Consider the spring-mass-dashpot system mounted on a massless cart as shown in Figure 3-3. Let us obtain mathematical models of this system by assuming that the cart is standing still for 1 < 0 and the spring-mass-dashpot system on the cart is also standing still for t< 0. In this system, u(t) is the displacement of the cart and is the input to the system. Att = 0, the cart is moved at a constant speed, or ú = constant. The displacement y(1) of the mass is the output. (The displacement is relative to the ground.) In this system, m denotes the mass, b denotes the viscous-friction coefficient, and k de- notes the spring constant. We assume that the friction force of the dashpot is proportional to ý - ů and that the spring is a linear spring; that is, the spring force is proportional to y – u. For translational systems, Newton's second law states that

Consider the spring-mass-dashpot system mounted on a massless cart as shown in Figure 3-3. Let us obtain mathematical models of this system by assuming that the cart is standing still for 1 < 0 and the spring-mass-dashpot system on the cart is also standing still for t< 0. In this system, u(t) is the displacement of the cart and is the input to the system. Att = 0, the cart is moved at a constant speed, or ú = constant. The displacement y(1) of the mass is the output. (The displacement is relative to the ground.) In this system, m denotes the mass, b denotes the viscous-friction coefficient, and k de- notes the spring constant. We assume that the friction force of the dashpot is proportional to ý - ů and that the spring is a linear spring; that is, the spring force is proportional to y – u. For translational systems, Newton's second law states that

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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Question

Consider the spring-mass-dashpot system mounted on a massless cart as shown in Figure 3-3. Let
us obtain mathematical models of this system by assuming that the cart is standing still for t < 0and
the spring-mass-dashpot system on the cart is also standing still for t < 0. In this system, u(t) is the
displacement of the cart and is the input to the system. Att = 0, the cart is moved at a constant speed,
or ú = constant. The displacement y(t) of the mass is the output. (The displacement is relative to
the ground.) In this system, m denotes the mass, b denotes the viscous-friction coefficient, and k de-
notes the spring constant. We assume that the friction force of the dashpot is proportional to y - ů
and that the spring is a linear spring; that is, the spring force is proportional to y – u.
For translational systems, Newton's second law states that
ma = SF
where m is a mass, a is the acceleration of the mass, and EF is the sum of the forces acting on the
mass in the direction of the acceleration a. Applying Newton's second law to the present system
and noting that the cart is massless, we obtain
d'y
dy
du
- k(y - u)
%3D
dr?
dt
or
d'y
dy
du
+ ku
dt
+ b
+ky = b
m
dr?
dt
This equation represents a mathematical model of the system considered. Taking the Laplace
transform of this last equation, assuming zero initial condition, gives
(ms? + bs + k)Y(s) = (bs + k)U (s)
Taking the ratio of Y(s) to U(s), we find the transfer function of the system to be
bs + k
Y(s)
Transfer function = G(s) =
U(s)
ms? + bs + k
Such a transfer-function representation of a mathematical model is used very frequently in
control engineering.
Massless cart
ww

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