Consider the pictured mass-spring system excited by a mov- ing base y(t) and with a drag force = -e times the velocity of the mass relative to the lab. Find the steady state motion r(t) = X(t) – X-q representing the displacement from equi- librium (X = natural spring length) relative to the moving base when the cart's motion is sinusoidal y(t) = Y, sin(wt). There are two stages to this problem: y(t) (given) X(1) m a) Derivation of the differential equation. It should come out the form indicated here: Megë +Cgi + KffI = Dj + Eÿ You are advised to use F = ma to derive it. The sum of the forces has two terms, with magnitudes: k times the stretch of the spring, and c times the velocity of the mass relative to the lab. Be careful in your derivation: z and X represent motion relative to the moving base; the mass's acceleration in the frame of the lab is therefore NOT ďr/dt², nor is its velocity relative to the lab equal to dr/dt (think about the position of the mass as described from the lab, this will involve the position of the cart relative to the lab, y(t), as well as the position of the mass relative to the cart, X (t)). Check the differential equation you derived. What are the five coefficients M, C, K, D and E?: Does the effective force (the right hand side) scale sensibly with dy/dt and d²y/dt?? Is your equation dimensionally consistent? Are the effective stiffness and damping positive? Does the equation reduce to what you should have if y(t) = 0? b) Obtaining the particular solution. Substitute y(t) = Y, sin(wt) and use the standard formulas for harmonic responses in terms of the quantities G(w) and ó(w). You do not need to re-derive them, but you will have to substitute for all the forms of the coefficients. (You need not consider initial conditions; they only affect the constants in the homogeneous part of the general solution rh, which dies away after enough time due to the damping; the question only asks for the steady state part of the solution.) You may save a lot of algebra if you remember that the particular solution associated with the sum of two forces is the sum of the particular solutions.
Consider the pictured mass-spring system excited by a mov- ing base y(t) and with a drag force = -e times the velocity of the mass relative to the lab. Find the steady state motion r(t) = X(t) – X-q representing the displacement from equi- librium (X = natural spring length) relative to the moving base when the cart's motion is sinusoidal y(t) = Y, sin(wt). There are two stages to this problem: y(t) (given) X(1) m a) Derivation of the differential equation. It should come out the form indicated here: Megë +Cgi + KffI = Dj + Eÿ You are advised to use F = ma to derive it. The sum of the forces has two terms, with magnitudes: k times the stretch of the spring, and c times the velocity of the mass relative to the lab. Be careful in your derivation: z and X represent motion relative to the moving base; the mass's acceleration in the frame of the lab is therefore NOT ďr/dt², nor is its velocity relative to the lab equal to dr/dt (think about the position of the mass as described from the lab, this will involve the position of the cart relative to the lab, y(t), as well as the position of the mass relative to the cart, X (t)). Check the differential equation you derived. What are the five coefficients M, C, K, D and E?: Does the effective force (the right hand side) scale sensibly with dy/dt and d²y/dt?? Is your equation dimensionally consistent? Are the effective stiffness and damping positive? Does the equation reduce to what you should have if y(t) = 0? b) Obtaining the particular solution. Substitute y(t) = Y, sin(wt) and use the standard formulas for harmonic responses in terms of the quantities G(w) and ó(w). You do not need to re-derive them, but you will have to substitute for all the forms of the coefficients. (You need not consider initial conditions; they only affect the constants in the homogeneous part of the general solution rh, which dies away after enough time due to the damping; the question only asks for the steady state part of the solution.) You may save a lot of algebra if you remember that the particular solution associated with the sum of two forces is the sum of the particular solutions.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question
Transcribed Image Text:Consider the pictured mass-spring system excited by a mov-
ing base y(t) and with a drag force = -e times the velocity
of the mass relative to the lab. Find the steady state motion
x(t) = X (t) – Xeg representing the displacement from equi-
librium (X = natural spring length) relative to the moving
base when the cart's motion is sinusoidal y(t) = Y, sin(wt).
y(t) (given)
X(t)
k
There are two stages to this problem:
a) Derivation of the differential equation. It should come out the form indicated here:
Meffë + Ceffå + KeffT = Dÿ+ Eÿ
You are advised to use F = ma to derive it. The sum of the forces has two terms, with
magnitudes: k times the stretch of the spring, and e times the velocity of the mass relative to
the lab. Be careful in your derivation: x and X represent motion relative to the moving base;
the mass's acceleration in the frame of the lab is therefore NOT d²r/dt, nor is its velocity
relative to the lab equal to dr/dt (think about the position of the mass as described from the
lab, this will involve the position of the cart relative to the lab, y(t), as well as the position
of the mass relative to the cart, X(t)). Check the differential equation you derived. What
are the five coefficients M, C, K, D and E?: Does the effective force (the right hand side)
scale sensibly with dy/dt and dy/dt?²? Is your equation dimensionally consistent? Are the
effective stiffness and damping positive? Does the equation reduce to what you should have
if y(t) = 0?
b) Obtaining the particular solution. Substitute y(t) = Yo sin(wt) and use the standard
formulas for harmonic responses in terms of the quantities G(w) and ø(w). You do not need
to re-derive them, but you will have to substitute for all the forms of the coefficients. (You
need not consider initial conditions; they only affect the constants in the homogeneous part of
the general solution xh, which dies away after enough time due to the damping; the question
only asks for the steady state part of the solution.) You may save a lot of algebra if you
remember that the particular solution associated with the sum of two forces is the sum of the
particular solutions.
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