The mass-spring-damper system shown in the following figure represents : 
a second-order system
The order of a system is the order of the differential equation that describes it, and for a mechanical system that is set by the number of independent energy stores.
Write the force balance. Newton's second law applied to the mass, with the spring and damper opposing the applied force F, gives
\(M\dfrac{d^{2}x}{dt^{2}}+B\dfrac{dx}{dt}+Kx=F(t)\)
The highest derivative is the second, so this is a second-order system — option 3.
Identify each term with its element :
| Element | Force | Energy stored |
|---|---|---|
| Mass M | \(M\ddot{x}\) | Kinetic, \(\tfrac{1}{2}Mv^{2}\) |
| Damper B | \(B\dot{x}\) | None — it dissipates |
| Spring K | \(Kx\) | Potential, \(\tfrac{1}{2}Kx^{2}\) |
Two energy stores, therefore second order. The damper contributes a first-derivative term but stores nothing, so it does not raise the order — exactly as a resistor does not raise the order of an electrical circuit. That is the point worth carrying away: it is the storage elements that count.
The transfer function and its standard form :
\(\dfrac{X(s)}{F(s)}=\dfrac{1}{Ms^{2}+Bs+K}=\dfrac{1/M}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}\)
with
\(\omega_{n}=\sqrt{\dfrac{K}{M}}\qquad \zeta=\dfrac{B}{2\sqrt{KM}}\)
— the mass and spring set the natural frequency, and the damper alone sets the damping ratio.
The electrical analogue makes the correspondence exact. A series RLC circuit obeys the identical equation with mass ↔ inductance, damper ↔ resistance and spring compliance ↔ capacitance; force corresponds to voltage and velocity to current. That is why an RLC circuit and a suspension both overshoot and ring in the same way, and why the whole vocabulary of overshoot, settling time and damping ratio transfers between the two domains.
Why the other orders would need different hardware : a zero-order system has no dynamics at all (a plain potentiometer), a first-order one has a single store (a thermometer, an RC circuit), and third order would require a further independent store beyond the two present here.
Hence, the mass-spring-damper is a second-order system.
Similar Questions
Read the following statements about the transfer function of a system :
(a) The transfer function provides complete insight into the structure of the system.
(b) It offers a symbolic picture about the dynamic characteristics of the system.
(c) It does not give any insight into the structure of the system.
(d) If the transfer functions of individual components of the system are known, the overall characteristics of the system can be determined just by taking their product.
Which of the above statements are correct ?
The error detector element in a control system gives
Three blocks having transfer functions \(G_1(s) = \frac{1}{s+2}\), \(G_2(s) = \frac{1}{s+5}\) and \(G_3(s) = \frac{s+1}{s+3}\) are cascaded. The equivalent transfer function is:
Important Questions from Basics of Control Systems
- Match the LIST-I with LIST-IIChoose the correct answer from the options given below:
LIST-I LIST-II A. Nyquist Criterion I. Polar Plots B. Bode Plot II. Rectangular Plots C. Phase Margin III. Amount of additional phase lag at the gain crossover frequency required to bring the system to the verge of instability D. Gain Margin IV. It is the reciprocal of the magnitude $|G(j\omega)|$ at the frequency at which the phase angle is $-180^\circ$. - Which of the following statement is not correct in respect of Transfer Function of linear Time Invariant System
A. In any open loop control system, the output is not compared with the reference input.
B. Traffic control by means of signals operated on a time basis is an example of closed loop control.
C. Closed loop control always implies the use of feedback control action in order to reduce system error.
D. Washing machine is an example of open loop system.
E. Stability is a major concern in open loop control system.
Choose the correct answer from the options given below:
Read the following statements about the transfer function of a system :
(a) The transfer function provides complete insight into the structure of the system.
(b) It offers a symbolic picture about the dynamic characteristics of the system.
(c) It does not give any insight into the structure of the system.
(d) If the transfer functions of individual components of the system are known, the overall characteristics of the system can be determined just by taking their product.
Which of the above statements are correct ?
The error detector element in a control system gives
