A convenient way to measure damping in a single-degree-of-freedom system is called logarith- mic decrement of free response. Consider the free response given by (4). Let's denote t₁ and t₂ denote the times corresponding to two consecutive displacements ₁ and 2 measured one cycle apart, i.e., where t₂ = t₁ + T T = 2π Wd According to (4), show that the logarithmic decrement dis x1 8 = log = (WnT X2 where log has base e. Combine (5), (7), and (8) to obtain 8 √47² +8² Will this method work for other initial conditions? why?

A convenient way to measure damping in a single-degree-of-freedom system is called logarith- mic decrement of free response. Consider the free response given by (4). Let's denote t₁ and t₂ denote the times corresponding to two consecutive displacements ₁ and 2 measured one cycle apart, i.e., where t₂ = t₁ + T T = 2π Wd According to (4), show that the logarithmic decrement dis x1 8 = log = (WnT X2 where log has base e. Combine (5), (7), and (8) to obtain 8 √47² +8² Will this method work for other initial conditions? why?

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

Given the vibrating system below: K4 Y(t) =Ysin30t where for = 30 and Y=20mm Find the following K1 K2 m C3 H C2 C1 C5 C4 1. Frequency Ratio 2. Displacement Transmissibility Ratio 3. Absolute displacement of the mass 4. Type of Damping 5. Equation of motion x(t). Assume Initial conditions for displacement and velocity 6. Graph 2 cycles of the vibrating system. You can use third party app for this. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2=40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m
Given the vibrating system below: K1 K3 K4 Find the following 1. Keq 2. Ceq 3. Natural angular velocity K2 m C1 C3 C5 C2 C4 4. Damped angular velocity 5. Type of Damping 6. Equation of motion x(t). Assume Initial conditions for displacement and velocity Graph 2 cycles of the vibrating system. You can use third party app for this. 7. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2= 40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m
1. Suppose you are riding your bicycle on a bumpy road having a surface profile that varies harmonically with +/- 6 cm undulations. The distance between consecutive peaks of these undulations is 2 m. When you sit on the seat of your bike for your casual ride, the springs deflect 5 cm. When you are seated, the damper under the seat provides an equivalent linear viscous damping of 10% of the critical damping. A simple representation of your ride on the "never-ending" rough road is shown below. (a) If you are riding your bicycle at a horizontal speed of 2.5 m/sec, how much bumping up and down will you experience? (b) Next day, you are carrying a backpack which increases your on-seat weight by 20%. Assuming that you are still able to ride with same speed, will this "loaded" ride be more or less comfortable than your previous, "no backpack" ride? M k/23 k/2 6 cm 2 m
An object attached to a spring undergoes simple harmonic motion modeled by the differential equation d²x = 0 where x (t) is the displacement of the mass (relative to equilibrium) at time t, m is the mass of the object, and k is the spring constant. A mass of 3 kilograms stretches the spring 0.2 meters. dt² Use this information to find the spring constant. (Use g = 9.8 meters/second²) m k = + kx The previous mass is detached from the spring and a mass of 17 kilograms is attached. This mass is displaced 0.45 meters below equilibrium and then launched with an initial velocity of 2 meters/second. Write the equation of motion in the form x(t) = c₁ cos(wt) + c₂ sin(wt). Do not leave unknown constants in your equation. x(t) = Rewrite the equation of motion in the form ä(t) = A sin(wt + ), where 0 ≤ ¢ < 2π. Do not leave unknown constants in your equation. x(t) =
For the system shown in Figure 2, u (t) and y (t) denote the absolute displacements of Building A and Building B, respectively. The two buildings are connected using a linear viscous damper with damping coefficient c. Due to construction activity, the floor mass of Building B was estimated that vibrates with harmonic displacement that is described by the following function: y(t) = y,cos (wt). k/2 Building A H k/2 M(t) C 73(1) Building B Equivalent (Building A) 00000000 Figure 2: Single-degree-of-freedom system in Problem 2. 111 uft) Please compute the following related to Building A: (a) Derive the equation of motion of the mass m. (b) Find the expression of the amplitude of the steady-state displacement of the mass m. (1) (Building B)
Let e be the counterclockwise angular displacement of the disk in the following system. a) Derive the governing differential equation of the system by using the Newton 2d law. b) Find the natural frequency of the system c) For what value of c is the damping ratio of the system equal to 1.15? fy-10 cm f30 cm m2 m1 Jputey10 kg-m m;-10 kg my-25 kg k-1x10' N/m
2- A free vibrations test is run to determine the stiffness and damping properties of an elastic element. A 20 kg block is attached to the element. The block is displaced 1 cm and released. The resulting oscillations are monitored with the results shown in Figure (1). Determine k and c for this element. 0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 0 0.06 0.12 0.18 Figure (1) 0.24 0.30 0.3
An undamped simple harmonic oscillator has mass 2.0 kg and spring constant 50 N/m. The initial displacement from equilibrium (at time t = 0) is 0.30 m and the initial velocity is 2.0 m/s. %3D 1. Determine the angular frequency (in rad/s), the frequency (in Hz), and the period (in s). Determine numerical values (not in terms of .) Determine the displacement in the form x(t) = Acos(mot) + Bsin(@ot). That is, solve for A and B in meters, and write oo in radians per second.
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?
Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…
The damped frequency of oscillation of an open loop underdamped system = 12rad/sec The magnitude of the real part of the underdamped system = 16 Determine the natural frequency (Answer to 0 decimal places)
A structure has a mass of 10 kip/g and a stiffness of 10 kip/in. The acceleration of gravity, g, is 386.1 in/s2. What is the value of damping assumed by the code?