/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 190 Using the MATLAB Program2 \(m,\)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 190

Using the MATLAB Program2 \(m,\) plot the free-vibration response of a viscously damped system with \(m=4 \mathrm{~kg}, k=2500 \mathrm{~N} / \mathrm{m}, x_{0}=100 \mathrm{~mm}, \dot{x}_{0}=-10 \mathrm{~m} / \mathrm{s}, \Delta t=0.01 \mathrm{~s}, n=50\) for the following values of the damping constant: a. \(c=0\) b. \(c=100 \mathrm{~N}-\mathrm{s} / \mathrm{m}\) c. \(c=200 \mathrm{~N}-\mathrm{s} / \mathrm{m}\) d. \(c=400 \mathrm{~N}-\mathrm{s} / \mathrm{m}\)

Short Answer

Expert verified
The free-vibration response of a viscously damped system with \( m = 4 \; kg, k = 2500 \; N/m, x_0 = 0.1 \; m, v_0 = -10 \; m/s, \Delta t = 0.01 \; s, \) and \( n = 50 \) is plotted using MATLAB for four damping constants: \( c = 0 \; N鈰卻/m \),\( c = 100 \; N鈰卻/m \),\( c = 200 \; N鈰卻/m \), and \( c = 400 \; N鈰卻/m \). The plots show the effect of different damping values on the system's behavior, with higher damping leading to faster decay of oscillations.

Step by step solution

01

Set up required variables

Create variables for the given data: mass (m), spring constant (k), initial displacement (x0), initial velocity (v0), time step (dt), number of steps (n), and an array containing damping constants (c_vals). ```matlab m = 4; % kg k = 2500; % N/m x0 = 0.1; % m (converted from 100 mm) v0 = -10; % m/s dt = 0.01; % s n = 50; c_vals = [0, 100, 200, 400]; % N鈰卻/m ```
02

Define the system characteristics

Calculate the natural frequency (wn) and the damping ratio (zeta) for each value of the damping constant. Store these values in arrays. ```matlab wn = sqrt(k/m); % Natural frequency zeta_vals = c_vals ./ (2 * sqrt(m * k)); % Damping ratio for each c value ```
03

Define a time vector and a function to calculate displacement

Create a time vector (t) from 0 to n*dt and a function (displacement) to calculate the displacement (x) based on the damping ratio (zeta) and natural frequency (wn) at each time value. ```matlab t = 0:dt:(n*dt); % Time vector displacement = @(zeta, wn, x0, v0, t) exp(-zeta .* wn .* t) .* (x0 .* cos(wn .* sqrt(1 - zeta.^2) .* t) + ((v0 + zeta .* wn .* x0) ./ sqrt(1-zeta.^2)).* sin(wn .* sqrt(1 - zeta.^2) .* t)); ```
04

Calculate and plot the displacement for each damping constant

For each value of the damping constant (and damping ratio), calculate the displacement using the displacement function and create a subplot showing the free-vibration response. ```matlab figure; for i = 1:length(c_vals) zeta = zeta_vals(i); x = displacement(zeta, wn, x0, v0, t); subplot(2, 2, i); plot(t, x); ylabel('Displacement (m)'); xlabel('Time (s)'); title(['Free-vibration response: c = ', num2str(c_vals(i)), ' N鈰卻/m']); end ``` Run this MATLAB script to obtain the plots for the free-vibration response of the viscously damped system for the given damping constants.

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Most popular questions from this chapter

A weight of \(25 \mathrm{~N}\) is suspended from a spring that has a stiffness of \(1000 \mathrm{~N} / \mathrm{m}\). The weight vibrates in the vertical direction under a constant damping force. When the weight is initially pulled downward a distance of \(10 \mathrm{~cm}\) from its static equilibrium position and released, it comes to rest after exactly two complete cycles. Find the magnitude of the damping force.

An automobile having a mass of \(2000 \mathrm{~kg}\) deflects its suspension springs \(0.02 \mathrm{~m}\) under static conditions. Determine the natural frequency of the automobile in the vertical direction by assuming damping to be negligible.

A metal block, placed on a rough surface, is attached to a spring and is given an initial displacement of \(10 \mathrm{~cm}\) from its equilibrium position. It is found that the natural time period of motion is \(1.0 \mathrm{~s}\) and that the amplitude reduces by \(0.5 \mathrm{~cm}\) in each cycle. Find (a) the kinetic coefficient of friction between the metal block and the surface and (b) the number of cycles of motion executed by the block before it stops.

The mass of a spring-mass system vibrates on a dry surface inclined at \(30^{\circ}\) to the horizontal as shown in Fig. \(2.123 .\) a. Derive the equation of motion. b. Find the response of the system for the following data: $$ m=20 \mathrm{~kg}, \quad k=1000 \mathrm{~N} / \mathrm{m}, \quad \mu=0.1, \quad x_{0}=0.1 \mathrm{~m}, \quad \dot{x}_{0}=5 \mathrm{~m} / \mathrm{s} $$

A mass of \(5 \mathrm{~kg}\) is attached to the top of a helical spring, and the system is made to vibrate by giving to the mass an initial deflection of \(25 \mathrm{~mm}\). The amplitude of the mass is found to reduce to \(10 \mathrm{~mm}\) after 100 cycles of vibration. Assuming a spring rate of \(200 \mathrm{~N} / \mathrm{m}\) for the helical spring, find the value of the hysteretic-damping coefficient \((h)\) of the spring.
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