the change in length of each spring from its equilibrium length L. When the mass is at its equilibrium position x = Thus in the rest state both springs are either elongated and under tension (h > 0) or at their natural rest length and under zero tension (h = 0). %3| 0, both springs are assumed to have length L +h with h > 0. х т L+ Ay FIGURE 4.P..2 Horizontal motion of a mass attached to two identical springs.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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Question
Show that the
motion of the mass in Figure 4.P.2 is
mx" + 2kx[ 1 - (L / sqrt((L + h)^2 + x^2)) = 0
Transcribed Image Text:Project 2 Linearization of a Nonlinear Mechanical System
An undamped, one degree-of-freedom mechanical system with forces dependent on position
is modeled by a second order differential equation
тx" + f(x) — 0,
(1)
where x denotes the position coordinate of a particle of mass m and - f(x) denotes the
force acting on the mass. We assume that f(0)
the equivalent dynamical system. Iff(x) has two derivatives at x = 0, then its second degree
Taylor formula with remainder at x = 0 is
= 0 so x = 0i + 0j is a critical point of
f(x) = f'(0)x + f"(z)x²,
(2)
where 0 < z < x and we have used the fact that f(0) = 0.
If the operating range of the system is such that f"(z)x²/2 is always negligible com-
pared to the linear term f'(0)x, then the nonlinear equation (1) may be approximated by its
linearization,
mx" + f'(0)x = 0.
(3)
Under these conditions Eq. (3) may provide valuable information about the motion of the
physical system. There are instances, however, where nonlinear behaviorof f(x), represented
by the remainder term f"(z)x²/2 in Eq. (2), is not negligible relative to f'(0)x and Eq. (3)
is a poor approximation to the actual system. In such cases, it is necessary to study the
Transcribed Image Text:nonlinear system directly. In this project, we explore some of these questions in the context
of a simple nonlinear system consisting of a mass attached to a pair of identical springs
and confined to motion in the horizontal direction on a frictionless surface, as shown in
Figure 4.P.2. The springs are assumed to obey Hooke's law, F,(Ay) = -kAy where Ay
is the change in length of each spring from its equilibrium length L. When the mass is at
its equilibrium position x =
Thus in the rest state both springs are either elongated and under tension (h > 0) or at their
natural rest length and under zero tension (h = 0).
%3|
0, both springs are assumed to have length L +h with h > 0.
х
т
L+ Ay
FIGURE 4.P..2
Horizontal motion of a mass attached to two identical springs.
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