# Accelerated mass and spring system

• donorcard
In summary, the conversation discusses a problem involving an accelerated mass and spring system, where the unsprung mass is given an instantaneous acceleration. The goal is to find the force exerted by the spring on the vehicle body. The conversation also mentions the use of conservation of energy and the relationship between acceleration and position. The solution involves writing a second order differential equation to solve for the force.
donorcard

## Homework Statement

Imagine a mass attached to a spring, equivalent to a vehicle suspension system, where the mass we are considering is the unsprung mass of the wheel/axle assembly. We are told the unsprung mass is given an instantaneous acceleration of 50m/s². We are given the spring constant of the suspension springs (k = 500kN/mm) and we are given the mass of the wheel/axle assembly (m = 200kg). The system is assumed to be undamped.

I am trying to find the force which is exerted by the spring upward onto the vehicle body.

## Homework Equations

Accelerated mass and spring system
1. Homework Statement

Imagine a mass attached to a spring, equivalent to a vehicle suspension system, where the mass we are considering is the unsprung mass of the wheel/axle assembly. We are told the unsprung mass is given an instantaneous acceleration of 50m/s². We are given the spring constant of the suspension springs (k = 500kN/mm) and we are given the mass of the wheel/axle assembly (m = 200kg). The system is assumed to be undamped.

I am trying to find the force which is exerted by the spring upward onto the vehicle body.

2. Homework Equations

This is directly proportional to the compression of the spring but as I see it I just cannot make it work. Conservation of energy seems best suited, however I require the velocity of the wheel assembly (for ke = ½mv²) which I do not know. I have no time step increment with which to work out the velocity either.

## The Attempt at a Solution

The acceleration of the wheel/axle assembly is specified in a load case. The problem as I see it is relating this acceleration to the deceleration caused by the spring. I am hopefully just missing something but this problem really has me stuck. If the spring weren’t there then the force on the vehicle body would be a simple F = ma where the acceleration is 50m/s² I think, but this is a much worse than real case.

The mass is accelerated (say +ve direction) at the 'start' of the action and is immediately deccelerated (-ve direction) by the spring. But how this relationship is expressed I can't fathom.

Any advice or views are appreciated, thank you.

donorcard said:

## Homework Statement

Imagine a mass attached to a spring, equivalent to a vehicle suspension system, where the mass we are considering is the unsprung mass of the wheel/axle assembly. We are told the unsprung mass is given an instantaneous acceleration of 50m/s². We are given the spring constant of the suspension springs (k = 500kN/mm) and we are given the mass of the wheel/axle assembly (m = 200kg). The system is assumed to be undamped.

I am trying to find the force which is exerted by the spring upward onto the vehicle body.

## Homework Equations

Accelerated mass and spring system
1. Homework Statement

Imagine a mass attached to a spring, equivalent to a vehicle suspension system, where the mass we are considering is the unsprung mass of the wheel/axle assembly. We are told the unsprung mass is given an instantaneous acceleration of 50m/s². We are given the spring constant of the suspension springs (k = 500kN/mm) and we are given the mass of the wheel/axle assembly (m = 200kg). The system is assumed to be undamped.

I am trying to find the force which is exerted by the spring upward onto the vehicle body.

2. Homework Equations

This is directly proportional to the compression of the spring but as I see it I just cannot make it work. Conservation of energy seems best suited, however I require the velocity of the wheel assembly (for ke = ½mv²) which I do not know. I have no time step increment with which to work out the velocity either.

## The Attempt at a Solution

The acceleration of the wheel/axle assembly is specified in a load case. The problem as I see it is relating this acceleration to the deceleration caused by the spring. I am hopefully just missing something but this problem really has me stuck. If the spring weren’t there then the force on the vehicle body would be a simple F = ma where the acceleration is 50m/s² I think, but this is a much worse than real case.

The mass is accelerated (say +ve direction) at the 'start' of the action and is immediately deccelerated (-ve direction) by the spring. But how this relationship is expressed I can't fathom.

Any advice or views are appreciated, thank you.

The force acting on the mass is the combination of the constant force F and the spring force kx.
You know that the resulting force accelerates the body. You know also that acceleration is the second derivative of position. You can write a second order differential equation and solve it.

I would approach this problem by first analyzing the forces acting on the system. In this case, the only external force acting on the wheel/axle assembly is the acceleration of 50m/s². However, once the spring is compressed, there will also be a force exerted by the spring on the wheel/axle assembly in the opposite direction to counteract the acceleration. This is known as Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

To determine the force exerted by the spring, we can use the equation F = kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the displacement is the compression of the spring, which can be calculated using the acceleration and the time it takes for the spring to compress.

As for the velocity of the wheel/axle assembly, it is not necessary to know the exact value in order to calculate the force exerted by the spring. Instead, we can use the conservation of energy to determine the velocity. The initial kinetic energy of the wheel/axle assembly is equal to the work done by the spring in compressing it, which can be calculated using the force and the displacement.

In summary, to find the force exerted by the spring on the vehicle body, we need to first calculate the compression of the spring using the acceleration and time, then use Hooke's Law to determine the force, and finally use the conservation of energy to determine the velocity. I hope this helps in solving the problem.

## 1. What is an accelerated mass and spring system?

An accelerated mass and spring system is a physical system that consists of a mass attached to a spring, which is then subjected to an external force or acceleration. This system is commonly used in physics experiments to study the relationship between force, mass, and acceleration.

## 2. How does an accelerated mass and spring system work?

When a force is applied to the mass attached to the spring, the spring will stretch or compress depending on the direction of the force. This causes the spring to exert a restoring force on the mass, which will accelerate the mass back towards its equilibrium position. This back-and-forth motion of the mass is known as simple harmonic motion.

## 3. What is the equation for calculating the period of an accelerated mass and spring system?

The period (T) of an accelerated mass and spring system can be calculated using the equation T = 2π √(m/k), where m is the mass attached to the spring and k is the spring constant. This equation assumes that there is no friction or damping in the system.

## 4. How does the mass and spring constant affect the behavior of an accelerated mass and spring system?

The mass and spring constant have a direct impact on the period and frequency of the system. A higher mass or a stiffer spring (higher spring constant) will result in a longer period and lower frequency, while a lower mass or a less stiff spring (lower spring constant) will result in a shorter period and higher frequency.

## 5. What are some real-life applications of accelerated mass and spring systems?

Accelerated mass and spring systems are commonly used in various technologies and industries, including mechanical engineering, aerospace engineering, and seismology. They are also used in everyday objects such as car suspensions, shock absorbers, and pogo sticks.

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