A mass of 1.0 g is attached to the end of a spring with a stiffness of 20 N/m. a. Determine the critical damping coefficient, c (i.e., the coefficient at which the system transitions between overdamped and underdamped conditions). b. A dashpot is chosen with a damping coefficient equal to one fifth of that determined in part a. This system is to be used as part of a high speed switch system in which a driving force opens and shuts a switch at fast speeds. fim m -X(1) F(t) The gap between the two electrodes is 1.0 mm. In order to close the switch, a sinusoidal force is applied equal to: F(t) = (16 mN) cos t An engineer is designing the switch such that the on/off function is as fast as possible. What is the highest driving frequency that will result in closing of the switch?

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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### Spring-Mass System Analysis

**Problem Statement:**

A mass of 1.0 g is attached to the end of a spring with a stiffness of 20 N/m.

**Tasks:**

a. **Determine the critical damping coefficient**, \( c \) (i.e., the coefficient at which the system transitions between overdamped and underdamped conditions).

b. A dashpot is chosen with a damping coefficient equal to one-fifth of that determined in part a. This system is to be used as part of a high-speed switch system in which a driving force opens and shuts a switch at fast speeds.

- **Diagram Explanation**:
  - The diagram shows a mass-spring-dashpot system denoted by:
    - \( m \): Mass
    - \( k \): Spring constant
    - \( X(t) \): Displacement over time
    - \( F(t) \): Applied force
  - The spring and dashpot are attached to a mass, which interacts with a surface on the right.

The gap between the two electrodes is 1.0 mm.

In order to close the switch, a sinusoidal force is applied equal to:

\[ F(t) = (16 \text{ mN}) \cos \omega t \]

An engineer is designing the switch such that the on/off function is as fast as possible. What is the highest driving frequency that will result in closing of the switch?
Transcribed Image Text:### Spring-Mass System Analysis **Problem Statement:** A mass of 1.0 g is attached to the end of a spring with a stiffness of 20 N/m. **Tasks:** a. **Determine the critical damping coefficient**, \( c \) (i.e., the coefficient at which the system transitions between overdamped and underdamped conditions). b. A dashpot is chosen with a damping coefficient equal to one-fifth of that determined in part a. This system is to be used as part of a high-speed switch system in which a driving force opens and shuts a switch at fast speeds. - **Diagram Explanation**: - The diagram shows a mass-spring-dashpot system denoted by: - \( m \): Mass - \( k \): Spring constant - \( X(t) \): Displacement over time - \( F(t) \): Applied force - The spring and dashpot are attached to a mass, which interacts with a surface on the right. The gap between the two electrodes is 1.0 mm. In order to close the switch, a sinusoidal force is applied equal to: \[ F(t) = (16 \text{ mN}) \cos \omega t \] An engineer is designing the switch such that the on/off function is as fast as possible. What is the highest driving frequency that will result in closing of the switch?
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