Consider a cam follower machine be modeled as a mass-spring-damper system. The base of thi system is subjected to a periodic force as shown, determine the response of the system by expanding the periodic force as a Fourier's series. u(t) 201 Zo |Z(2) 2T
Consider a cam follower machine be modeled as a mass-spring-damper system. The base of thi system is subjected to a periodic force as shown, determine the response of the system by expanding the periodic force as a Fourier's series. u(t) 201 Zo |Z(2) 2T
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
 *(Note: Include actual schematic in your website)*
### Diagram 2: Input Force Function, z(t)
The graph depicts the periodic force function, \( Z(t) \), that is applied to the mass-spring-damper system:
- **X-axis (t):** Represents time.
- **Y-axis (Z(t)):** Represents the displacement due to the periodic force.
- The force function is a sawtooth wave that rises linearly with a period \( T_n \) before dropping back to \( Z_0 \).
Graph explanation:
- The periodic force starts at \( Z_0 \) and increases linearly to a peak, then drops back to \( Z_0 \).
- This cycle repeats with the same period \( T_n \).
## Mathematical Approach
To determine the system's response:
1. **Fourier Series Expansion:**
- Expand the periodic force \( Z(t) \) into its Fourier series to express it as a sum of sinusoidal components.
2. **System Response:**
- Analyze the response of the mass-spring-damper system to each sinusoidal component of the Fourier series, utilizing the properties of linear systems.
This approach allows for understanding the behavior and dynamic response of the system under the given periodic force.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c227c7-5e7d-4c97-86f7-329995c469cc%2Fb6805ec0-61eb-4714-8afe-bc077e726092%2Fgtl1v8i_processed.png&w=3840&q=75)
Transcribed Image Text:# Modeling a Cam Follower Machine as a Mass-Spring-Damper System
## Problem Statement
Consider a cam follower machine to be modeled as a mass-spring-damper system. The base of this system is subjected to a periodic force as shown in the diagrams. Determine the response of the system by expanding the periodic force as a Fourier series.
## System Description
### Mechanical Representation
The system is composed of the following components:
- **Mass (m):** The mass of the system.
- **Spring (k):** Provides the restoring force proportional to the displacement.
- **Damper (C):** Provides the damping force proportional to the velocity.
- **Period Force (u(t)):** The force applied periodically to the system.
### Diagram 1: System Setup
The simple schematic illustrates the components of the mass-spring-damper system with the following elements:
- **z(t):** The displacement of the mass.
- **ω (omega):** Angular velocity of the camshaft driving the periodic force.
- **Periodic Force (u(t)):** Applied due to the cam mechanism.
 *(Note: Include actual schematic in your website)*
### Diagram 2: Input Force Function, z(t)
The graph depicts the periodic force function, \( Z(t) \), that is applied to the mass-spring-damper system:
- **X-axis (t):** Represents time.
- **Y-axis (Z(t)):** Represents the displacement due to the periodic force.
- The force function is a sawtooth wave that rises linearly with a period \( T_n \) before dropping back to \( Z_0 \).
Graph explanation:
- The periodic force starts at \( Z_0 \) and increases linearly to a peak, then drops back to \( Z_0 \).
- This cycle repeats with the same period \( T_n \).
## Mathematical Approach
To determine the system's response:
1. **Fourier Series Expansion:**
- Expand the periodic force \( Z(t) \) into its Fourier series to express it as a sum of sinusoidal components.
2. **System Response:**
- Analyze the response of the mass-spring-damper system to each sinusoidal component of the Fourier series, utilizing the properties of linear systems.
This approach allows for understanding the behavior and dynamic response of the system under the given periodic force.
![### Fourier Series and Steady-State Solution in Vibrational Systems
#### Definitions and Formulas
Given the quantity:
\[
\Omega = \frac{2\pi}{T}
\]
We define \( Z(t) \) as:
\[
Z(t) = \frac{Z_0}{T_0}t
\]
Expanding \( Z(t) \) using a Fourier series, we can write:
\[
Z(t) = \frac{a_0}{2} + \sum a_n \cos n\Omega t + \sum b_n \sin n\Omega t
\]
Here,
\[
a_0 = \frac{2}{T} \int_{0}^{T} Z(t) \, dt
\]
The coefficients \( a_n \) and \( b_n \) are given by:
\[
a_n = \frac{2}{T} \int_{0}^{T} Z(t) \cos n\Omega t \, dt
\]
\[
b_n = \frac{2}{T} \int_{0}^{T} Z(t) \sin n\Omega t \, dt
\]
For a specific \( Z(t) \):
\[
Z(t) = Z_0 \left( \frac{1}{2} + \frac{1}{\pi} \sum_{n=1}^{\infty} \frac{\sin 2n\pi t / T_0}{n} \right)
\]
Steady-state solution (Eq. 6.8 in the referenced text) is:
\[
u_s = \frac{a_0}{2k} + \sum \left[ \frac{a_n/k}{\sqrt{(1-n^2r^2)^2 + (2\xi nr)^2}} \cos(n\Omega t - \alpha_n) + \frac{b_n/k}{\sqrt{(1-n^2 r^2)^2 + (2\xi nr)^2}} \sin(n\Omega t - \alpha_n) \right]
\]
Where the phase shift \( \alpha_n \) and other terms are defined as:
\[
\alpha_n = \tan^{-1} \left( \frac{2\xi nr}{1-n^2r^2} \right)
\]
\[
r = \frac{\Omega}{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00c227c7-5e7d-4c97-86f7-329995c469cc%2Fb6805ec0-61eb-4714-8afe-bc077e726092%2F0dnkb18_processed.png&w=3840&q=75)
Transcribed Image Text:### Fourier Series and Steady-State Solution in Vibrational Systems
#### Definitions and Formulas
Given the quantity:
\[
\Omega = \frac{2\pi}{T}
\]
We define \( Z(t) \) as:
\[
Z(t) = \frac{Z_0}{T_0}t
\]
Expanding \( Z(t) \) using a Fourier series, we can write:
\[
Z(t) = \frac{a_0}{2} + \sum a_n \cos n\Omega t + \sum b_n \sin n\Omega t
\]
Here,
\[
a_0 = \frac{2}{T} \int_{0}^{T} Z(t) \, dt
\]
The coefficients \( a_n \) and \( b_n \) are given by:
\[
a_n = \frac{2}{T} \int_{0}^{T} Z(t) \cos n\Omega t \, dt
\]
\[
b_n = \frac{2}{T} \int_{0}^{T} Z(t) \sin n\Omega t \, dt
\]
For a specific \( Z(t) \):
\[
Z(t) = Z_0 \left( \frac{1}{2} + \frac{1}{\pi} \sum_{n=1}^{\infty} \frac{\sin 2n\pi t / T_0}{n} \right)
\]
Steady-state solution (Eq. 6.8 in the referenced text) is:
\[
u_s = \frac{a_0}{2k} + \sum \left[ \frac{a_n/k}{\sqrt{(1-n^2r^2)^2 + (2\xi nr)^2}} \cos(n\Omega t - \alpha_n) + \frac{b_n/k}{\sqrt{(1-n^2 r^2)^2 + (2\xi nr)^2}} \sin(n\Omega t - \alpha_n) \right]
\]
Where the phase shift \( \alpha_n \) and other terms are defined as:
\[
\alpha_n = \tan^{-1} \left( \frac{2\xi nr}{1-n^2r^2} \right)
\]
\[
r = \frac{\Omega}{\
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