Fixture Design for Spring and Damping Constants Determination

Abstract

This project is about Fixture design for finding spring constant and damping constant for suspension system. The determination of spring constant is done from the equation F=k⸹, where F= force acting on spring, k =spring constant and ⸹=displacement. For finding spring constant experimentally, we have designed the fixture with the help of mechanical devices like Scale hydraulic cylinder and hydraulic Jack.

The determination of damping coefficient is done by some equipments like load cell, hydraulic actuator, data acquisition system and function generator.

The damping coefficient will be found by calculation and graphical representation of excited amplitude and frequency of damper. Alternately the objective of project is to design such fixture which has less cost and it will be able to give output with maximum efficiency.

Spring Testing Machine for Finding Spring Constant

Above figure represents fixture design for spring but it was used earlier and it is an old technique for finding spring constant for springs. The components in the fixtures are load cell, frame, spring and scale.

But here in this project we have decided to design fixture with the help of hydraulic devices like hydraulic cylinder and hydraulic jack. Because it has great advantages than that old fixture. The new hydraulic fixture is available at low cost and even it is efficient in terms of measuring spring constant. The suspension system of our project is as below:

In this suspension for measuring spring constant, spring is detached from the system and put between hydraulic cylinder and hydraulic Jack. The pressure gauge is attached with the hydraulic cylinder for pressure measuring purpose. The whole procedure for measuring spring constant is as below:

First of all adjust the spring in between hydraulic jack and hydraulic cylinder. Hydraulic jack is operated by the handle provided in it. So produce force with help of hydraulic jack. Thus by manual pumping, piston in hydraulic cylinder moves forward and spring is compressed. Force is applied on the hydraulic cylinder and due to this deflection of spring is occurred. This pressure is displayed on pressure gauge in kg/cm². Take this pressure gauge reading and measure spring change in length. Thus by measuring spring deflection and pressure, stiffness is calculated from equation of F = k⸹.

Theoretical calculation of spring stiffness

Dₒ = outer diameter = 57.6 mm

d = wire diameter = 9.3 mm

Dᵢ = inner diameter = Dₒ - 2d = 39 mm

Mean coil diameter = D = (Dₒ+ Dᵢ )/2 = 48.3 mm

Spring index = D/d = 5.2

Wahl factor = K = (4C-1 )/(4C-4) + (0.615 )/C = 1.297 which is resultant factor for torsion shear stress and direct shear stress.

Force acting on spring in suspension system, F = mg

Total mass of 5 steel slab in suspension m = ρVg

Density of steel block is assumed as 7850 kg/m^3 .

Volume of steel slab = 1705.2 cm3 (from measurement of dimensions of steel slab)

= 1.7052 * 10-3 m3.

So mass of steel slab = ρVg = 7850*1.7052*10-3*9.8

= 13.385 kg

= 15 kg

There are 5 steel slab. So total mass = 15*5 = 75 kg. but we take as 80 kg.

So total load acting on spring = mg

= (80) (9.8)

= 784 N

Spring material is considered as oil hardened and tempered spring steel. The value of modulus of elasticity for it is G = 81370 N/mm2.

Permissible shear stress value for this spring,

Sut = 1855/(9.3)^0.187 = 1222.47 N/mm2

τ = 0.3*Sut = 366.74 N/mm2

So shear stress acting on spring,

τ = K (8PD/(πd^3 ))

= 1.297 ((8*784*48.3)/(π〖9.3〗^3 ))

= 155.5 N/mm2 < 366.74 N/mm2

Spring constant theoretical,

Ktheoretical = (Gd^4)/(8D^3 N) = (81370*(9.3)^4)/(8(*48.3)^3*N) , (where N = number of active turns = 15)

= 45016.68 N/m.

Advantages:

• Spring of different diameters can be checked
• Spring can be check without damaging the spring.
• The testing is carried out in very less time, so production rate is very high.
• One man effort is enough to check the spring.
• Semi-skilled and unskilled labor can operate this machine easily.
• The system is self-lubricating.
• The system is noiseless.
• It is portable and could be carried anywhere.

Disadvantages:

• As in this setup hydraulic jack is used there may be a chances of hydraulic leakages hence the periodic inspection, maintenance and refilling of oil is necessary.
• Proper reading of load and displacement is necessary.

Costing of Fixture

• Hydraulic cylinder Cost Estimation:

Specifications of hydraulic cylinder are,

Stroke length L = 100 mm

So bore diameter is in between L/1.5 to L  D = 7 mm

Max pressure inside cylinder = 150 bar to 450 bar

Material: stainless steel

Ports in cylinder = 25 mm

Calculating maximum Pressure generated in our suspension system,

Maximum force generated is 784 N but we take it as 900 N for safety purpose.

So maximum pressure generated inside cylinder ,

= 900/(π/(4 ) 〖0.07〗^2 )

= 233860.32 N/mm2

= 2.34 bar

Cost for this specification of cylinder is around Rs. 3000

• Hydraulic Jack Cost Estimation:

We need 10 ton of hydraulic jack which can applied maximum force of 98066.5 N

The price of jack is approximately Rs.1500 to Rs.2000.

• Pressure gauge Cost Estimation:

Here, we need pressure gauge having range around 0-300 psi or 0-200 psi which is made of stainless steel.

Price of pressure gauge is approximately is Rs.800 to Rs.900

So total expanses in making this assembly = 3000 + 2000 + 900 + other expanses

= 7000 to 8000 Rs.

Determination of Damping Co-efficient Using Sinusoidal Testing

The oil in the cylinder of Damper flows from one chamber to other chamber through some narrow gaps of base valve and piston valve as piston is moving back and forward in the cylinder tube. As a result, the damping force is produced because of the friction generated among oil molecules because of flow restriction due to narrow gaps in valve. The damping force is proportional to Velocity however; the damping force is always designed greater in the extension than in the compression stage to quickly accommodate shocks and provide damping, in order to improve vehicle dynamic behaviors.

As shown in above figure, it is the whole suspension system and its components are there. The damper is mounted in between load cell (which is basically 5 steel slabs of having certain weight) and hydraulic actuator. Hydraulic actuator is connected to data acquisition system and to motor which gives power to the suspension system.

For loading the damper excitation frequency and sampling frequency are set into the computer software and the signal sinusoidal excitation is produced through function generator. The amplification of signal is done by the help of digital A/D interface to make servo actuator to excite the Damper according to input data. The force produced across damper is calculated by the load cell (steel slabs) and dynamometer displacement is recorded by linear variable differential transformer (LVDT).

Here in our project in our laboratory we don’t have function generator and computer software facility like Matlab. So in this project we have to add function generator in order to excitation of damper as per desired frequency and amplitude. Because displacement driving is the excitation method used, the displacement data signal of the Damper piston can be directly recorded into system and the force generated across the damper is then recorded through the load cell fitted on the rigid beam.

The computer software collects this output displacement and force data then it is put into Microsoft Excel and Matlab software interface for analysis where various operations like data smoothening, determination of velocity using time and displacement data. Finally, Time -Displacement, Time-velocity, characteristic diagram (velocity - force) and work diagrams (Displacement -Force) are obtained for single cycle and for whole Test run.

Experimental Process for Damping Coefficient

The process of evaluation of Equivalent Damping Coefficient in which the real nonlinear MDOF system of Damper is replaced by simple elastic SDOF system having equivalent damping coefficient (Ceq) which is approximate value of damping coefficient corresponding to real nonlinear system, that to be use to evaluate damping force in system in simplest analysis.

The Equivalent elastic system is supposed to have same response as real nonlinear system under given same sinusoidal excitation. The equivalent viscous damping coefficient is determined on the assumptions that the real nonlinear Damper model and its equivalent linear system dissipates same amount of energy per cycle of response to sinusoidal excitation.

As system is assumed to be subjected to harmonic excitation it can be expressed as:

mx ̈ + (cx) ̇ + kx = X sin⁡ωt

For Steady state system, the energy lost per cycle in a damper in a harmonically forced system may be expressed as :

Wd = ∮(F_d*dx)

Where, F_d = (cx) ̇ and x = X sin⁡ωt

Wd = ∮((cx) ̇*dx) = ∮((cx²) ̇*dt

Where, dx = x ̇ dt

So, Wd = ω² x2 ∫_0^2π cos²⁡ωt * dt = π C ω X2

Ceq = W_d/(ω X²π) where, ω = 2 πf

Experimental Data Example

It is said that the Equivalent Damping Coefficient is vary in small range of value with slight increasing order towards higher frequency except for very low frequency where sudden increased value is shown. This is because friction damping effect as excitation frequency is very low damping force which is velocity dependent force is negligible and hence the friction force present is predominant.