Let us consider a simple spring-mass-damper system, given by: mx + ci+kx = 0 (1) where m, c, k (all positive scalars) are the mass, damper coefficient, and spring coefficient, respectively. x = R represents the displacement of the mass. M be the largest invariant set in Q. Theorem 1 (LaSalle's Invariance Principle Theorem [1]). Suppose that there exists a Lyapunov function V(x) of the system = f(x). Let be the non-empty set of state vectors such that ⇒ V(x) = 0. Let Then, the system is asymptotically stable if M only contains the origin. Theorem 2 (Higher-order derivative approach [2]). Let be the same set as discussed in Theorem 1. The system is asymptotically stable if both of the following conditions hold for an odd integer k: the first k - 1 derivatives of V evaluated on N are zero, i.e., d'³V/dt₁ = 0, Væ € N, i = 1, 2, ..., k − 1 the k-th derivative is negative definite on N, i.e., dkV/dt* <0, Væ € N.

Let us consider a simple spring-mass-damper system, given by: mx + ci+kx = 0 (1) where m, c, k (all positive scalars) are the mass, damper coefficient, and spring coefficient, respectively. x = R represents the displacement of the mass. M be the largest invariant set in Q. Theorem 1 (LaSalle's Invariance Principle Theorem [1]). Suppose that there exists a Lyapunov function V(x) of the system = f(x). Let be the non-empty set of state vectors such that ⇒ V(x) = 0. Let Then, the system is asymptotically stable if M only contains the origin. Theorem 2 (Higher-order derivative approach [2]). Let be the same set as discussed in Theorem 1. The system is asymptotically stable if both of the following conditions hold for an odd integer k: the first k - 1 derivatives of V evaluated on N are zero, i.e., d'³V/dt₁ = 0, Væ € N, i = 1, 2, ..., k − 1 the k-th derivative is negative definite on N, i.e., dkV/dt* <0, Væ € N.

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

See similar textbooks

Similar questions

Forcing Function Spring Constant (2)4 Mass k Friction Constant b m Mass Displacement y(t) 3. (20 pts.) Consider the following spring-mass-damper mechanical system (it is placed sideways, so that you won't need to consider gravity). The input is given by f(t), and the output is y(t). Find an equation in frequency domain (s domain) that defines the relationship between the input F(s) and the output Y(s).
Whats the first equation (circled) and how did we get it ?
Consider the spring mass damper system shown below. k m x(t) The forces of the spring and damper are represented by Fspring = -kx and Fdamper = -bx' respectively, where k is the spring constant and b is the damping coefficient. The mass has an initial displacement +1 m and velocity 0 m/s. 1. Use Newton's 2nd Law of Motion to derive an ODE to describe the mass's motion. 2. Find x (t) for m = 2 kg, b = 1 N s/m and k = {0.1,0.3,0.5} N. m. What happens when the stiffness of the spring is being increased?
Q5:- in the fig M= 20 kg, c=7 N.s/m, k=10 N/M, the excitation is 10 sin5t, find natural frequency, excition frequency Excitation force, is the device in the resonse Frequency, Is the device under damping? Find the equation of motion
Hello, please solve this problem. Thank you very much
A single degree of freedom mass-spring-viscous damper system with mass m, spring constantk and viscous damping coefficient q is critically damped. The correct relation among m, k, and q is q = 2km q = 2/ km 2k = b m k q =2
2. Consider a car suspension, modeled as a mass/spring/damper system (mass m, stiffness k, damping b). Suppose the height of the chassis is lo at rest, the height of the terrain below the driver varies as h(t), and the height of the chassis is denoted lo + y(t). (i.e., spring deflection away from rest is y(t) – h(t)). 2 (a) Give the transfer function G(s) = H(s) · = (b) Suppose the ground follows an oscillatory profile h(t) A cos(wx (t)) with magnitude A (in meters) and frequency w (measured in radians per meter). Suppose the car is traveling at a constant forward speed v. Using a frequency response analysis strategy, give the amplitude of oscillations experienced by the driver at steady state as a function of m, k, b, A, w, and v. Hint: You can't simply consider |G(iw)| to get the amplification in this case. (c) Suppose the ground varies by A = 5cm, w = 2 rad/m, and you are driving at v = 15 m/s. Using your answer to part (b), what amplitude of oscillation is felt by the driver when m…
2.10 Derive the equation for the displacement response of a viscously damped SDF system due to ini- tial velocity u(0) for three cases: (a) underdamped systems; (b) critically damped systems; and (c) overdamped systems. Plot u(t) : u(0)/w, against t/T, for 5 =0.1, 1, and 2.
I need help with b, c and d. if you can you can include 'a' also but i mainly need b, c and d. Thanks.
1) For the mass-damper system at right, the model in terms of velocity is mv + cv = f. The time constant of this first order system is t = m/c. For this system, a larger value of the damping constant c causes the time constant to decrease - which is counterintuitive. It seems that a small amount of damping should correspond to a system with a fast response. How can you reconcile this? m CA
Please try to give type solution fast i will rate for sure
For the shown spring-mass-damper system. Let m =1 kg, b = 2 N.s/m, k = 2 N/m and t) = sin2t N. (a) Write the equation of motion. (b) Find the amplitude of the steady state oscillations. (c) In steady state, find the time delay in seconds between the mass position and (t).