A usual model that pops out in engineering involves the response of a system to an external force. Take for instance a mechanical spring system modeled by the equation (ay" +by' + cy = f(t) y(0) = 0 y'(0) = 0 Here, y(t) denotes the displacement of the spring at time t, while the initial conditions tell us that the system is at rest (or is at equilibrium) and has zero velocity at the start t = 0 of the observation period. The forcing term f represents the external force applied to the spring at various time values. Hence, the solution y(t) represents the behavior of the system as a response to the external force applied to it. Imagine the spring being hit by a very forceful blow at t = 0. It can be thought of as like a hammer hitting a nail. The impact is instantaneous; the force is applied only at the moment t = 0. Hence, the forcing term is zero whenever t = 0 and has a very huge absolute value at t = 0 so that the total force is a finite constant. This force is usually denoted by the Dirac Delta function (even though it is not a function in the mathematical sense) 8. The solution y of (1) when f(t) = 8(t) is called the unit impulse response function. Recall that in the lecture, I mentioned the informal characterization of 8 as (+∞o, lo, and in a more formal sense, & is the function such that 8(t) = ∞0+. ·∞ +∞0 -∞0 t=0 t = 0' (1) 8(t) dt = 1 f(t) 8(ta) dt = f(a) Moreover, L{8(t)} = 1. Now, use the Method of Laplace transforms to find the unite response function of a mechanical system governed by (1) when a = 1 and b = c = 2. Then plot its graph over the interval [0, 100]. What happens to the system as time goes by? Does this behavior makes sense (from the physical point of view? Why say so?
A usual model that pops out in engineering involves the response of a system to an external force. Take for instance a mechanical spring system modeled by the equation (ay" +by' + cy = f(t) y(0) = 0 y'(0) = 0 Here, y(t) denotes the displacement of the spring at time t, while the initial conditions tell us that the system is at rest (or is at equilibrium) and has zero velocity at the start t = 0 of the observation period. The forcing term f represents the external force applied to the spring at various time values. Hence, the solution y(t) represents the behavior of the system as a response to the external force applied to it. Imagine the spring being hit by a very forceful blow at t = 0. It can be thought of as like a hammer hitting a nail. The impact is instantaneous; the force is applied only at the moment t = 0. Hence, the forcing term is zero whenever t = 0 and has a very huge absolute value at t = 0 so that the total force is a finite constant. This force is usually denoted by the Dirac Delta function (even though it is not a function in the mathematical sense) 8. The solution y of (1) when f(t) = 8(t) is called the unit impulse response function. Recall that in the lecture, I mentioned the informal characterization of 8 as (+∞o, lo, and in a more formal sense, & is the function such that 8(t) = ∞0+. ·∞ +∞0 -∞0 t=0 t = 0' (1) 8(t) dt = 1 f(t) 8(ta) dt = f(a) Moreover, L{8(t)} = 1. Now, use the Method of Laplace transforms to find the unite response function of a mechanical system governed by (1) when a = 1 and b = c = 2. Then plot its graph over the interval [0, 100]. What happens to the system as time goes by? Does this behavior makes sense (from the physical point of view? Why say so?
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![A usual model that pops out in engineering involves the response of a system to an external force. Take for
instance a mechanical spring system modeled by the equation
(ay" +by' + cy= = f(t)
y(0) = 0
y'(0) = 0
Here, y(t) denotes the displacement of the spring at time t, while the initial conditions tell us that the
system is at rest (or is at equilibrium) and has zero velocity at the start t = 0 of the observation period. The
forcing term f represents the external force applied to the spring at various time values. Hence, the
solution y(t) represents the behavior of the system as a response to the external force applied to it.
Imagine the spring being hit by a very forceful blow at t = 0. It can be thought of as like a hammer hitting a
nail. The impact is instantaneous; the force is applied only at the moment t = 0. Hence, the forcing term is
zero whenever t = 0 and has a very huge absolute value at t = 0 so that the total force is a finite constant.
This force is usually denoted the Dirac Delta function (even though it is not a function in the
mathematical sense) 8. The solution y of (1) when f(t) = 8(t) is called the unit impulse response function.
Recall that in the lecture, I mentioned the informal characterization of 8 as
t = 0
t = 0'
8(t) =
and in a more formal sense, & is the function such that
+00
s+∞o,
(0,
-00
(1)
+∞
f(t) dt = 1
-0
f(t) 8(ta) dt = f(a)
Moreover, L{8(t)} = 1.
Now, use the Method of Laplace transforms to find the unite response function of a mechanical system
governed by (1) when a = 1 and b = c = 2. Then plot its graph over the interval [0, 100]. What happens to
the system as time goes by? Does this behavior makes sense (from the physical point of view? Why say so?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35fb113f-0b3f-4785-ab69-a168a85af31c%2Fe12a448e-9d58-49b0-ba3d-64a78eb4088d%2F693ygq3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A usual model that pops out in engineering involves the response of a system to an external force. Take for
instance a mechanical spring system modeled by the equation
(ay" +by' + cy= = f(t)
y(0) = 0
y'(0) = 0
Here, y(t) denotes the displacement of the spring at time t, while the initial conditions tell us that the
system is at rest (or is at equilibrium) and has zero velocity at the start t = 0 of the observation period. The
forcing term f represents the external force applied to the spring at various time values. Hence, the
solution y(t) represents the behavior of the system as a response to the external force applied to it.
Imagine the spring being hit by a very forceful blow at t = 0. It can be thought of as like a hammer hitting a
nail. The impact is instantaneous; the force is applied only at the moment t = 0. Hence, the forcing term is
zero whenever t = 0 and has a very huge absolute value at t = 0 so that the total force is a finite constant.
This force is usually denoted the Dirac Delta function (even though it is not a function in the
mathematical sense) 8. The solution y of (1) when f(t) = 8(t) is called the unit impulse response function.
Recall that in the lecture, I mentioned the informal characterization of 8 as
t = 0
t = 0'
8(t) =
and in a more formal sense, & is the function such that
+00
s+∞o,
(0,
-00
(1)
+∞
f(t) dt = 1
-0
f(t) 8(ta) dt = f(a)
Moreover, L{8(t)} = 1.
Now, use the Method of Laplace transforms to find the unite response function of a mechanical system
governed by (1) when a = 1 and b = c = 2. Then plot its graph over the interval [0, 100]. What happens to
the system as time goes by? Does this behavior makes sense (from the physical point of view? Why say so?
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