A displacement–time plot for a free vibration analysis of a SDOF system gives a displacement of 7.5 mm at time [latex]t_{1}[/latex] and 1 mm after 7 cycles. Determine logarithmic decrement and damping ratio.

Logarithmic decrement is given by [latex]\pmb{\delta} =\frac{1}{j} \log _{e}\frac{x_{i}}{x_{i+j}} =2j \pi \xi \frac{\omega }{\omega _{D}}[/latex] [latex]\pmb{\delta} = \frac{1}{7} \log _{e}\frac{7.5}{1.0} =0.288[/latex] ∴ Damping ratio x = δ/2π = 0.288/2 × π = 0.046 or 4.6%

Question 2.7: A displacement–time plot for a free vibration analysis of a ...

Dynamics of Structures with MATLAB Applications

A displacement–time plot for a free vibration analysis of a SDOF system gives a displacement of 7.5 mm at time $t_{1}$ and 1 mm after 7 cycles. Determine logarithmic decrement and damping ratio.

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Consider a SDOF system shown in Figure 2.1. It undergoes undamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3500 kN/m. Determine the natural frequency and natural period of vibration. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Hence, plot a graph of ...

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The complete solution is given by Equations (2.6) ...

Question: 2.8

A SDOF system with a friction damper vibrates with an initial displacement of 50 mm. It completes 4 cycles in 0.8 second when its amplitude reduces to 5 mm. Estimate the coefficient of friction and when does the system come to rest. ...

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Period of vibration T = 0.8/4 = 0.2 sec Frequency ...

Question: 2.6

A SDOF frame has a mass = 5000 kg, lateral stiffness k = 4 × 10^6 N/m and a damping ratio of 4%. Determine its undamped and damped natural frequency. If the frame is displaced by 25 mm and released suddenly, determine the peak displacements of first five cycles of vibrations. ...

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Undamped natural frequency

\omega =\sqrt{\f...

Question: 2.5

Consider a SDOF system shown in Figure 2.2. It undergoes underdamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3750 kN/m and c = 50 kNs/m. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Plot the inertia, damping and spring forces. ...

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Mass m = 40 kN s²/m Stiffness k = 3750 kN/m Dampin...

Question: 2.4

Consider a SDOF system shown in Figure 2.2. It undergoes underdamped free vibrations. The mass = 40 kNs²/m and stiffness k = 3750 kN/m. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Plot the influence of 0, 5% and 10% damping on the displacement–time history. ...

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The response of a SDOF system can be determined us...

Question: 2.2

In the undamped SDOF system of Example 2.1, plot the response for different initial conditions as follows: Case 1: x(0) = 0.01 m, x(0) = 0.1 m/s Case 2: x(0) = 0, x(0) = 0.1 m/s Case 3: x(0) = 0.01 m, x(0) = 0 ...

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The solution can be found either using MS EXCEL or...

Question: 2.3

Consider a SDOF system shown in Figure 2.1. It undergoes underdamped free vibrations (damping ξ = 5%). The mass = 5 kNs²/m, stiffness k = 100 kN/m. The initial conditions are given as: x(0) = 0.02 m, x(0) = 0.05 m/s. Plot the displacement–time history and velocity time history using MATLAB. ...

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The plots of the displacement response and velocit...

Question 2.7: A displacement–time plot for a free vibration analysis of a ...

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Consider a SDOF system shown in Figure 2.1. It undergoes undamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3500 kN/m. Determine the natural frequency and natural period of vibration. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Hence, plot a graph of ...

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A SDOF system with a friction damper vibrates with an initial displacement of 50 mm. It completes 4 cycles in 0.8 second when its amplitude reduces to 5 mm. Estimate the coefficient of friction and when does the system come to rest. ...

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A SDOF frame has a mass = 5000 kg, lateral stiffness k = 4 × 10^6 N/m and a damping ratio of 4%. Determine its undamped and damped natural frequency. If the frame is displaced by 25 mm and released suddenly, determine the peak displacements of first five cycles of vibrations. ...

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Consider a SDOF system shown in Figure 2.2. It undergoes underdamped free vibrations. The mass = 40 kNs²/m, stiffness k = 3750 kN/m and c = 50 kNs/m. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Plot the inertia, damping and spring forces. ...

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Consider a SDOF system shown in Figure 2.2. It undergoes underdamped free vibrations. The mass = 40 kNs²/m and stiffness k = 3750 kN/m. The initial conditions are given as: x(0) = 0.01 m, x(0) = 0.1 m/s. Plot the influence of 0, 5% and 10% damping on the displacement–time history. ...

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In the undamped SDOF system of Example 2.1, plot the response for different initial conditions as follows: Case 1: x(0) = 0.01 m, x(0) = 0.1 m/s Case 2: x(0) = 0, x(0) = 0.1 m/s Case 3: x(0) = 0.01 m, x(0) = 0 ...

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Consider a SDOF system shown in Figure 2.1. It undergoes underdamped free vibrations (damping ξ = 5%). The mass = 5 kNs²/m, stiffness k = 100 kN/m. The initial conditions are given as: x(0) = 0.02 m, x(0) = 0.05 m/s. Plot the displacement–time history and velocity time history using MATLAB. ...

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