# Spring Stiffness and Amplitude, not understanding

• SA_Eng
In summary, a mass-spring system subject to a harmonic force can be described by the equation ma+kx=Fsin(ωt), where a is the second derivative of position. After solving for the steady state amplitude, it is found that increasing the stiffness (k) will result in a larger amplitude. This may seem counterintuitive, as a stiffer spring should resist motion more, but the equation shows that a higher stiffness results in a higher amplitude. However, the accuracy of this equation has been questioned and it is important to check the working and consider other factors such as the damping ratio.
SA_Eng
Hi,

For an undamped mass, spring system subject to a harmonic force, the equation is: ma+kx=Fsin(ωt)
where a=d^2/dx^2

After solving the diff eqns, the steady state amplitude is:
X=(F/m)/(ω^2-ω0^2)
where ω is the frequency and ω0 is the natural frequency =sqrt(k/m)

according to the amplitude equation, the amplitude will increase if you increase the stiffness.

I am struggling to understand why this is true, as i understand it, the spring should be resisting the motion of the mass and hence the stiffer the spring the less it moves. Think about if you increase the stiffness to infinity, equivelant to placing the mass on a rigid floor, the displacement of the mass should surely reduce to zero. Am I missing something?

Anyone able to explain?

according to the amplitude equation, the amplitude will increase if you increase the stiffness.
You mean this equation:
X=(F/m)/(ω^2-ω0^2)
... let's test this out ... a stiff spring would mean a higher k, so the relationship between ampltude and stiffness would be:

$$X=\frac{F}{m}\left ( \omega^2 - \frac{k}{m} \right )$$

Initially it looks like the bigger k, the smaller the term in brackets due to the subtraction. However, it kinda looks like if k is very big, then $\omega_0 > \omega$ ... making the amplitude negative: what could this mean?

[later] Taking a further look:
http://en.wikipedia.org/wiki/Harmonic_oscillator#Sinusoidal_driving_force
... your system has a damping ratio of zero, which simplifies the solution.

Using this, their amplitude comes out different from yours.
I'd check your working.

Last edited:

## 1. What is spring stiffness?

Spring stiffness refers to the amount of force required to compress or stretch a spring by a certain distance. It is a measure of the spring's resistance to deformation.

## 2. How is spring stiffness measured?

Spring stiffness is typically measured in units of force per unit of distance, such as pounds per inch or newtons per meter. This can be determined experimentally by applying a known force to the spring and measuring the resulting displacement.

## 3. What is amplitude in relation to springs?

Amplitude in relation to springs refers to the maximum displacement or distance that a spring can be compressed or stretched from its resting position. It is a measure of the spring's potential energy.

## 4. How does spring stiffness affect amplitude?

Higher spring stiffness means that more force is required to compress or stretch the spring, resulting in a smaller amplitude. Conversely, a lower spring stiffness allows for a larger amplitude.

## 5. Why is it important to understand spring stiffness and amplitude?

Understanding spring stiffness and amplitude is important in various fields of science and engineering, including mechanics, materials science, and design. It allows for the prediction and control of the behavior of springs in various applications, such as in suspension systems, shock absorbers, and mechanical devices.

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