To approximate a second-order undamped system with a First Order Plus Dead Time (FOPDT) model, you shall aim to find an equivalent system that captures the essential dynamics of the original system with a simpler representation. The FOPDT model is given by: 
where K is the process gain, 
is time constant and, θ is dead time.
where K is the process gain, is time constant and, θ is dead time.As stated in the question, your system is repereseted as: 
. This is a second-order system without damping as 
does not have a 
term (where ξ is the damping ratio and 
is the natural frequency). The natural frequency is 
in your case.
. This is a second-order system without damping as does not have a term (where ξ is the damping ratio and is the natural frequency). The natural frequency is in your case.You can consider the following steps to approximate to FOPDT
Identify Key Characteristics - For the original system, identify characteristics like the time to reach first peak, settling time, or steady-state gain. For a second-order undamped system, the steady-state gain is the coefficient of the transfer function, which is 7 in this case.
Approximate Parameters
Gain (K) : The gain of the FOPDT model should match the steady-state gain of the original system. Thus, 
.
.Time Constant (τ) : The time constant should represent the time it takes for the system to reach a significant portion of its final value. For a second-order undamped system, you might use the time it takes to reach the first peak or some empirical methods for estimation.
Dead Time (θ): This could represent the delay before the system starts responding. For systems without an explicit delay, this might be approximated as a fraction of time constant or based on system's response characteristics.
Approximation Method
A generic approach aimed towards approximating involves comparing step response of original system to that of FOPDT model and adjusting τ and θ to match key features of the response, such as the time to reach first peak or the half-rise time. However, for a purely undamped second-order system, there is no perfect match because the original system oscillates indefinitely while the FOPDT system does not.
Given lack of a standard method for directly converting a second-order undamped system to FOPDT, consider using empirical methods or judgment based on the system's step response, you might consider the following :
- Estimate τ based on when the system reaches a significant portion of its steady-state value for the first time.
- θ is based on when the system output first starts to rise, but for an undamped second-order system (as in your case), this is effectively zero.
As a footnote, this approach is highly approximate and relies on matching key characteristics rather than a direct mathematical transformation. For precise transformation, consider using simulations to iteratively adjust τ and θ until the FOPDT model's response closely aligns with that of the original system.
I hope this helps.
