Derivation of the Frequency Characteristics
Natural Frequency of a Spring-Mass Oscillator
Spring-Mass Oscillator without Damping
The base of the spring-mass oscillator model is considered fixed in space. In the equilibrium position let x = 0. The mass is m [kg] and the spring constant is k [N/m]. The static length of the spring is L0.
To obtain an equation of motion one identifies all static and dynamic forces that occur. The weight force m·g is invariant along x and in time and can therefore be left out.
The force acting on the mass due to the acceleration is m· ẍ .
The second force is caused by the spring and is k·x.
This is a homogeneous differential equation.
For the solution one makes the ansatz
The characteristic equation is
Undamped spring-mass oscillator
or
So for the equilibrium of the forces we write
with the solutions
With we find for the solution of the differential equation
A and B are given by the initial conditions. If for instance the deflection or displacement x at time 0 is equal to X 0, then follows
This is a continuous, harmonic oscillation with the frequency ω0.
is called the natural frequency of the system.
Spring-Mass Oscillator with Damping
We add a damping "dashpot" with the damping coefficient c.
The damping force is assumed to be proportional to the velocity and thus becomes c·ẋ.
The dimension of [c] is N/m/s or kg/s.
The summation of all forces leads to
or
Here it is useful to introduce the relative damping ζ which is defined as
Spring-mass oscillator with damping
I.e. the damping c is set in relation to m and ω0. The "2" is an arbitrary factor.
With , the differential equation can be written in the form
The differential equation is homogenous again and we make the same ansatz
The characteristic equation is then
with the solutions
For ζ = 0, the result for the undamped oscillator is obtained again.
We are more interested here in the case of a weakly damped oscillator, i.e. ζ < 1. The root expression then becomes negative and the solutions of the characteristic equation will be
They are conjugate complex
The total solution is equal to the sum of the particular solutions and becomes
If we set the imaginary part
so we get
A and B are again determined by the initial conditions. With x(t=0)=0 for example, the solution is
This is a sinusoidal oscillation with the frequency ωD and the initial amplitude A.
The oscillation frequency ωD is the damped natural frequency. Note that it is slightly different from ω0 of the undamped oscillation, depending on ζ .
Frequency Characteristics of the Seismic Transducer
Derivation of the Transfer Function
In contrast to the above we apply now a movement u=g(t) to the base. This means we force the base to follow a defined movement.
The position of the mass m as a function of time is given by u+x.
The acceleration force therefore becomes
For the equilibrium of the forces (we can leave out the static weight force again) we find the equation
or with separated variables
dividing by m , and introducing
and
Spring-mass oscillator under forced vibration
we obtain the same differential equation as for the free oscillation, but now it is inhomogeneous due to the term on the right
The solution consists of the superposition of the solution of the homogeneous differential equation and a particular solution of the inhomogeneous differential equation.
Hence, we get a decaying first part, as in the case of the free oscillation, plus a residual stationary part, which follows the excitation u=g(t).
As excitation function we choose a harmonic oscillation with constant displacement amplitude
with the derivations
and
The differential equation therefore becomes
We guess that the movement of the mass will be also a harmonic oscillation and therefore we take the following ansatz for the solution:
By deriving and substituting one obtains
A comparison of coefficients leads to the following system of equations for A and B
The determinant of the coefficients is calculated as folows:
The two solutions of the system of equations become
and
and therfore
We transform
into the form
For the amplitude and phase angle we get
and thereby the final result of the
general transfer function:
The negative sign in cos(ωt−φ) indicates, that the output sinal is lagging the input.
Frequency Response of Displacement, Velocity and Acceleration
To make it easier and for transferable results it is better to use the frequency response representation. The frequency response is a complex function of the frequency and describes the output signal in relation to a sinusoidal input signal in terms of amplitude and phase.
For the moment we only consider the amplitude response i.e. the ratio of the output amplitude with respect to the input amplitude as a function of the frequency.
Displacement Frequency Response
Still with the output amplitude X0 and the input amplitude U0 we get from the general transfer function the amplitude response
This function provides an amplification factor for every frequency and is therefore also called amplification function.
We introduce the dimensionless relative frequency ωR i.e. the excitation frequency in relation to the natural frequency
To do this we expand the expression of the frequency response with 1/ω0² to obtain the (dimensionless) response function of the displacement
Both the input U and output X in our model above are displacements and the frequency response is therefore denoted by displacement amplitude response Φd.
The double logarithmic diagram shows the function for different values of the relative damping ζ .
We can see that for frequencies well above the natural frequency the transfer ratio becomes 1.
Displacement amplitude response
This means in this region we have a 1 to 1 displacement behaviour. Near to the natural frequency we find a more ore less pronounced amplification depending on ζ with a maximum atthe resonance frequency. Below the resonance the curve leads into slope of 40dB / decade.
Acceleration Frequency Response
For the frequency response with an acceleration signal as input we have to derive the input function twice.
The negative sign actually concerns the phase angle and we could also write
Because we are not interested in the phase at this time, we can set it to zero and obtain for the
acceleration amplitude response Φa .
Acceleration amplitude response
The dimension of the function is s² because it is [displacement X0 / acceleration Ü0 ].
Due to the double derivative the function now tilts to the other side. I.e. in acceleration terms we find a direct transfer in the frequency domain below the natural frequency, while we find a roll-off rate of -40dB per decade for high frequencies. The resonance region looks similar to the displacement response.
Velocity Frequency Response
Finally we can calculate and plot the amplification function for an input in velocity terms and we get the
velocity amplitude response Φv .
The function is now symmetric around the natural frequency (logarthmic frequency axis).
The dimension of the function is
[displacement / velocity] = s
Velocity amplitude response
Response of the Phase Angle φ
In the general transfer function we find the phase angle φ in the output function.
The behaviour of the phase angle is the same in all three cases, displacement velocity and acceleration:
The change in phase through the resonance must not be confused with the fact that displacement, velocity and acceleration are each offset by π/2.
Phase progression of the general transfer function
Phase Response
The phase response as part of the frequency response is defined as the phase shift of the output signal with respect to the input signal and is therefore equal to -φ.
Resonance Frequency
In sensor technology, the resonance is defined as the maximum of the output amplitude for a constant imput amplitude.
Therefore, in view of the three different amplitude response functions, we also obtain different resonant frequencies for displacement, velocity and acceleration. We can obtain the values by deriving the function Φ and setting it to 0.
For the acceleration resonance as an example we obtain
and we find the zero at
Thus one obtains for the different resonance frequencies in an analogous way:
Acceleration resonance frequency:
Velocity resonance frequency:
Displacement resonance frequency:
How can it be that you get three different solutions for the same physical phenomenon?
At the resonance frequency, the oscillator absorbs the maximum power. Since the power is equal to force times velocity, this maximum is found when the phase angle of the output signal x is equal to π/2. In this situation, the amplitude of the system increases continuously until the energy absorption is in equilibrium with the energy loss due to damping.
This means that the physical phenomenon of the resonance is always at a phase angle of π/2 and this is always the case at ω=ω0 .
So in order to see the "correct" (physical) resonance we need to keep the velocity amplitude of the input signal constant. We note that then the resonance is at ω0.
With constant acceleration, the velocity amplitude decreases with increasing frequency with the factor 10 per decade. This means that the energy applied becomes greater at frequencies below ω0 and therefore the maximum-amplitude resonance shifts downwards.
The same, but in opposite, is true for constant displacement. Here, the maximum-amplitude resonance is shifted upwards.