Differential equation: Spring/Mass system of driven motion with damping

In summary, a 32 pound weight stretches a spring 2 feet and is then released from an initial position of 1 foot below the equilibrium position. The surrounding medium offers a damping force of 8 times the instantaneous velocity. The equation of motion can be found using the equation m\frac{d^{2}x}{dt^{2}}+\beta\frac{dx}{dt}+kx=f(t), where k=16\frac{lb}{ft} and m=1 slug. The external force is given as 2cos(5t). It was determined that β=8, making the problem easy to solve. Initial conditions should be carefully considered.
  • #1
TeenieBopper
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Homework Statement


A 32 pound weight stretches a spring 2 feet. The mass is then released from an initial position of 1 foot below the equilibrium position. The surrounding medium offers a damping force of 8 times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force of 2cos(5t).

Homework Equations


F=kx
m=W/g

m[itex]\frac{d^{2}x}{dt^{2}}[/itex]+[itex]\beta[/itex][itex]\frac{dx}{dt}[/itex]+kx=f(t)

The Attempt at a Solution



I found that k=16[itex]\frac{lb}{ft}[/itex] and m=1 slug. This gets me the following equation:

[itex]\frac{d^{2}x}{dt^{2}}[/itex]+[itex]\beta[/itex][itex]\frac{dx}{dt}[/itex]+16x=2cos(5t)

I'm at a loss for how to determine [itex]\beta[/itex], which is the damping force of 8 times the instantaneous velocity. I don't know how to determine instantaneous velocity. I know that once I have [itex]\beta[/itex], I can just use a LaPlace transform to find x(t). But [itex]\beta[/itex] is my stumbling block right now.

As I was writing this, it occurred to me that [itex]\frac{dx}{dt}[/itex]=instantaneous velocity and that would make [itex]\beta[/itex]=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?

We kind of rushed through this application in class the other day. Thanks in advance for any help you're able to provide.
 
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  • #2
TeenieBopper said:
As I was writing this, it occurred to me that [itex]\frac{dx}{dt}[/itex]=instantaneous velocity and that would make [itex]\beta[/itex]=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?.

Yeah I would agree that β=8 here. Just make sure your initial conditions are correct with the proper signs.
 

FAQ: Differential equation: Spring/Mass system of driven motion with damping

1. What is a spring/mass system of driven motion with damping?

A spring/mass system of driven motion with damping is a physical system that consists of a mass attached to a spring and subject to an external driving force, while also experiencing damping due to friction or other forces. This system can be described using differential equations, which relate the position, velocity, and acceleration of the mass to the parameters of the system.

2. How is the motion of a spring/mass system with damping different from one without damping?

In a spring/mass system without damping, the mass will continue to oscillate indefinitely when subjected to an external driving force. However, in a system with damping, the amplitude of the oscillations will decrease over time due to the dissipative effects of damping forces. This results in a system with a smaller amplitude and a different frequency of oscillation compared to one without damping.

3. What is the role of differential equations in studying spring/mass systems with damping?

Differential equations are used to describe the behavior of a spring/mass system with damping by relating the position, velocity, and acceleration of the mass to the parameters of the system such as the spring constant, mass, and damping coefficient. Solving these differential equations allows us to predict the motion of the system and understand how it responds to different driving forces and damping forces.

4. How does damping affect the resonance of a spring/mass system?

Damping affects the resonance of a spring/mass system by changing its natural frequency. In a system without damping, the natural frequency is solely determined by the mass and spring constant. However, in a system with damping, the natural frequency is reduced due to the dissipative effects of damping forces. This results in a shift in the resonance frequency and a decrease in the amplitude of oscillations at resonance.

5. Can a spring/mass system with damping ever reach a state of equilibrium?

Yes, a spring/mass system with damping can reach a state of equilibrium where the force of the spring and the damping forces balance out the external driving force. In this state, the mass will not experience any net force and will remain at rest. This state is known as the equilibrium position and can be calculated using the parameters of the system and the differential equations that describe its motion.

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