Linear Second-Order Equations

Pendulum Equation. To derive the pendulum equation (21), complete the following steps.

a. The angular momentum of the pendulum mass m measured about the support O in Figure 4.18 on page 208 is given by the product of the “lever arm” length $\ell$   and the component of the vector momentum mv perpendicular to the lever arm. Show that this gives angular momentum $=\mathrm{m}{\ell }^2\frac{\mathrm{d\theta }}{\mathrm{dt}}$ .

Use the energy integral lemma to show that pendulum motion obeys $\frac{{\left(\mathrm{\theta }\text{'}\right)}^2}{2}-\frac{\mathrm{g}}{\mathrm{l}}\mathrm{cos\theta }=\mathrm{constant}$

Use the result of Problem 8 to find the value of $\mathrm{\theta }\text{'}\left(0\right)$, the initial velocity, that must be imparted to a pendulum at rest to make it approach (but not cross over) the apex of its motion. Take l = g for simplicity.

Use the result of Problem 8 to prove that if the pendulum in Figure 4.18 on page 208 is released from rest at the angle $\mathbf{0}\mathbf{<}\mathbf{\alpha }\mathbf{<}\mathbf{\pi }$, then $\left|\mathbf{\theta }\left(\mathbf{t}\right)\right|\le \mathbf{\alpha }$for all t.

Use the mass-spring analogy to explain the qualitative nature of the solutions to the Rayleigh equation  (22) $\mathrm{y}-\left[1-{\left(\mathrm{y}\text{'}\right)}^2\right]\mathrm{y}\text{'}+\mathrm{y}=0$depicted in Figures 4.24 and 4.25.

Use reduction of order to derive the solution ${\mathrm{y}}_2\left(\mathrm{t}\right)$ in equation (5) for Legendre’s equation.

Figure 4.26 contains graphs of solutions to the Duffing, Airy, and van der Pol equations. Try to match the solution to the equation.

Verify that the formulas for the Bessel functions ${\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{t}\right),{\mathrm{Y}}_{\frac{1}{2}}\left(\mathrm{t}\right)$ do indeed solve equation (16).

Use the mass-spring oscillator analogy to decide whether all solutions to each of the following differential equations are bounded as $\mathbf{t}\to \mathbf{+}\mathrm{\infty }$

1. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{t}}^2\mathbf{y}\mathbf{=}\mathbf{0}$
   
2. $\mathbf{y}\mathbf{"}\mathbf{-}{\mathbf{t}}^2\mathbf{y}\mathbf{=}\mathbf{0}$
   
3. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{y}}^5\mathbf{=}\mathbf{0}$
   
4. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{y}}^6\mathbf{=}\mathbf{0}$
   
5. $\mathbf{y}\mathbf{"}\mathbf{+}(\mathbf{4}\mathbf{+}\mathbf{2}\mathbf{cost})\mathbf{y}\mathbf{=}\mathbf{0}$(Mathieu’s equation)
6. $\mathbf{y}\mathbf{"}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$
7. $\mathbf{y}\mathbf{"}\mathbf{-}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}$

Use the energy integral lemma to show that every solution to the Duffing equation (18) is bounded; that is $\left|\mathrm{y}\left(\mathrm{t}\right)\right|⩽\mathrm{M}$, for some M. [Hint: First argue that for $\frac{{\mathrm{y}}^2}{2}+\frac{{\mathrm{y}}^4}{4}⩽\mathrm{K}$ some K.]

Armageddon. Earth revolves around the sun in an approximately circular orbit with radius r = a, completing a revolution in the time $\mathrm{T}=2\mathrm{\pi }{\left(\frac{{\mathrm{a}}^3}{\mathrm{GM}}\right)}^{\frac{1}{2}}$, which is one Earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on Earth is given by $\frac{\mathrm{GMm}}{{\mathrm{r}}^2}$, where m is the mass of Earth. Therefore, if Earth “stood still,” losing its orbital velocity, it would fall on a straight line into the sun in accordance with Newton’s second law: $\mathrm{m}\frac{{\mathrm{d}}^2\mathrm{r}}{{\mathrm{dt}}^2}=-\frac{\mathrm{GMm}}{{\mathrm{r}}^2}$.

If this calamity occurred, what fraction of the normal year T would it take for Earth to splash into the sun (i.e., achieve r = 0)? [Hint: Use the energy integral lemma and the initial conditions $\mathrm{r}\left(0\right)=\mathrm{a},\mathrm{r}\text{'}\left(0\right)=0$ .]

A 2 – kg mass is attached to a spring with stiffness k = 50 N/m. The mass is displaced $\frac{1}{4}\mathbf{m}$to the left of the equilibrium point and given a velocity of 1 m/sec to the left. Neglecting damping, find the equation of motion of the mass along with the amplitude, period, and frequency. How long after release does the mass pass through the equilibrium position? 

A 3 – kg mass is attached to a spring with stiffness k = 48 N/m. The mass is displaced$\frac{1}{2}\mathbf{m}$ to the left of the equilibrium point and given a velocity of 2 m/sec to the right. The damping force is negligible. Find the equation of motion of the mass along with the amplitude, period, and frequency. How long after release does the mass pass through the equilibrium position?

The motion of a mass-spring system with damping is governed by 

$\begin{array}{l}\mathbf{y}\mathbf{"}\left(t\right)\mathbf{+}\mathbf{by}\mathbf{\text{'}}\left(t\right)\mathbf{+}\mathbf{16}\mathbf{y}\left(t\right)\mathbf{=}\mathbf{0}\mathbf{;}\\ \mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{.}\end{array}$

Find the equation of motion and sketch its graph for b = 0,6,8, and 10.

A 20 – kg mass is attached to a spring with a stiffness 200 N/m. The damping constant for the system is 140 N-sec/m. If the mass is pulled 25 cm to the right of the equilibrium and given an initial leftward velocity of 1 m/sec, when will it first return to its equilibrium position?

Show that for the underdamped system of Example 3, the times when the solution curve y(t) in (33) touches the exponential curves $\pm \sqrt{\frac{7}{12}}{\mathbf{e}}^{-2t}$are not the same values of t for which the function y(t) attains its relative extrema.

The motion of a mass-spring system with damping is governed by 

$\begin{array}{l}\mathbf{y}\mathbf{"}\left(t\right)\mathbf{+}4\mathbf{y}\mathbf{\text{'}}\left(t\right)\mathbf{+}\mathbf{ky}\left(t\right)\mathbf{=}\mathbf{0}\mathbf{;}\\ \mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{.}\end{array}$

Find the equation of motion and sketch its graph for k = 2, 4, and 6.

The motion of a mass-spring system with damping is governed by 

$$\begin{gathered} \mathrm{y}\left(\mathrm{t}\right)+\mathrm{by}\text{'}\left(\mathrm{t}\right)+64\mathrm{y}\left(\mathrm{t}\right)=0; \\ \mathrm{y}\left(0\right)=1, \mathrm{y}\text{'}\left(0\right)=0. \end{gathered}$$

Find the equation of motion and sketch its graph for b = 0,10,16, and 20.

The motion of a mass-spring system with damping is governed by

$$\begin{gathered} \mathrm{y}\left(\mathrm{t}\right)+10\mathrm{y}\text{'}\left(\mathrm{t}\right)+\mathrm{ky}\left(\mathrm{t}\right)=0; \\ \mathrm{y}\left(0\right)=1, \mathrm{y}\text{'}\left(0\right)=0. \end{gathered}$$ 

Find the equation of motion and sketch its graph for k = 20, 25, and 30.

A $\frac{1}{8}$ – kg mass is attached to a spring with stiffness 16 N/m. The damping constant for the system is 2 N-sec/m. If the mass is moved $\frac{3}{4}\mathrm{m}$to the left of the equilibrium and given an initial leftward velocity of 2 m/sec, determine the equation of motion of the mass and give its damping factor, quasiperiodic, and quasi frequency.

A 2 – kg mass is attached to a spring with a stiffness of 40 N/m. The damping constant for the system is $8\sqrt{5}$N-sec/m. If the mass is pulled 10 cm to the right of the equilibrium and given an initial rightward velocity of 2 m/sec, what is the maximum displacement from equilibrium that it will attain?

A $\frac{1}{4}$– kg mass is attached to a spring with stiffness 8 N/m. The damping constant for the system is $\frac{1}{4}$N-sec/m. If the mass is moved 1 m to the left of equilibrium and released, what is the maximum displacement to the right that it will attain?

A 1 – kg mass is attached to a spring with a stiffness of 100 N/m. The damping constant for the system is 0.2 N-sec/m. If the mass is pushed rightward from the equilibrium position with the velocity of 1 m/sec, when will it attain its maximum displacement to the right?

A $\frac{1}{4}$– kg mass is attached to a spring with stiffness 8 N/m. The damping constant for the system is 2 N-sec/m. If the mass is pushed 50 cm to the left of equilibrium and given a leftward velocity of 2 m/sec, when will the mass attains its maximum displacement to the left?

Consider the equation for free mechanical vibration, $\mathrm{my}+\mathrm{by}\text{'}+\mathrm{ky}=0$ , and assume the motion is critically damped. Let $\mathrm{y}\left(0\right)={\mathrm{y}}_0,\mathrm{y}\text{'}\left(0\right)={\mathrm{v}}_0$and assume ${\mathrm{y}}_0\ne 0$.

1. Prove that the mass will pass through its equilibrium at exactly one positive time if and only if $\frac{-2{\mathrm{my}}_0}{2{\mathrm{mv}}_0+{\mathrm{by}}_0}>0$.
2. Use computer software to illustrate part (a) for a specific choice of m, b, k, ${\mathrm{y}}_0$ , and ${\mathrm{v}}_0$. Be sure to include an appropriate graph in your illustration.

For an underdamped system, verify that$\mathbf{b}\to \mathbf{0}$ as the damping factor approaches the constant A and the quasi frequency approaches the natural frequency.$\frac{\sqrt{\frac{k}{m}}}{2\pi }$

How can one deduce the value of the damping constant b by observing the motion of an underdamped system? Assume that the mass m is known. 

A mass attached to a spring oscillates with a period of 3 sec. After 2 kg are added, the period becomes 4 sec. Assuming that we can neglect any damping or external forces, determine how much mass was originally attached to the spring. 

Consider the equation for free mechanical vibration, $\mathrm{my}+\mathrm{by}\text{'}+\mathrm{ky}=0$ , and assume the motion is over-damped. Suppose $\mathrm{y}\left(0\right)>0$and $\mathrm{y}\text{'}\left(0\right)>0$. Prove that the mass will never pass through its equilibrium at any positive time.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

Sketch the frequency response curve (13) for the system in which m = 4, k = 1, b = 2.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ and for the MKS system.

Sketch the frequency response curve (13) for the system in which m = 2, k = 3, b = 3.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

Determine the equation of motion for an undamped system at resonance governed by

$$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dt}}^2}+9\mathrm{y}=2\cos 3\mathrm{t}; \\ \mathrm{y}\left(0\right)=1,\mathrm{y}\text{'}\left(0\right)=0. \end{gathered}$$

Sketch the solution.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

Determine the equation of motion for an undamped system at resonance governed by

$$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dt}}^2}+\mathrm{y}=5\mathrm{cost} \\ \mathrm{y}\left(0\right)=0,\mathrm{y}\text{'}\left(0\right)=1 \end{gathered}$$

Sketch the solution.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

An undamped system is governed by

$$\begin{gathered} \mathrm{m}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dt}}^2}+\mathrm{ky}={\mathrm{F}}_0\cos \gamma \mathrm{t}; \\ \mathrm{y}\left(0\right)=\mathrm{y}\text{'}\left(0\right)=0, \end{gathered}$$

Where, $\gamma \ne \omega :=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$

1. Find the equation of motion of the system.
2. Use trigonometric identities to show that the solution can be written in the form data-custom-editor="chemistry" $\mathrm{y}\left(\mathrm{t}\right)=\frac{2{\mathrm{F}}_0}{\mathrm{m}\left({\omega }^2-{\gamma }^2\right)}\sin \left(\frac{\omega +\gamma }{2}\mathrm{t}\right)\sin \left(\frac{\omega -\gamma }{2}\mathrm{t}\right)$.
3. When is near, then is small, while is relatively large compared with. Hence, y(t) can be viewed as the product of a slowly varying sine function and a rapidly varying sine function. The net effect is a sine function y(t) with frequency, which serves as the time-varying amplitude of a sine function with frequency. This vibration phenomenon is referred to as beats and is used in turning stringed instruments. This same phenomenon in electronics is called amplitude modulation. To illustrate this phenomenon, sketch the curve y(t) for ${\mathrm{F}}_0=32,\mathrm{m}=2,\omega =9$ and $\gamma =7$ .

In the following problems, take $g=32ft/se{c}^2$ for the U.S. Customary System and $g=9.8m/se{c}^2$ the MKS system.

Derive the formula for given in (21).

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

Shock absorbers in automobiles and aircraft can be described as forced overdamped mass-spring systems. Derive an expression analogous to equation (8) for the general solution to the differential equation (1) when ${\mathrm{b}}^2>4\mathrm{mk}$.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

The response of an overdamped system to a constant force is governed by equation (1) with m = 2, b = 8, k = 6, and $F_o=18 and \gamma =0$ . If the system starts from rest, $\left[y\left(0\right)=y\text{'}\left(0\right)=0\right],$compute and sketch the displacement y(t). What is the limit of y(t) as $t\to +\infty$? Interpret this physically.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$  for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

Show that the period of the simple harmonic motion of a mass hanging from a spring is $2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$, where l denotes the amount (beyond its natural length) that the spring is stretched when the mass is at equilibrium.

A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At t = 0, an external force F(t) = 2 cos 2t lb is applied to the system. If the spring constant is 10 lb/ft and the damping constant is 1 lb-sec/ft, find the equation of motion of the mass. What is the resonance frequency for the system?

A 2 kg mass is attached to a spring hanging from the ceiling, thereby causing the spring to stretch 20 cm upon coming to rest at equilibrium. At time t = 0, the mass is displaced 5 cm below the equilibrium position and released. At this same instant, an external force F(t) = 0.3 cos t N is applied to the systems. If the damping constant for the system is 5 N-sec/m, determine the equation of the motion for the mass. What is the resonance frequency for the system?

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

A mass weighing 32 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At time t = 0, an external force F(t) = 3 cos 4t lb is applied to the system. If the spring constant is 5 lb/ft and the damping constant is 2 lb-sec/ft, find the steady-state solution for the system.

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 3 N/sec. At time t = 0, an external force $2\sin \left(2\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)$N is applied to the system. Determine the amplitude and frequency of the steady-state solution.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$ for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 3 N/sec. At time t = 0, an external force $2\sin \left(2\mathrm{t}\right)\cos \left(2\mathrm{t}\right)$ N is applied to the system. Determine the amplitude and frequency of the steady-state solution.

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

A helium-filled balloon on a cord, hanging y km above a level surface, is subjected to three forces:

1. The (constant) buoyant force B exerted by the external air pressure;
2. The weight of the balloon is mg, where m is the mass of the balloon;
3. The weight of the part of the cord that has been lifted off the surface, dgy, where d is the linear density (kg/m) of the cord. (We assume that the cord is longer than the height of the balloon.) 

Express Newton’s second law for the balloon and show that it is exactly analogous to the governing equation for a mass vertically hung from a spring (page 226). Find the equation of motion. What is the frequency of the balloon’s oscillations?

Find a general solution to the given differential equation $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{0}$.

Find a general solution to the given differential equation.$\mathbf{49}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{14}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.$\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.$\mathbf{9}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{30}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{25}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation. $\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{11}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{14}\mathbf{y}\mathbf{=}\mathbf{0}$