Question

In the spring-mass system of Problem \(31,\) suppose that the spring force is not given by Hooke's law but instead satisfies the relation $$ F_{s}=-\left(k u+\epsilon u^{3}\right) $$ where \(k>0\) and \(\epsilon\) is small but may be of either sign. The spring is called a hardening spring if \(\epsilon>0\) and a softening spring if \(\epsilon<0 .\) Why are these terms appropriate? (a) Show that the displacement \(u(t)\) of the mass from its equilibrium position satisfies the differential equation $$ m u^{\prime \prime}+\gamma u^{\prime}+k u+\epsilon u^{3}=0 $$ Suppose that the initial conditions are $$ u(0)=0, \quad u^{\prime}(0)=1 $$ In the remainder of this problem assume that \(m=1, k=1,\) and \(\gamma=0\). (b) Find \(u(t)\) when \(\epsilon=0\) and also determine the amplitude and period of the motion. (c) Let \(\epsilon=0.1 .\) Plot (a numerical approximation to) the solution. Does the motion appear to be periodic? Estimate the amplitude and period. (d) Repeat part (c) for \(\epsilon=0.2\) and \(\epsilon=0.3\) (e) Plot your estimated values of the amplitude \(A\) and the period \(T\) versus \(\epsilon\). Describe the way in which \(A\) and \(T\), respectively, depend on \(\epsilon\). (f) Repeat parts (c), (d), and (e) for negative values of \(\epsilon .\)

Short answer

#Answer# For the given spring-mass system with modified force equation, the differential equation for the displacement of the mass from its equilibrium position is: $$ m u^{\prime\prime} + \gamma u^{\prime} + ku + \epsilon u^3 = 0 $$ When ε = 0, the solution for the displacement is: $$ u(t) = \sin(t) $$ with amplitude (A) equal to 1 and period (T) equal to 2π. For the cases when ε is not equal to 0 (both positive and negative values), a numerical solver should be used to find approximations of the solutions and analyze the relationship between amplitude, period, and ε. The motion's periodicity, amplitude, and period depend on the specific value of ε.

Step-by-step solution

Part (a): Derive the differential equation for displacement

Using Newton's second law, we can write the equation for the force acting on the mass: $$ F = ma = m u^{\prime\prime} $$ The total force acting on the mass is the sum of the spring force and damping force: $$ F = - (k u + \epsilon u^3) - \gamma u^{\prime} $$ Therefore, $$ m u^{\prime\prime} + \gamma u^{\prime} + ku + \epsilon u^3 = 0 $$ This is the differential equation for the displacement of the mass from its equilibrium position.

Part (b): Solve for displacement when ε = 0; find amplitude and period

When ε = 0, the differential equation becomes: $$ m u^{\prime\prime} + \gamma u^{\prime} + ku = 0 $$ Given that \(m=1, k=1\), and \(\gamma=0\), the differential equation simplifies to: $$ u^{\prime\prime} + u = 0 $$ The general solution for the above differential equation can be written as: $$ u(t) = A \sin(\omega_0 t) + B \cos(\omega_0 t) $$ where \(\omega_0 = \sqrt{k/m} = 1\). Using the initial conditions: $$ u(0) = 0 \implies A \sin(0) + B \cos(0) = B = 0 $$ $$ u^{\prime}(0) = 1 \implies A \cos(0) - B \sin(0) = A = 1 $$ Therefore, the solution for \(u(t)\) is: $$ u(t) = \sin(t) $$ The amplitude and period of this sinusoidal motion are: Amplitude: \(A = 1\) Period: \(T = \frac{2 \pi}{\omega_0} = 2 \pi\)

Part (c), (d), and (e): Plots and amplitude/period relationship

For this section, a numerical solver (e.g., Euler's method or Runge-Kutta method) should be used to find approximations of the solutions for various values of ε (both positive and negative). Then, by plotting the results, we can observe the periodic behavior and estimate the amplitude and period. The specific instructions for numerical computations might be outside the scope of the problem, and it is advisable to use software (Python, Matlab, etc.) to obtain the plots and data for amplitude and period.

Part (f): Repeating steps for negative values of ε

Follow the same procedure as in Part (c), (d), and (e) to analyze the system's behavior with negative ε values. Use numerical methods and plots to estimate the amplitude and period and observe the relationship between ε, amplitude, and period for these values.

Key concepts

Spring-Mass System

A spring-mass system is a common physical system in mechanics used to study how an object connected to a spring oscillates when displaced from its equilibrium. This system consists of a mass attached to a spring, which can stretch or compress. The behavior of the spring is often initially described by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement:

• For force, Hooke's Law is expressed as: \[ F_s = -ku \] where \( k \) is the spring constant (stiffness) and \( u \) is the displacement.
• For the problem at hand, the spring's force is modified with an additional cubic term to introduce nonlinearity: \[ F_s = -(ku + \epsilon u^3) \] This extra term complicates the system's behavior by making it either harder or softer as \( u \) increases.

Understanding this system is crucial for analyzing how forces and motion interact, providing insights into oscillatory and dynamic behavior.

Nonlinear Oscillation

The introduction of a cubic term in the spring force equation causes the spring-mass system to exhibit nonlinear oscillations. Unlike linear systems, which have constant amplitude and frequency, nonlinear systems can change these properties depending on the displacement. Some key aspects include:

• Nonlinear Forces with Cubic Terms: When \( \epsilon > 0 \) or \( \epsilon < 0 \), we encounter nonlinear terms affecting the force direction, making the behavior of oscillations more complex than simple sinusoidal waves.
• Hardening and Softening Springs: The terms hardening (if \( \epsilon > 0 \)) and softening (if \( \epsilon < 0 \)) refer to how springs behave as they are deformed. \( \epsilon > 0 \) results in stiffer springs as \( u \) increases, while \( \epsilon < 0 \) indicates springs that become softer.
• Alternative Behavior: Nonlinear oscillations require more sophisticated tools for analysis, often needing numerical methods for accurate predictions of motion over time.

This complexity illustrates why differential equations with nonlinear terms are interesting yet challenging in the study of dynamic systems.

Periodic Motion

Periodic motion refers to motion that repeats after a regular interval. In oscillating systems like spring-mass, periodic motion is often sinusoidal, described by functions such as sine and cosine. However, when dealing with nonlinear oscillations, determining periodicity becomes less straightforward:

• Linear Oscillations: In a simple system, without nonlinear terms, periodic motion can be perfectly characterized using trigonometric functions with clearly defined amplitude and period.
• Effect of Nonlinearity: Nonlinear terms (\( \epsilon u^3 \)) affect the periodicity, potentially leading to shifts in amplitude and frequency as seen in real-world cases where force-displacement relationships are not purely linear.
• Estimate Amplitude and Period: Analytical methods might fail in providing exact values, requiring numerical approximations instead. Nonlinearity may also introduce phenomena like harmonic generation and amplitude modulation, further complicating periodicity.

Thus, understanding periodic motion in a nonlinear context requires handling complexities that go beyond traditional approaches.

Numerical Approximation

Numerical approximation techniques are essential when analytic solutions of differential equations are difficult or impossible to find, such as in systems with nonlinear terms. These methods approximate the solution at discrete points to understand system behavior:

• Euler's Method: A straightforward initial tool providing a first-order approximation. Though simple, it can accumulate errors over larger steps or longer simulations.
• Runge-Kutta Method: An improved technique that provides more accurate results by considering increments at multiple sub-intervals within each step.
• Software Tools: Computers implementing numerical methods can visualize the behavior of complex systems, as seen with plotting relationships like amplitude and period in nonlinear dynamics.

By leveraging these techniques, we can explore the impact of parameters like \( \epsilon \) in the spring-mass problem, examining its effect on motion characteristics like amplitude and period for both positive and negative values. Such numerical tools are invaluable in modern science and engineering for handling complex, real-world problems.