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• Ting Kee Chuong

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Contents OBJECTIVE:.................................................................................................................1 THEORY:.......................................................................................................................1 APPARATUS:................................................................................................................3 PROCEDURES:............................................................................................................4 RESULTS AND DISCUSSIONS:.................................................................................5 CONCLUSION:............................................................................................................9 REFERENCE................................................................................................................9

DM6: SPRING MASS SYSTEM OBJECTIVE: 1. To obtain the spring rate from experimental data. 2. To obtain actual and theoretical frequencies of oscillation for a free system with varying mass. THEORY: Part 1 The stiffness of spring or spring rate is defined as the force, or torque, per unit of corresponding deformation. In instance, the corresponding deformation would be the extension produced by the tensile force in a simple tension member. The formula of stiffness is presented as follow: Force( N ) EquationTensile 1 Stiffness= Extension Produced (m) The unit of stiffness is N/m (Newton per Meter). As the stiffness of the spring is also referred to as ‘spring rate’, the spring rate can be calculated by using Equation 1. For example, if a spring extends 15mm when a force of 45N is applied to it, the spring rate can be calculated as follow: Stiffness=

45 =3000 N /m=3 kN /m 15 ×10−3

1

Part 2

Figure 1: Elastic system represented by the spring of stiffness, S and the body of mass, m. The restoring force due to the spring stiffness is Sx if the mass is given a displacement x from the equilibrium position. When the spring is released, the force provides the mass an acceleration a as shown below: Sx=ma Equation 2

From the equation above, the acceleration is proportional to the displacement and is always directed towards the equilibrium position so that the mass travels in simple harmonic motion. The equation of periodic time is as follows:

√

x Equation 3 T =2 π ( )=2 π a

(√ ms )

Where: T= Periodic Time (sec) m= Mass of hanger and any additional masses 2

s= Spring Stiffness or Spring Rate The periodic time, T is the time taken for one complete oscillation and is given by: T=

1 fEquation 4

Where: f= Frequency of oscillation, Hz (Hertz), or number of cycles per second. Alternatively, Equation 4 can be rearranged to give: 1 f = Equation 5 T

APPARATUS: Part 1 Adjusting Stud

HVT12f Vibration Frame Helical Spring Hanger Clevis Load Hanger

Weights

Figure 2.1: Arrangement of Spring Mass System in Part 1

3

Metre Rule

Part 2

HVT12f Vibration Frame

Helical Spring Hanger Clevis Load Hanger Nylon String Weights

Figure 2.2: Arrangement of Spring Mass System in Part 2 PROCEDURES: Part 1 1. The hanger assembly was weighted and the mass was recorded in table 1 below. 2. The spring was attached to the HVT12f Vibration frame and the hanger assembly was suspended to the hanger clevis. 3. The tape measure supplied was being used to measure the unloaded length of the spring to the nearest 1mm. The flat ends of the spring were measured. This length was recorded in table 1 next to the NO LOAD condition.

4

4. 100g mass was incrementally added to the load hanger and the new length of the spring (the extension) between the same two reference points as before was recorded. The new length was recorded continuously until all the masses have been used. Part 2 1. The 4x100g masses (0.4kg) were kept onto the load hanger assembly. 2. The base of the load hanger was pull down with a suitable distance and at the same time with a stopwatch in one hand. 3. The stopwatch was started upon releasing the load hanger assembly and a full 10 oscillations was timed. The results were recorded into table 2. 4. The number of masses on the load hanger was adjusted as desired and retested, all results were recorded into table 2 each time a new test undertaken. RESULTS AND DISCUSSIONS: Part 1 Table 1 Hanger

Added

Total

Total

Spring

Spring

Assembly

Mass (kg)

Added

Applied

Extension

Extension

NO LOAD

Mass (kg) 0.1

Force (N) 0.981

(mm) 5.0

(m) 0.005

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.962 2.943 3.924 4.905 5.886 6.867 7.848 8.829

10.0 14.0 18.0 21.0 25.0 28.0 32.0 36.0

0.010 0.014 0.018 0.021 0.025 0.028 0.032 0.036

Mass (kg) 0.1

(0) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

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Graph of Applied Load against Spring Extension

Applied Load (N)

10 9 8 7 6 5 4 3 2 1 0

f(x) = 0.26x - 0.57

0

5

10

15

20

25

30

35

40

Spring Extension (mm)

From the graph, gradient

=

0.2609N/mm

=

0.2609N/mm

Manufacturer’s Spring Rate =

0.2850N/mm

Thus, Theoretical Spring Rate

Percentage Difference between Theoretical and Manufacturing Spring Rate =

Value−ManufacturingValue |TheoreticalManufacturer |×100 ' s Value

=

|0.2609−0.2850 |×100 0.2850

= 8.46%

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Table 2 Hanger

Adde

Total

Time Taken Actual

Theoretica

Assembl

d

Adde

for

l Periodic s

d

Complete

c Time, Time, Tthr Periodic Time,

Mass

Oscillations

Tact(s)

(s)

Tmnf (s)

(kg)

, T10 (s)

y

Mass Mass

(kg)

(kg)

10 Periodi

Manufacturer’

0.1

0.4

0.5

3.41

0.341

0.2751

0.2632

0.1

0.5

0.6

3.71

0.371

0.3013

0.2883

0.1

0.6

0.7

4.16

0.416

0.3255

0.3114

The calculation of actual periodic time, Tact is based on Equation 4. T act=

T 10 10

The calculation of theoretical periodic time, Tthr and manufacturer’s periodic time, Tmnf is based on Equation 3 but with different spring rate respectively. Tthr is calculated by using the value of theoretical spring rate obtained from graph in Part 1, whereas Tmnf is calculated from the spring rate given by manufacturer. T thr =2 π

(√ sm ) thr

where Sthr = 0.2609N/mm or 260.9N/m T mnf =2 π

(√ sm ) mnf

where Smnf = 0.2850N/mm or 285N/m

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Graph of Periodic Time against Total Added Mass 0.45 0.4 0.35 0.3 0.25

Periodic Time (s)

0.2 0.15 0.1 0.05 0 0.45

0.5

0.55

0.6

0.65

0.7

0.75

Total Applied Mass (kg) Actual Periodic Time (s)

Theoretical Periodic Time (s)

Manufacturer's Periodic Time (s)

According to the calculation and plotted graph, it can be deduced that our experiment encountered only a minimal error, which is 8.46%. There are several possible factors which contribute to the error in experiment: 1. Parallax Error -Observation error might occur when taking reading on the measuring tape to measure the extension of the spring. It happens when the eye is not perpendicular to the scale reading on the measuring tape. -Way to overcome: The experiment should be repeated 3 times or more to optimize the parallax error by obtaining the average of the readings. 2. Time Reaction Error -The time taken for the spring to complete 10 oscillations might be deviated as the observer might not start the stopwatch exactly at the point when the spring starts to oscillate. Besides, it is very difficult to count the 10 complete oscillations as the oscillating speed of the spring is too fast. -Way to overcome: The experiment needs to be repeated a few more times to ensure a consistency is obtained in the result. Indicator like a small ribbon can be tied on the spring to ease the counting of 10 complete oscillations. 8

3. Vibration of loads -The pieces of load added on the hanger will vibrate at different frequency when the spring oscillates. Due to large inertia of the load, the oscillation of the spring will be obstructed. -Way to overcome: The added loads are tied tightly so that the whole spring system can be considered as a rigid body. The loads are assured to oscillate together with the spring at same frequency and thus the oscillation process will not be obstructed by the inertia caused by the loads.

CONCLUSION: From this experiment, the Hooke’s law and principles of simple harmonic motion (SHM) are proven. The extension of the spring is proven to be directly proportional to the applied force with the spring rate/stiffness as the constant, within the condition that the elasticity limit of the spring is not exceeded. If the spring is deformed, the Hooke’s law will never be obeyed and the behavior of the spring as well as the extension cannot be predicted anymore from the linear graph. REFERENCE Hibbeler, R. (2011). Mechanics of Materials 8th Edition. New York: Pearson. PECKHAM, D. C. (2005). HOOKE'S LAW AND A SIMPLE SPRING.

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