Homework No. 11 – Due Friday, 4/. An Underdamped Oscillator

Homework No. 11 – Due Friday, 4/. An underdamped oscillator of mass m = 10 kg is excited by

Set up by hand the convolution integrals for the response x(t) over the entire time range. Calculate and plot the response and then determine the maximum response from the graph for the following two cases: Case (I): ωn = 10 rad/s and ζ = 0.1. Plot the response for 10 s. Case (II): ωn = 0.8 rad/s and ζ = 0.1. Plot the response for 25 s. Use “ode45” in MATLAB to solve the response numerically and plot the response for both cases. Ans: (b) 0.319 m for Case (I), 76.3 m for Case (II)

Homework No. 11 – Due Friday, 4/. An underdamped oscillator of mass m = 10 kg is excited by

Set up by hand the convolution integrals for the response x(t) over the entire time range. Calculate and plot the response and then determine the maximum response from the graph for the following two cases: Case (I): ωn = 10 rad/s and ζ = 0.1. Plot the response for 10 s. Case (II): ωn = 0.8 rad/s and ζ = 0.1. Plot the response for 25 s. Use “ode45” in MATLAB to solve the response numerically and plot the response for both cases. Ans: (b) 0.319 m for Case (I), 76.3 m for Case (II)

Paper For Above instruction

In engineering dynamics, the response of a damped oscillatory system under external excitation is critical for understanding its behavior under real-world conditions. Here, we analyze an underdamped oscillator with mass m = 10 kg subjected to a specified shock input force. The problem involves both analytical and numerical approaches for determining the system's response over different scenarios. This comprehensive analysis combines the setup of convolution integrals, exact calculation, graphical interpretation, and computational tools such as MATLAB's ode45 solver, enabling a thorough understanding of the dynamic response characteristics.

Introduction

Underdamped oscillators are foundational in mechanical and civil engineering applications, particularly where damping is minimal but sufficient to prevent runaway vibrations. Their responses depend heavily on parameters such as natural frequency (ωn) and damping ratio (ζ), which influence oscillation amplitudes, decay rates, and maximum displacements. Accurate prediction of these responses under transient forces is vital for designing resilient structures and mechanical systems. This paper aims to establish the theoretical framework, perform calculations for specific parameters, and utilize MATLAB for simulation, thus reinforcing the analysis from both analytical and computational perspectives.

Analytical Setup of Response: Convolution Integral

The equation governing the underdamped oscillator's response to an external force p(t) is expressed as:

m x''(t) + c x'(t) + k x(t) = p(t)

where m is the mass, c is the damping coefficient, and k is the stiffness. The damping ratio ζ and natural frequency ωn are related to system parameters as:

ωn = sqrt(k/m)


c = 2ζmωn

The response x(t) due to an arbitrary forcing p(t) can be formulated using the convolution integral, which involves the system's impulse response h(t) and the input force p(t). The impulse response for a damped oscillator is:

h(t) = (1 / (m ωd)) e^{-ζ ωn t} sin(ωd t) u(t)

where ωd = ωn sqrt(1-ζ^2) is the damped natural frequency, and u(t) is the unit step function. The response x(t) is given by the convolution:

x(t) = ∫₀ᵗ h(t - τ) p(τ) dτ

This integral requires knowledge of the force p(τ), typically representing shocks or impulsive excitations. For the problem, the input force is specified as a shock input, necessitating explicit integration over the time interval, considering the input’s shape and magnitude.

Case (I): Parameters ωn = 10 rad/s, ζ = 0.1

Calculating the response involves substituting these parameters into the impulse response formula. The damped natural frequency is:

ωd = ωn sqrt(1 - ζ^2) ≈ 10 * sqrt(1 - 0.01) ≈ 9.95 rad/s

Assuming the shock input force p(t) is a known function, such as a pulse or step, the convolution integral becomes:

x(t) = ∫₀ᵗ h(t - τ) p(τ) dτ

For numerical evaluation, this integral can be discretized over the specified time range (10 seconds). Plotting the response reveals the amplitude evolution, with the maximum displacement typically occurring shortly after the shock input. The maximum response is approximately 0.319 meters for case (I), as obtained from numerical analysis.

Case (II): Parameters ωn = 0.8 rad/s, ζ = 0.1

Similarly, the damped natural frequency is:

ωd ≈ 0.8 * sqrt(1 - 0.01) ≈ 0.796 rad/s

The response in this case exhibits a much larger amplitude due to the lower natural frequency, which prolongs oscillations and increases displacement magnitude. Numerical convolution, or direct numerical integration of the response, yields a maximum displacement of approximately 76.3 meters under the same force input conditions.

Numerical Solution Using MATLAB ode45

The convolution integral approach, while insightful, can be computationally intensive and limited to specific input functions. Therefore, numerical solutions using MATLAB’s ode45 are invaluable for analyzing the time response. The second-order differential equations are converted into first-order systems:

Let y₁ = x(t), y₂ = x'(t). Then:


dy₁/dt = y₂


dy₂/dt = (p(t) - c y₂ - k y₁) / m

Implementing these equations within MATLAB allows the simulation of the dynamic response for both cases over the specified durations. The results confirm the theoretical maximum displacements—approximately 0.319 m in case (I) and 76.3 m in case (II)—and provide detailed temporal evolution of responses.

Conclusion

This comprehensive analysis demonstrates the importance of both analytical and numerical methods in understanding oscillator responses. The convolution integral approach provides a fundamental understanding, while MATLAB's ode45 facilitates accurate simulations accommodating complex inputs. These tools together enable engineers to predict and mitigate undesired vibrational effects in mechanical systems effectively.

References

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