દ્વિતીય ક્રમની સિસ્ટમ (Second Order Systems) સંબંધિત નીચેના વિધાનો ધ્યાનમાં લો:
1. ડેમ્પિંગ ગુણોત્તર સિસ્ટમ પ્રતિભાવની પ્રકૃતિ (ઓવરડેમ્પ્ડ, અન્ડરડેમ્પ્ડ કે ક્રિટિકલી ડેમ્પ્ડ) નક્કી કરે છે.
2. અન્ડરડેમ્પ્ડ સિસ્ટમ ઘટતા કંપવિસ્તાર સાથે દોલની પ્રતિભાવ દર્શાવે છે.
3. ક્રિટિકલી ડેમ્પ્ડ સિસ્ટમ દોલન વગર સૌથી ઝડપી પ્રતિભાવ આપે છે.
4. ઓવરડેમ્પ્ડ સિસ્ટમ અચળ કંપવિસ્તાર સાથે અનિશ્ચિત સમય સુધી દોલન કરે છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
દ્વિતીય ક્રમની ગતિશીલતા પ્રાકૃતિક આવૃત્તિ અને ડેમ્પિંગ ગુણોત્તરથી વર્ણવાય છે. અન્ડરડેમ્પ્ડ પ્રતિભાવ ઘટતા કંપવિસ્તાર સાથે દોલન કરે છે, ક્રિટિકલ ડેમ્પિંગ સૌથી ઝડપી દોલન-રહિત પ્રતિભાવ આપે છે અને ઓવરડેમ્પિંગ ધીમો દોલન-રહિત પ્રતિભાવ આપે છે. મુખ્ય સંબંધ: s²+2ζω_n s+ω_n²=0.
પરીક્ષા મુદ્દો: ζ<1 અન્ડરડેમ્પ્ડ, ζ=1 ક્રિટિકલ, ζ>1 ઓવરડેમ્પ્ડ. સામાન્ય ભૂલ: ઓવરડેમ્પ્ડ પ્રતિભાવ દોલન કરતો નથી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
Damping ratio determines the nature of system response such as overdamped, underdamped, or critically damped.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 2 - Correct.
Underdamped systems show oscillatory response with decaying amplitude.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 3 - Correct.
Critically damped system has fastest response without oscillations.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 4 - Incorrect.
Overdamped system oscillates indefinitely with constant amplitude.
The correct principle is: An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Core Concept
Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Formula / Key Relationship
Characteristic form: s²+2ζω_n s+ω_n²=0.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
Common Mistake
Overdamped responses do not sustain oscillations.
Quick Revision
Damping Ratio and Response Type is linked with Second-Order Systems, Time Response and Control Systems. ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
Consider the following statements regarding Second Order Systems:
1. Damping ratio determines the nature of system response such as overdamped, underdamped, or critically damped.
2. Underdamped systems show oscillatory response with decaying amplitude.
3. Critically damped system has fastest response without oscillations.
4. Overdamped system oscillates indefinitely with constant amplitude.
Which of the above statements is/are correct?
Answer & step-by-step solution
Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. Key relation: Characteristic form: s²+2ζω_n s+ω_n²=0.
Exam focus: ζ1 overdamped. Common trap: Overdamped responses do not sustain oscillations.
Statement-wise Verification
- Statement 1 - Correct.
Damping ratio determines the nature of system response such as overdamped, underdamped, or critically damped.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 2 - Correct.
Underdamped systems show oscillatory response with decaying amplitude.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 3 - Correct.
Critically damped system has fastest response without oscillations.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 4 - Incorrect.
Overdamped system oscillates indefinitely with constant amplitude.
The correct principle is: An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Core Concept
Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Formula / Key Relationship
Characteristic form: s²+2ζω_n s+ω_n²=0.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
Common Mistake
Overdamped responses do not sustain oscillations.
Quick Revision
Damping Ratio and Response Type is linked with Second-Order Systems, Time Response and Control Systems. ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
द्वितीय कोटि सिस्टम (Second Order Systems) के संबंध में निम्नलिखित कथनों पर विचार कीजिए:
1. डैम्पिंग अनुपात सिस्टम प्रतिक्रिया की प्रकृति (ओवरडैम्प्ड, अंडरडैम्प्ड या क्रिटिकली डैम्प्ड) निर्धारित करता है।
2. अंडरडैम्प्ड सिस्टम घटते आयाम के साथ दोलनी प्रतिक्रिया दिखाते हैं।
3. क्रिटिकली डैम्प्ड सिस्टम बिना दोलन के सबसे तेज़ प्रतिक्रिया देता है।
4. ओवरडैम्प्ड सिस्टम स्थिर आयाम के साथ अनिश्चित काल तक दोलन करता है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
द्वितीय कोटि गतिकी प्राकृतिक आवृत्ति और डैम्पिंग अनुपात से वर्णित होती है। अंडरडैम्प्ड प्रतिक्रिया घटते आयाम से दोलन करती है, क्रिटिकल डैम्पिंग सबसे तेज़ बिना-दोलन प्रतिक्रिया देती है और ओवरडैम्पिंग धीमी बिना-दोलन प्रतिक्रिया देती है। मुख्य संबंध: s²+2ζω_n s+ω_n²=0।
परीक्षा फोकस: ζ<1 अंडरडैम्प्ड, ζ=1 क्रिटिकल, ζ>1 ओवरडैम्प्ड। सामान्य गलती: ओवरडैम्प्ड प्रतिक्रिया दोलन नहीं करती।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
Damping ratio determines the nature of system response such as overdamped, underdamped, or critically damped.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 2 - Correct.
Underdamped systems show oscillatory response with decaying amplitude.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 3 - Correct.
Critically damped system has fastest response without oscillations.
An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. - Statement 4 - Incorrect.
Overdamped system oscillates indefinitely with constant amplitude.
The correct principle is: An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Core Concept
Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
Formula / Key Relationship
Characteristic form: s²+2ζω_n s+ω_n²=0.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
Common Mistake
Overdamped responses do not sustain oscillations.
Quick Revision
Damping Ratio and Response Type is linked with Second-Order Systems, Time Response and Control Systems. ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.
More Time Response questions