Homework 5

Introduction: provides general background about the suspension of motorcycle system of , its applications, and its components.

Response using Laplace transformation.

solve    autocontrols

Consider the damped mass-spring system for mass of 1.5 kg, spring constant 10 N/m, damping 3 kg/s and an oscillating force 4.7 cos(@t) Newtons. That is, 1.5x" + 3x' + 10x = 4.7 cos(@t). What positive angular frequency o leads to maximum practical resonance? W = help (numbers) What is the maximum displacement of the mass in the steady state solution when the we are at practical resonance: C(@) = help

3. Consider the following system described below (parameters and inputs are in SI units): Im = 0.4 R = 0.18 k=30 b = 0.2 M m = 6 b=20 www Im T(t) = 2 R Mr Bammy Note: b is the damping coefficient between the mass and the ground (it is shown as c in the diagram, but we will use b). Complete the following: a) Find an equation of motion for the simplified equivalent mass system. Assume that the shaft between the inertias is rigid. b) Find the equations of motion in state variable form for this system. You should end up with 2 state variable equations. c) Numerically solve for the state variables and plot each state variable versus time. Assume, that the mass is initially displaced a distance of 0.1 in the positive direction and the mass velocity is initially zero. Also plot the spring force and damping force. d) This system is required to have a maximum displacement overshoot that is less than 15% of the steady state value to avoid the rack impacting another portion of the machine.…

Mass 15 kg. C=150 Ns/m, k = 200 N/m, a=0.5m, b = 0.75 m. Find: 1. Static equilibrium condition 2. Equation of motion of total resultant system 3. Natural frequency and damping ratio 4. Is system underdamped or overdamped? 5. What is equation to find transient response of this system? Just write equation. Do not find coefficients. 6. What is required value of mass m to make this system critically damped? QA b m B H

Sub- Mechanical vibrations

part b only please

Question is image. Only part ii) ?

Problem 2. For the single-degree-of-freedom mechanical system shown in the figure, the bar has mass m and length l (so that Io spring is unstretched when 0 = 0: ml² 피). If the %3D k a) find the linearized governing equations of motion for small 0; (т,1) b) find the natural frequency, damping ration, and damped natural frequency of oscillation for the free response; Neglect gravity.

Mechanical Vibration Question is image A 100-kg tumbler produces a harmonic force of magnitude 5000 N. The stiffness coefficient of the support is 10×10^4N/m and damping coefficient ζ=0.2 If the maximum deflection is to be less than 0.06m find the operating frequency range?

For small positive damping, c, in the sinusoidally forced mass-spring system ma" + ca' + ka = Ggcos (wt) there is a value of the forcing frequency w for which the amplitude of the response is greatest - a resonance peak. very light darmping moderate damping very heavy darmping forcing frequency, w Derive a formula for the location of the peak in terms of the parameters in the differential equation (m, c, k; Go). So the location of the peak depends on the damping . Where does the peak move to as the damping constant c is reduced to zero? Does the above frequency have a name? amplitude of response

Q.2 Two springs, damper and a mass are attached to a rigid, weightless bar PQ as shown below. Write the equation of motion of the system using D'Alemberts principle. Find the natural frequency of vibration of the system. J6+ (kde) ds. C 0000 m

Given the vibrating system below: K4 Y(t) =Ysin30t where fos = 30 and Y=20mm Find the following K3 K1 K2 1. Frequency Ratio 2. Displacement Transmissibility Ratio 3. Absolute displacement of the mass m C3 C2 C1 C5 C4 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

Q.2 Two springs, damper and a mass are attached to a rigid, weightless bar PQ as shown below. Write the equation of motion of the system using D'Alemberts principle. Find the natural frequency of vibration of the system. J6+ (kdo) de. C 0000 m

Consider an underdamped mass spring system with mass m = 1 kg spring constant k = 8 N/m, and damping coefficient, stretched past equilibrium by 1 m and released with an initial velocity of 0 m/s. N.S a = 4 m www. Equilibrium Position (a) Write the solution of the initial value problem. x(t) = (b) Identify the pseudo-frequency. Find the circular frequency if damping is removed. m N s m (c) Determine the first time in seconds in which the mass returns to equilibrium position after being set in motion. (Round your answer to two decimal places.)